From 2d3070f14f7ca70da8b6cea967f9b40d26763d37 Mon Sep 17 00:00:00 2001
From: krahets <krahets@163.com>
Date: Thu, 7 Dec 2023 20:12:31 +0800
Subject: [PATCH] deploy

---
 404.html                                      | 252 +++++++++---
 assets/javascripts/bundle.4e0fa4ba.min.js     |  29 --
 assets/javascripts/bundle.4e0fa4ba.min.js.map |   8 -
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 index.html                                    | 252 +++++++++---
 search/search_index.json                      |   2 +-
 sitemap.xml                                   | 208 +++++-----
 sitemap.xml.gz                                | Bin 999 -> 999 bytes
 123 files changed, 23104 insertions(+), 7031 deletions(-)
 delete mode 100644 assets/javascripts/bundle.4e0fa4ba.min.js
 delete mode 100644 assets/javascripts/bundle.4e0fa4ba.min.js.map
 create mode 100644 assets/javascripts/bundle.d7c377c4.min.js
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 create mode 100644 assets/javascripts/lunr/min/lunr.el.min.js
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 rename assets/javascripts/workers/{search.a264c092.min.js.map => search.f886a092.min.js.map} (80%)
 create mode 100644 assets/stylesheets/main.45e1311d.min.css
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 create mode 100644 assets/stylesheets/palette.06af60db.min.css
 create mode 100644 assets/stylesheets/palette.06af60db.min.css.map
 delete mode 100644 assets/stylesheets/palette.a5377069.min.css
 delete mode 100644 assets/stylesheets/palette.a5377069.min.css.map

diff --git a/404.html b/404.html
index 058b2413c..1757c3e3e 100644
--- a/404.html
+++ b/404.html
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+      <link rel="stylesheet" href="/assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="/assets/stylesheets/palette.a5377069.min.css">
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circle{fill:var(--md-mermaid-label-bg-color)}.actor{fill:var(--md-mermaid-sequence-actor-bg-color);stroke:var(--md-mermaid-sequence-actor-border-color)}text.actor>tspan{fill:var(--md-mermaid-sequence-actor-fg-color);font-family:var(--md-mermaid-font-family)}line{stroke:var(--md-mermaid-sequence-actor-line-color)}.actor-man circle,.actor-man line{fill:var(--md-mermaid-sequence-actorman-bg-color);stroke:var(--md-mermaid-sequence-actorman-line-color)}.messageLine0,.messageLine1{stroke:var(--md-mermaid-sequence-message-line-color)}.note{fill:var(--md-mermaid-sequence-note-bg-color);stroke:var(--md-mermaid-sequence-note-border-color)}.loopText,.loopText>tspan,.messageText,.noteText>tspan{stroke:none;font-family:var(--md-mermaid-font-family)!important}.messageText{fill:var(--md-mermaid-sequence-message-fg-color)}.loopText,.loopText>tspan{fill:var(--md-mermaid-sequence-loop-fg-color)}.noteText>tspan{fill:var(--md-mermaid-sequence-note-fg-color)}#arrowhead 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diff --git a/assets/javascripts/bundle.4e0fa4ba.min.js.map b/assets/javascripts/bundle.4e0fa4ba.min.js.map
deleted file mode 100644
index b3e2c4ef6..000000000
--- a/assets/javascripts/bundle.4e0fa4ba.min.js.map
+++ /dev/null
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-{
-  "version": 3,
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-  "sourcesContent": ["(function (global, factory) {\n  typeof exports === 'object' && typeof module !== 'undefined' ? factory() :\n  typeof define === 'function' && define.amd ? define(factory) :\n  (factory());\n}(this, (function () { 'use strict';\n\n  /**\n   * Applies the :focus-visible polyfill at the given scope.\n   * A scope in this case is either the top-level Document or a Shadow Root.\n   *\n   * @param {(Document|ShadowRoot)} scope\n   * @see https://github.com/WICG/focus-visible\n   */\n  function applyFocusVisiblePolyfill(scope) {\n    var hadKeyboardEvent = true;\n    var hadFocusVisibleRecently = false;\n    var hadFocusVisibleRecentlyTimeout = null;\n\n    var inputTypesAllowlist = {\n      text: true,\n      search: true,\n      url: true,\n      tel: true,\n      email: true,\n      password: true,\n      number: true,\n      date: true,\n      month: true,\n      week: true,\n      time: true,\n      datetime: true,\n      'datetime-local': true\n    };\n\n    /**\n     * Helper function for legacy browsers and iframes which sometimes focus\n     * elements like document, body, and non-interactive SVG.\n     * @param {Element} el\n     */\n    function isValidFocusTarget(el) {\n      if (\n        el &&\n        el !== document &&\n        el.nodeName !== 'HTML' &&\n        el.nodeName !== 'BODY' &&\n        'classList' in el &&\n        'contains' in el.classList\n      ) {\n        return true;\n      }\n      return false;\n    }\n\n    /**\n     * Computes whether the given element should automatically trigger the\n     * `focus-visible` class being added, i.e. whether it should always match\n     * `:focus-visible` when focused.\n     * @param {Element} el\n     * @return {boolean}\n     */\n    function focusTriggersKeyboardModality(el) {\n      var type = el.type;\n      var tagName = el.tagName;\n\n      if (tagName === 'INPUT' && inputTypesAllowlist[type] && !el.readOnly) {\n        return true;\n      }\n\n      if (tagName === 'TEXTAREA' && !el.readOnly) {\n        return true;\n      }\n\n      if (el.isContentEditable) {\n        return true;\n      }\n\n      return false;\n    }\n\n    /**\n     * Add the `focus-visible` class to the given element if it was not added by\n     * the author.\n     * @param {Element} el\n     */\n    function addFocusVisibleClass(el) {\n      if (el.classList.contains('focus-visible')) {\n        return;\n      }\n      el.classList.add('focus-visible');\n      el.setAttribute('data-focus-visible-added', '');\n    }\n\n    /**\n     * Remove the `focus-visible` class from the given element if it was not\n     * originally added by the author.\n     * @param {Element} el\n     */\n    function removeFocusVisibleClass(el) {\n      if (!el.hasAttribute('data-focus-visible-added')) {\n        return;\n      }\n      el.classList.remove('focus-visible');\n      el.removeAttribute('data-focus-visible-added');\n    }\n\n    /**\n     * If the most recent user interaction was via the keyboard;\n     * and the key press did not include a meta, alt/option, or control key;\n     * then the modality is keyboard. Otherwise, the modality is not keyboard.\n     * Apply `focus-visible` to any current active element and keep track\n     * of our keyboard modality state with `hadKeyboardEvent`.\n     * @param {KeyboardEvent} e\n     */\n    function onKeyDown(e) {\n      if (e.metaKey || e.altKey || e.ctrlKey) {\n        return;\n      }\n\n      if (isValidFocusTarget(scope.activeElement)) {\n        addFocusVisibleClass(scope.activeElement);\n      }\n\n      hadKeyboardEvent = true;\n    }\n\n    /**\n     * If at any point a user clicks with a pointing device, ensure that we change\n     * the modality away from keyboard.\n     * This avoids the situation where a user presses a key on an already focused\n     * element, and then clicks on a different element, focusing it with a\n     * pointing device, while we still think we're in keyboard modality.\n     * @param {Event} e\n     */\n    function onPointerDown(e) {\n      hadKeyboardEvent = false;\n    }\n\n    /**\n     * On `focus`, add the `focus-visible` class to the target if:\n     * - the target received focus as a result of keyboard navigation, or\n     * - the event target is an element that will likely require interaction\n     *   via the keyboard (e.g. a text box)\n     * @param {Event} e\n     */\n    function onFocus(e) {\n      // Prevent IE from focusing the document or HTML element.\n      if (!isValidFocusTarget(e.target)) {\n        return;\n      }\n\n      if (hadKeyboardEvent || focusTriggersKeyboardModality(e.target)) {\n        addFocusVisibleClass(e.target);\n      }\n    }\n\n    /**\n     * On `blur`, remove the `focus-visible` class from the target.\n     * @param {Event} e\n     */\n    function onBlur(e) {\n      if (!isValidFocusTarget(e.target)) {\n        return;\n      }\n\n      if (\n        e.target.classList.contains('focus-visible') ||\n        e.target.hasAttribute('data-focus-visible-added')\n      ) {\n        // To detect a tab/window switch, we look for a blur event followed\n        // rapidly by a visibility change.\n        // If we don't see a visibility change within 100ms, it's probably a\n        // regular focus change.\n        hadFocusVisibleRecently = true;\n        window.clearTimeout(hadFocusVisibleRecentlyTimeout);\n        hadFocusVisibleRecentlyTimeout = window.setTimeout(function() {\n          hadFocusVisibleRecently = false;\n        }, 100);\n        removeFocusVisibleClass(e.target);\n      }\n    }\n\n    /**\n     * If the user changes tabs, keep track of whether or not the previously\n     * focused element had .focus-visible.\n     * @param {Event} e\n     */\n    function onVisibilityChange(e) {\n      if (document.visibilityState === 'hidden') {\n        // If the tab becomes active again, the browser will handle calling focus\n        // on the element (Safari actually calls it twice).\n        // If this tab change caused a blur on an element with focus-visible,\n        // re-apply the class when the user switches back to the tab.\n        if (hadFocusVisibleRecently) {\n          hadKeyboardEvent = true;\n        }\n        addInitialPointerMoveListeners();\n      }\n    }\n\n    /**\n     * Add a group of listeners to detect usage of any pointing devices.\n     * These listeners will be added when the polyfill first loads, and anytime\n     * the window is blurred, so that they are active when the window regains\n     * focus.\n     */\n    function addInitialPointerMoveListeners() {\n      document.addEventListener('mousemove', onInitialPointerMove);\n      document.addEventListener('mousedown', onInitialPointerMove);\n      document.addEventListener('mouseup', onInitialPointerMove);\n      document.addEventListener('pointermove', onInitialPointerMove);\n      document.addEventListener('pointerdown', onInitialPointerMove);\n      document.addEventListener('pointerup', onInitialPointerMove);\n      document.addEventListener('touchmove', onInitialPointerMove);\n      document.addEventListener('touchstart', onInitialPointerMove);\n      document.addEventListener('touchend', onInitialPointerMove);\n    }\n\n    function removeInitialPointerMoveListeners() {\n      document.removeEventListener('mousemove', onInitialPointerMove);\n      document.removeEventListener('mousedown', onInitialPointerMove);\n      document.removeEventListener('mouseup', onInitialPointerMove);\n      document.removeEventListener('pointermove', onInitialPointerMove);\n      document.removeEventListener('pointerdown', onInitialPointerMove);\n      document.removeEventListener('pointerup', onInitialPointerMove);\n      document.removeEventListener('touchmove', onInitialPointerMove);\n      document.removeEventListener('touchstart', onInitialPointerMove);\n      document.removeEventListener('touchend', onInitialPointerMove);\n    }\n\n    /**\n     * When the polfyill first loads, assume the user is in keyboard modality.\n     * If any event is received from a pointing device (e.g. mouse, pointer,\n     * touch), turn off keyboard modality.\n     * This accounts for situations where focus enters the page from the URL bar.\n     * @param {Event} e\n     */\n    function onInitialPointerMove(e) {\n      // Work around a Safari quirk that fires a mousemove on <html> whenever the\n      // window blurs, even if you're tabbing out of the page. \u00AF\\_(\u30C4)_/\u00AF\n      if (e.target.nodeName && e.target.nodeName.toLowerCase() === 'html') {\n        return;\n      }\n\n      hadKeyboardEvent = false;\n      removeInitialPointerMoveListeners();\n    }\n\n    // For some kinds of state, we are interested in changes at the global scope\n    // only. For example, global pointer input, global key presses and global\n    // visibility change should affect the state at every scope:\n    document.addEventListener('keydown', onKeyDown, true);\n    document.addEventListener('mousedown', onPointerDown, true);\n    document.addEventListener('pointerdown', onPointerDown, true);\n    document.addEventListener('touchstart', onPointerDown, true);\n    document.addEventListener('visibilitychange', onVisibilityChange, true);\n\n    addInitialPointerMoveListeners();\n\n    // For focus and blur, we specifically care about state changes in the local\n    // scope. This is because focus / blur events that originate from within a\n    // shadow root are not re-dispatched from the host element if it was already\n    // the active element in its own scope:\n    scope.addEventListener('focus', onFocus, true);\n    scope.addEventListener('blur', onBlur, true);\n\n    // We detect that a node is a ShadowRoot by ensuring that it is a\n    // DocumentFragment and also has a host property. This check covers native\n    // implementation and polyfill implementation transparently. If we only cared\n    // about the native implementation, we could just check if the scope was\n    // an instance of a ShadowRoot.\n    if (scope.nodeType === Node.DOCUMENT_FRAGMENT_NODE && scope.host) {\n      // Since a ShadowRoot is a special kind of DocumentFragment, it does not\n      // have a root element to add a class to. So, we add this attribute to the\n      // host element instead:\n      scope.host.setAttribute('data-js-focus-visible', '');\n    } else if (scope.nodeType === Node.DOCUMENT_NODE) {\n      document.documentElement.classList.add('js-focus-visible');\n      document.documentElement.setAttribute('data-js-focus-visible', '');\n    }\n  }\n\n  // It is important to wrap all references to global window and document in\n  // these checks to support server-side rendering use cases\n  // @see https://github.com/WICG/focus-visible/issues/199\n  if (typeof window !== 'undefined' && typeof document !== 'undefined') {\n    // Make the polyfill helper globally available. This can be used as a signal\n    // to interested libraries that wish to coordinate with the polyfill for e.g.,\n    // applying the polyfill to a shadow root:\n    window.applyFocusVisiblePolyfill = applyFocusVisiblePolyfill;\n\n    // Notify interested libraries of the polyfill's presence, in case the\n    // polyfill was loaded lazily:\n    var event;\n\n    try {\n      event = new CustomEvent('focus-visible-polyfill-ready');\n    } catch (error) {\n      // IE11 does not support using CustomEvent as a constructor directly:\n      event = document.createEvent('CustomEvent');\n      event.initCustomEvent('focus-visible-polyfill-ready', false, false, {});\n    }\n\n    window.dispatchEvent(event);\n  }\n\n  if (typeof document !== 'undefined') {\n    // Apply the polyfill to the global document, so that no JavaScript\n    // coordination is required to use the polyfill in the top-level document:\n    applyFocusVisiblePolyfill(document);\n  }\n\n})));\n", "/*!\n * clipboard.js v2.0.11\n * https://clipboardjs.com/\n *\n * Licensed MIT \u00A9 Zeno Rocha\n */\n(function webpackUniversalModuleDefinition(root, factory) {\n\tif(typeof exports === 'object' && typeof module === 'object')\n\t\tmodule.exports = factory();\n\telse if(typeof define === 'function' && define.amd)\n\t\tdefine([], factory);\n\telse if(typeof exports === 'object')\n\t\texports[\"ClipboardJS\"] = factory();\n\telse\n\t\troot[\"ClipboardJS\"] = factory();\n})(this, function() {\nreturn /******/ (function() { // webpackBootstrap\n/******/ \tvar __webpack_modules__ = ({\n\n/***/ 686:\n/***/ (function(__unused_webpack_module, __webpack_exports__, __webpack_require__) {\n\n\"use strict\";\n\n// EXPORTS\n__webpack_require__.d(__webpack_exports__, {\n  \"default\": function() { return /* binding */ clipboard; }\n});\n\n// EXTERNAL MODULE: ./node_modules/tiny-emitter/index.js\nvar tiny_emitter = __webpack_require__(279);\nvar tiny_emitter_default = /*#__PURE__*/__webpack_require__.n(tiny_emitter);\n// EXTERNAL MODULE: ./node_modules/good-listener/src/listen.js\nvar listen = __webpack_require__(370);\nvar listen_default = /*#__PURE__*/__webpack_require__.n(listen);\n// EXTERNAL MODULE: ./node_modules/select/src/select.js\nvar src_select = __webpack_require__(817);\nvar select_default = /*#__PURE__*/__webpack_require__.n(src_select);\n;// CONCATENATED MODULE: ./src/common/command.js\n/**\n * Executes a given operation type.\n * @param {String} type\n * @return {Boolean}\n */\nfunction command(type) {\n  try {\n    return document.execCommand(type);\n  } catch (err) {\n    return false;\n  }\n}\n;// CONCATENATED MODULE: ./src/actions/cut.js\n\n\n/**\n * Cut action wrapper.\n * @param {String|HTMLElement} target\n * @return {String}\n */\n\nvar ClipboardActionCut = function ClipboardActionCut(target) {\n  var selectedText = select_default()(target);\n  command('cut');\n  return selectedText;\n};\n\n/* harmony default export */ var actions_cut = (ClipboardActionCut);\n;// CONCATENATED MODULE: ./src/common/create-fake-element.js\n/**\n * Creates a fake textarea element with a value.\n * @param {String} value\n * @return {HTMLElement}\n */\nfunction createFakeElement(value) {\n  var isRTL = document.documentElement.getAttribute('dir') === 'rtl';\n  var fakeElement = document.createElement('textarea'); // Prevent zooming on iOS\n\n  fakeElement.style.fontSize = '12pt'; // Reset box model\n\n  fakeElement.style.border = '0';\n  fakeElement.style.padding = '0';\n  fakeElement.style.margin = '0'; // Move element out of screen horizontally\n\n  fakeElement.style.position = 'absolute';\n  fakeElement.style[isRTL ? 'right' : 'left'] = '-9999px'; // Move element to the same position vertically\n\n  var yPosition = window.pageYOffset || document.documentElement.scrollTop;\n  fakeElement.style.top = \"\".concat(yPosition, \"px\");\n  fakeElement.setAttribute('readonly', '');\n  fakeElement.value = value;\n  return fakeElement;\n}\n;// CONCATENATED MODULE: ./src/actions/copy.js\n\n\n\n/**\n * Create fake copy action wrapper using a fake element.\n * @param {String} target\n * @param {Object} options\n * @return {String}\n */\n\nvar fakeCopyAction = function fakeCopyAction(value, options) {\n  var fakeElement = createFakeElement(value);\n  options.container.appendChild(fakeElement);\n  var selectedText = select_default()(fakeElement);\n  command('copy');\n  fakeElement.remove();\n  return selectedText;\n};\n/**\n * Copy action wrapper.\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @return {String}\n */\n\n\nvar ClipboardActionCopy = function ClipboardActionCopy(target) {\n  var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n    container: document.body\n  };\n  var selectedText = '';\n\n  if (typeof target === 'string') {\n    selectedText = fakeCopyAction(target, options);\n  } else if (target instanceof HTMLInputElement && !['text', 'search', 'url', 'tel', 'password'].includes(target === null || target === void 0 ? void 0 : target.type)) {\n    // If input type doesn't support `setSelectionRange`. Simulate it. https://developer.mozilla.org/en-US/docs/Web/API/HTMLInputElement/setSelectionRange\n    selectedText = fakeCopyAction(target.value, options);\n  } else {\n    selectedText = select_default()(target);\n    command('copy');\n  }\n\n  return selectedText;\n};\n\n/* harmony default export */ var actions_copy = (ClipboardActionCopy);\n;// CONCATENATED MODULE: ./src/actions/default.js\nfunction _typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { _typeof = function _typeof(obj) { return typeof obj; }; } else { _typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return _typeof(obj); }\n\n\n\n/**\n * Inner function which performs selection from either `text` or `target`\n * properties and then executes copy or cut operations.\n * @param {Object} options\n */\n\nvar ClipboardActionDefault = function ClipboardActionDefault() {\n  var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n  // Defines base properties passed from constructor.\n  var _options$action = options.action,\n      action = _options$action === void 0 ? 'copy' : _options$action,\n      container = options.container,\n      target = options.target,\n      text = options.text; // Sets the `action` to be performed which can be either 'copy' or 'cut'.\n\n  if (action !== 'copy' && action !== 'cut') {\n    throw new Error('Invalid \"action\" value, use either \"copy\" or \"cut\"');\n  } // Sets the `target` property using an element that will be have its content copied.\n\n\n  if (target !== undefined) {\n    if (target && _typeof(target) === 'object' && target.nodeType === 1) {\n      if (action === 'copy' && target.hasAttribute('disabled')) {\n        throw new Error('Invalid \"target\" attribute. Please use \"readonly\" instead of \"disabled\" attribute');\n      }\n\n      if (action === 'cut' && (target.hasAttribute('readonly') || target.hasAttribute('disabled'))) {\n        throw new Error('Invalid \"target\" attribute. You can\\'t cut text from elements with \"readonly\" or \"disabled\" attributes');\n      }\n    } else {\n      throw new Error('Invalid \"target\" value, use a valid Element');\n    }\n  } // Define selection strategy based on `text` property.\n\n\n  if (text) {\n    return actions_copy(text, {\n      container: container\n    });\n  } // Defines which selection strategy based on `target` property.\n\n\n  if (target) {\n    return action === 'cut' ? actions_cut(target) : actions_copy(target, {\n      container: container\n    });\n  }\n};\n\n/* harmony default export */ var actions_default = (ClipboardActionDefault);\n;// CONCATENATED MODULE: ./src/clipboard.js\nfunction clipboard_typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { clipboard_typeof = function _typeof(obj) { return typeof obj; }; } else { clipboard_typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return clipboard_typeof(obj); }\n\nfunction _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError(\"Cannot call a class as a function\"); } }\n\nfunction _defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if (\"value\" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } }\n\nfunction _createClass(Constructor, protoProps, staticProps) { if (protoProps) _defineProperties(Constructor.prototype, protoProps); if (staticProps) _defineProperties(Constructor, staticProps); return Constructor; }\n\nfunction _inherits(subClass, superClass) { if (typeof superClass !== \"function\" && superClass !== null) { throw new TypeError(\"Super expression must either be null or a function\"); } subClass.prototype = Object.create(superClass && superClass.prototype, { constructor: { value: subClass, writable: true, configurable: true } }); if (superClass) _setPrototypeOf(subClass, superClass); }\n\nfunction _setPrototypeOf(o, p) { _setPrototypeOf = Object.setPrototypeOf || function _setPrototypeOf(o, p) { o.__proto__ = p; return o; }; return _setPrototypeOf(o, p); }\n\nfunction _createSuper(Derived) { var hasNativeReflectConstruct = _isNativeReflectConstruct(); return function _createSuperInternal() { var Super = _getPrototypeOf(Derived), result; if (hasNativeReflectConstruct) { var NewTarget = _getPrototypeOf(this).constructor; result = Reflect.construct(Super, arguments, NewTarget); } else { result = Super.apply(this, arguments); } return _possibleConstructorReturn(this, result); }; }\n\nfunction _possibleConstructorReturn(self, call) { if (call && (clipboard_typeof(call) === \"object\" || typeof call === \"function\")) { return call; } return _assertThisInitialized(self); }\n\nfunction _assertThisInitialized(self) { if (self === void 0) { throw new ReferenceError(\"this hasn't been initialised - super() hasn't been called\"); } return self; }\n\nfunction _isNativeReflectConstruct() { if (typeof Reflect === \"undefined\" || !Reflect.construct) return false; if (Reflect.construct.sham) return false; if (typeof Proxy === \"function\") return true; try { Date.prototype.toString.call(Reflect.construct(Date, [], function () {})); return true; } catch (e) { return false; } }\n\nfunction _getPrototypeOf(o) { _getPrototypeOf = Object.setPrototypeOf ? Object.getPrototypeOf : function _getPrototypeOf(o) { return o.__proto__ || Object.getPrototypeOf(o); }; return _getPrototypeOf(o); }\n\n\n\n\n\n\n/**\n * Helper function to retrieve attribute value.\n * @param {String} suffix\n * @param {Element} element\n */\n\nfunction getAttributeValue(suffix, element) {\n  var attribute = \"data-clipboard-\".concat(suffix);\n\n  if (!element.hasAttribute(attribute)) {\n    return;\n  }\n\n  return element.getAttribute(attribute);\n}\n/**\n * Base class which takes one or more elements, adds event listeners to them,\n * and instantiates a new `ClipboardAction` on each click.\n */\n\n\nvar Clipboard = /*#__PURE__*/function (_Emitter) {\n  _inherits(Clipboard, _Emitter);\n\n  var _super = _createSuper(Clipboard);\n\n  /**\n   * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n   * @param {Object} options\n   */\n  function Clipboard(trigger, options) {\n    var _this;\n\n    _classCallCheck(this, Clipboard);\n\n    _this = _super.call(this);\n\n    _this.resolveOptions(options);\n\n    _this.listenClick(trigger);\n\n    return _this;\n  }\n  /**\n   * Defines if attributes would be resolved using internal setter functions\n   * or custom functions that were passed in the constructor.\n   * @param {Object} options\n   */\n\n\n  _createClass(Clipboard, [{\n    key: \"resolveOptions\",\n    value: function resolveOptions() {\n      var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n      this.action = typeof options.action === 'function' ? options.action : this.defaultAction;\n      this.target = typeof options.target === 'function' ? options.target : this.defaultTarget;\n      this.text = typeof options.text === 'function' ? options.text : this.defaultText;\n      this.container = clipboard_typeof(options.container) === 'object' ? options.container : document.body;\n    }\n    /**\n     * Adds a click event listener to the passed trigger.\n     * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n     */\n\n  }, {\n    key: \"listenClick\",\n    value: function listenClick(trigger) {\n      var _this2 = this;\n\n      this.listener = listen_default()(trigger, 'click', function (e) {\n        return _this2.onClick(e);\n      });\n    }\n    /**\n     * Defines a new `ClipboardAction` on each click event.\n     * @param {Event} e\n     */\n\n  }, {\n    key: \"onClick\",\n    value: function onClick(e) {\n      var trigger = e.delegateTarget || e.currentTarget;\n      var action = this.action(trigger) || 'copy';\n      var text = actions_default({\n        action: action,\n        container: this.container,\n        target: this.target(trigger),\n        text: this.text(trigger)\n      }); // Fires an event based on the copy operation result.\n\n      this.emit(text ? 'success' : 'error', {\n        action: action,\n        text: text,\n        trigger: trigger,\n        clearSelection: function clearSelection() {\n          if (trigger) {\n            trigger.focus();\n          }\n\n          window.getSelection().removeAllRanges();\n        }\n      });\n    }\n    /**\n     * Default `action` lookup function.\n     * @param {Element} trigger\n     */\n\n  }, {\n    key: \"defaultAction\",\n    value: function defaultAction(trigger) {\n      return getAttributeValue('action', trigger);\n    }\n    /**\n     * Default `target` lookup function.\n     * @param {Element} trigger\n     */\n\n  }, {\n    key: \"defaultTarget\",\n    value: function defaultTarget(trigger) {\n      var selector = getAttributeValue('target', trigger);\n\n      if (selector) {\n        return document.querySelector(selector);\n      }\n    }\n    /**\n     * Allow fire programmatically a copy action\n     * @param {String|HTMLElement} target\n     * @param {Object} options\n     * @returns Text copied.\n     */\n\n  }, {\n    key: \"defaultText\",\n\n    /**\n     * Default `text` lookup function.\n     * @param {Element} trigger\n     */\n    value: function defaultText(trigger) {\n      return getAttributeValue('text', trigger);\n    }\n    /**\n     * Destroy lifecycle.\n     */\n\n  }, {\n    key: \"destroy\",\n    value: function destroy() {\n      this.listener.destroy();\n    }\n  }], [{\n    key: \"copy\",\n    value: function copy(target) {\n      var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n        container: document.body\n      };\n      return actions_copy(target, options);\n    }\n    /**\n     * Allow fire programmatically a cut action\n     * @param {String|HTMLElement} target\n     * @returns Text cutted.\n     */\n\n  }, {\n    key: \"cut\",\n    value: function cut(target) {\n      return actions_cut(target);\n    }\n    /**\n     * Returns the support of the given action, or all actions if no action is\n     * given.\n     * @param {String} [action]\n     */\n\n  }, {\n    key: \"isSupported\",\n    value: function isSupported() {\n      var action = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : ['copy', 'cut'];\n      var actions = typeof action === 'string' ? [action] : action;\n      var support = !!document.queryCommandSupported;\n      actions.forEach(function (action) {\n        support = support && !!document.queryCommandSupported(action);\n      });\n      return support;\n    }\n  }]);\n\n  return Clipboard;\n}((tiny_emitter_default()));\n\n/* harmony default export */ var clipboard = (Clipboard);\n\n/***/ }),\n\n/***/ 828:\n/***/ (function(module) {\n\nvar DOCUMENT_NODE_TYPE = 9;\n\n/**\n * A polyfill for Element.matches()\n */\nif (typeof Element !== 'undefined' && !Element.prototype.matches) {\n    var proto = Element.prototype;\n\n    proto.matches = proto.matchesSelector ||\n                    proto.mozMatchesSelector ||\n                    proto.msMatchesSelector ||\n                    proto.oMatchesSelector ||\n                    proto.webkitMatchesSelector;\n}\n\n/**\n * Finds the closest parent that matches a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @return {Function}\n */\nfunction closest (element, selector) {\n    while (element && element.nodeType !== DOCUMENT_NODE_TYPE) {\n        if (typeof element.matches === 'function' &&\n            element.matches(selector)) {\n          return element;\n        }\n        element = element.parentNode;\n    }\n}\n\nmodule.exports = closest;\n\n\n/***/ }),\n\n/***/ 438:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar closest = __webpack_require__(828);\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction _delegate(element, selector, type, callback, useCapture) {\n    var listenerFn = listener.apply(this, arguments);\n\n    element.addEventListener(type, listenerFn, useCapture);\n\n    return {\n        destroy: function() {\n            element.removeEventListener(type, listenerFn, useCapture);\n        }\n    }\n}\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element|String|Array} [elements]\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction delegate(elements, selector, type, callback, useCapture) {\n    // Handle the regular Element usage\n    if (typeof elements.addEventListener === 'function') {\n        return _delegate.apply(null, arguments);\n    }\n\n    // Handle Element-less usage, it defaults to global delegation\n    if (typeof type === 'function') {\n        // Use `document` as the first parameter, then apply arguments\n        // This is a short way to .unshift `arguments` without running into deoptimizations\n        return _delegate.bind(null, document).apply(null, arguments);\n    }\n\n    // Handle Selector-based usage\n    if (typeof elements === 'string') {\n        elements = document.querySelectorAll(elements);\n    }\n\n    // Handle Array-like based usage\n    return Array.prototype.map.call(elements, function (element) {\n        return _delegate(element, selector, type, callback, useCapture);\n    });\n}\n\n/**\n * Finds closest match and invokes callback.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Function}\n */\nfunction listener(element, selector, type, callback) {\n    return function(e) {\n        e.delegateTarget = closest(e.target, selector);\n\n        if (e.delegateTarget) {\n            callback.call(element, e);\n        }\n    }\n}\n\nmodule.exports = delegate;\n\n\n/***/ }),\n\n/***/ 879:\n/***/ (function(__unused_webpack_module, exports) {\n\n/**\n * Check if argument is a HTML element.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.node = function(value) {\n    return value !== undefined\n        && value instanceof HTMLElement\n        && value.nodeType === 1;\n};\n\n/**\n * Check if argument is a list of HTML elements.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.nodeList = function(value) {\n    var type = Object.prototype.toString.call(value);\n\n    return value !== undefined\n        && (type === '[object NodeList]' || type === '[object HTMLCollection]')\n        && ('length' in value)\n        && (value.length === 0 || exports.node(value[0]));\n};\n\n/**\n * Check if argument is a string.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.string = function(value) {\n    return typeof value === 'string'\n        || value instanceof String;\n};\n\n/**\n * Check if argument is a function.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.fn = function(value) {\n    var type = Object.prototype.toString.call(value);\n\n    return type === '[object Function]';\n};\n\n\n/***/ }),\n\n/***/ 370:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar is = __webpack_require__(879);\nvar delegate = __webpack_require__(438);\n\n/**\n * Validates all params and calls the right\n * listener function based on its target type.\n *\n * @param {String|HTMLElement|HTMLCollection|NodeList} target\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listen(target, type, callback) {\n    if (!target && !type && !callback) {\n        throw new Error('Missing required arguments');\n    }\n\n    if (!is.string(type)) {\n        throw new TypeError('Second argument must be a String');\n    }\n\n    if (!is.fn(callback)) {\n        throw new TypeError('Third argument must be a Function');\n    }\n\n    if (is.node(target)) {\n        return listenNode(target, type, callback);\n    }\n    else if (is.nodeList(target)) {\n        return listenNodeList(target, type, callback);\n    }\n    else if (is.string(target)) {\n        return listenSelector(target, type, callback);\n    }\n    else {\n        throw new TypeError('First argument must be a String, HTMLElement, HTMLCollection, or NodeList');\n    }\n}\n\n/**\n * Adds an event listener to a HTML element\n * and returns a remove listener function.\n *\n * @param {HTMLElement} node\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNode(node, type, callback) {\n    node.addEventListener(type, callback);\n\n    return {\n        destroy: function() {\n            node.removeEventListener(type, callback);\n        }\n    }\n}\n\n/**\n * Add an event listener to a list of HTML elements\n * and returns a remove listener function.\n *\n * @param {NodeList|HTMLCollection} nodeList\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNodeList(nodeList, type, callback) {\n    Array.prototype.forEach.call(nodeList, function(node) {\n        node.addEventListener(type, callback);\n    });\n\n    return {\n        destroy: function() {\n            Array.prototype.forEach.call(nodeList, function(node) {\n                node.removeEventListener(type, callback);\n            });\n        }\n    }\n}\n\n/**\n * Add an event listener to a selector\n * and returns a remove listener function.\n *\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenSelector(selector, type, callback) {\n    return delegate(document.body, selector, type, callback);\n}\n\nmodule.exports = listen;\n\n\n/***/ }),\n\n/***/ 817:\n/***/ (function(module) {\n\nfunction select(element) {\n    var selectedText;\n\n    if (element.nodeName === 'SELECT') {\n        element.focus();\n\n        selectedText = element.value;\n    }\n    else if (element.nodeName === 'INPUT' || element.nodeName === 'TEXTAREA') {\n        var isReadOnly = element.hasAttribute('readonly');\n\n        if (!isReadOnly) {\n            element.setAttribute('readonly', '');\n        }\n\n        element.select();\n        element.setSelectionRange(0, element.value.length);\n\n        if (!isReadOnly) {\n            element.removeAttribute('readonly');\n        }\n\n        selectedText = element.value;\n    }\n    else {\n        if (element.hasAttribute('contenteditable')) {\n            element.focus();\n        }\n\n        var selection = window.getSelection();\n        var range = document.createRange();\n\n        range.selectNodeContents(element);\n        selection.removeAllRanges();\n        selection.addRange(range);\n\n        selectedText = selection.toString();\n    }\n\n    return selectedText;\n}\n\nmodule.exports = select;\n\n\n/***/ }),\n\n/***/ 279:\n/***/ (function(module) {\n\nfunction E () {\n  // Keep this empty so it's easier to inherit from\n  // (via https://github.com/lipsmack from https://github.com/scottcorgan/tiny-emitter/issues/3)\n}\n\nE.prototype = {\n  on: function (name, callback, ctx) {\n    var e = this.e || (this.e = {});\n\n    (e[name] || (e[name] = [])).push({\n      fn: callback,\n      ctx: ctx\n    });\n\n    return this;\n  },\n\n  once: function (name, callback, ctx) {\n    var self = this;\n    function listener () {\n      self.off(name, listener);\n      callback.apply(ctx, arguments);\n    };\n\n    listener._ = callback\n    return this.on(name, listener, ctx);\n  },\n\n  emit: function (name) {\n    var data = [].slice.call(arguments, 1);\n    var evtArr = ((this.e || (this.e = {}))[name] || []).slice();\n    var i = 0;\n    var len = evtArr.length;\n\n    for (i; i < len; i++) {\n      evtArr[i].fn.apply(evtArr[i].ctx, data);\n    }\n\n    return this;\n  },\n\n  off: function (name, callback) {\n    var e = this.e || (this.e = {});\n    var evts = e[name];\n    var liveEvents = [];\n\n    if (evts && callback) {\n      for (var i = 0, len = evts.length; i < len; i++) {\n        if (evts[i].fn !== callback && evts[i].fn._ !== callback)\n          liveEvents.push(evts[i]);\n      }\n    }\n\n    // Remove event from queue to prevent memory leak\n    // Suggested by https://github.com/lazd\n    // Ref: https://github.com/scottcorgan/tiny-emitter/commit/c6ebfaa9bc973b33d110a84a307742b7cf94c953#commitcomment-5024910\n\n    (liveEvents.length)\n      ? e[name] = liveEvents\n      : delete e[name];\n\n    return this;\n  }\n};\n\nmodule.exports = E;\nmodule.exports.TinyEmitter = E;\n\n\n/***/ })\n\n/******/ \t});\n/************************************************************************/\n/******/ \t// The module cache\n/******/ \tvar __webpack_module_cache__ = {};\n/******/ \t\n/******/ \t// The require function\n/******/ \tfunction __webpack_require__(moduleId) {\n/******/ \t\t// Check if module is in cache\n/******/ \t\tif(__webpack_module_cache__[moduleId]) {\n/******/ \t\t\treturn __webpack_module_cache__[moduleId].exports;\n/******/ \t\t}\n/******/ \t\t// Create a new module (and put it into the cache)\n/******/ \t\tvar module = __webpack_module_cache__[moduleId] = {\n/******/ \t\t\t// no module.id needed\n/******/ \t\t\t// no module.loaded needed\n/******/ \t\t\texports: {}\n/******/ \t\t};\n/******/ \t\n/******/ \t\t// Execute the module function\n/******/ \t\t__webpack_modules__[moduleId](module, module.exports, __webpack_require__);\n/******/ \t\n/******/ \t\t// Return the exports of the module\n/******/ \t\treturn module.exports;\n/******/ \t}\n/******/ \t\n/************************************************************************/\n/******/ \t/* webpack/runtime/compat get default export */\n/******/ \t!function() {\n/******/ \t\t// getDefaultExport function for compatibility with non-harmony modules\n/******/ \t\t__webpack_require__.n = function(module) {\n/******/ \t\t\tvar getter = module && module.__esModule ?\n/******/ \t\t\t\tfunction() { return module['default']; } :\n/******/ \t\t\t\tfunction() { return module; };\n/******/ \t\t\t__webpack_require__.d(getter, { a: getter });\n/******/ \t\t\treturn getter;\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/define property getters */\n/******/ \t!function() {\n/******/ \t\t// define getter functions for harmony exports\n/******/ \t\t__webpack_require__.d = function(exports, definition) {\n/******/ \t\t\tfor(var key in definition) {\n/******/ \t\t\t\tif(__webpack_require__.o(definition, key) && !__webpack_require__.o(exports, key)) {\n/******/ \t\t\t\t\tObject.defineProperty(exports, key, { enumerable: true, get: definition[key] });\n/******/ \t\t\t\t}\n/******/ \t\t\t}\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/hasOwnProperty shorthand */\n/******/ \t!function() {\n/******/ \t\t__webpack_require__.o = function(obj, prop) { return Object.prototype.hasOwnProperty.call(obj, prop); }\n/******/ \t}();\n/******/ \t\n/************************************************************************/\n/******/ \t// module exports must be returned from runtime so entry inlining is disabled\n/******/ \t// startup\n/******/ \t// Load entry module and return exports\n/******/ \treturn __webpack_require__(686);\n/******/ })()\n.default;\n});", "/*!\n * escape-html\n * Copyright(c) 2012-2013 TJ Holowaychuk\n * Copyright(c) 2015 Andreas Lubbe\n * Copyright(c) 2015 Tiancheng \"Timothy\" Gu\n * MIT Licensed\n */\n\n'use strict';\n\n/**\n * Module variables.\n * @private\n */\n\nvar matchHtmlRegExp = /[\"'&<>]/;\n\n/**\n * Module exports.\n * @public\n */\n\nmodule.exports = escapeHtml;\n\n/**\n * Escape special characters in the given string of html.\n *\n * @param  {string} string The string to escape for inserting into HTML\n * @return {string}\n * @public\n */\n\nfunction escapeHtml(string) {\n  var str = '' + string;\n  var match = matchHtmlRegExp.exec(str);\n\n  if (!match) {\n    return str;\n  }\n\n  var escape;\n  var html = '';\n  var index = 0;\n  var lastIndex = 0;\n\n  for (index = match.index; index < str.length; index++) {\n    switch (str.charCodeAt(index)) {\n      case 34: // \"\n        escape = '&quot;';\n        break;\n      case 38: // &\n        escape = '&amp;';\n        break;\n      case 39: // '\n        escape = '&#39;';\n        break;\n      case 60: // <\n        escape = '&lt;';\n        break;\n      case 62: // >\n        escape = '&gt;';\n        break;\n      default:\n        continue;\n    }\n\n    if (lastIndex !== index) {\n      html += str.substring(lastIndex, index);\n    }\n\n    lastIndex = index + 1;\n    html += escape;\n  }\n\n  return lastIndex !== index\n    ? html + str.substring(lastIndex, index)\n    : html;\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport \"focus-visible\"\n\nimport {\n  EMPTY,\n  NEVER,\n  Observable,\n  Subject,\n  defer,\n  delay,\n  filter,\n  map,\n  merge,\n  mergeWith,\n  shareReplay,\n  switchMap\n} from \"rxjs\"\n\nimport { configuration, feature } from \"./_\"\nimport {\n  at,\n  getActiveElement,\n  getOptionalElement,\n  requestJSON,\n  setLocation,\n  setToggle,\n  watchDocument,\n  watchKeyboard,\n  watchLocation,\n  watchLocationTarget,\n  watchMedia,\n  watchPrint,\n  watchScript,\n  watchViewport\n} from \"./browser\"\nimport {\n  getComponentElement,\n  getComponentElements,\n  mountAnnounce,\n  mountBackToTop,\n  mountConsent,\n  mountContent,\n  mountDialog,\n  mountHeader,\n  mountHeaderTitle,\n  mountPalette,\n  mountSearch,\n  mountSearchHiglight,\n  mountSidebar,\n  mountSource,\n  mountTableOfContents,\n  mountTabs,\n  watchHeader,\n  watchMain\n} from \"./components\"\nimport {\n  SearchIndex,\n  setupClipboardJS,\n  setupInstantLoading,\n  setupVersionSelector\n} from \"./integrations\"\nimport {\n  patchIndeterminate,\n  patchScrollfix,\n  patchScrolllock\n} from \"./patches\"\nimport \"./polyfills\"\n\n/* ----------------------------------------------------------------------------\n * Functions - @todo refactor\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch search index\n *\n * @returns Search index observable\n */\nfunction fetchSearchIndex(): Observable<SearchIndex> {\n  if (location.protocol === \"file:\") {\n    return watchScript(\n      `${new URL(\"search/search_index.js\", config.base)}`\n    )\n      .pipe(\n        // @ts-ignore - @todo fix typings\n        map(() => __index),\n        shareReplay(1)\n      )\n  } else {\n    return requestJSON<SearchIndex>(\n      new URL(\"search/search_index.json\", config.base)\n    )\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Application\n * ------------------------------------------------------------------------- */\n\n/* Yay, JavaScript is available */\ndocument.documentElement.classList.remove(\"no-js\")\ndocument.documentElement.classList.add(\"js\")\n\n/* Set up navigation observables and subjects */\nconst document$ = watchDocument()\nconst location$ = watchLocation()\nconst target$   = watchLocationTarget(location$)\nconst keyboard$ = watchKeyboard()\n\n/* Set up media observables */\nconst viewport$ = watchViewport()\nconst tablet$   = watchMedia(\"(min-width: 960px)\")\nconst screen$   = watchMedia(\"(min-width: 1220px)\")\nconst print$    = watchPrint()\n\n/* Retrieve search index, if search is enabled */\nconst config = configuration()\nconst index$ = document.forms.namedItem(\"search\")\n  ? fetchSearchIndex()\n  : NEVER\n\n/* Set up Clipboard.js integration */\nconst alert$ = new Subject<string>()\nsetupClipboardJS({ alert$ })\n\n/* Set up instant loading, if enabled */\nif (feature(\"navigation.instant\"))\n  setupInstantLoading({ location$, viewport$ })\n    .subscribe(document$)\n\n/* Set up version selector */\nif (config.version?.provider === \"mike\")\n  setupVersionSelector({ document$ })\n\n/* Always close drawer and search on navigation */\nmerge(location$, target$)\n  .pipe(\n    delay(125)\n  )\n    .subscribe(() => {\n      setToggle(\"drawer\", false)\n      setToggle(\"search\", false)\n    })\n\n/* Set up global keyboard handlers */\nkeyboard$\n  .pipe(\n    filter(({ mode }) => mode === \"global\")\n  )\n    .subscribe(key => {\n      switch (key.type) {\n\n        /* Go to previous page */\n        case \"p\":\n        case \",\":\n          const prev = getOptionalElement<HTMLLinkElement>(\"link[rel=prev]\")\n          if (typeof prev !== \"undefined\")\n            setLocation(prev)\n          break\n\n        /* Go to next page */\n        case \"n\":\n        case \".\":\n          const next = getOptionalElement<HTMLLinkElement>(\"link[rel=next]\")\n          if (typeof next !== \"undefined\")\n            setLocation(next)\n          break\n\n        /* Expand navigation, see https://bit.ly/3ZjG5io */\n        case \"Enter\":\n          const active = getActiveElement()\n          if (active instanceof HTMLLabelElement)\n            active.click()\n      }\n    })\n\n/* Set up patches */\npatchIndeterminate({ document$, tablet$ })\npatchScrollfix({ document$ })\npatchScrolllock({ viewport$, tablet$ })\n\n/* Set up header and main area observable */\nconst header$ = watchHeader(getComponentElement(\"header\"), { viewport$ })\nconst main$ = document$\n  .pipe(\n    map(() => getComponentElement(\"main\")),\n    switchMap(el => watchMain(el, { viewport$, header$ })),\n    shareReplay(1)\n  )\n\n/* Set up control component observables */\nconst control$ = merge(\n\n  /* Consent */\n  ...getComponentElements(\"consent\")\n    .map(el => mountConsent(el, { target$ })),\n\n  /* Dialog */\n  ...getComponentElements(\"dialog\")\n    .map(el => mountDialog(el, { alert$ })),\n\n  /* Header */\n  ...getComponentElements(\"header\")\n    .map(el => mountHeader(el, { viewport$, header$, main$ })),\n\n  /* Color palette */\n  ...getComponentElements(\"palette\")\n    .map(el => mountPalette(el)),\n\n  /* Search */\n  ...getComponentElements(\"search\")\n    .map(el => mountSearch(el, { index$, keyboard$ })),\n\n  /* Repository information */\n  ...getComponentElements(\"source\")\n    .map(el => mountSource(el))\n)\n\n/* Set up content component observables */\nconst content$ = defer(() => merge(\n\n  /* Announcement bar */\n  ...getComponentElements(\"announce\")\n    .map(el => mountAnnounce(el)),\n\n  /* Content */\n  ...getComponentElements(\"content\")\n    .map(el => mountContent(el, { viewport$, target$, print$ })),\n\n  /* Search highlighting */\n  ...getComponentElements(\"content\")\n    .map(el => feature(\"search.highlight\")\n      ? mountSearchHiglight(el, { index$, location$ })\n      : EMPTY\n    ),\n\n  /* Header title */\n  ...getComponentElements(\"header-title\")\n    .map(el => mountHeaderTitle(el, { viewport$, header$ })),\n\n  /* Sidebar */\n  ...getComponentElements(\"sidebar\")\n    .map(el => el.getAttribute(\"data-md-type\") === \"navigation\"\n      ? at(screen$, () => mountSidebar(el, { viewport$, header$, main$ }))\n      : at(tablet$, () => mountSidebar(el, { viewport$, header$, main$ }))\n    ),\n\n  /* Navigation tabs */\n  ...getComponentElements(\"tabs\")\n    .map(el => mountTabs(el, { viewport$, header$ })),\n\n  /* Table of contents */\n  ...getComponentElements(\"toc\")\n    .map(el => mountTableOfContents(el, {\n      viewport$, header$, main$, target$\n    })),\n\n  /* Back-to-top button */\n  ...getComponentElements(\"top\")\n    .map(el => mountBackToTop(el, { viewport$, header$, main$, target$ }))\n))\n\n/* Set up component observables */\nconst component$ = document$\n  .pipe(\n    switchMap(() => content$),\n    mergeWith(control$),\n    shareReplay(1)\n  )\n\n/* Subscribe to all components */\ncomponent$.subscribe()\n\n/* ----------------------------------------------------------------------------\n * Exports\n * ------------------------------------------------------------------------- */\n\nwindow.document$  = document$          /* Document observable */\nwindow.location$  = location$          /* Location subject */\nwindow.target$    = target$            /* Location target observable */\nwindow.keyboard$  = keyboard$          /* Keyboard observable */\nwindow.viewport$  = viewport$          /* Viewport observable */\nwindow.tablet$    = tablet$            /* Media tablet observable */\nwindow.screen$    = screen$            /* Media screen observable */\nwindow.print$     = print$             /* Media print observable */\nwindow.alert$     = alert$             /* Alert subject */\nwindow.component$ = component$         /* Component observable */\n", "/*! *****************************************************************************\r\nCopyright (c) Microsoft Corporation.\r\n\r\nPermission to use, copy, modify, and/or distribute this software for any\r\npurpose with or without fee is hereby granted.\r\n\r\nTHE SOFTWARE IS PROVIDED \"AS IS\" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH\r\nREGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY\r\nAND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT,\r\nINDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM\r\nLOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR\r\nOTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR\r\nPERFORMANCE OF THIS SOFTWARE.\r\n***************************************************************************** */\r\n/* global Reflect, Promise */\r\n\r\nvar extendStatics = function(d, b) {\r\n    extendStatics = Object.setPrototypeOf ||\r\n        ({ __proto__: [] } instanceof Array && function (d, b) { d.__proto__ = b; }) ||\r\n        function (d, b) { for (var p in b) if (Object.prototype.hasOwnProperty.call(b, p)) d[p] = b[p]; };\r\n    return extendStatics(d, b);\r\n};\r\n\r\nexport function __extends(d, b) {\r\n    if (typeof b !== \"function\" && b !== null)\r\n        throw new TypeError(\"Class extends value \" + String(b) + \" is not a constructor or null\");\r\n    extendStatics(d, b);\r\n    function __() { this.constructor = d; }\r\n    d.prototype = b === null ? Object.create(b) : (__.prototype = b.prototype, new __());\r\n}\r\n\r\nexport var __assign = function() {\r\n    __assign = Object.assign || function __assign(t) {\r\n        for (var s, i = 1, n = arguments.length; i < n; i++) {\r\n            s = arguments[i];\r\n            for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p)) t[p] = s[p];\r\n        }\r\n        return t;\r\n    }\r\n    return __assign.apply(this, arguments);\r\n}\r\n\r\nexport function __rest(s, e) {\r\n    var t = {};\r\n    for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p) && e.indexOf(p) < 0)\r\n        t[p] = s[p];\r\n    if (s != null && typeof Object.getOwnPropertySymbols === \"function\")\r\n        for (var i = 0, p = Object.getOwnPropertySymbols(s); i < p.length; i++) {\r\n            if (e.indexOf(p[i]) < 0 && Object.prototype.propertyIsEnumerable.call(s, p[i]))\r\n                t[p[i]] = s[p[i]];\r\n        }\r\n    return t;\r\n}\r\n\r\nexport function __decorate(decorators, target, key, desc) {\r\n    var c = arguments.length, r = c < 3 ? target : desc === null ? desc = Object.getOwnPropertyDescriptor(target, key) : desc, d;\r\n    if (typeof Reflect === \"object\" && typeof Reflect.decorate === \"function\") r = Reflect.decorate(decorators, target, key, desc);\r\n    else for (var i = decorators.length - 1; i >= 0; i--) if (d = decorators[i]) r = (c < 3 ? d(r) : c > 3 ? d(target, key, r) : d(target, key)) || r;\r\n    return c > 3 && r && Object.defineProperty(target, key, r), r;\r\n}\r\n\r\nexport function __param(paramIndex, decorator) {\r\n    return function (target, key) { decorator(target, key, paramIndex); }\r\n}\r\n\r\nexport function __metadata(metadataKey, metadataValue) {\r\n    if (typeof Reflect === \"object\" && typeof Reflect.metadata === \"function\") return Reflect.metadata(metadataKey, metadataValue);\r\n}\r\n\r\nexport function __awaiter(thisArg, _arguments, P, generator) {\r\n    function adopt(value) { return value instanceof P ? value : new P(function (resolve) { resolve(value); }); }\r\n    return new (P || (P = Promise))(function (resolve, reject) {\r\n        function fulfilled(value) { try { step(generator.next(value)); } catch (e) { reject(e); } }\r\n        function rejected(value) { try { step(generator[\"throw\"](value)); } catch (e) { reject(e); } }\r\n        function step(result) { result.done ? resolve(result.value) : adopt(result.value).then(fulfilled, rejected); }\r\n        step((generator = generator.apply(thisArg, _arguments || [])).next());\r\n    });\r\n}\r\n\r\nexport function __generator(thisArg, body) {\r\n    var _ = { label: 0, sent: function() { if (t[0] & 1) throw t[1]; return t[1]; }, trys: [], ops: [] }, f, y, t, g;\r\n    return g = { next: verb(0), \"throw\": verb(1), \"return\": verb(2) }, typeof Symbol === \"function\" && (g[Symbol.iterator] = function() { return this; }), g;\r\n    function verb(n) { return function (v) { return step([n, v]); }; }\r\n    function step(op) {\r\n        if (f) throw new TypeError(\"Generator is already executing.\");\r\n        while (_) try {\r\n            if (f = 1, y && (t = op[0] & 2 ? y[\"return\"] : op[0] ? y[\"throw\"] || ((t = y[\"return\"]) && t.call(y), 0) : y.next) && !(t = t.call(y, op[1])).done) return t;\r\n            if (y = 0, t) op = [op[0] & 2, t.value];\r\n            switch (op[0]) {\r\n                case 0: case 1: t = op; break;\r\n                case 4: _.label++; return { value: op[1], done: false };\r\n                case 5: _.label++; y = op[1]; op = [0]; continue;\r\n                case 7: op = _.ops.pop(); _.trys.pop(); continue;\r\n                default:\r\n                    if (!(t = _.trys, t = t.length > 0 && t[t.length - 1]) && (op[0] === 6 || op[0] === 2)) { _ = 0; continue; }\r\n                    if (op[0] === 3 && (!t || (op[1] > t[0] && op[1] < t[3]))) { _.label = op[1]; break; }\r\n                    if (op[0] === 6 && _.label < t[1]) { _.label = t[1]; t = op; break; }\r\n                    if (t && _.label < t[2]) { _.label = t[2]; _.ops.push(op); break; }\r\n                    if (t[2]) _.ops.pop();\r\n                    _.trys.pop(); continue;\r\n            }\r\n            op = body.call(thisArg, _);\r\n        } catch (e) { op = [6, e]; y = 0; } finally { f = t = 0; }\r\n        if (op[0] & 5) throw op[1]; return { value: op[0] ? op[1] : void 0, done: true };\r\n    }\r\n}\r\n\r\nexport var __createBinding = Object.create ? (function(o, m, k, k2) {\r\n    if (k2 === undefined) k2 = k;\r\n    Object.defineProperty(o, k2, { enumerable: true, get: function() { return m[k]; } });\r\n}) : (function(o, m, k, k2) {\r\n    if (k2 === undefined) k2 = k;\r\n    o[k2] = m[k];\r\n});\r\n\r\nexport function __exportStar(m, o) {\r\n    for (var p in m) if (p !== \"default\" && !Object.prototype.hasOwnProperty.call(o, p)) __createBinding(o, m, p);\r\n}\r\n\r\nexport function __values(o) {\r\n    var s = typeof Symbol === \"function\" && Symbol.iterator, m = s && o[s], i = 0;\r\n    if (m) return m.call(o);\r\n    if (o && typeof o.length === \"number\") return {\r\n        next: function () {\r\n            if (o && i >= o.length) o = void 0;\r\n            return { value: o && o[i++], done: !o };\r\n        }\r\n    };\r\n    throw new TypeError(s ? \"Object is not iterable.\" : \"Symbol.iterator is not defined.\");\r\n}\r\n\r\nexport function __read(o, n) {\r\n    var m = typeof Symbol === \"function\" && o[Symbol.iterator];\r\n    if (!m) return o;\r\n    var i = m.call(o), r, ar = [], e;\r\n    try {\r\n        while ((n === void 0 || n-- > 0) && !(r = i.next()).done) ar.push(r.value);\r\n    }\r\n    catch (error) { e = { error: error }; }\r\n    finally {\r\n        try {\r\n            if (r && !r.done && (m = i[\"return\"])) m.call(i);\r\n        }\r\n        finally { if (e) throw e.error; }\r\n    }\r\n    return ar;\r\n}\r\n\r\n/** @deprecated */\r\nexport function __spread() {\r\n    for (var ar = [], i = 0; i < arguments.length; i++)\r\n        ar = ar.concat(__read(arguments[i]));\r\n    return ar;\r\n}\r\n\r\n/** @deprecated */\r\nexport function __spreadArrays() {\r\n    for (var s = 0, i = 0, il = arguments.length; i < il; i++) s += arguments[i].length;\r\n    for (var r = Array(s), k = 0, i = 0; i < il; i++)\r\n        for (var a = arguments[i], j = 0, jl = a.length; j < jl; j++, k++)\r\n            r[k] = a[j];\r\n    return r;\r\n}\r\n\r\nexport function __spreadArray(to, from, pack) {\r\n    if (pack || arguments.length === 2) for (var i = 0, l = from.length, ar; i < l; i++) {\r\n        if (ar || !(i in from)) {\r\n            if (!ar) ar = Array.prototype.slice.call(from, 0, i);\r\n            ar[i] = from[i];\r\n        }\r\n    }\r\n    return to.concat(ar || Array.prototype.slice.call(from));\r\n}\r\n\r\nexport function __await(v) {\r\n    return this instanceof __await ? (this.v = v, this) : new __await(v);\r\n}\r\n\r\nexport function __asyncGenerator(thisArg, _arguments, generator) {\r\n    if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\r\n    var g = generator.apply(thisArg, _arguments || []), i, q = [];\r\n    return i = {}, verb(\"next\"), verb(\"throw\"), verb(\"return\"), i[Symbol.asyncIterator] = function () { return this; }, i;\r\n    function verb(n) { if (g[n]) i[n] = function (v) { return new Promise(function (a, b) { q.push([n, v, a, b]) > 1 || resume(n, v); }); }; }\r\n    function resume(n, v) { try { step(g[n](v)); } catch (e) { settle(q[0][3], e); } }\r\n    function step(r) { r.value instanceof __await ? Promise.resolve(r.value.v).then(fulfill, reject) : settle(q[0][2], r); }\r\n    function fulfill(value) { resume(\"next\", value); }\r\n    function reject(value) { resume(\"throw\", value); }\r\n    function settle(f, v) { if (f(v), q.shift(), q.length) resume(q[0][0], q[0][1]); }\r\n}\r\n\r\nexport function __asyncDelegator(o) {\r\n    var i, p;\r\n    return i = {}, verb(\"next\"), verb(\"throw\", function (e) { throw e; }), verb(\"return\"), i[Symbol.iterator] = function () { return this; }, i;\r\n    function verb(n, f) { i[n] = o[n] ? function (v) { return (p = !p) ? { value: __await(o[n](v)), done: n === \"return\" } : f ? f(v) : v; } : f; }\r\n}\r\n\r\nexport function __asyncValues(o) {\r\n    if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\r\n    var m = o[Symbol.asyncIterator], i;\r\n    return m ? m.call(o) : (o = typeof __values === \"function\" ? __values(o) : o[Symbol.iterator](), i = {}, verb(\"next\"), verb(\"throw\"), verb(\"return\"), i[Symbol.asyncIterator] = function () { return this; }, i);\r\n    function verb(n) { i[n] = o[n] && function (v) { return new Promise(function (resolve, reject) { v = o[n](v), settle(resolve, reject, v.done, v.value); }); }; }\r\n    function settle(resolve, reject, d, v) { Promise.resolve(v).then(function(v) { resolve({ value: v, done: d }); }, reject); }\r\n}\r\n\r\nexport function __makeTemplateObject(cooked, raw) {\r\n    if (Object.defineProperty) { Object.defineProperty(cooked, \"raw\", { value: raw }); } else { cooked.raw = raw; }\r\n    return cooked;\r\n};\r\n\r\nvar __setModuleDefault = Object.create ? (function(o, v) {\r\n    Object.defineProperty(o, \"default\", { enumerable: true, value: v });\r\n}) : function(o, v) {\r\n    o[\"default\"] = v;\r\n};\r\n\r\nexport function __importStar(mod) {\r\n    if (mod && mod.__esModule) return mod;\r\n    var result = {};\r\n    if (mod != null) for (var k in mod) if (k !== \"default\" && Object.prototype.hasOwnProperty.call(mod, k)) __createBinding(result, mod, k);\r\n    __setModuleDefault(result, mod);\r\n    return result;\r\n}\r\n\r\nexport function __importDefault(mod) {\r\n    return (mod && mod.__esModule) ? mod : { default: mod };\r\n}\r\n\r\nexport function __classPrivateFieldGet(receiver, state, kind, f) {\r\n    if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a getter\");\r\n    if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot read private member from an object whose class did not declare it\");\r\n    return kind === \"m\" ? f : kind === \"a\" ? f.call(receiver) : f ? f.value : state.get(receiver);\r\n}\r\n\r\nexport function __classPrivateFieldSet(receiver, state, value, kind, f) {\r\n    if (kind === \"m\") throw new TypeError(\"Private method is not writable\");\r\n    if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a setter\");\r\n    if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot write private member to an object whose class did not declare it\");\r\n    return (kind === \"a\" ? f.call(receiver, value) : f ? f.value = value : state.set(receiver, value)), value;\r\n}\r\n", "/**\n * Returns true if the object is a function.\n * @param value The value to check\n */\nexport function isFunction(value: any): value is (...args: any[]) => any {\n  return typeof value === 'function';\n}\n", "/**\n * Used to create Error subclasses until the community moves away from ES5.\n *\n * This is because compiling from TypeScript down to ES5 has issues with subclassing Errors\n * as well as other built-in types: https://github.com/Microsoft/TypeScript/issues/12123\n *\n * @param createImpl A factory function to create the actual constructor implementation. The returned\n * function should be a named function that calls `_super` internally.\n */\nexport function createErrorClass<T>(createImpl: (_super: any) => any): T {\n  const _super = (instance: any) => {\n    Error.call(instance);\n    instance.stack = new Error().stack;\n  };\n\n  const ctorFunc = createImpl(_super);\n  ctorFunc.prototype = Object.create(Error.prototype);\n  ctorFunc.prototype.constructor = ctorFunc;\n  return ctorFunc;\n}\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface UnsubscriptionError extends Error {\n  readonly errors: any[];\n}\n\nexport interface UnsubscriptionErrorCtor {\n  /**\n   * @deprecated Internal implementation detail. Do not construct error instances.\n   * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n   */\n  new (errors: any[]): UnsubscriptionError;\n}\n\n/**\n * An error thrown when one or more errors have occurred during the\n * `unsubscribe` of a {@link Subscription}.\n */\nexport const UnsubscriptionError: UnsubscriptionErrorCtor = createErrorClass(\n  (_super) =>\n    function UnsubscriptionErrorImpl(this: any, errors: (Error | string)[]) {\n      _super(this);\n      this.message = errors\n        ? `${errors.length} errors occurred during unsubscription:\n${errors.map((err, i) => `${i + 1}) ${err.toString()}`).join('\\n  ')}`\n        : '';\n      this.name = 'UnsubscriptionError';\n      this.errors = errors;\n    }\n);\n", "/**\n * Removes an item from an array, mutating it.\n * @param arr The array to remove the item from\n * @param item The item to remove\n */\nexport function arrRemove<T>(arr: T[] | undefined | null, item: T) {\n  if (arr) {\n    const index = arr.indexOf(item);\n    0 <= index && arr.splice(index, 1);\n  }\n}\n", "import { isFunction } from './util/isFunction';\nimport { UnsubscriptionError } from './util/UnsubscriptionError';\nimport { SubscriptionLike, TeardownLogic, Unsubscribable } from './types';\nimport { arrRemove } from './util/arrRemove';\n\n/**\n * Represents a disposable resource, such as the execution of an Observable. A\n * Subscription has one important method, `unsubscribe`, that takes no argument\n * and just disposes the resource held by the subscription.\n *\n * Additionally, subscriptions may be grouped together through the `add()`\n * method, which will attach a child Subscription to the current Subscription.\n * When a Subscription is unsubscribed, all its children (and its grandchildren)\n * will be unsubscribed as well.\n *\n * @class Subscription\n */\nexport class Subscription implements SubscriptionLike {\n  /** @nocollapse */\n  public static EMPTY = (() => {\n    const empty = new Subscription();\n    empty.closed = true;\n    return empty;\n  })();\n\n  /**\n   * A flag to indicate whether this Subscription has already been unsubscribed.\n   */\n  public closed = false;\n\n  private _parentage: Subscription[] | Subscription | null = null;\n\n  /**\n   * The list of registered finalizers to execute upon unsubscription. Adding and removing from this\n   * list occurs in the {@link #add} and {@link #remove} methods.\n   */\n  private _finalizers: Exclude<TeardownLogic, void>[] | null = null;\n\n  /**\n   * @param initialTeardown A function executed first as part of the finalization\n   * process that is kicked off when {@link #unsubscribe} is called.\n   */\n  constructor(private initialTeardown?: () => void) {}\n\n  /**\n   * Disposes the resources held by the subscription. May, for instance, cancel\n   * an ongoing Observable execution or cancel any other type of work that\n   * started when the Subscription was created.\n   * @return {void}\n   */\n  unsubscribe(): void {\n    let errors: any[] | undefined;\n\n    if (!this.closed) {\n      this.closed = true;\n\n      // Remove this from it's parents.\n      const { _parentage } = this;\n      if (_parentage) {\n        this._parentage = null;\n        if (Array.isArray(_parentage)) {\n          for (const parent of _parentage) {\n            parent.remove(this);\n          }\n        } else {\n          _parentage.remove(this);\n        }\n      }\n\n      const { initialTeardown: initialFinalizer } = this;\n      if (isFunction(initialFinalizer)) {\n        try {\n          initialFinalizer();\n        } catch (e) {\n          errors = e instanceof UnsubscriptionError ? e.errors : [e];\n        }\n      }\n\n      const { _finalizers } = this;\n      if (_finalizers) {\n        this._finalizers = null;\n        for (const finalizer of _finalizers) {\n          try {\n            execFinalizer(finalizer);\n          } catch (err) {\n            errors = errors ?? [];\n            if (err instanceof UnsubscriptionError) {\n              errors = [...errors, ...err.errors];\n            } else {\n              errors.push(err);\n            }\n          }\n        }\n      }\n\n      if (errors) {\n        throw new UnsubscriptionError(errors);\n      }\n    }\n  }\n\n  /**\n   * Adds a finalizer to this subscription, so that finalization will be unsubscribed/called\n   * when this subscription is unsubscribed. If this subscription is already {@link #closed},\n   * because it has already been unsubscribed, then whatever finalizer is passed to it\n   * will automatically be executed (unless the finalizer itself is also a closed subscription).\n   *\n   * Closed Subscriptions cannot be added as finalizers to any subscription. Adding a closed\n   * subscription to a any subscription will result in no operation. (A noop).\n   *\n   * Adding a subscription to itself, or adding `null` or `undefined` will not perform any\n   * operation at all. (A noop).\n   *\n   * `Subscription` instances that are added to this instance will automatically remove themselves\n   * if they are unsubscribed. Functions and {@link Unsubscribable} objects that you wish to remove\n   * will need to be removed manually with {@link #remove}\n   *\n   * @param teardown The finalization logic to add to this subscription.\n   */\n  add(teardown: TeardownLogic): void {\n    // Only add the finalizer if it's not undefined\n    // and don't add a subscription to itself.\n    if (teardown && teardown !== this) {\n      if (this.closed) {\n        // If this subscription is already closed,\n        // execute whatever finalizer is handed to it automatically.\n        execFinalizer(teardown);\n      } else {\n        if (teardown instanceof Subscription) {\n          // We don't add closed subscriptions, and we don't add the same subscription\n          // twice. Subscription unsubscribe is idempotent.\n          if (teardown.closed || teardown._hasParent(this)) {\n            return;\n          }\n          teardown._addParent(this);\n        }\n        (this._finalizers = this._finalizers ?? []).push(teardown);\n      }\n    }\n  }\n\n  /**\n   * Checks to see if a this subscription already has a particular parent.\n   * This will signal that this subscription has already been added to the parent in question.\n   * @param parent the parent to check for\n   */\n  private _hasParent(parent: Subscription) {\n    const { _parentage } = this;\n    return _parentage === parent || (Array.isArray(_parentage) && _parentage.includes(parent));\n  }\n\n  /**\n   * Adds a parent to this subscription so it can be removed from the parent if it\n   * unsubscribes on it's own.\n   *\n   * NOTE: THIS ASSUMES THAT {@link _hasParent} HAS ALREADY BEEN CHECKED.\n   * @param parent The parent subscription to add\n   */\n  private _addParent(parent: Subscription) {\n    const { _parentage } = this;\n    this._parentage = Array.isArray(_parentage) ? (_parentage.push(parent), _parentage) : _parentage ? [_parentage, parent] : parent;\n  }\n\n  /**\n   * Called on a child when it is removed via {@link #remove}.\n   * @param parent The parent to remove\n   */\n  private _removeParent(parent: Subscription) {\n    const { _parentage } = this;\n    if (_parentage === parent) {\n      this._parentage = null;\n    } else if (Array.isArray(_parentage)) {\n      arrRemove(_parentage, parent);\n    }\n  }\n\n  /**\n   * Removes a finalizer from this subscription that was previously added with the {@link #add} method.\n   *\n   * Note that `Subscription` instances, when unsubscribed, will automatically remove themselves\n   * from every other `Subscription` they have been added to. This means that using the `remove` method\n   * is not a common thing and should be used thoughtfully.\n   *\n   * If you add the same finalizer instance of a function or an unsubscribable object to a `Subscription` instance\n   * more than once, you will need to call `remove` the same number of times to remove all instances.\n   *\n   * All finalizer instances are removed to free up memory upon unsubscription.\n   *\n   * @param teardown The finalizer to remove from this subscription\n   */\n  remove(teardown: Exclude<TeardownLogic, void>): void {\n    const { _finalizers } = this;\n    _finalizers && arrRemove(_finalizers, teardown);\n\n    if (teardown instanceof Subscription) {\n      teardown._removeParent(this);\n    }\n  }\n}\n\nexport const EMPTY_SUBSCRIPTION = Subscription.EMPTY;\n\nexport function isSubscription(value: any): value is Subscription {\n  return (\n    value instanceof Subscription ||\n    (value && 'closed' in value && isFunction(value.remove) && isFunction(value.add) && isFunction(value.unsubscribe))\n  );\n}\n\nfunction execFinalizer(finalizer: Unsubscribable | (() => void)) {\n  if (isFunction(finalizer)) {\n    finalizer();\n  } else {\n    finalizer.unsubscribe();\n  }\n}\n", "import { Subscriber } from './Subscriber';\nimport { ObservableNotification } from './types';\n\n/**\n * The {@link GlobalConfig} object for RxJS. It is used to configure things\n * like how to react on unhandled errors.\n */\nexport const config: GlobalConfig = {\n  onUnhandledError: null,\n  onStoppedNotification: null,\n  Promise: undefined,\n  useDeprecatedSynchronousErrorHandling: false,\n  useDeprecatedNextContext: false,\n};\n\n/**\n * The global configuration object for RxJS, used to configure things\n * like how to react on unhandled errors. Accessible via {@link config}\n * object.\n */\nexport interface GlobalConfig {\n  /**\n   * A registration point for unhandled errors from RxJS. These are errors that\n   * cannot were not handled by consuming code in the usual subscription path. For\n   * example, if you have this configured, and you subscribe to an observable without\n   * providing an error handler, errors from that subscription will end up here. This\n   * will _always_ be called asynchronously on another job in the runtime. This is because\n   * we do not want errors thrown in this user-configured handler to interfere with the\n   * behavior of the library.\n   */\n  onUnhandledError: ((err: any) => void) | null;\n\n  /**\n   * A registration point for notifications that cannot be sent to subscribers because they\n   * have completed, errored or have been explicitly unsubscribed. By default, next, complete\n   * and error notifications sent to stopped subscribers are noops. However, sometimes callers\n   * might want a different behavior. For example, with sources that attempt to report errors\n   * to stopped subscribers, a caller can configure RxJS to throw an unhandled error instead.\n   * This will _always_ be called asynchronously on another job in the runtime. This is because\n   * we do not want errors thrown in this user-configured handler to interfere with the\n   * behavior of the library.\n   */\n  onStoppedNotification: ((notification: ObservableNotification<any>, subscriber: Subscriber<any>) => void) | null;\n\n  /**\n   * The promise constructor used by default for {@link Observable#toPromise toPromise} and {@link Observable#forEach forEach}\n   * methods.\n   *\n   * @deprecated As of version 8, RxJS will no longer support this sort of injection of a\n   * Promise constructor. If you need a Promise implementation other than native promises,\n   * please polyfill/patch Promise as you see appropriate. Will be removed in v8.\n   */\n  Promise?: PromiseConstructorLike;\n\n  /**\n   * If true, turns on synchronous error rethrowing, which is a deprecated behavior\n   * in v6 and higher. This behavior enables bad patterns like wrapping a subscribe\n   * call in a try/catch block. It also enables producer interference, a nasty bug\n   * where a multicast can be broken for all observers by a downstream consumer with\n   * an unhandled error. DO NOT USE THIS FLAG UNLESS IT'S NEEDED TO BUY TIME\n   * FOR MIGRATION REASONS.\n   *\n   * @deprecated As of version 8, RxJS will no longer support synchronous throwing\n   * of unhandled errors. All errors will be thrown on a separate call stack to prevent bad\n   * behaviors described above. Will be removed in v8.\n   */\n  useDeprecatedSynchronousErrorHandling: boolean;\n\n  /**\n   * If true, enables an as-of-yet undocumented feature from v5: The ability to access\n   * `unsubscribe()` via `this` context in `next` functions created in observers passed\n   * to `subscribe`.\n   *\n   * This is being removed because the performance was severely problematic, and it could also cause\n   * issues when types other than POJOs are passed to subscribe as subscribers, as they will likely have\n   * their `this` context overwritten.\n   *\n   * @deprecated As of version 8, RxJS will no longer support altering the\n   * context of next functions provided as part of an observer to Subscribe. Instead,\n   * you will have access to a subscription or a signal or token that will allow you to do things like\n   * unsubscribe and test closed status. Will be removed in v8.\n   */\n  useDeprecatedNextContext: boolean;\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetTimeoutFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearTimeoutFunction = (handle: TimerHandle) => void;\n\ninterface TimeoutProvider {\n  setTimeout: SetTimeoutFunction;\n  clearTimeout: ClearTimeoutFunction;\n  delegate:\n    | {\n        setTimeout: SetTimeoutFunction;\n        clearTimeout: ClearTimeoutFunction;\n      }\n    | undefined;\n}\n\nexport const timeoutProvider: TimeoutProvider = {\n  // When accessing the delegate, use the variable rather than `this` so that\n  // the functions can be called without being bound to the provider.\n  setTimeout(handler: () => void, timeout?: number, ...args) {\n    const { delegate } = timeoutProvider;\n    if (delegate?.setTimeout) {\n      return delegate.setTimeout(handler, timeout, ...args);\n    }\n    return setTimeout(handler, timeout, ...args);\n  },\n  clearTimeout(handle) {\n    const { delegate } = timeoutProvider;\n    return (delegate?.clearTimeout || clearTimeout)(handle as any);\n  },\n  delegate: undefined,\n};\n", "import { config } from '../config';\nimport { timeoutProvider } from '../scheduler/timeoutProvider';\n\n/**\n * Handles an error on another job either with the user-configured {@link onUnhandledError},\n * or by throwing it on that new job so it can be picked up by `window.onerror`, `process.on('error')`, etc.\n *\n * This should be called whenever there is an error that is out-of-band with the subscription\n * or when an error hits a terminal boundary of the subscription and no error handler was provided.\n *\n * @param err the error to report\n */\nexport function reportUnhandledError(err: any) {\n  timeoutProvider.setTimeout(() => {\n    const { onUnhandledError } = config;\n    if (onUnhandledError) {\n      // Execute the user-configured error handler.\n      onUnhandledError(err);\n    } else {\n      // Throw so it is picked up by the runtime's uncaught error mechanism.\n      throw err;\n    }\n  });\n}\n", "/* tslint:disable:no-empty */\nexport function noop() { }\n", "import { CompleteNotification, NextNotification, ErrorNotification } from './types';\n\n/**\n * A completion object optimized for memory use and created to be the\n * same \"shape\" as other notifications in v8.\n * @internal\n */\nexport const COMPLETE_NOTIFICATION = (() => createNotification('C', undefined, undefined) as CompleteNotification)();\n\n/**\n * Internal use only. Creates an optimized error notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function errorNotification(error: any): ErrorNotification {\n  return createNotification('E', undefined, error) as any;\n}\n\n/**\n * Internal use only. Creates an optimized next notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function nextNotification<T>(value: T) {\n  return createNotification('N', value, undefined) as NextNotification<T>;\n}\n\n/**\n * Ensures that all notifications created internally have the same \"shape\" in v8.\n *\n * TODO: This is only exported to support a crazy legacy test in `groupBy`.\n * @internal\n */\nexport function createNotification(kind: 'N' | 'E' | 'C', value: any, error: any) {\n  return {\n    kind,\n    value,\n    error,\n  };\n}\n", "import { config } from '../config';\n\nlet context: { errorThrown: boolean; error: any } | null = null;\n\n/**\n * Handles dealing with errors for super-gross mode. Creates a context, in which\n * any synchronously thrown errors will be passed to {@link captureError}. Which\n * will record the error such that it will be rethrown after the call back is complete.\n * TODO: Remove in v8\n * @param cb An immediately executed function.\n */\nexport function errorContext(cb: () => void) {\n  if (config.useDeprecatedSynchronousErrorHandling) {\n    const isRoot = !context;\n    if (isRoot) {\n      context = { errorThrown: false, error: null };\n    }\n    cb();\n    if (isRoot) {\n      const { errorThrown, error } = context!;\n      context = null;\n      if (errorThrown) {\n        throw error;\n      }\n    }\n  } else {\n    // This is the general non-deprecated path for everyone that\n    // isn't crazy enough to use super-gross mode (useDeprecatedSynchronousErrorHandling)\n    cb();\n  }\n}\n\n/**\n * Captures errors only in super-gross mode.\n * @param err the error to capture\n */\nexport function captureError(err: any) {\n  if (config.useDeprecatedSynchronousErrorHandling && context) {\n    context.errorThrown = true;\n    context.error = err;\n  }\n}\n", "import { isFunction } from './util/isFunction';\nimport { Observer, ObservableNotification } from './types';\nimport { isSubscription, Subscription } from './Subscription';\nimport { config } from './config';\nimport { reportUnhandledError } from './util/reportUnhandledError';\nimport { noop } from './util/noop';\nimport { nextNotification, errorNotification, COMPLETE_NOTIFICATION } from './NotificationFactories';\nimport { timeoutProvider } from './scheduler/timeoutProvider';\nimport { captureError } from './util/errorContext';\n\n/**\n * Implements the {@link Observer} interface and extends the\n * {@link Subscription} class. While the {@link Observer} is the public API for\n * consuming the values of an {@link Observable}, all Observers get converted to\n * a Subscriber, in order to provide Subscription-like capabilities such as\n * `unsubscribe`. Subscriber is a common type in RxJS, and crucial for\n * implementing operators, but it is rarely used as a public API.\n *\n * @class Subscriber<T>\n */\nexport class Subscriber<T> extends Subscription implements Observer<T> {\n  /**\n   * A static factory for a Subscriber, given a (potentially partial) definition\n   * of an Observer.\n   * @param next The `next` callback of an Observer.\n   * @param error The `error` callback of an\n   * Observer.\n   * @param complete The `complete` callback of an\n   * Observer.\n   * @return A Subscriber wrapping the (partially defined)\n   * Observer represented by the given arguments.\n   * @nocollapse\n   * @deprecated Do not use. Will be removed in v8. There is no replacement for this\n   * method, and there is no reason to be creating instances of `Subscriber` directly.\n   * If you have a specific use case, please file an issue.\n   */\n  static create<T>(next?: (x?: T) => void, error?: (e?: any) => void, complete?: () => void): Subscriber<T> {\n    return new SafeSubscriber(next, error, complete);\n  }\n\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  protected isStopped: boolean = false;\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  protected destination: Subscriber<any> | Observer<any>; // this `any` is the escape hatch to erase extra type param (e.g. R)\n\n  /**\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   * There is no reason to directly create an instance of Subscriber. This type is exported for typings reasons.\n   */\n  constructor(destination?: Subscriber<any> | Observer<any>) {\n    super();\n    if (destination) {\n      this.destination = destination;\n      // Automatically chain subscriptions together here.\n      // if destination is a Subscription, then it is a Subscriber.\n      if (isSubscription(destination)) {\n        destination.add(this);\n      }\n    } else {\n      this.destination = EMPTY_OBSERVER;\n    }\n  }\n\n  /**\n   * The {@link Observer} callback to receive notifications of type `next` from\n   * the Observable, with a value. The Observable may call this method 0 or more\n   * times.\n   * @param {T} [value] The `next` value.\n   * @return {void}\n   */\n  next(value?: T): void {\n    if (this.isStopped) {\n      handleStoppedNotification(nextNotification(value), this);\n    } else {\n      this._next(value!);\n    }\n  }\n\n  /**\n   * The {@link Observer} callback to receive notifications of type `error` from\n   * the Observable, with an attached `Error`. Notifies the Observer that\n   * the Observable has experienced an error condition.\n   * @param {any} [err] The `error` exception.\n   * @return {void}\n   */\n  error(err?: any): void {\n    if (this.isStopped) {\n      handleStoppedNotification(errorNotification(err), this);\n    } else {\n      this.isStopped = true;\n      this._error(err);\n    }\n  }\n\n  /**\n   * The {@link Observer} callback to receive a valueless notification of type\n   * `complete` from the Observable. Notifies the Observer that the Observable\n   * has finished sending push-based notifications.\n   * @return {void}\n   */\n  complete(): void {\n    if (this.isStopped) {\n      handleStoppedNotification(COMPLETE_NOTIFICATION, this);\n    } else {\n      this.isStopped = true;\n      this._complete();\n    }\n  }\n\n  unsubscribe(): void {\n    if (!this.closed) {\n      this.isStopped = true;\n      super.unsubscribe();\n      this.destination = null!;\n    }\n  }\n\n  protected _next(value: T): void {\n    this.destination.next(value);\n  }\n\n  protected _error(err: any): void {\n    try {\n      this.destination.error(err);\n    } finally {\n      this.unsubscribe();\n    }\n  }\n\n  protected _complete(): void {\n    try {\n      this.destination.complete();\n    } finally {\n      this.unsubscribe();\n    }\n  }\n}\n\n/**\n * This bind is captured here because we want to be able to have\n * compatibility with monoid libraries that tend to use a method named\n * `bind`. In particular, a library called Monio requires this.\n */\nconst _bind = Function.prototype.bind;\n\nfunction bind<Fn extends (...args: any[]) => any>(fn: Fn, thisArg: any): Fn {\n  return _bind.call(fn, thisArg);\n}\n\n/**\n * Internal optimization only, DO NOT EXPOSE.\n * @internal\n */\nclass ConsumerObserver<T> implements Observer<T> {\n  constructor(private partialObserver: Partial<Observer<T>>) {}\n\n  next(value: T): void {\n    const { partialObserver } = this;\n    if (partialObserver.next) {\n      try {\n        partialObserver.next(value);\n      } catch (error) {\n        handleUnhandledError(error);\n      }\n    }\n  }\n\n  error(err: any): void {\n    const { partialObserver } = this;\n    if (partialObserver.error) {\n      try {\n        partialObserver.error(err);\n      } catch (error) {\n        handleUnhandledError(error);\n      }\n    } else {\n      handleUnhandledError(err);\n    }\n  }\n\n  complete(): void {\n    const { partialObserver } = this;\n    if (partialObserver.complete) {\n      try {\n        partialObserver.complete();\n      } catch (error) {\n        handleUnhandledError(error);\n      }\n    }\n  }\n}\n\nexport class SafeSubscriber<T> extends Subscriber<T> {\n  constructor(\n    observerOrNext?: Partial<Observer<T>> | ((value: T) => void) | null,\n    error?: ((e?: any) => void) | null,\n    complete?: (() => void) | null\n  ) {\n    super();\n\n    let partialObserver: Partial<Observer<T>>;\n    if (isFunction(observerOrNext) || !observerOrNext) {\n      // The first argument is a function, not an observer. The next\n      // two arguments *could* be observers, or they could be empty.\n      partialObserver = {\n        next: (observerOrNext ?? undefined) as (((value: T) => void) | undefined),\n        error: error ?? undefined,\n        complete: complete ?? undefined,\n      };\n    } else {\n      // The first argument is a partial observer.\n      let context: any;\n      if (this && config.useDeprecatedNextContext) {\n        // This is a deprecated path that made `this.unsubscribe()` available in\n        // next handler functions passed to subscribe. This only exists behind a flag\n        // now, as it is *very* slow.\n        context = Object.create(observerOrNext);\n        context.unsubscribe = () => this.unsubscribe();\n        partialObserver = {\n          next: observerOrNext.next && bind(observerOrNext.next, context),\n          error: observerOrNext.error && bind(observerOrNext.error, context),\n          complete: observerOrNext.complete && bind(observerOrNext.complete, context),\n        };\n      } else {\n        // The \"normal\" path. Just use the partial observer directly.\n        partialObserver = observerOrNext;\n      }\n    }\n\n    // Wrap the partial observer to ensure it's a full observer, and\n    // make sure proper error handling is accounted for.\n    this.destination = new ConsumerObserver(partialObserver);\n  }\n}\n\nfunction handleUnhandledError(error: any) {\n  if (config.useDeprecatedSynchronousErrorHandling) {\n    captureError(error);\n  } else {\n    // Ideal path, we report this as an unhandled error,\n    // which is thrown on a new call stack.\n    reportUnhandledError(error);\n  }\n}\n\n/**\n * An error handler used when no error handler was supplied\n * to the SafeSubscriber -- meaning no error handler was supplied\n * do the `subscribe` call on our observable.\n * @param err The error to handle\n */\nfunction defaultErrorHandler(err: any) {\n  throw err;\n}\n\n/**\n * A handler for notifications that cannot be sent to a stopped subscriber.\n * @param notification The notification being sent\n * @param subscriber The stopped subscriber\n */\nfunction handleStoppedNotification(notification: ObservableNotification<any>, subscriber: Subscriber<any>) {\n  const { onStoppedNotification } = config;\n  onStoppedNotification && timeoutProvider.setTimeout(() => onStoppedNotification(notification, subscriber));\n}\n\n/**\n * The observer used as a stub for subscriptions where the user did not\n * pass any arguments to `subscribe`. Comes with the default error handling\n * behavior.\n */\nexport const EMPTY_OBSERVER: Readonly<Observer<any>> & { closed: true } = {\n  closed: true,\n  next: noop,\n  error: defaultErrorHandler,\n  complete: noop,\n};\n", "/**\n * Symbol.observable or a string \"@@observable\". Used for interop\n *\n * @deprecated We will no longer be exporting this symbol in upcoming versions of RxJS.\n * Instead polyfill and use Symbol.observable directly *or* use https://www.npmjs.com/package/symbol-observable\n */\nexport const observable: string | symbol = (() => (typeof Symbol === 'function' && Symbol.observable) || '@@observable')();\n", "/**\n * This function takes one parameter and just returns it. Simply put,\n * this is like `<T>(x: T): T => x`.\n *\n * ## Examples\n *\n * This is useful in some cases when using things like `mergeMap`\n *\n * ```ts\n * import { interval, take, map, range, mergeMap, identity } from 'rxjs';\n *\n * const source$ = interval(1000).pipe(take(5));\n *\n * const result$ = source$.pipe(\n *   map(i => range(i)),\n *   mergeMap(identity) // same as mergeMap(x => x)\n * );\n *\n * result$.subscribe({\n *   next: console.log\n * });\n * ```\n *\n * Or when you want to selectively apply an operator\n *\n * ```ts\n * import { interval, take, identity } from 'rxjs';\n *\n * const shouldLimit = () => Math.random() < 0.5;\n *\n * const source$ = interval(1000);\n *\n * const result$ = source$.pipe(shouldLimit() ? take(5) : identity);\n *\n * result$.subscribe({\n *   next: console.log\n * });\n * ```\n *\n * @param x Any value that is returned by this function\n * @returns The value passed as the first parameter to this function\n */\nexport function identity<T>(x: T): T {\n  return x;\n}\n", "import { identity } from './identity';\nimport { UnaryFunction } from '../types';\n\nexport function pipe(): typeof identity;\nexport function pipe<T, A>(fn1: UnaryFunction<T, A>): UnaryFunction<T, A>;\nexport function pipe<T, A, B>(fn1: UnaryFunction<T, A>, fn2: UnaryFunction<A, B>): UnaryFunction<T, B>;\nexport function pipe<T, A, B, C>(fn1: UnaryFunction<T, A>, fn2: UnaryFunction<A, B>, fn3: UnaryFunction<B, C>): UnaryFunction<T, C>;\nexport function pipe<T, A, B, C, D>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>\n): UnaryFunction<T, D>;\nexport function pipe<T, A, B, C, D, E>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>\n): UnaryFunction<T, E>;\nexport function pipe<T, A, B, C, D, E, F>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>\n): UnaryFunction<T, F>;\nexport function pipe<T, A, B, C, D, E, F, G>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>\n): UnaryFunction<T, G>;\nexport function pipe<T, A, B, C, D, E, F, G, H>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>,\n  fn8: UnaryFunction<G, H>\n): UnaryFunction<T, H>;\nexport function pipe<T, A, B, C, D, E, F, G, H, I>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>,\n  fn8: UnaryFunction<G, H>,\n  fn9: UnaryFunction<H, I>\n): UnaryFunction<T, I>;\nexport function pipe<T, A, B, C, D, E, F, G, H, I>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>,\n  fn8: UnaryFunction<G, H>,\n  fn9: UnaryFunction<H, I>,\n  ...fns: UnaryFunction<any, any>[]\n): UnaryFunction<T, unknown>;\n\n/**\n * pipe() can be called on one or more functions, each of which can take one argument (\"UnaryFunction\")\n * and uses it to return a value.\n * It returns a function that takes one argument, passes it to the first UnaryFunction, and then\n * passes the result to the next one, passes that result to the next one, and so on.  \n */\nexport function pipe(...fns: Array<UnaryFunction<any, any>>): UnaryFunction<any, any> {\n  return pipeFromArray(fns);\n}\n\n/** @internal */\nexport function pipeFromArray<T, R>(fns: Array<UnaryFunction<T, R>>): UnaryFunction<T, R> {\n  if (fns.length === 0) {\n    return identity as UnaryFunction<any, any>;\n  }\n\n  if (fns.length === 1) {\n    return fns[0];\n  }\n\n  return function piped(input: T): R {\n    return fns.reduce((prev: any, fn: UnaryFunction<T, R>) => fn(prev), input as any);\n  };\n}\n", "import { Operator } from './Operator';\nimport { SafeSubscriber, Subscriber } from './Subscriber';\nimport { isSubscription, Subscription } from './Subscription';\nimport { TeardownLogic, OperatorFunction, Subscribable, Observer } from './types';\nimport { observable as Symbol_observable } from './symbol/observable';\nimport { pipeFromArray } from './util/pipe';\nimport { config } from './config';\nimport { isFunction } from './util/isFunction';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A representation of any set of values over any amount of time. This is the most basic building block\n * of RxJS.\n *\n * @class Observable<T>\n */\nexport class Observable<T> implements Subscribable<T> {\n  /**\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   */\n  source: Observable<any> | undefined;\n\n  /**\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   */\n  operator: Operator<any, T> | undefined;\n\n  /**\n   * @constructor\n   * @param {Function} subscribe the function that is called when the Observable is\n   * initially subscribed to. This function is given a Subscriber, to which new values\n   * can be `next`ed, or an `error` method can be called to raise an error, or\n   * `complete` can be called to notify of a successful completion.\n   */\n  constructor(subscribe?: (this: Observable<T>, subscriber: Subscriber<T>) => TeardownLogic) {\n    if (subscribe) {\n      this._subscribe = subscribe;\n    }\n  }\n\n  // HACK: Since TypeScript inherits static properties too, we have to\n  // fight against TypeScript here so Subject can have a different static create signature\n  /**\n   * Creates a new Observable by calling the Observable constructor\n   * @owner Observable\n   * @method create\n   * @param {Function} subscribe? the subscriber function to be passed to the Observable constructor\n   * @return {Observable} a new observable\n   * @nocollapse\n   * @deprecated Use `new Observable()` instead. Will be removed in v8.\n   */\n  static create: (...args: any[]) => any = <T>(subscribe?: (subscriber: Subscriber<T>) => TeardownLogic) => {\n    return new Observable<T>(subscribe);\n  };\n\n  /**\n   * Creates a new Observable, with this Observable instance as the source, and the passed\n   * operator defined as the new observable's operator.\n   * @method lift\n   * @param operator the operator defining the operation to take on the observable\n   * @return a new observable with the Operator applied\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   * If you have implemented an operator using `lift`, it is recommended that you create an\n   * operator by simply returning `new Observable()` directly. See \"Creating new operators from\n   * scratch\" section here: https://rxjs.dev/guide/operators\n   */\n  lift<R>(operator?: Operator<T, R>): Observable<R> {\n    const observable = new Observable<R>();\n    observable.source = this;\n    observable.operator = operator;\n    return observable;\n  }\n\n  subscribe(observerOrNext?: Partial<Observer<T>> | ((value: T) => void)): Subscription;\n  /** @deprecated Instead of passing separate callback arguments, use an observer argument. Signatures taking separate callback arguments will be removed in v8. Details: https://rxjs.dev/deprecations/subscribe-arguments */\n  subscribe(next?: ((value: T) => void) | null, error?: ((error: any) => void) | null, complete?: (() => void) | null): Subscription;\n  /**\n   * Invokes an execution of an Observable and registers Observer handlers for notifications it will emit.\n   *\n   * <span class=\"informal\">Use it when you have all these Observables, but still nothing is happening.</span>\n   *\n   * `subscribe` is not a regular operator, but a method that calls Observable's internal `subscribe` function. It\n   * might be for example a function that you passed to Observable's constructor, but most of the time it is\n   * a library implementation, which defines what will be emitted by an Observable, and when it be will emitted. This means\n   * that calling `subscribe` is actually the moment when Observable starts its work, not when it is created, as it is often\n   * the thought.\n   *\n   * Apart from starting the execution of an Observable, this method allows you to listen for values\n   * that an Observable emits, as well as for when it completes or errors. You can achieve this in two\n   * of the following ways.\n   *\n   * The first way is creating an object that implements {@link Observer} interface. It should have methods\n   * defined by that interface, but note that it should be just a regular JavaScript object, which you can create\n   * yourself in any way you want (ES6 class, classic function constructor, object literal etc.). In particular, do\n   * not attempt to use any RxJS implementation details to create Observers - you don't need them. Remember also\n   * that your object does not have to implement all methods. If you find yourself creating a method that doesn't\n   * do anything, you can simply omit it. Note however, if the `error` method is not provided and an error happens,\n   * it will be thrown asynchronously. Errors thrown asynchronously cannot be caught using `try`/`catch`. Instead,\n   * use the {@link onUnhandledError} configuration option or use a runtime handler (like `window.onerror` or\n   * `process.on('error)`) to be notified of unhandled errors. Because of this, it's recommended that you provide\n   * an `error` method to avoid missing thrown errors.\n   *\n   * The second way is to give up on Observer object altogether and simply provide callback functions in place of its methods.\n   * This means you can provide three functions as arguments to `subscribe`, where the first function is equivalent\n   * of a `next` method, the second of an `error` method and the third of a `complete` method. Just as in case of an Observer,\n   * if you do not need to listen for something, you can omit a function by passing `undefined` or `null`,\n   * since `subscribe` recognizes these functions by where they were placed in function call. When it comes\n   * to the `error` function, as with an Observer, if not provided, errors emitted by an Observable will be thrown asynchronously.\n   *\n   * You can, however, subscribe with no parameters at all. This may be the case where you're not interested in terminal events\n   * and you also handled emissions internally by using operators (e.g. using `tap`).\n   *\n   * Whichever style of calling `subscribe` you use, in both cases it returns a Subscription object.\n   * This object allows you to call `unsubscribe` on it, which in turn will stop the work that an Observable does and will clean\n   * up all resources that an Observable used. Note that cancelling a subscription will not call `complete` callback\n   * provided to `subscribe` function, which is reserved for a regular completion signal that comes from an Observable.\n   *\n   * Remember that callbacks provided to `subscribe` are not guaranteed to be called asynchronously.\n   * It is an Observable itself that decides when these functions will be called. For example {@link of}\n   * by default emits all its values synchronously. Always check documentation for how given Observable\n   * will behave when subscribed and if its default behavior can be modified with a `scheduler`.\n   *\n   * #### Examples\n   *\n   * Subscribe with an {@link guide/observer Observer}\n   *\n   * ```ts\n   * import { of } from 'rxjs';\n   *\n   * const sumObserver = {\n   *   sum: 0,\n   *   next(value) {\n   *     console.log('Adding: ' + value);\n   *     this.sum = this.sum + value;\n   *   },\n   *   error() {\n   *     // We actually could just remove this method,\n   *     // since we do not really care about errors right now.\n   *   },\n   *   complete() {\n   *     console.log('Sum equals: ' + this.sum);\n   *   }\n   * };\n   *\n   * of(1, 2, 3) // Synchronously emits 1, 2, 3 and then completes.\n   *   .subscribe(sumObserver);\n   *\n   * // Logs:\n   * // 'Adding: 1'\n   * // 'Adding: 2'\n   * // 'Adding: 3'\n   * // 'Sum equals: 6'\n   * ```\n   *\n   * Subscribe with functions ({@link deprecations/subscribe-arguments deprecated})\n   *\n   * ```ts\n   * import { of } from 'rxjs'\n   *\n   * let sum = 0;\n   *\n   * of(1, 2, 3).subscribe(\n   *   value => {\n   *     console.log('Adding: ' + value);\n   *     sum = sum + value;\n   *   },\n   *   undefined,\n   *   () => console.log('Sum equals: ' + sum)\n   * );\n   *\n   * // Logs:\n   * // 'Adding: 1'\n   * // 'Adding: 2'\n   * // 'Adding: 3'\n   * // 'Sum equals: 6'\n   * ```\n   *\n   * Cancel a subscription\n   *\n   * ```ts\n   * import { interval } from 'rxjs';\n   *\n   * const subscription = interval(1000).subscribe({\n   *   next(num) {\n   *     console.log(num)\n   *   },\n   *   complete() {\n   *     // Will not be called, even when cancelling subscription.\n   *     console.log('completed!');\n   *   }\n   * });\n   *\n   * setTimeout(() => {\n   *   subscription.unsubscribe();\n   *   console.log('unsubscribed!');\n   * }, 2500);\n   *\n   * // Logs:\n   * // 0 after 1s\n   * // 1 after 2s\n   * // 'unsubscribed!' after 2.5s\n   * ```\n   *\n   * @param {Observer|Function} observerOrNext (optional) Either an observer with methods to be called,\n   * or the first of three possible handlers, which is the handler for each value emitted from the subscribed\n   * Observable.\n   * @param {Function} error (optional) A handler for a terminal event resulting from an error. If no error handler is provided,\n   * the error will be thrown asynchronously as unhandled.\n   * @param {Function} complete (optional) A handler for a terminal event resulting from successful completion.\n   * @return {Subscription} a subscription reference to the registered handlers\n   * @method subscribe\n   */\n  subscribe(\n    observerOrNext?: Partial<Observer<T>> | ((value: T) => void) | null,\n    error?: ((error: any) => void) | null,\n    complete?: (() => void) | null\n  ): Subscription {\n    const subscriber = isSubscriber(observerOrNext) ? observerOrNext : new SafeSubscriber(observerOrNext, error, complete);\n\n    errorContext(() => {\n      const { operator, source } = this;\n      subscriber.add(\n        operator\n          ? // We're dealing with a subscription in the\n            // operator chain to one of our lifted operators.\n            operator.call(subscriber, source)\n          : source\n          ? // If `source` has a value, but `operator` does not, something that\n            // had intimate knowledge of our API, like our `Subject`, must have\n            // set it. We're going to just call `_subscribe` directly.\n            this._subscribe(subscriber)\n          : // In all other cases, we're likely wrapping a user-provided initializer\n            // function, so we need to catch errors and handle them appropriately.\n            this._trySubscribe(subscriber)\n      );\n    });\n\n    return subscriber;\n  }\n\n  /** @internal */\n  protected _trySubscribe(sink: Subscriber<T>): TeardownLogic {\n    try {\n      return this._subscribe(sink);\n    } catch (err) {\n      // We don't need to return anything in this case,\n      // because it's just going to try to `add()` to a subscription\n      // above.\n      sink.error(err);\n    }\n  }\n\n  /**\n   * Used as a NON-CANCELLABLE means of subscribing to an observable, for use with\n   * APIs that expect promises, like `async/await`. You cannot unsubscribe from this.\n   *\n   * **WARNING**: Only use this with observables you *know* will complete. If the source\n   * observable does not complete, you will end up with a promise that is hung up, and\n   * potentially all of the state of an async function hanging out in memory. To avoid\n   * this situation, look into adding something like {@link timeout}, {@link take},\n   * {@link takeWhile}, or {@link takeUntil} amongst others.\n   *\n   * #### Example\n   *\n   * ```ts\n   * import { interval, take } from 'rxjs';\n   *\n   * const source$ = interval(1000).pipe(take(4));\n   *\n   * async function getTotal() {\n   *   let total = 0;\n   *\n   *   await source$.forEach(value => {\n   *     total += value;\n   *     console.log('observable -> ' + value);\n   *   });\n   *\n   *   return total;\n   * }\n   *\n   * getTotal().then(\n   *   total => console.log('Total: ' + total)\n   * );\n   *\n   * // Expected:\n   * // 'observable -> 0'\n   * // 'observable -> 1'\n   * // 'observable -> 2'\n   * // 'observable -> 3'\n   * // 'Total: 6'\n   * ```\n   *\n   * @param next a handler for each value emitted by the observable\n   * @return a promise that either resolves on observable completion or\n   *  rejects with the handled error\n   */\n  forEach(next: (value: T) => void): Promise<void>;\n\n  /**\n   * @param next a handler for each value emitted by the observable\n   * @param promiseCtor a constructor function used to instantiate the Promise\n   * @return a promise that either resolves on observable completion or\n   *  rejects with the handled error\n   * @deprecated Passing a Promise constructor will no longer be available\n   * in upcoming versions of RxJS. This is because it adds weight to the library, for very\n   * little benefit. If you need this functionality, it is recommended that you either\n   * polyfill Promise, or you create an adapter to convert the returned native promise\n   * to whatever promise implementation you wanted. Will be removed in v8.\n   */\n  forEach(next: (value: T) => void, promiseCtor: PromiseConstructorLike): Promise<void>;\n\n  forEach(next: (value: T) => void, promiseCtor?: PromiseConstructorLike): Promise<void> {\n    promiseCtor = getPromiseCtor(promiseCtor);\n\n    return new promiseCtor<void>((resolve, reject) => {\n      const subscriber = new SafeSubscriber<T>({\n        next: (value) => {\n          try {\n            next(value);\n          } catch (err) {\n            reject(err);\n            subscriber.unsubscribe();\n          }\n        },\n        error: reject,\n        complete: resolve,\n      });\n      this.subscribe(subscriber);\n    }) as Promise<void>;\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<any>): TeardownLogic {\n    return this.source?.subscribe(subscriber);\n  }\n\n  /**\n   * An interop point defined by the es7-observable spec https://github.com/zenparsing/es-observable\n   * @method Symbol.observable\n   * @return {Observable} this instance of the observable\n   */\n  [Symbol_observable]() {\n    return this;\n  }\n\n  /* tslint:disable:max-line-length */\n  pipe(): Observable<T>;\n  pipe<A>(op1: OperatorFunction<T, A>): Observable<A>;\n  pipe<A, B>(op1: OperatorFunction<T, A>, op2: OperatorFunction<A, B>): Observable<B>;\n  pipe<A, B, C>(op1: OperatorFunction<T, A>, op2: OperatorFunction<A, B>, op3: OperatorFunction<B, C>): Observable<C>;\n  pipe<A, B, C, D>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>\n  ): Observable<D>;\n  pipe<A, B, C, D, E>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>\n  ): Observable<E>;\n  pipe<A, B, C, D, E, F>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>\n  ): Observable<F>;\n  pipe<A, B, C, D, E, F, G>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>\n  ): Observable<G>;\n  pipe<A, B, C, D, E, F, G, H>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>,\n    op8: OperatorFunction<G, H>\n  ): Observable<H>;\n  pipe<A, B, C, D, E, F, G, H, I>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>,\n    op8: OperatorFunction<G, H>,\n    op9: OperatorFunction<H, I>\n  ): Observable<I>;\n  pipe<A, B, C, D, E, F, G, H, I>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>,\n    op8: OperatorFunction<G, H>,\n    op9: OperatorFunction<H, I>,\n    ...operations: OperatorFunction<any, any>[]\n  ): Observable<unknown>;\n  /* tslint:enable:max-line-length */\n\n  /**\n   * Used to stitch together functional operators into a chain.\n   * @method pipe\n   * @return {Observable} the Observable result of all of the operators having\n   * been called in the order they were passed in.\n   *\n   * ## Example\n   *\n   * ```ts\n   * import { interval, filter, map, scan } from 'rxjs';\n   *\n   * interval(1000)\n   *   .pipe(\n   *     filter(x => x % 2 === 0),\n   *     map(x => x + x),\n   *     scan((acc, x) => acc + x)\n   *   )\n   *   .subscribe(x => console.log(x));\n   * ```\n   */\n  pipe(...operations: OperatorFunction<any, any>[]): Observable<any> {\n    return pipeFromArray(operations)(this);\n  }\n\n  /* tslint:disable:max-line-length */\n  /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n  toPromise(): Promise<T | undefined>;\n  /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n  toPromise(PromiseCtor: typeof Promise): Promise<T | undefined>;\n  /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n  toPromise(PromiseCtor: PromiseConstructorLike): Promise<T | undefined>;\n  /* tslint:enable:max-line-length */\n\n  /**\n   * Subscribe to this Observable and get a Promise resolving on\n   * `complete` with the last emission (if any).\n   *\n   * **WARNING**: Only use this with observables you *know* will complete. If the source\n   * observable does not complete, you will end up with a promise that is hung up, and\n   * potentially all of the state of an async function hanging out in memory. To avoid\n   * this situation, look into adding something like {@link timeout}, {@link take},\n   * {@link takeWhile}, or {@link takeUntil} amongst others.\n   *\n   * @method toPromise\n   * @param [promiseCtor] a constructor function used to instantiate\n   * the Promise\n   * @return A Promise that resolves with the last value emit, or\n   * rejects on an error. If there were no emissions, Promise\n   * resolves with undefined.\n   * @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise\n   */\n  toPromise(promiseCtor?: PromiseConstructorLike): Promise<T | undefined> {\n    promiseCtor = getPromiseCtor(promiseCtor);\n\n    return new promiseCtor((resolve, reject) => {\n      let value: T | undefined;\n      this.subscribe(\n        (x: T) => (value = x),\n        (err: any) => reject(err),\n        () => resolve(value)\n      );\n    }) as Promise<T | undefined>;\n  }\n}\n\n/**\n * Decides between a passed promise constructor from consuming code,\n * A default configured promise constructor, and the native promise\n * constructor and returns it. If nothing can be found, it will throw\n * an error.\n * @param promiseCtor The optional promise constructor to passed by consuming code\n */\nfunction getPromiseCtor(promiseCtor: PromiseConstructorLike | undefined) {\n  return promiseCtor ?? config.Promise ?? Promise;\n}\n\nfunction isObserver<T>(value: any): value is Observer<T> {\n  return value && isFunction(value.next) && isFunction(value.error) && isFunction(value.complete);\n}\n\nfunction isSubscriber<T>(value: any): value is Subscriber<T> {\n  return (value && value instanceof Subscriber) || (isObserver(value) && isSubscription(value));\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { OperatorFunction } from '../types';\nimport { isFunction } from './isFunction';\n\n/**\n * Used to determine if an object is an Observable with a lift function.\n */\nexport function hasLift(source: any): source is { lift: InstanceType<typeof Observable>['lift'] } {\n  return isFunction(source?.lift);\n}\n\n/**\n * Creates an `OperatorFunction`. Used to define operators throughout the library in a concise way.\n * @param init The logic to connect the liftedSource to the subscriber at the moment of subscription.\n */\nexport function operate<T, R>(\n  init: (liftedSource: Observable<T>, subscriber: Subscriber<R>) => (() => void) | void\n): OperatorFunction<T, R> {\n  return (source: Observable<T>) => {\n    if (hasLift(source)) {\n      return source.lift(function (this: Subscriber<R>, liftedSource: Observable<T>) {\n        try {\n          return init(liftedSource, this);\n        } catch (err) {\n          this.error(err);\n        }\n      });\n    }\n    throw new TypeError('Unable to lift unknown Observable type');\n  };\n}\n", "import { Subscriber } from '../Subscriber';\n\n/**\n * Creates an instance of an `OperatorSubscriber`.\n * @param destination The downstream subscriber.\n * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n * and send to the `destination` error handler.\n * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n * this handler are sent to the `destination` error handler.\n * @param onFinalize Additional teardown logic here. This will only be called on teardown if the\n * subscriber itself is not already closed. This is called after all other teardown logic is executed.\n */\nexport function createOperatorSubscriber<T>(\n  destination: Subscriber<any>,\n  onNext?: (value: T) => void,\n  onComplete?: () => void,\n  onError?: (err: any) => void,\n  onFinalize?: () => void\n): Subscriber<T> {\n  return new OperatorSubscriber(destination, onNext, onComplete, onError, onFinalize);\n}\n\n/**\n * A generic helper for allowing operators to be created with a Subscriber and\n * use closures to capture necessary state from the operator function itself.\n */\nexport class OperatorSubscriber<T> extends Subscriber<T> {\n  /**\n   * Creates an instance of an `OperatorSubscriber`.\n   * @param destination The downstream subscriber.\n   * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n   * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n   * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n   * and send to the `destination` error handler.\n   * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n   * this handler are sent to the `destination` error handler.\n   * @param onFinalize Additional finalization logic here. This will only be called on finalization if the\n   * subscriber itself is not already closed. This is called after all other finalization logic is executed.\n   * @param shouldUnsubscribe An optional check to see if an unsubscribe call should truly unsubscribe.\n   * NOTE: This currently **ONLY** exists to support the strange behavior of {@link groupBy}, where unsubscription\n   * to the resulting observable does not actually disconnect from the source if there are active subscriptions\n   * to any grouped observable. (DO NOT EXPOSE OR USE EXTERNALLY!!!)\n   */\n  constructor(\n    destination: Subscriber<any>,\n    onNext?: (value: T) => void,\n    onComplete?: () => void,\n    onError?: (err: any) => void,\n    private onFinalize?: () => void,\n    private shouldUnsubscribe?: () => boolean\n  ) {\n    // It's important - for performance reasons - that all of this class's\n    // members are initialized and that they are always initialized in the same\n    // order. This will ensure that all OperatorSubscriber instances have the\n    // same hidden class in V8. This, in turn, will help keep the number of\n    // hidden classes involved in property accesses within the base class as\n    // low as possible. If the number of hidden classes involved exceeds four,\n    // the property accesses will become megamorphic and performance penalties\n    // will be incurred - i.e. inline caches won't be used.\n    //\n    // The reasons for ensuring all instances have the same hidden class are\n    // further discussed in this blog post from Benedikt Meurer:\n    // https://benediktmeurer.de/2018/03/23/impact-of-polymorphism-on-component-based-frameworks-like-react/\n    super(destination);\n    this._next = onNext\n      ? function (this: OperatorSubscriber<T>, value: T) {\n          try {\n            onNext(value);\n          } catch (err) {\n            destination.error(err);\n          }\n        }\n      : super._next;\n    this._error = onError\n      ? function (this: OperatorSubscriber<T>, err: any) {\n          try {\n            onError(err);\n          } catch (err) {\n            // Send any errors that occur down stream.\n            destination.error(err);\n          } finally {\n            // Ensure finalization.\n            this.unsubscribe();\n          }\n        }\n      : super._error;\n    this._complete = onComplete\n      ? function (this: OperatorSubscriber<T>) {\n          try {\n            onComplete();\n          } catch (err) {\n            // Send any errors that occur down stream.\n            destination.error(err);\n          } finally {\n            // Ensure finalization.\n            this.unsubscribe();\n          }\n        }\n      : super._complete;\n  }\n\n  unsubscribe() {\n    if (!this.shouldUnsubscribe || this.shouldUnsubscribe()) {\n      const { closed } = this;\n      super.unsubscribe();\n      // Execute additional teardown if we have any and we didn't already do so.\n      !closed && this.onFinalize?.();\n    }\n  }\n}\n", "import { Subscription } from '../Subscription';\n\ninterface AnimationFrameProvider {\n  schedule(callback: FrameRequestCallback): Subscription;\n  requestAnimationFrame: typeof requestAnimationFrame;\n  cancelAnimationFrame: typeof cancelAnimationFrame;\n  delegate:\n    | {\n        requestAnimationFrame: typeof requestAnimationFrame;\n        cancelAnimationFrame: typeof cancelAnimationFrame;\n      }\n    | undefined;\n}\n\nexport const animationFrameProvider: AnimationFrameProvider = {\n  // When accessing the delegate, use the variable rather than `this` so that\n  // the functions can be called without being bound to the provider.\n  schedule(callback) {\n    let request = requestAnimationFrame;\n    let cancel: typeof cancelAnimationFrame | undefined = cancelAnimationFrame;\n    const { delegate } = animationFrameProvider;\n    if (delegate) {\n      request = delegate.requestAnimationFrame;\n      cancel = delegate.cancelAnimationFrame;\n    }\n    const handle = request((timestamp) => {\n      // Clear the cancel function. The request has been fulfilled, so\n      // attempting to cancel the request upon unsubscription would be\n      // pointless.\n      cancel = undefined;\n      callback(timestamp);\n    });\n    return new Subscription(() => cancel?.(handle));\n  },\n  requestAnimationFrame(...args) {\n    const { delegate } = animationFrameProvider;\n    return (delegate?.requestAnimationFrame || requestAnimationFrame)(...args);\n  },\n  cancelAnimationFrame(...args) {\n    const { delegate } = animationFrameProvider;\n    return (delegate?.cancelAnimationFrame || cancelAnimationFrame)(...args);\n  },\n  delegate: undefined,\n};\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface ObjectUnsubscribedError extends Error {}\n\nexport interface ObjectUnsubscribedErrorCtor {\n  /**\n   * @deprecated Internal implementation detail. Do not construct error instances.\n   * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n   */\n  new (): ObjectUnsubscribedError;\n}\n\n/**\n * An error thrown when an action is invalid because the object has been\n * unsubscribed.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n *\n * @class ObjectUnsubscribedError\n */\nexport const ObjectUnsubscribedError: ObjectUnsubscribedErrorCtor = createErrorClass(\n  (_super) =>\n    function ObjectUnsubscribedErrorImpl(this: any) {\n      _super(this);\n      this.name = 'ObjectUnsubscribedError';\n      this.message = 'object unsubscribed';\n    }\n);\n", "import { Operator } from './Operator';\nimport { Observable } from './Observable';\nimport { Subscriber } from './Subscriber';\nimport { Subscription, EMPTY_SUBSCRIPTION } from './Subscription';\nimport { Observer, SubscriptionLike, TeardownLogic } from './types';\nimport { ObjectUnsubscribedError } from './util/ObjectUnsubscribedError';\nimport { arrRemove } from './util/arrRemove';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A Subject is a special type of Observable that allows values to be\n * multicasted to many Observers. Subjects are like EventEmitters.\n *\n * Every Subject is an Observable and an Observer. You can subscribe to a\n * Subject, and you can call next to feed values as well as error and complete.\n */\nexport class Subject<T> extends Observable<T> implements SubscriptionLike {\n  closed = false;\n\n  private currentObservers: Observer<T>[] | null = null;\n\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  observers: Observer<T>[] = [];\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  isStopped = false;\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  hasError = false;\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  thrownError: any = null;\n\n  /**\n   * Creates a \"subject\" by basically gluing an observer to an observable.\n   *\n   * @nocollapse\n   * @deprecated Recommended you do not use. Will be removed at some point in the future. Plans for replacement still under discussion.\n   */\n  static create: (...args: any[]) => any = <T>(destination: Observer<T>, source: Observable<T>): AnonymousSubject<T> => {\n    return new AnonymousSubject<T>(destination, source);\n  };\n\n  constructor() {\n    // NOTE: This must be here to obscure Observable's constructor.\n    super();\n  }\n\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  lift<R>(operator: Operator<T, R>): Observable<R> {\n    const subject = new AnonymousSubject(this, this);\n    subject.operator = operator as any;\n    return subject as any;\n  }\n\n  /** @internal */\n  protected _throwIfClosed() {\n    if (this.closed) {\n      throw new ObjectUnsubscribedError();\n    }\n  }\n\n  next(value: T) {\n    errorContext(() => {\n      this._throwIfClosed();\n      if (!this.isStopped) {\n        if (!this.currentObservers) {\n          this.currentObservers = Array.from(this.observers);\n        }\n        for (const observer of this.currentObservers) {\n          observer.next(value);\n        }\n      }\n    });\n  }\n\n  error(err: any) {\n    errorContext(() => {\n      this._throwIfClosed();\n      if (!this.isStopped) {\n        this.hasError = this.isStopped = true;\n        this.thrownError = err;\n        const { observers } = this;\n        while (observers.length) {\n          observers.shift()!.error(err);\n        }\n      }\n    });\n  }\n\n  complete() {\n    errorContext(() => {\n      this._throwIfClosed();\n      if (!this.isStopped) {\n        this.isStopped = true;\n        const { observers } = this;\n        while (observers.length) {\n          observers.shift()!.complete();\n        }\n      }\n    });\n  }\n\n  unsubscribe() {\n    this.isStopped = this.closed = true;\n    this.observers = this.currentObservers = null!;\n  }\n\n  get observed() {\n    return this.observers?.length > 0;\n  }\n\n  /** @internal */\n  protected _trySubscribe(subscriber: Subscriber<T>): TeardownLogic {\n    this._throwIfClosed();\n    return super._trySubscribe(subscriber);\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<T>): Subscription {\n    this._throwIfClosed();\n    this._checkFinalizedStatuses(subscriber);\n    return this._innerSubscribe(subscriber);\n  }\n\n  /** @internal */\n  protected _innerSubscribe(subscriber: Subscriber<any>) {\n    const { hasError, isStopped, observers } = this;\n    if (hasError || isStopped) {\n      return EMPTY_SUBSCRIPTION;\n    }\n    this.currentObservers = null;\n    observers.push(subscriber);\n    return new Subscription(() => {\n      this.currentObservers = null;\n      arrRemove(observers, subscriber);\n    });\n  }\n\n  /** @internal */\n  protected _checkFinalizedStatuses(subscriber: Subscriber<any>) {\n    const { hasError, thrownError, isStopped } = this;\n    if (hasError) {\n      subscriber.error(thrownError);\n    } else if (isStopped) {\n      subscriber.complete();\n    }\n  }\n\n  /**\n   * Creates a new Observable with this Subject as the source. You can do this\n   * to create custom Observer-side logic of the Subject and conceal it from\n   * code that uses the Observable.\n   * @return {Observable} Observable that the Subject casts to\n   */\n  asObservable(): Observable<T> {\n    const observable: any = new Observable<T>();\n    observable.source = this;\n    return observable;\n  }\n}\n\n/**\n * @class AnonymousSubject<T>\n */\nexport class AnonymousSubject<T> extends Subject<T> {\n  constructor(\n    /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n    public destination?: Observer<T>,\n    source?: Observable<T>\n  ) {\n    super();\n    this.source = source;\n  }\n\n  next(value: T) {\n    this.destination?.next?.(value);\n  }\n\n  error(err: any) {\n    this.destination?.error?.(err);\n  }\n\n  complete() {\n    this.destination?.complete?.();\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<T>): Subscription {\n    return this.source?.subscribe(subscriber) ?? EMPTY_SUBSCRIPTION;\n  }\n}\n", "import { TimestampProvider } from '../types';\n\ninterface DateTimestampProvider extends TimestampProvider {\n  delegate: TimestampProvider | undefined;\n}\n\nexport const dateTimestampProvider: DateTimestampProvider = {\n  now() {\n    // Use the variable rather than `this` so that the function can be called\n    // without being bound to the provider.\n    return (dateTimestampProvider.delegate || Date).now();\n  },\n  delegate: undefined,\n};\n", "import { Subject } from './Subject';\nimport { TimestampProvider } from './types';\nimport { Subscriber } from './Subscriber';\nimport { Subscription } from './Subscription';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * A variant of {@link Subject} that \"replays\" old values to new subscribers by emitting them when they first subscribe.\n *\n * `ReplaySubject` has an internal buffer that will store a specified number of values that it has observed. Like `Subject`,\n * `ReplaySubject` \"observes\" values by having them passed to its `next` method. When it observes a value, it will store that\n * value for a time determined by the configuration of the `ReplaySubject`, as passed to its constructor.\n *\n * When a new subscriber subscribes to the `ReplaySubject` instance, it will synchronously emit all values in its buffer in\n * a First-In-First-Out (FIFO) manner. The `ReplaySubject` will also complete, if it has observed completion; and it will\n * error if it has observed an error.\n *\n * There are two main configuration items to be concerned with:\n *\n * 1. `bufferSize` - This will determine how many items are stored in the buffer, defaults to infinite.\n * 2. `windowTime` - The amount of time to hold a value in the buffer before removing it from the buffer.\n *\n * Both configurations may exist simultaneously. So if you would like to buffer a maximum of 3 values, as long as the values\n * are less than 2 seconds old, you could do so with a `new ReplaySubject(3, 2000)`.\n *\n * ### Differences with BehaviorSubject\n *\n * `BehaviorSubject` is similar to `new ReplaySubject(1)`, with a couple of exceptions:\n *\n * 1. `BehaviorSubject` comes \"primed\" with a single value upon construction.\n * 2. `ReplaySubject` will replay values, even after observing an error, where `BehaviorSubject` will not.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n * @see {@link shareReplay}\n */\nexport class ReplaySubject<T> extends Subject<T> {\n  private _buffer: (T | number)[] = [];\n  private _infiniteTimeWindow = true;\n\n  /**\n   * @param bufferSize The size of the buffer to replay on subscription\n   * @param windowTime The amount of time the buffered items will stay buffered\n   * @param timestampProvider An object with a `now()` method that provides the current timestamp. This is used to\n   * calculate the amount of time something has been buffered.\n   */\n  constructor(\n    private _bufferSize = Infinity,\n    private _windowTime = Infinity,\n    private _timestampProvider: TimestampProvider = dateTimestampProvider\n  ) {\n    super();\n    this._infiniteTimeWindow = _windowTime === Infinity;\n    this._bufferSize = Math.max(1, _bufferSize);\n    this._windowTime = Math.max(1, _windowTime);\n  }\n\n  next(value: T): void {\n    const { isStopped, _buffer, _infiniteTimeWindow, _timestampProvider, _windowTime } = this;\n    if (!isStopped) {\n      _buffer.push(value);\n      !_infiniteTimeWindow && _buffer.push(_timestampProvider.now() + _windowTime);\n    }\n    this._trimBuffer();\n    super.next(value);\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<T>): Subscription {\n    this._throwIfClosed();\n    this._trimBuffer();\n\n    const subscription = this._innerSubscribe(subscriber);\n\n    const { _infiniteTimeWindow, _buffer } = this;\n    // We use a copy here, so reentrant code does not mutate our array while we're\n    // emitting it to a new subscriber.\n    const copy = _buffer.slice();\n    for (let i = 0; i < copy.length && !subscriber.closed; i += _infiniteTimeWindow ? 1 : 2) {\n      subscriber.next(copy[i] as T);\n    }\n\n    this._checkFinalizedStatuses(subscriber);\n\n    return subscription;\n  }\n\n  private _trimBuffer() {\n    const { _bufferSize, _timestampProvider, _buffer, _infiniteTimeWindow } = this;\n    // If we don't have an infinite buffer size, and we're over the length,\n    // use splice to truncate the old buffer values off. Note that we have to\n    // double the size for instances where we're not using an infinite time window\n    // because we're storing the values and the timestamps in the same array.\n    const adjustedBufferSize = (_infiniteTimeWindow ? 1 : 2) * _bufferSize;\n    _bufferSize < Infinity && adjustedBufferSize < _buffer.length && _buffer.splice(0, _buffer.length - adjustedBufferSize);\n\n    // Now, if we're not in an infinite time window, remove all values where the time is\n    // older than what is allowed.\n    if (!_infiniteTimeWindow) {\n      const now = _timestampProvider.now();\n      let last = 0;\n      // Search the array for the first timestamp that isn't expired and\n      // truncate the buffer up to that point.\n      for (let i = 1; i < _buffer.length && (_buffer[i] as number) <= now; i += 2) {\n        last = i;\n      }\n      last && _buffer.splice(0, last + 1);\n    }\n  }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Subscription } from '../Subscription';\nimport { SchedulerAction } from '../types';\n\n/**\n * A unit of work to be executed in a `scheduler`. An action is typically\n * created from within a {@link SchedulerLike} and an RxJS user does not need to concern\n * themselves about creating and manipulating an Action.\n *\n * ```ts\n * class Action<T> extends Subscription {\n *   new (scheduler: Scheduler, work: (state?: T) => void);\n *   schedule(state?: T, delay: number = 0): Subscription;\n * }\n * ```\n *\n * @class Action<T>\n */\nexport class Action<T> extends Subscription {\n  constructor(scheduler: Scheduler, work: (this: SchedulerAction<T>, state?: T) => void) {\n    super();\n  }\n  /**\n   * Schedules this action on its parent {@link SchedulerLike} for execution. May be passed\n   * some context object, `state`. May happen at some point in the future,\n   * according to the `delay` parameter, if specified.\n   * @param {T} [state] Some contextual data that the `work` function uses when\n   * called by the Scheduler.\n   * @param {number} [delay] Time to wait before executing the work, where the\n   * time unit is implicit and defined by the Scheduler.\n   * @return {void}\n   */\n  public schedule(state?: T, delay: number = 0): Subscription {\n    return this;\n  }\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetIntervalFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearIntervalFunction = (handle: TimerHandle) => void;\n\ninterface IntervalProvider {\n  setInterval: SetIntervalFunction;\n  clearInterval: ClearIntervalFunction;\n  delegate:\n    | {\n        setInterval: SetIntervalFunction;\n        clearInterval: ClearIntervalFunction;\n      }\n    | undefined;\n}\n\nexport const intervalProvider: IntervalProvider = {\n  // When accessing the delegate, use the variable rather than `this` so that\n  // the functions can be called without being bound to the provider.\n  setInterval(handler: () => void, timeout?: number, ...args) {\n    const { delegate } = intervalProvider;\n    if (delegate?.setInterval) {\n      return delegate.setInterval(handler, timeout, ...args);\n    }\n    return setInterval(handler, timeout, ...args);\n  },\n  clearInterval(handle) {\n    const { delegate } = intervalProvider;\n    return (delegate?.clearInterval || clearInterval)(handle as any);\n  },\n  delegate: undefined,\n};\n", "import { Action } from './Action';\nimport { SchedulerAction } from '../types';\nimport { Subscription } from '../Subscription';\nimport { AsyncScheduler } from './AsyncScheduler';\nimport { intervalProvider } from './intervalProvider';\nimport { arrRemove } from '../util/arrRemove';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncAction<T> extends Action<T> {\n  public id: TimerHandle | undefined;\n  public state?: T;\n  // @ts-ignore: Property has no initializer and is not definitely assigned\n  public delay: number;\n  protected pending: boolean = false;\n\n  constructor(protected scheduler: AsyncScheduler, protected work: (this: SchedulerAction<T>, state?: T) => void) {\n    super(scheduler, work);\n  }\n\n  public schedule(state?: T, delay: number = 0): Subscription {\n    if (this.closed) {\n      return this;\n    }\n\n    // Always replace the current state with the new state.\n    this.state = state;\n\n    const id = this.id;\n    const scheduler = this.scheduler;\n\n    //\n    // Important implementation note:\n    //\n    // Actions only execute once by default, unless rescheduled from within the\n    // scheduled callback. This allows us to implement single and repeat\n    // actions via the same code path, without adding API surface area, as well\n    // as mimic traditional recursion but across asynchronous boundaries.\n    //\n    // However, JS runtimes and timers distinguish between intervals achieved by\n    // serial `setTimeout` calls vs. a single `setInterval` call. An interval of\n    // serial `setTimeout` calls can be individually delayed, which delays\n    // scheduling the next `setTimeout`, and so on. `setInterval` attempts to\n    // guarantee the interval callback will be invoked more precisely to the\n    // interval period, regardless of load.\n    //\n    // Therefore, we use `setInterval` to schedule single and repeat actions.\n    // If the action reschedules itself with the same delay, the interval is not\n    // canceled. If the action doesn't reschedule, or reschedules with a\n    // different delay, the interval will be canceled after scheduled callback\n    // execution.\n    //\n    if (id != null) {\n      this.id = this.recycleAsyncId(scheduler, id, delay);\n    }\n\n    // Set the pending flag indicating that this action has been scheduled, or\n    // has recursively rescheduled itself.\n    this.pending = true;\n\n    this.delay = delay;\n    // If this action has already an async Id, don't request a new one.\n    this.id = this.id ?? this.requestAsyncId(scheduler, this.id, delay);\n\n    return this;\n  }\n\n  protected requestAsyncId(scheduler: AsyncScheduler, _id?: TimerHandle, delay: number = 0): TimerHandle {\n    return intervalProvider.setInterval(scheduler.flush.bind(scheduler, this), delay);\n  }\n\n  protected recycleAsyncId(_scheduler: AsyncScheduler, id?: TimerHandle, delay: number | null = 0): TimerHandle | undefined {\n    // If this action is rescheduled with the same delay time, don't clear the interval id.\n    if (delay != null && this.delay === delay && this.pending === false) {\n      return id;\n    }\n    // Otherwise, if the action's delay time is different from the current delay,\n    // or the action has been rescheduled before it's executed, clear the interval id\n    if (id != null) {\n      intervalProvider.clearInterval(id);\n    }\n\n    return undefined;\n  }\n\n  /**\n   * Immediately executes this action and the `work` it contains.\n   * @return {any}\n   */\n  public execute(state: T, delay: number): any {\n    if (this.closed) {\n      return new Error('executing a cancelled action');\n    }\n\n    this.pending = false;\n    const error = this._execute(state, delay);\n    if (error) {\n      return error;\n    } else if (this.pending === false && this.id != null) {\n      // Dequeue if the action didn't reschedule itself. Don't call\n      // unsubscribe(), because the action could reschedule later.\n      // For example:\n      // ```\n      // scheduler.schedule(function doWork(counter) {\n      //   /* ... I'm a busy worker bee ... */\n      //   var originalAction = this;\n      //   /* wait 100ms before rescheduling the action */\n      //   setTimeout(function () {\n      //     originalAction.schedule(counter + 1);\n      //   }, 100);\n      // }, 1000);\n      // ```\n      this.id = this.recycleAsyncId(this.scheduler, this.id, null);\n    }\n  }\n\n  protected _execute(state: T, _delay: number): any {\n    let errored: boolean = false;\n    let errorValue: any;\n    try {\n      this.work(state);\n    } catch (e) {\n      errored = true;\n      // HACK: Since code elsewhere is relying on the \"truthiness\" of the\n      // return here, we can't have it return \"\" or 0 or false.\n      // TODO: Clean this up when we refactor schedulers mid-version-8 or so.\n      errorValue = e ? e : new Error('Scheduled action threw falsy error');\n    }\n    if (errored) {\n      this.unsubscribe();\n      return errorValue;\n    }\n  }\n\n  unsubscribe() {\n    if (!this.closed) {\n      const { id, scheduler } = this;\n      const { actions } = scheduler;\n\n      this.work = this.state = this.scheduler = null!;\n      this.pending = false;\n\n      arrRemove(actions, this);\n      if (id != null) {\n        this.id = this.recycleAsyncId(scheduler, id, null);\n      }\n\n      this.delay = null!;\n      super.unsubscribe();\n    }\n  }\n}\n", "import { Action } from './scheduler/Action';\nimport { Subscription } from './Subscription';\nimport { SchedulerLike, SchedulerAction } from './types';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * An execution context and a data structure to order tasks and schedule their\n * execution. Provides a notion of (potentially virtual) time, through the\n * `now()` getter method.\n *\n * Each unit of work in a Scheduler is called an `Action`.\n *\n * ```ts\n * class Scheduler {\n *   now(): number;\n *   schedule(work, delay?, state?): Subscription;\n * }\n * ```\n *\n * @class Scheduler\n * @deprecated Scheduler is an internal implementation detail of RxJS, and\n * should not be used directly. Rather, create your own class and implement\n * {@link SchedulerLike}. Will be made internal in v8.\n */\nexport class Scheduler implements SchedulerLike {\n  public static now: () => number = dateTimestampProvider.now;\n\n  constructor(private schedulerActionCtor: typeof Action, now: () => number = Scheduler.now) {\n    this.now = now;\n  }\n\n  /**\n   * A getter method that returns a number representing the current time\n   * (at the time this function was called) according to the scheduler's own\n   * internal clock.\n   * @return {number} A number that represents the current time. May or may not\n   * have a relation to wall-clock time. May or may not refer to a time unit\n   * (e.g. milliseconds).\n   */\n  public now: () => number;\n\n  /**\n   * Schedules a function, `work`, for execution. May happen at some point in\n   * the future, according to the `delay` parameter, if specified. May be passed\n   * some context object, `state`, which will be passed to the `work` function.\n   *\n   * The given arguments will be processed an stored as an Action object in a\n   * queue of actions.\n   *\n   * @param {function(state: ?T): ?Subscription} work A function representing a\n   * task, or some unit of work to be executed by the Scheduler.\n   * @param {number} [delay] Time to wait before executing the work, where the\n   * time unit is implicit and defined by the Scheduler itself.\n   * @param {T} [state] Some contextual data that the `work` function uses when\n   * called by the Scheduler.\n   * @return {Subscription} A subscription in order to be able to unsubscribe\n   * the scheduled work.\n   */\n  public schedule<T>(work: (this: SchedulerAction<T>, state?: T) => void, delay: number = 0, state?: T): Subscription {\n    return new this.schedulerActionCtor<T>(this, work).schedule(state, delay);\n  }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Action } from './Action';\nimport { AsyncAction } from './AsyncAction';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncScheduler extends Scheduler {\n  public actions: Array<AsyncAction<any>> = [];\n  /**\n   * A flag to indicate whether the Scheduler is currently executing a batch of\n   * queued actions.\n   * @type {boolean}\n   * @internal\n   */\n  public _active: boolean = false;\n  /**\n   * An internal ID used to track the latest asynchronous task such as those\n   * coming from `setTimeout`, `setInterval`, `requestAnimationFrame`, and\n   * others.\n   * @type {any}\n   * @internal\n   */\n  public _scheduled: TimerHandle | undefined;\n\n  constructor(SchedulerAction: typeof Action, now: () => number = Scheduler.now) {\n    super(SchedulerAction, now);\n  }\n\n  public flush(action: AsyncAction<any>): void {\n    const { actions } = this;\n\n    if (this._active) {\n      actions.push(action);\n      return;\n    }\n\n    let error: any;\n    this._active = true;\n\n    do {\n      if ((error = action.execute(action.state, action.delay))) {\n        break;\n      }\n    } while ((action = actions.shift()!)); // exhaust the scheduler queue\n\n    this._active = false;\n\n    if (error) {\n      while ((action = actions.shift()!)) {\n        action.unsubscribe();\n      }\n      throw error;\n    }\n  }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\n/**\n *\n * Async Scheduler\n *\n * <span class=\"informal\">Schedule task as if you used setTimeout(task, duration)</span>\n *\n * `async` scheduler schedules tasks asynchronously, by putting them on the JavaScript\n * event loop queue. It is best used to delay tasks in time or to schedule tasks repeating\n * in intervals.\n *\n * If you just want to \"defer\" task, that is to perform it right after currently\n * executing synchronous code ends (commonly achieved by `setTimeout(deferredTask, 0)`),\n * better choice will be the {@link asapScheduler} scheduler.\n *\n * ## Examples\n * Use async scheduler to delay task\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * const task = () => console.log('it works!');\n *\n * asyncScheduler.schedule(task, 2000);\n *\n * // After 2 seconds logs:\n * // \"it works!\"\n * ```\n *\n * Use async scheduler to repeat task in intervals\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * function task(state) {\n *   console.log(state);\n *   this.schedule(state + 1, 1000); // `this` references currently executing Action,\n *                                   // which we reschedule with new state and delay\n * }\n *\n * asyncScheduler.schedule(task, 3000, 0);\n *\n * // Logs:\n * // 0 after 3s\n * // 1 after 4s\n * // 2 after 5s\n * // 3 after 6s\n * ```\n */\n\nexport const asyncScheduler = new AsyncScheduler(AsyncAction);\n\n/**\n * @deprecated Renamed to {@link asyncScheduler}. Will be removed in v8.\n */\nexport const async = asyncScheduler;\n", "import { AsyncAction } from './AsyncAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\nimport { SchedulerAction } from '../types';\nimport { animationFrameProvider } from './animationFrameProvider';\nimport { TimerHandle } from './timerHandle';\n\nexport class AnimationFrameAction<T> extends AsyncAction<T> {\n  constructor(protected scheduler: AnimationFrameScheduler, protected work: (this: SchedulerAction<T>, state?: T) => void) {\n    super(scheduler, work);\n  }\n\n  protected requestAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle {\n    // If delay is greater than 0, request as an async action.\n    if (delay !== null && delay > 0) {\n      return super.requestAsyncId(scheduler, id, delay);\n    }\n    // Push the action to the end of the scheduler queue.\n    scheduler.actions.push(this);\n    // If an animation frame has already been requested, don't request another\n    // one. If an animation frame hasn't been requested yet, request one. Return\n    // the current animation frame request id.\n    return scheduler._scheduled || (scheduler._scheduled = animationFrameProvider.requestAnimationFrame(() => scheduler.flush(undefined)));\n  }\n\n  protected recycleAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle | undefined {\n    // If delay exists and is greater than 0, or if the delay is null (the\n    // action wasn't rescheduled) but was originally scheduled as an async\n    // action, then recycle as an async action.\n    if (delay != null ? delay > 0 : this.delay > 0) {\n      return super.recycleAsyncId(scheduler, id, delay);\n    }\n    // If the scheduler queue has no remaining actions with the same async id,\n    // cancel the requested animation frame and set the scheduled flag to\n    // undefined so the next AnimationFrameAction will request its own.\n    const { actions } = scheduler;\n    if (id != null && actions[actions.length - 1]?.id !== id) {\n      animationFrameProvider.cancelAnimationFrame(id as number);\n      scheduler._scheduled = undefined;\n    }\n    // Return undefined so the action knows to request a new async id if it's rescheduled.\n    return undefined;\n  }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\nexport class AnimationFrameScheduler extends AsyncScheduler {\n  public flush(action?: AsyncAction<any>): void {\n    this._active = true;\n    // The async id that effects a call to flush is stored in _scheduled.\n    // Before executing an action, it's necessary to check the action's async\n    // id to determine whether it's supposed to be executed in the current\n    // flush.\n    // Previous implementations of this method used a count to determine this,\n    // but that was unsound, as actions that are unsubscribed - i.e. cancelled -\n    // are removed from the actions array and that can shift actions that are\n    // scheduled to be executed in a subsequent flush into positions at which\n    // they are executed within the current flush.\n    const flushId = this._scheduled;\n    this._scheduled = undefined;\n\n    const { actions } = this;\n    let error: any;\n    action = action || actions.shift()!;\n\n    do {\n      if ((error = action.execute(action.state, action.delay))) {\n        break;\n      }\n    } while ((action = actions[0]) && action.id === flushId && actions.shift());\n\n    this._active = false;\n\n    if (error) {\n      while ((action = actions[0]) && action.id === flushId && actions.shift()) {\n        action.unsubscribe();\n      }\n      throw error;\n    }\n  }\n}\n", "import { AnimationFrameAction } from './AnimationFrameAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\n\n/**\n *\n * Animation Frame Scheduler\n *\n * <span class=\"informal\">Perform task when `window.requestAnimationFrame` would fire</span>\n *\n * When `animationFrame` scheduler is used with delay, it will fall back to {@link asyncScheduler} scheduler\n * behaviour.\n *\n * Without delay, `animationFrame` scheduler can be used to create smooth browser animations.\n * It makes sure scheduled task will happen just before next browser content repaint,\n * thus performing animations as efficiently as possible.\n *\n * ## Example\n * Schedule div height animation\n * ```ts\n * // html: <div style=\"background: #0ff;\"></div>\n * import { animationFrameScheduler } from 'rxjs';\n *\n * const div = document.querySelector('div');\n *\n * animationFrameScheduler.schedule(function(height) {\n *   div.style.height = height + \"px\";\n *\n *   this.schedule(height + 1);  // `this` references currently executing Action,\n *                               // which we reschedule with new state\n * }, 0, 0);\n *\n * // You will see a div element growing in height\n * ```\n */\n\nexport const animationFrameScheduler = new AnimationFrameScheduler(AnimationFrameAction);\n\n/**\n * @deprecated Renamed to {@link animationFrameScheduler}. Will be removed in v8.\n */\nexport const animationFrame = animationFrameScheduler;\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\n\n/**\n * A simple Observable that emits no items to the Observer and immediately\n * emits a complete notification.\n *\n * <span class=\"informal\">Just emits 'complete', and nothing else.</span>\n *\n * ![](empty.png)\n *\n * A simple Observable that only emits the complete notification. It can be used\n * for composing with other Observables, such as in a {@link mergeMap}.\n *\n * ## Examples\n *\n * Log complete notification\n *\n * ```ts\n * import { EMPTY } from 'rxjs';\n *\n * EMPTY.subscribe({\n *   next: () => console.log('Next'),\n *   complete: () => console.log('Complete!')\n * });\n *\n * // Outputs\n * // Complete!\n * ```\n *\n * Emit the number 7, then complete\n *\n * ```ts\n * import { EMPTY, startWith } from 'rxjs';\n *\n * const result = EMPTY.pipe(startWith(7));\n * result.subscribe(x => console.log(x));\n *\n * // Outputs\n * // 7\n * ```\n *\n * Map and flatten only odd numbers to the sequence `'a'`, `'b'`, `'c'`\n *\n * ```ts\n * import { interval, mergeMap, of, EMPTY } from 'rxjs';\n *\n * const interval$ = interval(1000);\n * const result = interval$.pipe(\n *   mergeMap(x => x % 2 === 1 ? of('a', 'b', 'c') : EMPTY),\n * );\n * result.subscribe(x => console.log(x));\n *\n * // Results in the following to the console:\n * // x is equal to the count on the interval, e.g. (0, 1, 2, 3, ...)\n * // x will occur every 1000ms\n * // if x % 2 is equal to 1, print a, b, c (each on its own)\n * // if x % 2 is not equal to 1, nothing will be output\n * ```\n *\n * @see {@link Observable}\n * @see {@link NEVER}\n * @see {@link of}\n * @see {@link throwError}\n */\nexport const EMPTY = new Observable<never>((subscriber) => subscriber.complete());\n\n/**\n * @param scheduler A {@link SchedulerLike} to use for scheduling\n * the emission of the complete notification.\n * @deprecated Replaced with the {@link EMPTY} constant or {@link scheduled} (e.g. `scheduled([], scheduler)`). Will be removed in v8.\n */\nexport function empty(scheduler?: SchedulerLike) {\n  return scheduler ? emptyScheduled(scheduler) : EMPTY;\n}\n\nfunction emptyScheduled(scheduler: SchedulerLike) {\n  return new Observable<never>((subscriber) => scheduler.schedule(() => subscriber.complete()));\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport function isScheduler(value: any): value is SchedulerLike {\n  return value && isFunction(value.schedule);\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\nimport { isScheduler } from './isScheduler';\n\nfunction last<T>(arr: T[]): T | undefined {\n  return arr[arr.length - 1];\n}\n\nexport function popResultSelector(args: any[]): ((...args: unknown[]) => unknown) | undefined {\n  return isFunction(last(args)) ? args.pop() : undefined;\n}\n\nexport function popScheduler(args: any[]): SchedulerLike | undefined {\n  return isScheduler(last(args)) ? args.pop() : undefined;\n}\n\nexport function popNumber(args: any[], defaultValue: number): number {\n  return typeof last(args) === 'number' ? args.pop()! : defaultValue;\n}\n", "export const isArrayLike = (<T>(x: any): x is ArrayLike<T> => x && typeof x.length === 'number' && typeof x !== 'function');", "import { isFunction } from \"./isFunction\";\n\n/**\n * Tests to see if the object is \"thennable\".\n * @param value the object to test\n */\nexport function isPromise(value: any): value is PromiseLike<any> {\n  return isFunction(value?.then);\n}\n", "import { InteropObservable } from '../types';\nimport { observable as Symbol_observable } from '../symbol/observable';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being Observable (but not necessary an Rx Observable) */\nexport function isInteropObservable(input: any): input is InteropObservable<any> {\n  return isFunction(input[Symbol_observable]);\n}\n", "import { isFunction } from './isFunction';\n\nexport function isAsyncIterable<T>(obj: any): obj is AsyncIterable<T> {\n  return Symbol.asyncIterator && isFunction(obj?.[Symbol.asyncIterator]);\n}\n", "/**\n * Creates the TypeError to throw if an invalid object is passed to `from` or `scheduled`.\n * @param input The object that was passed.\n */\nexport function createInvalidObservableTypeError(input: any) {\n  // TODO: We should create error codes that can be looked up, so this can be less verbose.\n  return new TypeError(\n    `You provided ${\n      input !== null && typeof input === 'object' ? 'an invalid object' : `'${input}'`\n    } where a stream was expected. You can provide an Observable, Promise, ReadableStream, Array, AsyncIterable, or Iterable.`\n  );\n}\n", "export function getSymbolIterator(): symbol {\n  if (typeof Symbol !== 'function' || !Symbol.iterator) {\n    return '@@iterator' as any;\n  }\n\n  return Symbol.iterator;\n}\n\nexport const iterator = getSymbolIterator();\n", "import { iterator as Symbol_iterator } from '../symbol/iterator';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being an Iterable */\nexport function isIterable(input: any): input is Iterable<any> {\n  return isFunction(input?.[Symbol_iterator]);\n}\n", "import { ReadableStreamLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport async function* readableStreamLikeToAsyncGenerator<T>(readableStream: ReadableStreamLike<T>): AsyncGenerator<T> {\n  const reader = readableStream.getReader();\n  try {\n    while (true) {\n      const { value, done } = await reader.read();\n      if (done) {\n        return;\n      }\n      yield value!;\n    }\n  } finally {\n    reader.releaseLock();\n  }\n}\n\nexport function isReadableStreamLike<T>(obj: any): obj is ReadableStreamLike<T> {\n  // We don't want to use instanceof checks because they would return\n  // false for instances from another Realm, like an <iframe>.\n  return isFunction(obj?.getReader);\n}\n", "import { isArrayLike } from '../util/isArrayLike';\nimport { isPromise } from '../util/isPromise';\nimport { Observable } from '../Observable';\nimport { ObservableInput, ObservedValueOf, ReadableStreamLike } from '../types';\nimport { isInteropObservable } from '../util/isInteropObservable';\nimport { isAsyncIterable } from '../util/isAsyncIterable';\nimport { createInvalidObservableTypeError } from '../util/throwUnobservableError';\nimport { isIterable } from '../util/isIterable';\nimport { isReadableStreamLike, readableStreamLikeToAsyncGenerator } from '../util/isReadableStreamLike';\nimport { Subscriber } from '../Subscriber';\nimport { isFunction } from '../util/isFunction';\nimport { reportUnhandledError } from '../util/reportUnhandledError';\nimport { observable as Symbol_observable } from '../symbol/observable';\n\nexport function innerFrom<O extends ObservableInput<any>>(input: O): Observable<ObservedValueOf<O>>;\nexport function innerFrom<T>(input: ObservableInput<T>): Observable<T> {\n  if (input instanceof Observable) {\n    return input;\n  }\n  if (input != null) {\n    if (isInteropObservable(input)) {\n      return fromInteropObservable(input);\n    }\n    if (isArrayLike(input)) {\n      return fromArrayLike(input);\n    }\n    if (isPromise(input)) {\n      return fromPromise(input);\n    }\n    if (isAsyncIterable(input)) {\n      return fromAsyncIterable(input);\n    }\n    if (isIterable(input)) {\n      return fromIterable(input);\n    }\n    if (isReadableStreamLike(input)) {\n      return fromReadableStreamLike(input);\n    }\n  }\n\n  throw createInvalidObservableTypeError(input);\n}\n\n/**\n * Creates an RxJS Observable from an object that implements `Symbol.observable`.\n * @param obj An object that properly implements `Symbol.observable`.\n */\nexport function fromInteropObservable<T>(obj: any) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    const obs = obj[Symbol_observable]();\n    if (isFunction(obs.subscribe)) {\n      return obs.subscribe(subscriber);\n    }\n    // Should be caught by observable subscribe function error handling.\n    throw new TypeError('Provided object does not correctly implement Symbol.observable');\n  });\n}\n\n/**\n * Synchronously emits the values of an array like and completes.\n * This is exported because there are creation functions and operators that need to\n * make direct use of the same logic, and there's no reason to make them run through\n * `from` conditionals because we *know* they're dealing with an array.\n * @param array The array to emit values from\n */\nexport function fromArrayLike<T>(array: ArrayLike<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    // Loop over the array and emit each value. Note two things here:\n    // 1. We're making sure that the subscriber is not closed on each loop.\n    //    This is so we don't continue looping over a very large array after\n    //    something like a `take`, `takeWhile`, or other synchronous unsubscription\n    //    has already unsubscribed.\n    // 2. In this form, reentrant code can alter that array we're looping over.\n    //    This is a known issue, but considered an edge case. The alternative would\n    //    be to copy the array before executing the loop, but this has\n    //    performance implications.\n    for (let i = 0; i < array.length && !subscriber.closed; i++) {\n      subscriber.next(array[i]);\n    }\n    subscriber.complete();\n  });\n}\n\nexport function fromPromise<T>(promise: PromiseLike<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    promise\n      .then(\n        (value) => {\n          if (!subscriber.closed) {\n            subscriber.next(value);\n            subscriber.complete();\n          }\n        },\n        (err: any) => subscriber.error(err)\n      )\n      .then(null, reportUnhandledError);\n  });\n}\n\nexport function fromIterable<T>(iterable: Iterable<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    for (const value of iterable) {\n      subscriber.next(value);\n      if (subscriber.closed) {\n        return;\n      }\n    }\n    subscriber.complete();\n  });\n}\n\nexport function fromAsyncIterable<T>(asyncIterable: AsyncIterable<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    process(asyncIterable, subscriber).catch((err) => subscriber.error(err));\n  });\n}\n\nexport function fromReadableStreamLike<T>(readableStream: ReadableStreamLike<T>) {\n  return fromAsyncIterable(readableStreamLikeToAsyncGenerator(readableStream));\n}\n\nasync function process<T>(asyncIterable: AsyncIterable<T>, subscriber: Subscriber<T>) {\n  for await (const value of asyncIterable) {\n    subscriber.next(value);\n    // A side-effect may have closed our subscriber,\n    // check before the next iteration.\n    if (subscriber.closed) {\n      return;\n    }\n  }\n  subscriber.complete();\n}\n", "import { Subscription } from '../Subscription';\nimport { SchedulerAction, SchedulerLike } from '../types';\n\nexport function executeSchedule(\n  parentSubscription: Subscription,\n  scheduler: SchedulerLike,\n  work: () => void,\n  delay: number,\n  repeat: true\n): void;\nexport function executeSchedule(\n  parentSubscription: Subscription,\n  scheduler: SchedulerLike,\n  work: () => void,\n  delay?: number,\n  repeat?: false\n): Subscription;\n\nexport function executeSchedule(\n  parentSubscription: Subscription,\n  scheduler: SchedulerLike,\n  work: () => void,\n  delay = 0,\n  repeat = false\n): Subscription | void {\n  const scheduleSubscription = scheduler.schedule(function (this: SchedulerAction<any>) {\n    work();\n    if (repeat) {\n      parentSubscription.add(this.schedule(null, delay));\n    } else {\n      this.unsubscribe();\n    }\n  }, delay);\n\n  parentSubscription.add(scheduleSubscription);\n\n  if (!repeat) {\n    // Because user-land scheduler implementations are unlikely to properly reuse\n    // Actions for repeat scheduling, we can't trust that the returned subscription\n    // will control repeat subscription scenarios. So we're trying to avoid using them\n    // incorrectly within this library.\n    return scheduleSubscription;\n  }\n}\n", "/** @prettier */\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { executeSchedule } from '../util/executeSchedule';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Re-emits all notifications from source Observable with specified scheduler.\n *\n * <span class=\"informal\">Ensure a specific scheduler is used, from outside of an Observable.</span>\n *\n * `observeOn` is an operator that accepts a scheduler as a first parameter, which will be used to reschedule\n * notifications emitted by the source Observable. It might be useful, if you do not have control over\n * internal scheduler of a given Observable, but want to control when its values are emitted nevertheless.\n *\n * Returned Observable emits the same notifications (nexted values, complete and error events) as the source Observable,\n * but rescheduled with provided scheduler. Note that this doesn't mean that source Observables internal\n * scheduler will be replaced in any way. Original scheduler still will be used, but when the source Observable emits\n * notification, it will be immediately scheduled again - this time with scheduler passed to `observeOn`.\n * An anti-pattern would be calling `observeOn` on Observable that emits lots of values synchronously, to split\n * that emissions into asynchronous chunks. For this to happen, scheduler would have to be passed into the source\n * Observable directly (usually into the operator that creates it). `observeOn` simply delays notifications a\n * little bit more, to ensure that they are emitted at expected moments.\n *\n * As a matter of fact, `observeOn` accepts second parameter, which specifies in milliseconds with what delay notifications\n * will be emitted. The main difference between {@link delay} operator and `observeOn` is that `observeOn`\n * will delay all notifications - including error notifications - while `delay` will pass through error\n * from source Observable immediately when it is emitted. In general it is highly recommended to use `delay` operator\n * for any kind of delaying of values in the stream, while using `observeOn` to specify which scheduler should be used\n * for notification emissions in general.\n *\n * ## Example\n *\n * Ensure values in subscribe are called just before browser repaint\n *\n * ```ts\n * import { interval, observeOn, animationFrameScheduler } from 'rxjs';\n *\n * const someDiv = document.createElement('div');\n * someDiv.style.cssText = 'width: 200px;background: #09c';\n * document.body.appendChild(someDiv);\n * const intervals = interval(10);      // Intervals are scheduled\n *                                      // with async scheduler by default...\n * intervals.pipe(\n *   observeOn(animationFrameScheduler) // ...but we will observe on animationFrame\n * )                                    // scheduler to ensure smooth animation.\n * .subscribe(val => {\n *   someDiv.style.height = val + 'px';\n * });\n * ```\n *\n * @see {@link delay}\n *\n * @param scheduler Scheduler that will be used to reschedule notifications from source Observable.\n * @param delay Number of milliseconds that states with what delay every notification should be rescheduled.\n * @return A function that returns an Observable that emits the same\n * notifications as the source Observable, but with provided scheduler.\n */\nexport function observeOn<T>(scheduler: SchedulerLike, delay = 0): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => executeSchedule(subscriber, scheduler, () => subscriber.next(value), delay),\n        () => executeSchedule(subscriber, scheduler, () => subscriber.complete(), delay),\n        (err) => executeSchedule(subscriber, scheduler, () => subscriber.error(err), delay)\n      )\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { operate } from '../util/lift';\n\n/**\n * Asynchronously subscribes Observers to this Observable on the specified {@link SchedulerLike}.\n *\n * With `subscribeOn` you can decide what type of scheduler a specific Observable will be using when it is subscribed to.\n *\n * Schedulers control the speed and order of emissions to observers from an Observable stream.\n *\n * ![](subscribeOn.png)\n *\n * ## Example\n *\n * Given the following code:\n *\n * ```ts\n * import { of, merge } from 'rxjs';\n *\n * const a = of(1, 2, 3);\n * const b = of(4, 5, 6);\n *\n * merge(a, b).subscribe(console.log);\n *\n * // Outputs\n * // 1\n * // 2\n * // 3\n * // 4\n * // 5\n * // 6\n * ```\n *\n * Both Observable `a` and `b` will emit their values directly and synchronously once they are subscribed to.\n *\n * If we instead use the `subscribeOn` operator declaring that we want to use the {@link asyncScheduler} for values emitted by Observable `a`:\n *\n * ```ts\n * import { of, subscribeOn, asyncScheduler, merge } from 'rxjs';\n *\n * const a = of(1, 2, 3).pipe(subscribeOn(asyncScheduler));\n * const b = of(4, 5, 6);\n *\n * merge(a, b).subscribe(console.log);\n *\n * // Outputs\n * // 4\n * // 5\n * // 6\n * // 1\n * // 2\n * // 3\n * ```\n *\n * The reason for this is that Observable `b` emits its values directly and synchronously like before\n * but the emissions from `a` are scheduled on the event loop because we are now using the {@link asyncScheduler} for that specific Observable.\n *\n * @param scheduler The {@link SchedulerLike} to perform subscription actions on.\n * @param delay A delay to pass to the scheduler to delay subscriptions\n * @return A function that returns an Observable modified so that its\n * subscriptions happen on the specified {@link SchedulerLike}.\n */\nexport function subscribeOn<T>(scheduler: SchedulerLike, delay: number = 0): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    subscriber.add(scheduler.schedule(() => source.subscribe(subscriber), delay));\n  });\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { observeOn } from '../operators/observeOn';\nimport { subscribeOn } from '../operators/subscribeOn';\nimport { InteropObservable, SchedulerLike } from '../types';\n\nexport function scheduleObservable<T>(input: InteropObservable<T>, scheduler: SchedulerLike) {\n  return innerFrom(input).pipe(subscribeOn(scheduler), observeOn(scheduler));\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { observeOn } from '../operators/observeOn';\nimport { subscribeOn } from '../operators/subscribeOn';\nimport { SchedulerLike } from '../types';\n\nexport function schedulePromise<T>(input: PromiseLike<T>, scheduler: SchedulerLike) {\n  return innerFrom(input).pipe(subscribeOn(scheduler), observeOn(scheduler));\n}\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\n\nexport function scheduleArray<T>(input: ArrayLike<T>, scheduler: SchedulerLike) {\n  return new Observable<T>((subscriber) => {\n    // The current array index.\n    let i = 0;\n    // Start iterating over the array like on a schedule.\n    return scheduler.schedule(function () {\n      if (i === input.length) {\n        // If we have hit the end of the array like in the\n        // previous job, we can complete.\n        subscriber.complete();\n      } else {\n        // Otherwise let's next the value at the current index,\n        // then increment our index.\n        subscriber.next(input[i++]);\n        // If the last emission didn't cause us to close the subscriber\n        // (via take or some side effect), reschedule the job and we'll\n        // make another pass.\n        if (!subscriber.closed) {\n          this.schedule();\n        }\n      }\n    });\n  });\n}\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\nimport { iterator as Symbol_iterator } from '../symbol/iterator';\nimport { isFunction } from '../util/isFunction';\nimport { executeSchedule } from '../util/executeSchedule';\n\n/**\n * Used in {@link scheduled} to create an observable from an Iterable.\n * @param input The iterable to create an observable from\n * @param scheduler The scheduler to use\n */\nexport function scheduleIterable<T>(input: Iterable<T>, scheduler: SchedulerLike) {\n  return new Observable<T>((subscriber) => {\n    let iterator: Iterator<T, T>;\n\n    // Schedule the initial creation of the iterator from\n    // the iterable. This is so the code in the iterable is\n    // not called until the scheduled job fires.\n    executeSchedule(subscriber, scheduler, () => {\n      // Create the iterator.\n      iterator = (input as any)[Symbol_iterator]();\n\n      executeSchedule(\n        subscriber,\n        scheduler,\n        () => {\n          let value: T;\n          let done: boolean | undefined;\n          try {\n            // Pull the value out of the iterator\n            ({ value, done } = iterator.next());\n          } catch (err) {\n            // We got an error while pulling from the iterator\n            subscriber.error(err);\n            return;\n          }\n\n          if (done) {\n            // If it is \"done\" we just complete. This mimics the\n            // behavior of JavaScript's `for..of` consumption of\n            // iterables, which will not emit the value from an iterator\n            // result of `{ done: true: value: 'here' }`.\n            subscriber.complete();\n          } else {\n            // The iterable is not done, emit the value.\n            subscriber.next(value);\n          }\n        },\n        0,\n        true\n      );\n    });\n\n    // During finalization, if we see this iterator has a `return` method,\n    // then we know it is a Generator, and not just an Iterator. So we call\n    // the `return()` function. This will ensure that any `finally { }` blocks\n    // inside of the generator we can hit will be hit properly.\n    return () => isFunction(iterator?.return) && iterator.return();\n  });\n}\n", "import { SchedulerLike } from '../types';\nimport { Observable } from '../Observable';\nimport { executeSchedule } from '../util/executeSchedule';\n\nexport function scheduleAsyncIterable<T>(input: AsyncIterable<T>, scheduler: SchedulerLike) {\n  if (!input) {\n    throw new Error('Iterable cannot be null');\n  }\n  return new Observable<T>((subscriber) => {\n    executeSchedule(subscriber, scheduler, () => {\n      const iterator = input[Symbol.asyncIterator]();\n      executeSchedule(\n        subscriber,\n        scheduler,\n        () => {\n          iterator.next().then((result) => {\n            if (result.done) {\n              // This will remove the subscriptions from\n              // the parent subscription.\n              subscriber.complete();\n            } else {\n              subscriber.next(result.value);\n            }\n          });\n        },\n        0,\n        true\n      );\n    });\n  });\n}\n", "import { SchedulerLike, ReadableStreamLike } from '../types';\nimport { Observable } from '../Observable';\nimport { scheduleAsyncIterable } from './scheduleAsyncIterable';\nimport { readableStreamLikeToAsyncGenerator } from '../util/isReadableStreamLike';\n\nexport function scheduleReadableStreamLike<T>(input: ReadableStreamLike<T>, scheduler: SchedulerLike): Observable<T> {\n  return scheduleAsyncIterable(readableStreamLikeToAsyncGenerator(input), scheduler);\n}\n", "import { scheduleObservable } from './scheduleObservable';\nimport { schedulePromise } from './schedulePromise';\nimport { scheduleArray } from './scheduleArray';\nimport { scheduleIterable } from './scheduleIterable';\nimport { scheduleAsyncIterable } from './scheduleAsyncIterable';\nimport { isInteropObservable } from '../util/isInteropObservable';\nimport { isPromise } from '../util/isPromise';\nimport { isArrayLike } from '../util/isArrayLike';\nimport { isIterable } from '../util/isIterable';\nimport { ObservableInput, SchedulerLike } from '../types';\nimport { Observable } from '../Observable';\nimport { isAsyncIterable } from '../util/isAsyncIterable';\nimport { createInvalidObservableTypeError } from '../util/throwUnobservableError';\nimport { isReadableStreamLike } from '../util/isReadableStreamLike';\nimport { scheduleReadableStreamLike } from './scheduleReadableStreamLike';\n\n/**\n * Converts from a common {@link ObservableInput} type to an observable where subscription and emissions\n * are scheduled on the provided scheduler.\n *\n * @see {@link from}\n * @see {@link of}\n *\n * @param input The observable, array, promise, iterable, etc you would like to schedule\n * @param scheduler The scheduler to use to schedule the subscription and emissions from\n * the returned observable.\n */\nexport function scheduled<T>(input: ObservableInput<T>, scheduler: SchedulerLike): Observable<T> {\n  if (input != null) {\n    if (isInteropObservable(input)) {\n      return scheduleObservable(input, scheduler);\n    }\n    if (isArrayLike(input)) {\n      return scheduleArray(input, scheduler);\n    }\n    if (isPromise(input)) {\n      return schedulePromise(input, scheduler);\n    }\n    if (isAsyncIterable(input)) {\n      return scheduleAsyncIterable(input, scheduler);\n    }\n    if (isIterable(input)) {\n      return scheduleIterable(input, scheduler);\n    }\n    if (isReadableStreamLike(input)) {\n      return scheduleReadableStreamLike(input, scheduler);\n    }\n  }\n  throw createInvalidObservableTypeError(input);\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInput, SchedulerLike, ObservedValueOf } from '../types';\nimport { scheduled } from '../scheduled/scheduled';\nimport { innerFrom } from './innerFrom';\n\nexport function from<O extends ObservableInput<any>>(input: O): Observable<ObservedValueOf<O>>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function from<O extends ObservableInput<any>>(input: O, scheduler: SchedulerLike | undefined): Observable<ObservedValueOf<O>>;\n\n/**\n * Creates an Observable from an Array, an array-like object, a Promise, an iterable object, or an Observable-like object.\n *\n * <span class=\"informal\">Converts almost anything to an Observable.</span>\n *\n * ![](from.png)\n *\n * `from` converts various other objects and data types into Observables. It also converts a Promise, an array-like, or an\n * <a href=\"https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Iteration_protocols#iterable\" target=\"_blank\">iterable</a>\n * object into an Observable that emits the items in that promise, array, or iterable. A String, in this context, is treated\n * as an array of characters. Observable-like objects (contains a function named with the ES2015 Symbol for Observable) can also be\n * converted through this operator.\n *\n * ## Examples\n *\n * Converts an array to an Observable\n *\n * ```ts\n * import { from } from 'rxjs';\n *\n * const array = [10, 20, 30];\n * const result = from(array);\n *\n * result.subscribe(x => console.log(x));\n *\n * // Logs:\n * // 10\n * // 20\n * // 30\n * ```\n *\n * Convert an infinite iterable (from a generator) to an Observable\n *\n * ```ts\n * import { from, take } from 'rxjs';\n *\n * function* generateDoubles(seed) {\n *    let i = seed;\n *    while (true) {\n *      yield i;\n *      i = 2 * i; // double it\n *    }\n * }\n *\n * const iterator = generateDoubles(3);\n * const result = from(iterator).pipe(take(10));\n *\n * result.subscribe(x => console.log(x));\n *\n * // Logs:\n * // 3\n * // 6\n * // 12\n * // 24\n * // 48\n * // 96\n * // 192\n * // 384\n * // 768\n * // 1536\n * ```\n *\n * With `asyncScheduler`\n *\n * ```ts\n * import { from, asyncScheduler } from 'rxjs';\n *\n * console.log('start');\n *\n * const array = [10, 20, 30];\n * const result = from(array, asyncScheduler);\n *\n * result.subscribe(x => console.log(x));\n *\n * console.log('end');\n *\n * // Logs:\n * // 'start'\n * // 'end'\n * // 10\n * // 20\n * // 30\n * ```\n *\n * @see {@link fromEvent}\n * @see {@link fromEventPattern}\n *\n * @param {ObservableInput<T>} A subscription object, a Promise, an Observable-like,\n * an Array, an iterable, or an array-like object to be converted.\n * @param {SchedulerLike} An optional {@link SchedulerLike} on which to schedule the emission of values.\n * @return {Observable<T>}\n */\nexport function from<T>(input: ObservableInput<T>, scheduler?: SchedulerLike): Observable<T> {\n  return scheduler ? scheduled(input, scheduler) : innerFrom(input);\n}\n", "import { SchedulerLike, ValueFromArray } from '../types';\nimport { Observable } from '../Observable';\nimport { popScheduler } from '../util/args';\nimport { from } from './from';\n\n// Devs are more likely to pass null or undefined than they are a scheduler\n// without accompanying values. To make things easier for (naughty) devs who\n// use the `strictNullChecks: false` TypeScript compiler option, these\n// overloads with explicit null and undefined values are included.\n\nexport function of(value: null): Observable<null>;\nexport function of(value: undefined): Observable<undefined>;\n\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function of(scheduler: SchedulerLike): Observable<never>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function of<A extends readonly unknown[]>(...valuesAndScheduler: [...A, SchedulerLike]): Observable<ValueFromArray<A>>;\n\nexport function of(): Observable<never>;\n/** @deprecated Do not specify explicit type parameters. Signatures with type parameters that cannot be inferred will be removed in v8. */\nexport function of<T>(): Observable<T>;\nexport function of<T>(value: T): Observable<T>;\nexport function of<A extends readonly unknown[]>(...values: A): Observable<ValueFromArray<A>>;\n\n/**\n * Converts the arguments to an observable sequence.\n *\n * <span class=\"informal\">Each argument becomes a `next` notification.</span>\n *\n * ![](of.png)\n *\n * Unlike {@link from}, it does not do any flattening and emits each argument in whole\n * as a separate `next` notification.\n *\n * ## Examples\n *\n * Emit the values `10, 20, 30`\n *\n * ```ts\n * import { of } from 'rxjs';\n *\n * of(10, 20, 30)\n *   .subscribe({\n *     next: value => console.log('next:', value),\n *     error: err => console.log('error:', err),\n *     complete: () => console.log('the end'),\n *   });\n *\n * // Outputs\n * // next: 10\n * // next: 20\n * // next: 30\n * // the end\n * ```\n *\n * Emit the array `[1, 2, 3]`\n *\n * ```ts\n * import { of } from 'rxjs';\n *\n * of([1, 2, 3])\n *   .subscribe({\n *     next: value => console.log('next:', value),\n *     error: err => console.log('error:', err),\n *     complete: () => console.log('the end'),\n *   });\n *\n * // Outputs\n * // next: [1, 2, 3]\n * // the end\n * ```\n *\n * @see {@link from}\n * @see {@link range}\n *\n * @param {...T} values A comma separated list of arguments you want to be emitted\n * @return {Observable} An Observable that emits the arguments\n * described above and then completes.\n */\nexport function of<T>(...args: Array<T | SchedulerLike>): Observable<T> {\n  const scheduler = popScheduler(args);\n  return from(args as T[], scheduler);\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { SchedulerLike } from '../types';\nimport { isFunction } from '../util/isFunction';\n\n/**\n * Creates an observable that will create an error instance and push it to the consumer as an error\n * immediately upon subscription.\n *\n * <span class=\"informal\">Just errors and does nothing else</span>\n *\n * ![](throw.png)\n *\n * This creation function is useful for creating an observable that will create an error and error every\n * time it is subscribed to. Generally, inside of most operators when you might want to return an errored\n * observable, this is unnecessary. In most cases, such as in the inner return of {@link concatMap},\n * {@link mergeMap}, {@link defer}, and many others, you can simply throw the error, and RxJS will pick\n * that up and notify the consumer of the error.\n *\n * ## Example\n *\n * Create a simple observable that will create a new error with a timestamp and log it\n * and the message every time you subscribe to it\n *\n * ```ts\n * import { throwError } from 'rxjs';\n *\n * let errorCount = 0;\n *\n * const errorWithTimestamp$ = throwError(() => {\n *   const error: any = new Error(`This is error number ${ ++errorCount }`);\n *   error.timestamp = Date.now();\n *   return error;\n * });\n *\n * errorWithTimestamp$.subscribe({\n *   error: err => console.log(err.timestamp, err.message)\n * });\n *\n * errorWithTimestamp$.subscribe({\n *   error: err => console.log(err.timestamp, err.message)\n * });\n *\n * // Logs the timestamp and a new error message for each subscription\n * ```\n *\n * ### Unnecessary usage\n *\n * Using `throwError` inside of an operator or creation function\n * with a callback, is usually not necessary\n *\n * ```ts\n * import { of, concatMap, timer, throwError } from 'rxjs';\n *\n * const delays$ = of(1000, 2000, Infinity, 3000);\n *\n * delays$.pipe(\n *   concatMap(ms => {\n *     if (ms < 10000) {\n *       return timer(ms);\n *     } else {\n *       // This is probably overkill.\n *       return throwError(() => new Error(`Invalid time ${ ms }`));\n *     }\n *   })\n * )\n * .subscribe({\n *   next: console.log,\n *   error: console.error\n * });\n * ```\n *\n * You can just throw the error instead\n *\n * ```ts\n * import { of, concatMap, timer } from 'rxjs';\n *\n * const delays$ = of(1000, 2000, Infinity, 3000);\n *\n * delays$.pipe(\n *   concatMap(ms => {\n *     if (ms < 10000) {\n *       return timer(ms);\n *     } else {\n *       // Cleaner and easier to read for most folks.\n *       throw new Error(`Invalid time ${ ms }`);\n *     }\n *   })\n * )\n * .subscribe({\n *   next: console.log,\n *   error: console.error\n * });\n * ```\n *\n * @param errorFactory A factory function that will create the error instance that is pushed.\n */\nexport function throwError(errorFactory: () => any): Observable<never>;\n\n/**\n * Returns an observable that will error with the specified error immediately upon subscription.\n *\n * @param error The error instance to emit\n * @deprecated Support for passing an error value will be removed in v8. Instead, pass a factory function to `throwError(() => new Error('test'))`. This is\n * because it will create the error at the moment it should be created and capture a more appropriate stack trace. If\n * for some reason you need to create the error ahead of time, you can still do that: `const err = new Error('test'); throwError(() => err);`.\n */\nexport function throwError(error: any): Observable<never>;\n\n/**\n * Notifies the consumer of an error using a given scheduler by scheduling it at delay `0` upon subscription.\n *\n * @param errorOrErrorFactory An error instance or error factory\n * @param scheduler A scheduler to use to schedule the error notification\n * @deprecated The `scheduler` parameter will be removed in v8.\n * Use `throwError` in combination with {@link observeOn}: `throwError(() => new Error('test')).pipe(observeOn(scheduler));`.\n * Details: https://rxjs.dev/deprecations/scheduler-argument\n */\nexport function throwError(errorOrErrorFactory: any, scheduler: SchedulerLike): Observable<never>;\n\nexport function throwError(errorOrErrorFactory: any, scheduler?: SchedulerLike): Observable<never> {\n  const errorFactory = isFunction(errorOrErrorFactory) ? errorOrErrorFactory : () => errorOrErrorFactory;\n  const init = (subscriber: Subscriber<never>) => subscriber.error(errorFactory());\n  return new Observable(scheduler ? (subscriber) => scheduler.schedule(init as any, 0, subscriber) : init);\n}\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface EmptyError extends Error {}\n\nexport interface EmptyErrorCtor {\n  /**\n   * @deprecated Internal implementation detail. Do not construct error instances.\n   * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n   */\n  new (): EmptyError;\n}\n\n/**\n * An error thrown when an Observable or a sequence was queried but has no\n * elements.\n *\n * @see {@link first}\n * @see {@link last}\n * @see {@link single}\n * @see {@link firstValueFrom}\n * @see {@link lastValueFrom}\n *\n * @class EmptyError\n */\nexport const EmptyError: EmptyErrorCtor = createErrorClass((_super) => function EmptyErrorImpl(this: any) {\n  _super(this);\n  this.name = 'EmptyError';\n  this.message = 'no elements in sequence';\n});\n", "/**\n * Checks to see if a value is not only a `Date` object,\n * but a *valid* `Date` object that can be converted to a\n * number. For example, `new Date('blah')` is indeed an\n * `instanceof Date`, however it cannot be converted to a\n * number.\n */\nexport function isValidDate(value: any): value is Date {\n  return value instanceof Date && !isNaN(value as any);\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\nexport function map<T, R>(project: (value: T, index: number) => R): OperatorFunction<T, R>;\n/** @deprecated Use a closure instead of a `thisArg`. Signatures accepting a `thisArg` will be removed in v8. */\nexport function map<T, R, A>(project: (this: A, value: T, index: number) => R, thisArg: A): OperatorFunction<T, R>;\n\n/**\n * Applies a given `project` function to each value emitted by the source\n * Observable, and emits the resulting values as an Observable.\n *\n * <span class=\"informal\">Like [Array.prototype.map()](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/map),\n * it passes each source value through a transformation function to get\n * corresponding output values.</span>\n *\n * ![](map.png)\n *\n * Similar to the well known `Array.prototype.map` function, this operator\n * applies a projection to each value and emits that projection in the output\n * Observable.\n *\n * ## Example\n *\n * Map every click to the `clientX` position of that click\n *\n * ```ts\n * import { fromEvent, map } from 'rxjs';\n *\n * const clicks = fromEvent<PointerEvent>(document, 'click');\n * const positions = clicks.pipe(map(ev => ev.clientX));\n *\n * positions.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link mapTo}\n * @see {@link pluck}\n *\n * @param {function(value: T, index: number): R} project The function to apply\n * to each `value` emitted by the source Observable. The `index` parameter is\n * the number `i` for the i-th emission that has happened since the\n * subscription, starting from the number `0`.\n * @param {any} [thisArg] An optional argument to define what `this` is in the\n * `project` function.\n * @return A function that returns an Observable that emits the values from the\n * source Observable transformed by the given `project` function.\n */\nexport function map<T, R>(project: (value: T, index: number) => R, thisArg?: any): OperatorFunction<T, R> {\n  return operate((source, subscriber) => {\n    // The index of the value from the source. Used with projection.\n    let index = 0;\n    // Subscribe to the source, all errors and completions are sent along\n    // to the consumer.\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value: T) => {\n        // Call the projection function with the appropriate this context,\n        // and send the resulting value to the consumer.\n        subscriber.next(project.call(thisArg, value, index++));\n      })\n    );\n  });\n}\n", "import { OperatorFunction } from \"../types\";\nimport { map } from \"../operators/map\";\n\nconst { isArray } = Array;\n\nfunction callOrApply<T, R>(fn: ((...values: T[]) => R), args: T|T[]): R {\n    return isArray(args) ? fn(...args) : fn(args);\n}\n\n/**\n * Used in several -- mostly deprecated -- situations where we need to \n * apply a list of arguments or a single argument to a result selector.\n */\nexport function mapOneOrManyArgs<T, R>(fn: ((...values: T[]) => R)): OperatorFunction<T|T[], R> {\n    return map(args => callOrApply(fn, args))\n}", "const { isArray } = Array;\nconst { getPrototypeOf, prototype: objectProto, keys: getKeys } = Object;\n\n/**\n * Used in functions where either a list of arguments, a single array of arguments, or a\n * dictionary of arguments can be returned. Returns an object with an `args` property with\n * the arguments in an array, if it is a dictionary, it will also return the `keys` in another\n * property.\n */\nexport function argsArgArrayOrObject<T, O extends Record<string, T>>(args: T[] | [O] | [T[]]): { args: T[]; keys: string[] | null } {\n  if (args.length === 1) {\n    const first = args[0];\n    if (isArray(first)) {\n      return { args: first, keys: null };\n    }\n    if (isPOJO(first)) {\n      const keys = getKeys(first);\n      return {\n        args: keys.map((key) => first[key]),\n        keys,\n      };\n    }\n  }\n\n  return { args: args as T[], keys: null };\n}\n\nfunction isPOJO(obj: any): obj is object {\n  return obj && typeof obj === 'object' && getPrototypeOf(obj) === objectProto;\n}\n", "export function createObject(keys: string[], values: any[]) {\n  return keys.reduce((result, key, i) => ((result[key] = values[i]), result), {} as any);\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInput, SchedulerLike, ObservedValueOf, ObservableInputTuple } from '../types';\nimport { argsArgArrayOrObject } from '../util/argsArgArrayOrObject';\nimport { Subscriber } from '../Subscriber';\nimport { from } from './from';\nimport { identity } from '../util/identity';\nimport { Subscription } from '../Subscription';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\nimport { popResultSelector, popScheduler } from '../util/args';\nimport { createObject } from '../util/createObject';\nimport { createOperatorSubscriber } from '../operators/OperatorSubscriber';\nimport { AnyCatcher } from '../AnyCatcher';\nimport { executeSchedule } from '../util/executeSchedule';\n\n// combineLatest(any)\n// We put this first because we need to catch cases where the user has supplied\n// _exactly `any`_ as the argument. Since `any` literally matches _anything_,\n// we don't want it to randomly hit one of the other type signatures below,\n// as we have no idea at build-time what type we should be returning when given an any.\n\n/**\n * You have passed `any` here, we can't figure out if it is\n * an array or an object, so you're getting `unknown`. Use better types.\n * @param arg Something typed as `any`\n */\nexport function combineLatest<T extends AnyCatcher>(arg: T): Observable<unknown>;\n\n// combineLatest([a, b, c])\nexport function combineLatest(sources: []): Observable<never>;\nexport function combineLatest<A extends readonly unknown[]>(sources: readonly [...ObservableInputTuple<A>]): Observable<A>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[], R>(\n  sources: readonly [...ObservableInputTuple<A>],\n  resultSelector: (...values: A) => R,\n  scheduler: SchedulerLike\n): Observable<R>;\nexport function combineLatest<A extends readonly unknown[], R>(\n  sources: readonly [...ObservableInputTuple<A>],\n  resultSelector: (...values: A) => R\n): Observable<R>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[]>(\n  sources: readonly [...ObservableInputTuple<A>],\n  scheduler: SchedulerLike\n): Observable<A>;\n\n// combineLatest(a, b, c)\n/** @deprecated Pass an array of sources instead. The rest-parameters signature will be removed in v8. Details: https://rxjs.dev/deprecations/array-argument */\nexport function combineLatest<A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): Observable<A>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[], R>(\n  ...sourcesAndResultSelectorAndScheduler: [...ObservableInputTuple<A>, (...values: A) => R, SchedulerLike]\n): Observable<R>;\n/** @deprecated Pass an array of sources instead. The rest-parameters signature will be removed in v8. Details: https://rxjs.dev/deprecations/array-argument */\nexport function combineLatest<A extends readonly unknown[], R>(\n  ...sourcesAndResultSelector: [...ObservableInputTuple<A>, (...values: A) => R]\n): Observable<R>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[]>(\n  ...sourcesAndScheduler: [...ObservableInputTuple<A>, SchedulerLike]\n): Observable<A>;\n\n// combineLatest({a, b, c})\nexport function combineLatest(sourcesObject: { [K in any]: never }): Observable<never>;\nexport function combineLatest<T extends Record<string, ObservableInput<any>>>(\n  sourcesObject: T\n): Observable<{ [K in keyof T]: ObservedValueOf<T[K]> }>;\n\n/**\n * Combines multiple Observables to create an Observable whose values are\n * calculated from the latest values of each of its input Observables.\n *\n * <span class=\"informal\">Whenever any input Observable emits a value, it\n * computes a formula using the latest values from all the inputs, then emits\n * the output of that formula.</span>\n *\n * ![](combineLatest.png)\n *\n * `combineLatest` combines the values from all the Observables passed in the\n * observables array. This is done by subscribing to each Observable in order and,\n * whenever any Observable emits, collecting an array of the most recent\n * values from each Observable. So if you pass `n` Observables to this operator,\n * the returned Observable will always emit an array of `n` values, in an order\n * corresponding to the order of the passed Observables (the value from the first Observable\n * will be at index 0 of the array and so on).\n *\n * Static version of `combineLatest` accepts an array of Observables. Note that an array of\n * Observables is a good choice, if you don't know beforehand how many Observables\n * you will combine. Passing an empty array will result in an Observable that\n * completes immediately.\n *\n * To ensure the output array always has the same length, `combineLatest` will\n * actually wait for all input Observables to emit at least once,\n * before it starts emitting results. This means if some Observable emits\n * values before other Observables started emitting, all these values but the last\n * will be lost. On the other hand, if some Observable does not emit a value but\n * completes, resulting Observable will complete at the same moment without\n * emitting anything, since it will now be impossible to include a value from the\n * completed Observable in the resulting array. Also, if some input Observable does\n * not emit any value and never completes, `combineLatest` will also never emit\n * and never complete, since, again, it will wait for all streams to emit some\n * value.\n *\n * If at least one Observable was passed to `combineLatest` and all passed Observables\n * emitted something, the resulting Observable will complete when all combined\n * streams complete. So even if some Observable completes, the result of\n * `combineLatest` will still emit values when other Observables do. In case\n * of a completed Observable, its value from now on will always be the last\n * emitted value. On the other hand, if any Observable errors, `combineLatest`\n * will error immediately as well, and all other Observables will be unsubscribed.\n *\n * ## Examples\n *\n * Combine two timer Observables\n *\n * ```ts\n * import { timer, combineLatest } from 'rxjs';\n *\n * const firstTimer = timer(0, 1000); // emit 0, 1, 2... after every second, starting from now\n * const secondTimer = timer(500, 1000); // emit 0, 1, 2... after every second, starting 0,5s from now\n * const combinedTimers = combineLatest([firstTimer, secondTimer]);\n * combinedTimers.subscribe(value => console.log(value));\n * // Logs\n * // [0, 0] after 0.5s\n * // [1, 0] after 1s\n * // [1, 1] after 1.5s\n * // [2, 1] after 2s\n * ```\n *\n * Combine a dictionary of Observables\n *\n * ```ts\n * import { of, delay, startWith, combineLatest } from 'rxjs';\n *\n * const observables = {\n *   a: of(1).pipe(delay(1000), startWith(0)),\n *   b: of(5).pipe(delay(5000), startWith(0)),\n *   c: of(10).pipe(delay(10000), startWith(0))\n * };\n * const combined = combineLatest(observables);\n * combined.subscribe(value => console.log(value));\n * // Logs\n * // { a: 0, b: 0, c: 0 } immediately\n * // { a: 1, b: 0, c: 0 } after 1s\n * // { a: 1, b: 5, c: 0 } after 5s\n * // { a: 1, b: 5, c: 10 } after 10s\n * ```\n *\n * Combine an array of Observables\n *\n * ```ts\n * import { of, delay, startWith, combineLatest } from 'rxjs';\n *\n * const observables = [1, 5, 10].map(\n *   n => of(n).pipe(\n *     delay(n * 1000), // emit 0 and then emit n after n seconds\n *     startWith(0)\n *   )\n * );\n * const combined = combineLatest(observables);\n * combined.subscribe(value => console.log(value));\n * // Logs\n * // [0, 0, 0] immediately\n * // [1, 0, 0] after 1s\n * // [1, 5, 0] after 5s\n * // [1, 5, 10] after 10s\n * ```\n *\n * Use map operator to dynamically calculate the Body-Mass Index\n *\n * ```ts\n * import { of, combineLatest, map } from 'rxjs';\n *\n * const weight = of(70, 72, 76, 79, 75);\n * const height = of(1.76, 1.77, 1.78);\n * const bmi = combineLatest([weight, height]).pipe(\n *   map(([w, h]) => w / (h * h)),\n * );\n * bmi.subscribe(x => console.log('BMI is ' + x));\n *\n * // With output to console:\n * // BMI is 24.212293388429753\n * // BMI is 23.93948099205209\n * // BMI is 23.671253629592222\n * ```\n *\n * @see {@link combineLatestAll}\n * @see {@link merge}\n * @see {@link withLatestFrom}\n *\n * @param {ObservableInput} [observables] An array of input Observables to combine with each other.\n * An array of Observables must be given as the first argument.\n * @param {function} [project] An optional function to project the values from\n * the combined latest values into a new value on the output Observable.\n * @param {SchedulerLike} [scheduler=null] The {@link SchedulerLike} to use for subscribing to\n * each input Observable.\n * @return {Observable} An Observable of projected values from the most recent\n * values from each input Observable, or an array of the most recent values from\n * each input Observable.\n */\nexport function combineLatest<O extends ObservableInput<any>, R>(...args: any[]): Observable<R> | Observable<ObservedValueOf<O>[]> {\n  const scheduler = popScheduler(args);\n  const resultSelector = popResultSelector(args);\n\n  const { args: observables, keys } = argsArgArrayOrObject(args);\n\n  if (observables.length === 0) {\n    // If no observables are passed, or someone has passed an empty array\n    // of observables, or even an empty object POJO, we need to just\n    // complete (EMPTY), but we have to honor the scheduler provided if any.\n    return from([], scheduler as any);\n  }\n\n  const result = new Observable<ObservedValueOf<O>[]>(\n    combineLatestInit(\n      observables as ObservableInput<ObservedValueOf<O>>[],\n      scheduler,\n      keys\n        ? // A handler for scrubbing the array of args into a dictionary.\n          (values) => createObject(keys, values)\n        : // A passthrough to just return the array\n          identity\n    )\n  );\n\n  return resultSelector ? (result.pipe(mapOneOrManyArgs(resultSelector)) as Observable<R>) : result;\n}\n\nexport function combineLatestInit(\n  observables: ObservableInput<any>[],\n  scheduler?: SchedulerLike,\n  valueTransform: (values: any[]) => any = identity\n) {\n  return (subscriber: Subscriber<any>) => {\n    // The outer subscription. We're capturing this in a function\n    // because we may have to schedule it.\n    maybeSchedule(\n      scheduler,\n      () => {\n        const { length } = observables;\n        // A store for the values each observable has emitted so far. We match observable to value on index.\n        const values = new Array(length);\n        // The number of currently active subscriptions, as they complete, we decrement this number to see if\n        // we are all done combining values, so we can complete the result.\n        let active = length;\n        // The number of inner sources that still haven't emitted the first value\n        // We need to track this because all sources need to emit one value in order\n        // to start emitting values.\n        let remainingFirstValues = length;\n        // The loop to kick off subscription. We're keying everything on index `i` to relate the observables passed\n        // in to the slot in the output array or the key in the array of keys in the output dictionary.\n        for (let i = 0; i < length; i++) {\n          maybeSchedule(\n            scheduler,\n            () => {\n              const source = from(observables[i], scheduler as any);\n              let hasFirstValue = false;\n              source.subscribe(\n                createOperatorSubscriber(\n                  subscriber,\n                  (value) => {\n                    // When we get a value, record it in our set of values.\n                    values[i] = value;\n                    if (!hasFirstValue) {\n                      // If this is our first value, record that.\n                      hasFirstValue = true;\n                      remainingFirstValues--;\n                    }\n                    if (!remainingFirstValues) {\n                      // We're not waiting for any more\n                      // first values, so we can emit!\n                      subscriber.next(valueTransform(values.slice()));\n                    }\n                  },\n                  () => {\n                    if (!--active) {\n                      // We only complete the result if we have no more active\n                      // inner observables.\n                      subscriber.complete();\n                    }\n                  }\n                )\n              );\n            },\n            subscriber\n          );\n        }\n      },\n      subscriber\n    );\n  };\n}\n\n/**\n * A small utility to handle the couple of locations where we want to schedule if a scheduler was provided,\n * but we don't if there was no scheduler.\n */\nfunction maybeSchedule(scheduler: SchedulerLike | undefined, execute: () => void, subscription: Subscription) {\n  if (scheduler) {\n    executeSchedule(subscription, scheduler, execute);\n  } else {\n    execute();\n  }\n}\n", "import { Observable } from '../Observable';\nimport { innerFrom } from '../observable/innerFrom';\nimport { Subscriber } from '../Subscriber';\nimport { ObservableInput, SchedulerLike } from '../types';\nimport { executeSchedule } from '../util/executeSchedule';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * A process embodying the general \"merge\" strategy. This is used in\n * `mergeMap` and `mergeScan` because the logic is otherwise nearly identical.\n * @param source The original source observable\n * @param subscriber The consumer subscriber\n * @param project The projection function to get our inner sources\n * @param concurrent The number of concurrent inner subscriptions\n * @param onBeforeNext Additional logic to apply before nexting to our consumer\n * @param expand If `true` this will perform an \"expand\" strategy, which differs only\n * in that it recurses, and the inner subscription must be schedule-able.\n * @param innerSubScheduler A scheduler to use to schedule inner subscriptions,\n * this is to support the expand strategy, mostly, and should be deprecated\n */\nexport function mergeInternals<T, R>(\n  source: Observable<T>,\n  subscriber: Subscriber<R>,\n  project: (value: T, index: number) => ObservableInput<R>,\n  concurrent: number,\n  onBeforeNext?: (innerValue: R) => void,\n  expand?: boolean,\n  innerSubScheduler?: SchedulerLike,\n  additionalFinalizer?: () => void\n) {\n  // Buffered values, in the event of going over our concurrency limit\n  const buffer: T[] = [];\n  // The number of active inner subscriptions.\n  let active = 0;\n  // An index to pass to our accumulator function\n  let index = 0;\n  // Whether or not the outer source has completed.\n  let isComplete = false;\n\n  /**\n   * Checks to see if we can complete our result or not.\n   */\n  const checkComplete = () => {\n    // If the outer has completed, and nothing is left in the buffer,\n    // and we don't have any active inner subscriptions, then we can\n    // Emit the state and complete.\n    if (isComplete && !buffer.length && !active) {\n      subscriber.complete();\n    }\n  };\n\n  // If we're under our concurrency limit, just start the inner subscription, otherwise buffer and wait.\n  const outerNext = (value: T) => (active < concurrent ? doInnerSub(value) : buffer.push(value));\n\n  const doInnerSub = (value: T) => {\n    // If we're expanding, we need to emit the outer values and the inner values\n    // as the inners will \"become outers\" in a way as they are recursively fed\n    // back to the projection mechanism.\n    expand && subscriber.next(value as any);\n\n    // Increment the number of active subscriptions so we can track it\n    // against our concurrency limit later.\n    active++;\n\n    // A flag used to show that the inner observable completed.\n    // This is checked during finalization to see if we should\n    // move to the next item in the buffer, if there is on.\n    let innerComplete = false;\n\n    // Start our inner subscription.\n    innerFrom(project(value, index++)).subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (innerValue) => {\n          // `mergeScan` has additional handling here. For example\n          // taking the inner value and updating state.\n          onBeforeNext?.(innerValue);\n\n          if (expand) {\n            // If we're expanding, then just recurse back to our outer\n            // handler. It will emit the value first thing.\n            outerNext(innerValue as any);\n          } else {\n            // Otherwise, emit the inner value.\n            subscriber.next(innerValue);\n          }\n        },\n        () => {\n          // Flag that we have completed, so we know to check the buffer\n          // during finalization.\n          innerComplete = true;\n        },\n        // Errors are passed to the destination.\n        undefined,\n        () => {\n          // During finalization, if the inner completed (it wasn't errored or\n          // cancelled), then we want to try the next item in the buffer if\n          // there is one.\n          if (innerComplete) {\n            // We have to wrap this in a try/catch because it happens during\n            // finalization, possibly asynchronously, and we want to pass\n            // any errors that happen (like in a projection function) to\n            // the outer Subscriber.\n            try {\n              // INNER SOURCE COMPLETE\n              // Decrement the active count to ensure that the next time\n              // we try to call `doInnerSub`, the number is accurate.\n              active--;\n              // If we have more values in the buffer, try to process those\n              // Note that this call will increment `active` ahead of the\n              // next conditional, if there were any more inner subscriptions\n              // to start.\n              while (buffer.length && active < concurrent) {\n                const bufferedValue = buffer.shift()!;\n                // Particularly for `expand`, we need to check to see if a scheduler was provided\n                // for when we want to start our inner subscription. Otherwise, we just start\n                // are next inner subscription.\n                if (innerSubScheduler) {\n                  executeSchedule(subscriber, innerSubScheduler, () => doInnerSub(bufferedValue));\n                } else {\n                  doInnerSub(bufferedValue);\n                }\n              }\n              // Check to see if we can complete, and complete if so.\n              checkComplete();\n            } catch (err) {\n              subscriber.error(err);\n            }\n          }\n        }\n      )\n    );\n  };\n\n  // Subscribe to our source observable.\n  source.subscribe(\n    createOperatorSubscriber(subscriber, outerNext, () => {\n      // Outer completed, make a note of it, and check to see if we can complete everything.\n      isComplete = true;\n      checkComplete();\n    })\n  );\n\n  // Additional finalization (for when the destination is torn down).\n  // Other finalization is added implicitly via subscription above.\n  return () => {\n    additionalFinalizer?.();\n  };\n}\n", "import { ObservableInput, OperatorFunction, ObservedValueOf } from '../types';\nimport { map } from './map';\nimport { innerFrom } from '../observable/innerFrom';\nimport { operate } from '../util/lift';\nimport { mergeInternals } from './mergeInternals';\nimport { isFunction } from '../util/isFunction';\n\n/* tslint:disable:max-line-length */\nexport function mergeMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  concurrent?: number\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function mergeMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: undefined,\n  concurrent?: number\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function mergeMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: (outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R,\n  concurrent?: number\n): OperatorFunction<T, R>;\n/* tslint:enable:max-line-length */\n\n/**\n * Projects each source value to an Observable which is merged in the output\n * Observable.\n *\n * <span class=\"informal\">Maps each value to an Observable, then flattens all of\n * these inner Observables using {@link mergeAll}.</span>\n *\n * ![](mergeMap.png)\n *\n * Returns an Observable that emits items based on applying a function that you\n * supply to each item emitted by the source Observable, where that function\n * returns an Observable, and then merging those resulting Observables and\n * emitting the results of this merger.\n *\n * ## Example\n *\n * Map and flatten each letter to an Observable ticking every 1 second\n *\n * ```ts\n * import { of, mergeMap, interval, map } from 'rxjs';\n *\n * const letters = of('a', 'b', 'c');\n * const result = letters.pipe(\n *   mergeMap(x => interval(1000).pipe(map(i => x + i)))\n * );\n *\n * result.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // a0\n * // b0\n * // c0\n * // a1\n * // b1\n * // c1\n * // continues to list a, b, c every second with respective ascending integers\n * ```\n *\n * @see {@link concatMap}\n * @see {@link exhaustMap}\n * @see {@link merge}\n * @see {@link mergeAll}\n * @see {@link mergeMapTo}\n * @see {@link mergeScan}\n * @see {@link switchMap}\n *\n * @param {function(value: T, ?index: number): ObservableInput} project A function\n * that, when applied to an item emitted by the source Observable, returns an\n * Observable.\n * @param {number} [concurrent=Infinity] Maximum number of input\n * Observables being subscribed to concurrently.\n * @return A function that returns an Observable that emits the result of\n * applying the projection function (and the optional deprecated\n * `resultSelector`) to each item emitted by the source Observable and merging\n * the results of the Observables obtained from this transformation.\n */\nexport function mergeMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector?: ((outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R) | number,\n  concurrent: number = Infinity\n): OperatorFunction<T, ObservedValueOf<O> | R> {\n  if (isFunction(resultSelector)) {\n    // DEPRECATED PATH\n    return mergeMap((a, i) => map((b: any, ii: number) => resultSelector(a, b, i, ii))(innerFrom(project(a, i))), concurrent);\n  } else if (typeof resultSelector === 'number') {\n    concurrent = resultSelector;\n  }\n\n  return operate((source, subscriber) => mergeInternals(source, subscriber, project, concurrent));\n}\n", "import { mergeMap } from './mergeMap';\nimport { identity } from '../util/identity';\nimport { OperatorFunction, ObservableInput, ObservedValueOf } from '../types';\n\n/**\n * Converts a higher-order Observable into a first-order Observable which\n * concurrently delivers all values that are emitted on the inner Observables.\n *\n * <span class=\"informal\">Flattens an Observable-of-Observables.</span>\n *\n * ![](mergeAll.png)\n *\n * `mergeAll` subscribes to an Observable that emits Observables, also known as\n * a higher-order Observable. Each time it observes one of these emitted inner\n * Observables, it subscribes to that and delivers all the values from the\n * inner Observable on the output Observable. The output Observable only\n * completes once all inner Observables have completed. Any error delivered by\n * a inner Observable will be immediately emitted on the output Observable.\n *\n * ## Examples\n *\n * Spawn a new interval Observable for each click event, and blend their outputs as one Observable\n *\n * ```ts\n * import { fromEvent, map, interval, mergeAll } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const higherOrder = clicks.pipe(map(() => interval(1000)));\n * const firstOrder = higherOrder.pipe(mergeAll());\n *\n * firstOrder.subscribe(x => console.log(x));\n * ```\n *\n * Count from 0 to 9 every second for each click, but only allow 2 concurrent timers\n *\n * ```ts\n * import { fromEvent, map, interval, take, mergeAll } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const higherOrder = clicks.pipe(\n *   map(() => interval(1000).pipe(take(10)))\n * );\n * const firstOrder = higherOrder.pipe(mergeAll(2));\n *\n * firstOrder.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link combineLatestAll}\n * @see {@link concatAll}\n * @see {@link exhaustAll}\n * @see {@link merge}\n * @see {@link mergeMap}\n * @see {@link mergeMapTo}\n * @see {@link mergeScan}\n * @see {@link switchAll}\n * @see {@link switchMap}\n * @see {@link zipAll}\n *\n * @param {number} [concurrent=Infinity] Maximum number of inner\n * Observables being subscribed to concurrently.\n * @return A function that returns an Observable that emits values coming from\n * all the inner Observables emitted by the source Observable.\n */\nexport function mergeAll<O extends ObservableInput<any>>(concurrent: number = Infinity): OperatorFunction<O, ObservedValueOf<O>> {\n  return mergeMap(identity, concurrent);\n}\n", "import { mergeAll } from './mergeAll';\nimport { OperatorFunction, ObservableInput, ObservedValueOf } from '../types';\n\n/**\n * Converts a higher-order Observable into a first-order Observable by\n * concatenating the inner Observables in order.\n *\n * <span class=\"informal\">Flattens an Observable-of-Observables by putting one\n * inner Observable after the other.</span>\n *\n * ![](concatAll.svg)\n *\n * Joins every Observable emitted by the source (a higher-order Observable), in\n * a serial fashion. It subscribes to each inner Observable only after the\n * previous inner Observable has completed, and merges all of their values into\n * the returned observable.\n *\n * __Warning:__ If the source Observable emits Observables quickly and\n * endlessly, and the inner Observables it emits generally complete slower than\n * the source emits, you can run into memory issues as the incoming Observables\n * collect in an unbounded buffer.\n *\n * Note: `concatAll` is equivalent to `mergeAll` with concurrency parameter set\n * to `1`.\n *\n * ## Example\n *\n * For each click event, tick every second from 0 to 3, with no concurrency\n *\n * ```ts\n * import { fromEvent, map, interval, take, concatAll } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const higherOrder = clicks.pipe(\n *   map(() => interval(1000).pipe(take(4)))\n * );\n * const firstOrder = higherOrder.pipe(concatAll());\n * firstOrder.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // (results are not concurrent)\n * // For every click on the \"document\" it will emit values 0 to 3 spaced\n * // on a 1000ms interval\n * // one click = 1000ms-> 0 -1000ms-> 1 -1000ms-> 2 -1000ms-> 3\n * ```\n *\n * @see {@link combineLatestAll}\n * @see {@link concat}\n * @see {@link concatMap}\n * @see {@link concatMapTo}\n * @see {@link exhaustAll}\n * @see {@link mergeAll}\n * @see {@link switchAll}\n * @see {@link switchMap}\n * @see {@link zipAll}\n *\n * @return A function that returns an Observable emitting values from all the\n * inner Observables concatenated.\n */\nexport function concatAll<O extends ObservableInput<any>>(): OperatorFunction<O, ObservedValueOf<O>> {\n  return mergeAll(1);\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInputTuple, SchedulerLike } from '../types';\nimport { concatAll } from '../operators/concatAll';\nimport { popScheduler } from '../util/args';\nimport { from } from './from';\n\nexport function concat<T extends readonly unknown[]>(...inputs: [...ObservableInputTuple<T>]): Observable<T[number]>;\nexport function concat<T extends readonly unknown[]>(\n  ...inputsAndScheduler: [...ObservableInputTuple<T>, SchedulerLike]\n): Observable<T[number]>;\n\n/**\n * Creates an output Observable which sequentially emits all values from the first given\n * Observable and then moves on to the next.\n *\n * <span class=\"informal\">Concatenates multiple Observables together by\n * sequentially emitting their values, one Observable after the other.</span>\n *\n * ![](concat.png)\n *\n * `concat` joins multiple Observables together, by subscribing to them one at a time and\n * merging their results into the output Observable. You can pass either an array of\n * Observables, or put them directly as arguments. Passing an empty array will result\n * in Observable that completes immediately.\n *\n * `concat` will subscribe to first input Observable and emit all its values, without\n * changing or affecting them in any way. When that Observable completes, it will\n * subscribe to then next Observable passed and, again, emit its values. This will be\n * repeated, until the operator runs out of Observables. When last input Observable completes,\n * `concat` will complete as well. At any given moment only one Observable passed to operator\n * emits values. If you would like to emit values from passed Observables concurrently, check out\n * {@link merge} instead, especially with optional `concurrent` parameter. As a matter of fact,\n * `concat` is an equivalent of `merge` operator with `concurrent` parameter set to `1`.\n *\n * Note that if some input Observable never completes, `concat` will also never complete\n * and Observables following the one that did not complete will never be subscribed. On the other\n * hand, if some Observable simply completes immediately after it is subscribed, it will be\n * invisible for `concat`, which will just move on to the next Observable.\n *\n * If any Observable in chain errors, instead of passing control to the next Observable,\n * `concat` will error immediately as well. Observables that would be subscribed after\n * the one that emitted error, never will.\n *\n * If you pass to `concat` the same Observable many times, its stream of values\n * will be \"replayed\" on every subscription, which means you can repeat given Observable\n * as many times as you like. If passing the same Observable to `concat` 1000 times becomes tedious,\n * you can always use {@link repeat}.\n *\n * ## Examples\n *\n * Concatenate a timer counting from 0 to 3 with a synchronous sequence from 1 to 10\n *\n * ```ts\n * import { interval, take, range, concat } from 'rxjs';\n *\n * const timer = interval(1000).pipe(take(4));\n * const sequence = range(1, 10);\n * const result = concat(timer, sequence);\n * result.subscribe(x => console.log(x));\n *\n * // results in:\n * // 0 -1000ms-> 1 -1000ms-> 2 -1000ms-> 3 -immediate-> 1 ... 10\n * ```\n *\n * Concatenate 3 Observables\n *\n * ```ts\n * import { interval, take, concat } from 'rxjs';\n *\n * const timer1 = interval(1000).pipe(take(10));\n * const timer2 = interval(2000).pipe(take(6));\n * const timer3 = interval(500).pipe(take(10));\n *\n * const result = concat(timer1, timer2, timer3);\n * result.subscribe(x => console.log(x));\n *\n * // results in the following:\n * // (Prints to console sequentially)\n * // -1000ms-> 0 -1000ms-> 1 -1000ms-> ... 9\n * // -2000ms-> 0 -2000ms-> 1 -2000ms-> ... 5\n * // -500ms-> 0 -500ms-> 1 -500ms-> ... 9\n * ```\n *\n * Concatenate the same Observable to repeat it\n *\n * ```ts\n * import { interval, take, concat } from 'rxjs';\n *\n * const timer = interval(1000).pipe(take(2));\n *\n * concat(timer, timer) // concatenating the same Observable!\n *   .subscribe({\n *     next: value => console.log(value),\n *     complete: () => console.log('...and it is done!')\n *   });\n *\n * // Logs:\n * // 0 after 1s\n * // 1 after 2s\n * // 0 after 3s\n * // 1 after 4s\n * // '...and it is done!' also after 4s\n * ```\n *\n * @see {@link concatAll}\n * @see {@link concatMap}\n * @see {@link concatMapTo}\n * @see {@link startWith}\n * @see {@link endWith}\n *\n * @param args Input Observables to concatenate.\n */\nexport function concat(...args: any[]): Observable<unknown> {\n  return concatAll()(from(args, popScheduler(args)));\n}\n", "import { Observable } from '../Observable';\nimport { ObservedValueOf, ObservableInput } from '../types';\nimport { innerFrom } from './innerFrom';\n\n/**\n * Creates an Observable that, on subscribe, calls an Observable factory to\n * make an Observable for each new Observer.\n *\n * <span class=\"informal\">Creates the Observable lazily, that is, only when it\n * is subscribed.\n * </span>\n *\n * ![](defer.png)\n *\n * `defer` allows you to create an Observable only when the Observer\n * subscribes. It waits until an Observer subscribes to it, calls the given\n * factory function to get an Observable -- where a factory function typically\n * generates a new Observable -- and subscribes the Observer to this Observable.\n * In case the factory function returns a falsy value, then EMPTY is used as\n * Observable instead. Last but not least, an exception during the factory\n * function call is transferred to the Observer by calling `error`.\n *\n * ## Example\n *\n * Subscribe to either an Observable of clicks or an Observable of interval, at random\n *\n * ```ts\n * import { defer, fromEvent, interval } from 'rxjs';\n *\n * const clicksOrInterval = defer(() => {\n *   return Math.random() > 0.5\n *     ? fromEvent(document, 'click')\n *     : interval(1000);\n * });\n * clicksOrInterval.subscribe(x => console.log(x));\n *\n * // Results in the following behavior:\n * // If the result of Math.random() is greater than 0.5 it will listen\n * // for clicks anywhere on the \"document\"; when document is clicked it\n * // will log a MouseEvent object to the console. If the result is less\n * // than 0.5 it will emit ascending numbers, one every second(1000ms).\n * ```\n *\n * @see {@link Observable}\n *\n * @param {function(): ObservableInput} observableFactory The Observable\n * factory function to invoke for each Observer that subscribes to the output\n * Observable. May also return a Promise, which will be converted on the fly\n * to an Observable.\n * @return {Observable} An Observable whose Observers' subscriptions trigger\n * an invocation of the given Observable factory function.\n */\nexport function defer<R extends ObservableInput<any>>(observableFactory: () => R): Observable<ObservedValueOf<R>> {\n  return new Observable<ObservedValueOf<R>>((subscriber) => {\n    innerFrom(observableFactory()).subscribe(subscriber);\n  });\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { Observable } from '../Observable';\nimport { mergeMap } from '../operators/mergeMap';\nimport { isArrayLike } from '../util/isArrayLike';\nimport { isFunction } from '../util/isFunction';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\n\n// These constants are used to create handler registry functions using array mapping below.\nconst nodeEventEmitterMethods = ['addListener', 'removeListener'] as const;\nconst eventTargetMethods = ['addEventListener', 'removeEventListener'] as const;\nconst jqueryMethods = ['on', 'off'] as const;\n\nexport interface NodeStyleEventEmitter {\n  addListener(eventName: string | symbol, handler: NodeEventHandler): this;\n  removeListener(eventName: string | symbol, handler: NodeEventHandler): this;\n}\n\nexport type NodeEventHandler = (...args: any[]) => void;\n\n// For APIs that implement `addListener` and `removeListener` methods that may\n// not use the same arguments or return EventEmitter values\n// such as React Native\nexport interface NodeCompatibleEventEmitter {\n  addListener(eventName: string, handler: NodeEventHandler): void | {};\n  removeListener(eventName: string, handler: NodeEventHandler): void | {};\n}\n\n// Use handler types like those in @types/jquery. See:\n// https://github.com/DefinitelyTyped/DefinitelyTyped/blob/847731ba1d7fa6db6b911c0e43aa0afe596e7723/types/jquery/misc.d.ts#L6395\nexport interface JQueryStyleEventEmitter<TContext, T> {\n  on(eventName: string, handler: (this: TContext, t: T, ...args: any[]) => any): void;\n  off(eventName: string, handler: (this: TContext, t: T, ...args: any[]) => any): void;\n}\n\nexport interface EventListenerObject<E> {\n  handleEvent(evt: E): void;\n}\n\nexport interface HasEventTargetAddRemove<E> {\n  addEventListener(\n    type: string,\n    listener: ((evt: E) => void) | EventListenerObject<E> | null,\n    options?: boolean | AddEventListenerOptions\n  ): void;\n  removeEventListener(\n    type: string,\n    listener: ((evt: E) => void) | EventListenerObject<E> | null,\n    options?: EventListenerOptions | boolean\n  ): void;\n}\n\nexport interface EventListenerOptions {\n  capture?: boolean;\n  passive?: boolean;\n  once?: boolean;\n}\n\nexport interface AddEventListenerOptions extends EventListenerOptions {\n  once?: boolean;\n  passive?: boolean;\n}\n\nexport function fromEvent<T>(target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>, eventName: string): Observable<T>;\nexport function fromEvent<T, R>(\n  target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>,\n  eventName: string,\n  resultSelector: (event: T) => R\n): Observable<R>;\nexport function fromEvent<T>(\n  target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>,\n  eventName: string,\n  options: EventListenerOptions\n): Observable<T>;\nexport function fromEvent<T, R>(\n  target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>,\n  eventName: string,\n  options: EventListenerOptions,\n  resultSelector: (event: T) => R\n): Observable<R>;\n\nexport function fromEvent(target: NodeStyleEventEmitter | ArrayLike<NodeStyleEventEmitter>, eventName: string): Observable<unknown>;\n/** @deprecated Do not specify explicit type parameters. Signatures with type parameters that cannot be inferred will be removed in v8. */\nexport function fromEvent<T>(target: NodeStyleEventEmitter | ArrayLike<NodeStyleEventEmitter>, eventName: string): Observable<T>;\nexport function fromEvent<R>(\n  target: NodeStyleEventEmitter | ArrayLike<NodeStyleEventEmitter>,\n  eventName: string,\n  resultSelector: (...args: any[]) => R\n): Observable<R>;\n\nexport function fromEvent(\n  target: NodeCompatibleEventEmitter | ArrayLike<NodeCompatibleEventEmitter>,\n  eventName: string\n): Observable<unknown>;\n/** @deprecated Do not specify explicit type parameters. Signatures with type parameters that cannot be inferred will be removed in v8. */\nexport function fromEvent<T>(target: NodeCompatibleEventEmitter | ArrayLike<NodeCompatibleEventEmitter>, eventName: string): Observable<T>;\nexport function fromEvent<R>(\n  target: NodeCompatibleEventEmitter | ArrayLike<NodeCompatibleEventEmitter>,\n  eventName: string,\n  resultSelector: (...args: any[]) => R\n): Observable<R>;\n\nexport function fromEvent<T>(\n  target: JQueryStyleEventEmitter<any, T> | ArrayLike<JQueryStyleEventEmitter<any, T>>,\n  eventName: string\n): Observable<T>;\nexport function fromEvent<T, R>(\n  target: JQueryStyleEventEmitter<any, T> | ArrayLike<JQueryStyleEventEmitter<any, T>>,\n  eventName: string,\n  resultSelector: (value: T, ...args: any[]) => R\n): Observable<R>;\n\n/**\n * Creates an Observable that emits events of a specific type coming from the\n * given event target.\n *\n * <span class=\"informal\">Creates an Observable from DOM events, or Node.js\n * EventEmitter events or others.</span>\n *\n * ![](fromEvent.png)\n *\n * `fromEvent` accepts as a first argument event target, which is an object with methods\n * for registering event handler functions. As a second argument it takes string that indicates\n * type of event we want to listen for. `fromEvent` supports selected types of event targets,\n * which are described in detail below. If your event target does not match any of the ones listed,\n * you should use {@link fromEventPattern}, which can be used on arbitrary APIs.\n * When it comes to APIs supported by `fromEvent`, their methods for adding and removing event\n * handler functions have different names, but they all accept a string describing event type\n * and function itself, which will be called whenever said event happens.\n *\n * Every time resulting Observable is subscribed, event handler function will be registered\n * to event target on given event type. When that event fires, value\n * passed as a first argument to registered function will be emitted by output Observable.\n * When Observable is unsubscribed, function will be unregistered from event target.\n *\n * Note that if event target calls registered function with more than one argument, second\n * and following arguments will not appear in resulting stream. In order to get access to them,\n * you can pass to `fromEvent` optional project function, which will be called with all arguments\n * passed to event handler. Output Observable will then emit value returned by project function,\n * instead of the usual value.\n *\n * Remember that event targets listed below are checked via duck typing. It means that\n * no matter what kind of object you have and no matter what environment you work in,\n * you can safely use `fromEvent` on that object if it exposes described methods (provided\n * of course they behave as was described above). So for example if Node.js library exposes\n * event target which has the same method names as DOM EventTarget, `fromEvent` is still\n * a good choice.\n *\n * If the API you use is more callback then event handler oriented (subscribed\n * callback function fires only once and thus there is no need to manually\n * unregister it), you should use {@link bindCallback} or {@link bindNodeCallback}\n * instead.\n *\n * `fromEvent` supports following types of event targets:\n *\n * **DOM EventTarget**\n *\n * This is an object with `addEventListener` and `removeEventListener` methods.\n *\n * In the browser, `addEventListener` accepts - apart from event type string and event\n * handler function arguments - optional third parameter, which is either an object or boolean,\n * both used for additional configuration how and when passed function will be called. When\n * `fromEvent` is used with event target of that type, you can provide this values\n * as third parameter as well.\n *\n * **Node.js EventEmitter**\n *\n * An object with `addListener` and `removeListener` methods.\n *\n * **JQuery-style event target**\n *\n * An object with `on` and `off` methods\n *\n * **DOM NodeList**\n *\n * List of DOM Nodes, returned for example by `document.querySelectorAll` or `Node.childNodes`.\n *\n * Although this collection is not event target in itself, `fromEvent` will iterate over all Nodes\n * it contains and install event handler function in every of them. When returned Observable\n * is unsubscribed, function will be removed from all Nodes.\n *\n * **DOM HtmlCollection**\n *\n * Just as in case of NodeList it is a collection of DOM nodes. Here as well event handler function is\n * installed and removed in each of elements.\n *\n *\n * ## Examples\n *\n * Emit clicks happening on the DOM document\n *\n * ```ts\n * import { fromEvent } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * clicks.subscribe(x => console.log(x));\n *\n * // Results in:\n * // MouseEvent object logged to console every time a click\n * // occurs on the document.\n * ```\n *\n * Use `addEventListener` with capture option\n *\n * ```ts\n * import { fromEvent } from 'rxjs';\n *\n * const div = document.createElement('div');\n * div.style.cssText = 'width: 200px; height: 200px; background: #09c;';\n * document.body.appendChild(div);\n *\n * // note optional configuration parameter which will be passed to addEventListener\n * const clicksInDocument = fromEvent(document, 'click', { capture: true });\n * const clicksInDiv = fromEvent(div, 'click');\n *\n * clicksInDocument.subscribe(() => console.log('document'));\n * clicksInDiv.subscribe(() => console.log('div'));\n *\n * // By default events bubble UP in DOM tree, so normally\n * // when we would click on div in document\n * // \"div\" would be logged first and then \"document\".\n * // Since we specified optional `capture` option, document\n * // will catch event when it goes DOWN DOM tree, so console\n * // will log \"document\" and then \"div\".\n * ```\n *\n * @see {@link bindCallback}\n * @see {@link bindNodeCallback}\n * @see {@link fromEventPattern}\n *\n * @param {FromEventTarget<T>} target The DOM EventTarget, Node.js\n * EventEmitter, JQuery-like event target, NodeList or HTMLCollection to attach the event handler to.\n * @param {string} eventName The event name of interest, being emitted by the\n * `target`.\n * @param {EventListenerOptions} [options] Options to pass through to addEventListener\n * @return {Observable<T>}\n */\nexport function fromEvent<T>(\n  target: any,\n  eventName: string,\n  options?: EventListenerOptions | ((...args: any[]) => T),\n  resultSelector?: (...args: any[]) => T\n): Observable<T> {\n  if (isFunction(options)) {\n    resultSelector = options;\n    options = undefined;\n  }\n  if (resultSelector) {\n    return fromEvent<T>(target, eventName, options as EventListenerOptions).pipe(mapOneOrManyArgs(resultSelector));\n  }\n\n  // Figure out our add and remove methods. In order to do this,\n  // we are going to analyze the target in a preferred order, if\n  // the target matches a given signature, we take the two \"add\" and \"remove\"\n  // method names and apply them to a map to create opposite versions of the\n  // same function. This is because they all operate in duplicate pairs,\n  // `addListener(name, handler)`, `removeListener(name, handler)`, for example.\n  // The call only differs by method name, as to whether or not you're adding or removing.\n  const [add, remove] =\n    // If it is an EventTarget, we need to use a slightly different method than the other two patterns.\n    isEventTarget(target)\n      ? eventTargetMethods.map((methodName) => (handler: any) => target[methodName](eventName, handler, options as EventListenerOptions))\n      : // In all other cases, the call pattern is identical with the exception of the method names.\n      isNodeStyleEventEmitter(target)\n      ? nodeEventEmitterMethods.map(toCommonHandlerRegistry(target, eventName))\n      : isJQueryStyleEventEmitter(target)\n      ? jqueryMethods.map(toCommonHandlerRegistry(target, eventName))\n      : [];\n\n  // If add is falsy, it's because we didn't match a pattern above.\n  // Check to see if it is an ArrayLike, because if it is, we want to\n  // try to apply fromEvent to all of it's items. We do this check last,\n  // because there are may be some types that are both ArrayLike *and* implement\n  // event registry points, and we'd rather delegate to that when possible.\n  if (!add) {\n    if (isArrayLike(target)) {\n      return mergeMap((subTarget: any) => fromEvent(subTarget, eventName, options as EventListenerOptions))(\n        innerFrom(target)\n      ) as Observable<T>;\n    }\n  }\n\n  // If add is falsy and we made it here, it's because we didn't\n  // match any valid target objects above.\n  if (!add) {\n    throw new TypeError('Invalid event target');\n  }\n\n  return new Observable<T>((subscriber) => {\n    // The handler we are going to register. Forwards the event object, by itself, or\n    // an array of arguments to the event handler, if there is more than one argument,\n    // to the consumer.\n    const handler = (...args: any[]) => subscriber.next(1 < args.length ? args : args[0]);\n    // Do the work of adding the handler to the target.\n    add(handler);\n    // When we finalize, we want to remove the handler and free up memory.\n    return () => remove!(handler);\n  });\n}\n\n/**\n * Used to create `add` and `remove` functions to register and unregister event handlers\n * from a target in the most common handler pattern, where there are only two arguments.\n * (e.g.  `on(name, fn)`, `off(name, fn)`, `addListener(name, fn)`, or `removeListener(name, fn)`)\n * @param target The target we're calling methods on\n * @param eventName The event name for the event we're creating register or unregister functions for\n */\nfunction toCommonHandlerRegistry(target: any, eventName: string) {\n  return (methodName: string) => (handler: any) => target[methodName](eventName, handler);\n}\n\n/**\n * Checks to see if the target implements the required node-style EventEmitter methods\n * for adding and removing event handlers.\n * @param target the object to check\n */\nfunction isNodeStyleEventEmitter(target: any): target is NodeStyleEventEmitter {\n  return isFunction(target.addListener) && isFunction(target.removeListener);\n}\n\n/**\n * Checks to see if the target implements the required jQuery-style EventEmitter methods\n * for adding and removing event handlers.\n * @param target the object to check\n */\nfunction isJQueryStyleEventEmitter(target: any): target is JQueryStyleEventEmitter<any, any> {\n  return isFunction(target.on) && isFunction(target.off);\n}\n\n/**\n * Checks to see if the target implements the required EventTarget methods\n * for adding and removing event handlers.\n * @param target the object to check\n */\nfunction isEventTarget(target: any): target is HasEventTargetAddRemove<any> {\n  return isFunction(target.addEventListener) && isFunction(target.removeEventListener);\n}\n", "import { Observable } from '../Observable';\nimport { isFunction } from '../util/isFunction';\nimport { NodeEventHandler } from './fromEvent';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\n\n/* tslint:disable:max-line-length */\nexport function fromEventPattern<T>(\n  addHandler: (handler: NodeEventHandler) => any,\n  removeHandler?: (handler: NodeEventHandler, signal?: any) => void\n): Observable<T>;\nexport function fromEventPattern<T>(\n  addHandler: (handler: NodeEventHandler) => any,\n  removeHandler?: (handler: NodeEventHandler, signal?: any) => void,\n  resultSelector?: (...args: any[]) => T\n): Observable<T>;\n/* tslint:enable:max-line-length */\n\n/**\n * Creates an Observable from an arbitrary API for registering event handlers.\n *\n * <span class=\"informal\">When that method for adding event handler was something {@link fromEvent}\n * was not prepared for.</span>\n *\n * ![](fromEventPattern.png)\n *\n * `fromEventPattern` allows you to convert into an Observable any API that supports registering handler functions\n * for events. It is similar to {@link fromEvent}, but far\n * more flexible. In fact, all use cases of {@link fromEvent} could be easily handled by\n * `fromEventPattern` (although in slightly more verbose way).\n *\n * This operator accepts as a first argument an `addHandler` function, which will be injected with\n * handler parameter. That handler is actually an event handler function that you now can pass\n * to API expecting it. `addHandler` will be called whenever Observable\n * returned by the operator is subscribed, so registering handler in API will not\n * necessarily happen when `fromEventPattern` is called.\n *\n * After registration, every time an event that we listen to happens,\n * Observable returned by `fromEventPattern` will emit value that event handler\n * function was called with. Note that if event handler was called with more\n * than one argument, second and following arguments will not appear in the Observable.\n *\n * If API you are using allows to unregister event handlers as well, you can pass to `fromEventPattern`\n * another function - `removeHandler` - as a second parameter. It will be injected\n * with the same handler function as before, which now you can use to unregister\n * it from the API. `removeHandler` will be called when consumer of resulting Observable\n * unsubscribes from it.\n *\n * In some APIs unregistering is actually handled differently. Method registering an event handler\n * returns some kind of token, which is later used to identify which function should\n * be unregistered or it itself has method that unregisters event handler.\n * If that is the case with your API, make sure token returned\n * by registering method is returned by `addHandler`. Then it will be passed\n * as a second argument to `removeHandler`, where you will be able to use it.\n *\n * If you need access to all event handler parameters (not only the first one),\n * or you need to transform them in any way, you can call `fromEventPattern` with optional\n * third parameter - project function which will accept all arguments passed to\n * event handler when it is called. Whatever is returned from project function will appear on\n * resulting stream instead of usual event handlers first argument. This means\n * that default project can be thought of as function that takes its first parameter\n * and ignores the rest.\n *\n * ## Examples\n *\n * Emits clicks happening on the DOM document\n *\n * ```ts\n * import { fromEventPattern } from 'rxjs';\n *\n * function addClickHandler(handler) {\n *   document.addEventListener('click', handler);\n * }\n *\n * function removeClickHandler(handler) {\n *   document.removeEventListener('click', handler);\n * }\n *\n * const clicks = fromEventPattern(\n *   addClickHandler,\n *   removeClickHandler\n * );\n * clicks.subscribe(x => console.log(x));\n *\n * // Whenever you click anywhere in the browser, DOM MouseEvent\n * // object will be logged.\n * ```\n *\n * Use with API that returns cancellation token\n *\n * ```ts\n * import { fromEventPattern } from 'rxjs';\n *\n * const token = someAPI.registerEventHandler(function() {});\n * someAPI.unregisterEventHandler(token); // this APIs cancellation method accepts\n *                                        // not handler itself, but special token.\n *\n * const someAPIObservable = fromEventPattern(\n *   function(handler) { return someAPI.registerEventHandler(handler); }, // Note that we return the token here...\n *   function(handler, token) { someAPI.unregisterEventHandler(token); }  // ...to then use it here.\n * );\n * ```\n *\n * Use with project function\n *\n * ```ts\n * import { fromEventPattern } from 'rxjs';\n *\n * someAPI.registerEventHandler((eventType, eventMessage) => {\n *   console.log(eventType, eventMessage); // Logs 'EVENT_TYPE' 'EVENT_MESSAGE' to console.\n * });\n *\n * const someAPIObservable = fromEventPattern(\n *   handler => someAPI.registerEventHandler(handler),\n *   handler => someAPI.unregisterEventHandler(handler)\n *   (eventType, eventMessage) => eventType + ' --- ' + eventMessage // without that function only 'EVENT_TYPE'\n * );                                                                // would be emitted by the Observable\n *\n * someAPIObservable.subscribe(value => console.log(value));\n *\n * // Logs:\n * // 'EVENT_TYPE --- EVENT_MESSAGE'\n * ```\n *\n * @see {@link fromEvent}\n * @see {@link bindCallback}\n * @see {@link bindNodeCallback}\n *\n * @param {function(handler: Function): any} addHandler A function that takes\n * a `handler` function as argument and attaches it somehow to the actual\n * source of events.\n * @param {function(handler: Function, token?: any): void} [removeHandler] A function that\n * takes a `handler` function as an argument and removes it from the event source. If `addHandler`\n * returns some kind of token, `removeHandler` function will have it as a second parameter.\n * @param {function(...args: any): T} [project] A function to\n * transform results. It takes the arguments from the event handler and\n * should return a single value.\n * @return {Observable<T>} Observable which, when an event happens, emits first parameter\n * passed to registered event handler. Alternatively it emits whatever project function returns\n * at that moment.\n */\nexport function fromEventPattern<T>(\n  addHandler: (handler: NodeEventHandler) => any,\n  removeHandler?: (handler: NodeEventHandler, signal?: any) => void,\n  resultSelector?: (...args: any[]) => T\n): Observable<T | T[]> {\n  if (resultSelector) {\n    return fromEventPattern<T>(addHandler, removeHandler).pipe(mapOneOrManyArgs(resultSelector));\n  }\n\n  return new Observable<T | T[]>((subscriber) => {\n    const handler = (...e: T[]) => subscriber.next(e.length === 1 ? e[0] : e);\n    const retValue = addHandler(handler);\n    return isFunction(removeHandler) ? () => removeHandler(handler, retValue) : undefined;\n  });\n}\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\nimport { async as asyncScheduler } from '../scheduler/async';\nimport { isScheduler } from '../util/isScheduler';\nimport { isValidDate } from '../util/isDate';\n\n/**\n * Creates an observable that will wait for a specified time period, or exact date, before\n * emitting the number 0.\n *\n * <span class=\"informal\">Used to emit a notification after a delay.</span>\n *\n * This observable is useful for creating delays in code, or racing against other values\n * for ad-hoc timeouts.\n *\n * The `delay` is specified by default in milliseconds, however providing a custom scheduler could\n * create a different behavior.\n *\n * ## Examples\n *\n * Wait 3 seconds and start another observable\n *\n * You might want to use `timer` to delay subscription to an\n * observable by a set amount of time. Here we use a timer with\n * {@link concatMapTo} or {@link concatMap} in order to wait\n * a few seconds and start a subscription to a source.\n *\n * ```ts\n * import { of, timer, concatMap } from 'rxjs';\n *\n * // This could be any observable\n * const source = of(1, 2, 3);\n *\n * timer(3000)\n *   .pipe(concatMap(() => source))\n *   .subscribe(console.log);\n * ```\n *\n * Take all values until the start of the next minute\n *\n * Using a `Date` as the trigger for the first emission, you can\n * do things like wait until midnight to fire an event, or in this case,\n * wait until a new minute starts (chosen so the example wouldn't take\n * too long to run) in order to stop watching a stream. Leveraging\n * {@link takeUntil}.\n *\n * ```ts\n * import { interval, takeUntil, timer } from 'rxjs';\n *\n * // Build a Date object that marks the\n * // next minute.\n * const currentDate = new Date();\n * const startOfNextMinute = new Date(\n *   currentDate.getFullYear(),\n *   currentDate.getMonth(),\n *   currentDate.getDate(),\n *   currentDate.getHours(),\n *   currentDate.getMinutes() + 1\n * );\n *\n * // This could be any observable stream\n * const source = interval(1000);\n *\n * const result = source.pipe(\n *   takeUntil(timer(startOfNextMinute))\n * );\n *\n * result.subscribe(console.log);\n * ```\n *\n * ### Known Limitations\n *\n * - The {@link asyncScheduler} uses `setTimeout` which has limitations for how far in the future it can be scheduled.\n *\n * - If a `scheduler` is provided that returns a timestamp other than an epoch from `now()`, and\n * a `Date` object is passed to the `dueTime` argument, the calculation for when the first emission\n * should occur will be incorrect. In this case, it would be best to do your own calculations\n * ahead of time, and pass a `number` in as the `dueTime`.\n *\n * @param due If a `number`, the amount of time in milliseconds to wait before emitting.\n * If a `Date`, the exact time at which to emit.\n * @param scheduler The scheduler to use to schedule the delay. Defaults to {@link asyncScheduler}.\n */\nexport function timer(due: number | Date, scheduler?: SchedulerLike): Observable<0>;\n\n/**\n * Creates an observable that starts an interval after a specified delay, emitting incrementing numbers -- starting at `0` --\n * on each interval after words.\n *\n * The `delay` and `intervalDuration` are specified by default in milliseconds, however providing a custom scheduler could\n * create a different behavior.\n *\n * ## Example\n *\n * ### Start an interval that starts right away\n *\n * Since {@link interval} waits for the passed delay before starting,\n * sometimes that's not ideal. You may want to start an interval immediately.\n * `timer` works well for this. Here we have both side-by-side so you can\n * see them in comparison.\n *\n * Note that this observable will never complete.\n *\n * ```ts\n * import { timer, interval } from 'rxjs';\n *\n * timer(0, 1000).subscribe(n => console.log('timer', n));\n * interval(1000).subscribe(n => console.log('interval', n));\n * ```\n *\n * ### Known Limitations\n *\n * - The {@link asyncScheduler} uses `setTimeout` which has limitations for how far in the future it can be scheduled.\n *\n * - If a `scheduler` is provided that returns a timestamp other than an epoch from `now()`, and\n * a `Date` object is passed to the `dueTime` argument, the calculation for when the first emission\n * should occur will be incorrect. In this case, it would be best to do your own calculations\n * ahead of time, and pass a `number` in as the `startDue`.\n * @param startDue If a `number`, is the time to wait before starting the interval.\n * If a `Date`, is the exact time at which to start the interval.\n * @param intervalDuration The delay between each value emitted in the interval. Passing a\n * negative number here will result in immediate completion after the first value is emitted, as though\n * no `intervalDuration` was passed at all.\n * @param scheduler The scheduler to use to schedule the delay. Defaults to {@link asyncScheduler}.\n */\nexport function timer(startDue: number | Date, intervalDuration: number, scheduler?: SchedulerLike): Observable<number>;\n\n/**\n * @deprecated The signature allowing `undefined` to be passed for `intervalDuration` will be removed in v8. Use the `timer(dueTime, scheduler?)` signature instead.\n */\nexport function timer(dueTime: number | Date, unused: undefined, scheduler?: SchedulerLike): Observable<0>;\n\nexport function timer(\n  dueTime: number | Date = 0,\n  intervalOrScheduler?: number | SchedulerLike,\n  scheduler: SchedulerLike = asyncScheduler\n): Observable<number> {\n  // Since negative intervalDuration is treated as though no\n  // interval was specified at all, we start with a negative number.\n  let intervalDuration = -1;\n\n  if (intervalOrScheduler != null) {\n    // If we have a second argument, and it's a scheduler,\n    // override the scheduler we had defaulted. Otherwise,\n    // it must be an interval.\n    if (isScheduler(intervalOrScheduler)) {\n      scheduler = intervalOrScheduler;\n    } else {\n      // Note that this *could* be negative, in which case\n      // it's like not passing an intervalDuration at all.\n      intervalDuration = intervalOrScheduler;\n    }\n  }\n\n  return new Observable((subscriber) => {\n    // If a valid date is passed, calculate how long to wait before\n    // executing the first value... otherwise, if it's a number just schedule\n    // that many milliseconds (or scheduler-specified unit size) in the future.\n    let due = isValidDate(dueTime) ? +dueTime - scheduler!.now() : dueTime;\n\n    if (due < 0) {\n      // Ensure we don't schedule in the future.\n      due = 0;\n    }\n\n    // The incrementing value we emit.\n    let n = 0;\n\n    // Start the timer.\n    return scheduler.schedule(function () {\n      if (!subscriber.closed) {\n        // Emit the next value and increment.\n        subscriber.next(n++);\n\n        if (0 <= intervalDuration) {\n          // If we have a interval after the initial timer,\n          // reschedule with the period.\n          this.schedule(undefined, intervalDuration);\n        } else {\n          // We didn't have an interval. So just complete.\n          subscriber.complete();\n        }\n      }\n    }, due);\n  });\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInput, ObservableInputTuple, SchedulerLike } from '../types';\nimport { mergeAll } from '../operators/mergeAll';\nimport { innerFrom } from './innerFrom';\nimport { EMPTY } from './empty';\nimport { popNumber, popScheduler } from '../util/args';\nimport { from } from './from';\n\nexport function merge<A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): Observable<A[number]>;\nexport function merge<A extends readonly unknown[]>(...sourcesAndConcurrency: [...ObservableInputTuple<A>, number?]): Observable<A[number]>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `mergeAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function merge<A extends readonly unknown[]>(\n  ...sourcesAndScheduler: [...ObservableInputTuple<A>, SchedulerLike?]\n): Observable<A[number]>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `mergeAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function merge<A extends readonly unknown[]>(\n  ...sourcesAndConcurrencyAndScheduler: [...ObservableInputTuple<A>, number?, SchedulerLike?]\n): Observable<A[number]>;\n\n/**\n * Creates an output Observable which concurrently emits all values from every\n * given input Observable.\n *\n * <span class=\"informal\">Flattens multiple Observables together by blending\n * their values into one Observable.</span>\n *\n * ![](merge.png)\n *\n * `merge` subscribes to each given input Observable (as arguments), and simply\n * forwards (without doing any transformation) all the values from all the input\n * Observables to the output Observable. The output Observable only completes\n * once all input Observables have completed. Any error delivered by an input\n * Observable will be immediately emitted on the output Observable.\n *\n * ## Examples\n *\n * Merge together two Observables: 1s interval and clicks\n *\n * ```ts\n * import { merge, fromEvent, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const timer = interval(1000);\n * const clicksOrTimer = merge(clicks, timer);\n * clicksOrTimer.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // timer will emit ascending values, one every second(1000ms) to console\n * // clicks logs MouseEvents to console every time the \"document\" is clicked\n * // Since the two streams are merged you see these happening\n * // as they occur.\n * ```\n *\n * Merge together 3 Observables, but run only 2 concurrently\n *\n * ```ts\n * import { interval, take, merge } from 'rxjs';\n *\n * const timer1 = interval(1000).pipe(take(10));\n * const timer2 = interval(2000).pipe(take(6));\n * const timer3 = interval(500).pipe(take(10));\n *\n * const concurrent = 2; // the argument\n * const merged = merge(timer1, timer2, timer3, concurrent);\n * merged.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // - First timer1 and timer2 will run concurrently\n * // - timer1 will emit a value every 1000ms for 10 iterations\n * // - timer2 will emit a value every 2000ms for 6 iterations\n * // - after timer1 hits its max iteration, timer2 will\n * //   continue, and timer3 will start to run concurrently with timer2\n * // - when timer2 hits its max iteration it terminates, and\n * //   timer3 will continue to emit a value every 500ms until it is complete\n * ```\n *\n * @see {@link mergeAll}\n * @see {@link mergeMap}\n * @see {@link mergeMapTo}\n * @see {@link mergeScan}\n *\n * @param {...ObservableInput} observables Input Observables to merge together.\n * @param {number} [concurrent=Infinity] Maximum number of input\n * Observables being subscribed to concurrently.\n * @param {SchedulerLike} [scheduler=null] The {@link SchedulerLike} to use for managing\n * concurrency of input Observables.\n * @return {Observable} an Observable that emits items that are the result of\n * every input Observable.\n */\nexport function merge(...args: (ObservableInput<unknown> | number | SchedulerLike)[]): Observable<unknown> {\n  const scheduler = popScheduler(args);\n  const concurrent = popNumber(args, Infinity);\n  const sources = args as ObservableInput<unknown>[];\n  return !sources.length\n    ? // No source provided\n      EMPTY\n    : sources.length === 1\n    ? // One source? Just return it.\n      innerFrom(sources[0])\n    : // Merge all sources\n      mergeAll(concurrent)(from(sources, scheduler));\n}\n", "import { Observable } from '../Observable';\nimport { noop } from '../util/noop';\n\n/**\n * An Observable that emits no items to the Observer and never completes.\n *\n * ![](never.png)\n *\n * A simple Observable that emits neither values nor errors nor the completion\n * notification. It can be used for testing purposes or for composing with other\n * Observables. Please note that by never emitting a complete notification, this\n * Observable keeps the subscription from being disposed automatically.\n * Subscriptions need to be manually disposed.\n *\n * ##  Example\n *\n * Emit the number 7, then never emit anything else (not even complete)\n *\n * ```ts\n * import { NEVER, startWith } from 'rxjs';\n *\n * const info = () => console.log('Will not be called');\n *\n * const result = NEVER.pipe(startWith(7));\n * result.subscribe({\n *   next: x => console.log(x),\n *   error: info,\n *   complete: info\n * });\n * ```\n *\n * @see {@link Observable}\n * @see {@link EMPTY}\n * @see {@link of}\n * @see {@link throwError}\n */\nexport const NEVER = new Observable<never>(noop);\n\n/**\n * @deprecated Replaced with the {@link NEVER} constant. Will be removed in v8.\n */\nexport function never() {\n  return NEVER;\n}\n", "const { isArray } = Array;\n\n/**\n * Used in operators and functions that accept either a list of arguments, or an array of arguments\n * as a single argument.\n */\nexport function argsOrArgArray<T>(args: (T | T[])[]): T[] {\n  return args.length === 1 && isArray(args[0]) ? args[0] : (args as T[]);\n}\n", "import { OperatorFunction, MonoTypeOperatorFunction, TruthyTypesOf } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/** @deprecated Use a closure instead of a `thisArg`. Signatures accepting a `thisArg` will be removed in v8. */\nexport function filter<T, S extends T, A>(predicate: (this: A, value: T, index: number) => value is S, thisArg: A): OperatorFunction<T, S>;\nexport function filter<T, S extends T>(predicate: (value: T, index: number) => value is S): OperatorFunction<T, S>;\nexport function filter<T>(predicate: BooleanConstructor): OperatorFunction<T, TruthyTypesOf<T>>;\n/** @deprecated Use a closure instead of a `thisArg`. Signatures accepting a `thisArg` will be removed in v8. */\nexport function filter<T, A>(predicate: (this: A, value: T, index: number) => boolean, thisArg: A): MonoTypeOperatorFunction<T>;\nexport function filter<T>(predicate: (value: T, index: number) => boolean): MonoTypeOperatorFunction<T>;\n\n/**\n * Filter items emitted by the source Observable by only emitting those that\n * satisfy a specified predicate.\n *\n * <span class=\"informal\">Like\n * [Array.prototype.filter()](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/filter),\n * it only emits a value from the source if it passes a criterion function.</span>\n *\n * ![](filter.png)\n *\n * Similar to the well-known `Array.prototype.filter` method, this operator\n * takes values from the source Observable, passes them through a `predicate`\n * function and only emits those values that yielded `true`.\n *\n * ## Example\n *\n * Emit only click events whose target was a DIV element\n *\n * ```ts\n * import { fromEvent, filter } from 'rxjs';\n *\n * const div = document.createElement('div');\n * div.style.cssText = 'width: 200px; height: 200px; background: #09c;';\n * document.body.appendChild(div);\n *\n * const clicks = fromEvent(document, 'click');\n * const clicksOnDivs = clicks.pipe(filter(ev => (<HTMLElement>ev.target).tagName === 'DIV'));\n * clicksOnDivs.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link distinct}\n * @see {@link distinctUntilChanged}\n * @see {@link distinctUntilKeyChanged}\n * @see {@link ignoreElements}\n * @see {@link partition}\n * @see {@link skip}\n *\n * @param predicate A function that\n * evaluates each value emitted by the source Observable. If it returns `true`,\n * the value is emitted, if `false` the value is not passed to the output\n * Observable. The `index` parameter is the number `i` for the i-th source\n * emission that has happened since the subscription, starting from the number\n * `0`.\n * @param thisArg An optional argument to determine the value of `this`\n * in the `predicate` function.\n * @return A function that returns an Observable that emits items from the\n * source Observable that satisfy the specified `predicate`.\n */\nexport function filter<T>(predicate: (value: T, index: number) => boolean, thisArg?: any): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    // An index passed to our predicate function on each call.\n    let index = 0;\n\n    // Subscribe to the source, all errors and completions are\n    // forwarded to the consumer.\n    source.subscribe(\n      // Call the predicate with the appropriate `this` context,\n      // if the predicate returns `true`, then send the value\n      // to the consumer.\n      createOperatorSubscriber(subscriber, (value) => predicate.call(thisArg, value, index++) && subscriber.next(value))\n    );\n  });\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInputTuple } from '../types';\nimport { innerFrom } from './innerFrom';\nimport { argsOrArgArray } from '../util/argsOrArgArray';\nimport { EMPTY } from './empty';\nimport { createOperatorSubscriber } from '../operators/OperatorSubscriber';\nimport { popResultSelector } from '../util/args';\n\nexport function zip<A extends readonly unknown[]>(sources: [...ObservableInputTuple<A>]): Observable<A>;\nexport function zip<A extends readonly unknown[], R>(\n  sources: [...ObservableInputTuple<A>],\n  resultSelector: (...values: A) => R\n): Observable<R>;\nexport function zip<A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): Observable<A>;\nexport function zip<A extends readonly unknown[], R>(\n  ...sourcesAndResultSelector: [...ObservableInputTuple<A>, (...values: A) => R]\n): Observable<R>;\n\n/**\n * Combines multiple Observables to create an Observable whose values are calculated from the values, in order, of each\n * of its input Observables.\n *\n * If the last parameter is a function, this function is used to compute the created value from the input values.\n * Otherwise, an array of the input values is returned.\n *\n * ## Example\n *\n * Combine age and name from different sources\n *\n * ```ts\n * import { of, zip, map } from 'rxjs';\n *\n * const age$ = of(27, 25, 29);\n * const name$ = of('Foo', 'Bar', 'Beer');\n * const isDev$ = of(true, true, false);\n *\n * zip(age$, name$, isDev$).pipe(\n *   map(([age, name, isDev]) => ({ age, name, isDev }))\n * )\n * .subscribe(x => console.log(x));\n *\n * // Outputs\n * // { age: 27, name: 'Foo', isDev: true }\n * // { age: 25, name: 'Bar', isDev: true }\n * // { age: 29, name: 'Beer', isDev: false }\n * ```\n *\n * @param sources\n * @return {Observable<R>}\n */\nexport function zip(...args: unknown[]): Observable<unknown> {\n  const resultSelector = popResultSelector(args);\n\n  const sources = argsOrArgArray(args) as Observable<unknown>[];\n\n  return sources.length\n    ? new Observable<unknown[]>((subscriber) => {\n        // A collection of buffers of values from each source.\n        // Keyed by the same index with which the sources were passed in.\n        let buffers: unknown[][] = sources.map(() => []);\n\n        // An array of flags of whether or not the sources have completed.\n        // This is used to check to see if we should complete the result.\n        // Keyed by the same index with which the sources were passed in.\n        let completed = sources.map(() => false);\n\n        // When everything is done, release the arrays above.\n        subscriber.add(() => {\n          buffers = completed = null!;\n        });\n\n        // Loop over our sources and subscribe to each one. The index `i` is\n        // especially important here, because we use it in closures below to\n        // access the related buffers and completion properties\n        for (let sourceIndex = 0; !subscriber.closed && sourceIndex < sources.length; sourceIndex++) {\n          innerFrom(sources[sourceIndex]).subscribe(\n            createOperatorSubscriber(\n              subscriber,\n              (value) => {\n                buffers[sourceIndex].push(value);\n                // if every buffer has at least one value in it, then we\n                // can shift out the oldest value from each buffer and emit\n                // them as an array.\n                if (buffers.every((buffer) => buffer.length)) {\n                  const result: any = buffers.map((buffer) => buffer.shift()!);\n                  // Emit the array. If theres' a result selector, use that.\n                  subscriber.next(resultSelector ? resultSelector(...result) : result);\n                  // If any one of the sources is both complete and has an empty buffer\n                  // then we complete the result. This is because we cannot possibly have\n                  // any more values to zip together.\n                  if (buffers.some((buffer, i) => !buffer.length && completed[i])) {\n                    subscriber.complete();\n                  }\n                }\n              },\n              () => {\n                // This source completed. Mark it as complete so we can check it later\n                // if we have to.\n                completed[sourceIndex] = true;\n                // But, if this complete source has nothing in its buffer, then we\n                // can complete the result, because we can't possibly have any more\n                // values from this to zip together with the other values.\n                !buffers[sourceIndex].length && subscriber.complete();\n              }\n            )\n          );\n        }\n\n        // When everything is done, release the arrays above.\n        return () => {\n          buffers = completed = null!;\n        };\n      })\n    : EMPTY;\n}\n", "import { Subscriber } from '../Subscriber';\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\n\nimport { operate } from '../util/lift';\nimport { innerFrom } from '../observable/innerFrom';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Ignores source values for a duration determined by another Observable, then\n * emits the most recent value from the source Observable, then repeats this\n * process.\n *\n * <span class=\"informal\">It's like {@link auditTime}, but the silencing\n * duration is determined by a second Observable.</span>\n *\n * ![](audit.svg)\n *\n * `audit` is similar to `throttle`, but emits the last value from the silenced\n * time window, instead of the first value. `audit` emits the most recent value\n * from the source Observable on the output Observable as soon as its internal\n * timer becomes disabled, and ignores source values while the timer is enabled.\n * Initially, the timer is disabled. As soon as the first source value arrives,\n * the timer is enabled by calling the `durationSelector` function with the\n * source value, which returns the \"duration\" Observable. When the duration\n * Observable emits a value, the timer is disabled, then the most\n * recent source value is emitted on the output Observable, and this process\n * repeats for the next source value.\n *\n * ## Example\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, audit, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(audit(ev => interval(1000)));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link auditTime}\n * @see {@link debounce}\n * @see {@link delayWhen}\n * @see {@link sample}\n * @see {@link throttle}\n *\n * @param durationSelector A function\n * that receives a value from the source Observable, for computing the silencing\n * duration, returned as an Observable or a Promise.\n * @return A function that returns an Observable that performs rate-limiting of\n * emissions from the source Observable.\n */\nexport function audit<T>(durationSelector: (value: T) => ObservableInput<any>): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let hasValue = false;\n    let lastValue: T | null = null;\n    let durationSubscriber: Subscriber<any> | null = null;\n    let isComplete = false;\n\n    const endDuration = () => {\n      durationSubscriber?.unsubscribe();\n      durationSubscriber = null;\n      if (hasValue) {\n        hasValue = false;\n        const value = lastValue!;\n        lastValue = null;\n        subscriber.next(value);\n      }\n      isComplete && subscriber.complete();\n    };\n\n    const cleanupDuration = () => {\n      durationSubscriber = null;\n      isComplete && subscriber.complete();\n    };\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          hasValue = true;\n          lastValue = value;\n          if (!durationSubscriber) {\n            innerFrom(durationSelector(value)).subscribe(\n              (durationSubscriber = createOperatorSubscriber(subscriber, endDuration, cleanupDuration))\n            );\n          }\n        },\n        () => {\n          isComplete = true;\n          (!hasValue || !durationSubscriber || durationSubscriber.closed) && subscriber.complete();\n        }\n      )\n    );\n  });\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { audit } from './audit';\nimport { timer } from '../observable/timer';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\n\n/**\n * Ignores source values for `duration` milliseconds, then emits the most recent\n * value from the source Observable, then repeats this process.\n *\n * <span class=\"informal\">When it sees a source value, it ignores that plus\n * the next ones for `duration` milliseconds, and then it emits the most recent\n * value from the source.</span>\n *\n * ![](auditTime.png)\n *\n * `auditTime` is similar to `throttleTime`, but emits the last value from the\n * silenced time window, instead of the first value. `auditTime` emits the most\n * recent value from the source Observable on the output Observable as soon as\n * its internal timer becomes disabled, and ignores source values while the\n * timer is enabled. Initially, the timer is disabled. As soon as the first\n * source value arrives, the timer is enabled. After `duration` milliseconds (or\n * the time unit determined internally by the optional `scheduler`) has passed,\n * the timer is disabled, then the most recent source value is emitted on the\n * output Observable, and this process repeats for the next source value.\n * Optionally takes a {@link SchedulerLike} for managing timers.\n *\n * ## Example\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, auditTime } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(auditTime(1000));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link audit}\n * @see {@link debounceTime}\n * @see {@link delay}\n * @see {@link sampleTime}\n * @see {@link throttleTime}\n *\n * @param {number} duration Time to wait before emitting the most recent source\n * value, measured in milliseconds or the time unit determined internally\n * by the optional `scheduler`.\n * @param {SchedulerLike} [scheduler=async] The {@link SchedulerLike} to use for\n * managing the timers that handle the rate-limiting behavior.\n * @return A function that returns an Observable that performs rate-limiting of\n * emissions from the source Observable.\n */\nexport function auditTime<T>(duration: number, scheduler: SchedulerLike = asyncScheduler): MonoTypeOperatorFunction<T> {\n  return audit(() => timer(duration, scheduler));\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { arrRemove } from '../util/arrRemove';\n\n/**\n * Buffers the source Observable values until the size hits the maximum\n * `bufferSize` given.\n *\n * <span class=\"informal\">Collects values from the past as an array, and emits\n * that array only when its size reaches `bufferSize`.</span>\n *\n * ![](bufferCount.png)\n *\n * Buffers a number of values from the source Observable by `bufferSize` then\n * emits the buffer and clears it, and starts a new buffer each\n * `startBufferEvery` values. If `startBufferEvery` is not provided or is\n * `null`, then new buffers are started immediately at the start of the source\n * and when each buffer closes and is emitted.\n *\n * ## Examples\n *\n * Emit the last two click events as an array\n *\n * ```ts\n * import { fromEvent, bufferCount } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const buffered = clicks.pipe(bufferCount(2));\n * buffered.subscribe(x => console.log(x));\n * ```\n *\n * On every click, emit the last two click events as an array\n *\n * ```ts\n * import { fromEvent, bufferCount } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const buffered = clicks.pipe(bufferCount(2, 1));\n * buffered.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link buffer}\n * @see {@link bufferTime}\n * @see {@link bufferToggle}\n * @see {@link bufferWhen}\n * @see {@link pairwise}\n * @see {@link windowCount}\n *\n * @param {number} bufferSize The maximum size of the buffer emitted.\n * @param {number} [startBufferEvery] Interval at which to start a new buffer.\n * For example if `startBufferEvery` is `2`, then a new buffer will be started\n * on every other value from the source. A new buffer is started at the\n * beginning of the source by default.\n * @return A function that returns an Observable of arrays of buffered values.\n */\nexport function bufferCount<T>(bufferSize: number, startBufferEvery: number | null = null): OperatorFunction<T, T[]> {\n  // If no `startBufferEvery` value was supplied, then we're\n  // opening and closing on the bufferSize itself.\n  startBufferEvery = startBufferEvery ?? bufferSize;\n\n  return operate((source, subscriber) => {\n    let buffers: T[][] = [];\n    let count = 0;\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          let toEmit: T[][] | null = null;\n\n          // Check to see if we need to start a buffer.\n          // This will start one at the first value, and then\n          // a new one every N after that.\n          if (count++ % startBufferEvery! === 0) {\n            buffers.push([]);\n          }\n\n          // Push our value into our active buffers.\n          for (const buffer of buffers) {\n            buffer.push(value);\n            // Check to see if we're over the bufferSize\n            // if we are, record it so we can emit it later.\n            // If we emitted it now and removed it, it would\n            // mutate the `buffers` array while we're looping\n            // over it.\n            if (bufferSize <= buffer.length) {\n              toEmit = toEmit ?? [];\n              toEmit.push(buffer);\n            }\n          }\n\n          if (toEmit) {\n            // We have found some buffers that are over the\n            // `bufferSize`. Emit them, and remove them from our\n            // buffers list.\n            for (const buffer of toEmit) {\n              arrRemove(buffers, buffer);\n              subscriber.next(buffer);\n            }\n          }\n        },\n        () => {\n          // When the source completes, emit all of our\n          // active buffers.\n          for (const buffer of buffers) {\n            subscriber.next(buffer);\n          }\n          subscriber.complete();\n        },\n        // Pass all errors through to consumer.\n        undefined,\n        () => {\n          // Clean up our memory when we finalize\n          buffers = null!;\n        }\n      )\n    );\n  });\n}\n", "import { Observable } from '../Observable';\n\nimport { ObservableInput, OperatorFunction, ObservedValueOf } from '../types';\nimport { Subscription } from '../Subscription';\nimport { innerFrom } from '../observable/innerFrom';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { operate } from '../util/lift';\n\n/* tslint:disable:max-line-length */\nexport function catchError<T, O extends ObservableInput<any>>(\n  selector: (err: any, caught: Observable<T>) => O\n): OperatorFunction<T, T | ObservedValueOf<O>>;\n/* tslint:enable:max-line-length */\n\n/**\n * Catches errors on the observable to be handled by returning a new observable or throwing an error.\n *\n * <span class=\"informal\">\n * It only listens to the error channel and ignores notifications.\n * Handles errors from the source observable, and maps them to a new observable.\n * The error may also be rethrown, or a new error can be thrown to emit an error from the result.\n * </span>\n *\n * ![](catch.png)\n *\n * This operator handles errors, but forwards along all other events to the resulting observable.\n * If the source observable terminates with an error, it will map that error to a new observable,\n * subscribe to it, and forward all of its events to the resulting observable.\n *\n * ## Examples\n *\n * Continue with a different Observable when there's an error\n *\n * ```ts\n * import { of, map, catchError } from 'rxjs';\n *\n * of(1, 2, 3, 4, 5)\n *   .pipe(\n *     map(n => {\n *       if (n === 4) {\n *         throw 'four!';\n *       }\n *       return n;\n *     }),\n *     catchError(err => of('I', 'II', 'III', 'IV', 'V'))\n *   )\n *   .subscribe(x => console.log(x));\n *   // 1, 2, 3, I, II, III, IV, V\n * ```\n *\n * Retry the caught source Observable again in case of error, similar to `retry()` operator\n *\n * ```ts\n * import { of, map, catchError, take } from 'rxjs';\n *\n * of(1, 2, 3, 4, 5)\n *   .pipe(\n *     map(n => {\n *       if (n === 4) {\n *         throw 'four!';\n *       }\n *       return n;\n *     }),\n *     catchError((err, caught) => caught),\n *     take(30)\n *   )\n *   .subscribe(x => console.log(x));\n *   // 1, 2, 3, 1, 2, 3, ...\n * ```\n *\n * Throw a new error when the source Observable throws an error\n *\n * ```ts\n * import { of, map, catchError } from 'rxjs';\n *\n * of(1, 2, 3, 4, 5)\n *   .pipe(\n *     map(n => {\n *       if (n === 4) {\n *         throw 'four!';\n *       }\n *       return n;\n *     }),\n *     catchError(err => {\n *       throw 'error in source. Details: ' + err;\n *     })\n *   )\n *   .subscribe({\n *     next: x => console.log(x),\n *     error: err => console.log(err)\n *   });\n *   // 1, 2, 3, error in source. Details: four!\n * ```\n *\n * @see {@link onErrorResumeNext}\n * @see {@link repeat}\n * @see {@link repeatWhen}\n * @see {@link retry }\n * @see {@link retryWhen}\n *\n * @param {function} selector a function that takes as arguments `err`, which is the error, and `caught`, which\n * is the source observable, in case you'd like to \"retry\" that observable by returning it again. Whatever observable\n * is returned by the `selector` will be used to continue the observable chain.\n * @return A function that returns an Observable that originates from either\n * the source or the Observable returned by the `selector` function.\n */\nexport function catchError<T, O extends ObservableInput<any>>(\n  selector: (err: any, caught: Observable<T>) => O\n): OperatorFunction<T, T | ObservedValueOf<O>> {\n  return operate((source, subscriber) => {\n    let innerSub: Subscription | null = null;\n    let syncUnsub = false;\n    let handledResult: Observable<ObservedValueOf<O>>;\n\n    innerSub = source.subscribe(\n      createOperatorSubscriber(subscriber, undefined, undefined, (err) => {\n        handledResult = innerFrom(selector(err, catchError(selector)(source)));\n        if (innerSub) {\n          innerSub.unsubscribe();\n          innerSub = null;\n          handledResult.subscribe(subscriber);\n        } else {\n          // We don't have an innerSub yet, that means the error was synchronous\n          // because the subscribe call hasn't returned yet.\n          syncUnsub = true;\n        }\n      })\n    );\n\n    if (syncUnsub) {\n      // We have a synchronous error, we need to make sure to\n      // finalize right away. This ensures that callbacks in the `finalize` operator are called\n      // at the right time, and that finalization occurs at the expected\n      // time between the source error and the subscription to the\n      // next observable.\n      innerSub.unsubscribe();\n      innerSub = null;\n      handledResult!.subscribe(subscriber);\n    }\n  });\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * A basic scan operation. This is used for `scan` and `reduce`.\n * @param accumulator The accumulator to use\n * @param seed The seed value for the state to accumulate\n * @param hasSeed Whether or not a seed was provided\n * @param emitOnNext Whether or not to emit the state on next\n * @param emitBeforeComplete Whether or not to emit the before completion\n */\n\nexport function scanInternals<V, A, S>(\n  accumulator: (acc: V | A | S, value: V, index: number) => A,\n  seed: S,\n  hasSeed: boolean,\n  emitOnNext: boolean,\n  emitBeforeComplete?: undefined | true\n) {\n  return (source: Observable<V>, subscriber: Subscriber<any>) => {\n    // Whether or not we have state yet. This will only be\n    // false before the first value arrives if we didn't get\n    // a seed value.\n    let hasState = hasSeed;\n    // The state that we're tracking, starting with the seed,\n    // if there is one, and then updated by the return value\n    // from the accumulator on each emission.\n    let state: any = seed;\n    // An index to pass to the accumulator function.\n    let index = 0;\n\n    // Subscribe to our source. All errors and completions are passed through.\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          // Always increment the index.\n          const i = index++;\n          // Set the state\n          state = hasState\n            ? // We already have state, so we can get the new state from the accumulator\n              accumulator(state, value, i)\n            : // We didn't have state yet, a seed value was not provided, so\n\n              // we set the state to the first value, and mark that we have state now\n              ((hasState = true), value);\n\n          // Maybe send it to the consumer.\n          emitOnNext && subscriber.next(state);\n        },\n        // If an onComplete was given, call it, otherwise\n        // just pass through the complete notification to the consumer.\n        emitBeforeComplete &&\n          (() => {\n            hasState && subscriber.next(state);\n            subscriber.complete();\n          })\n      )\n    );\n  };\n}\n", "import { combineLatestInit } from '../observable/combineLatest';\nimport { ObservableInput, ObservableInputTuple, OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { argsOrArgArray } from '../util/argsOrArgArray';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\nimport { pipe } from '../util/pipe';\nimport { popResultSelector } from '../util/args';\n\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(\n  sources: [...ObservableInputTuple<A>],\n  project: (...values: [T, ...A]) => R\n): OperatorFunction<T, R>;\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(sources: [...ObservableInputTuple<A>]): OperatorFunction<T, [T, ...A]>;\n\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(\n  ...sourcesAndProject: [...ObservableInputTuple<A>, (...values: [T, ...A]) => R]\n): OperatorFunction<T, R>;\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(...sources: [...ObservableInputTuple<A>]): OperatorFunction<T, [T, ...A]>;\n\n/**\n * @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8.\n */\nexport function combineLatest<T, R>(...args: (ObservableInput<any> | ((...values: any[]) => R))[]): OperatorFunction<T, unknown> {\n  const resultSelector = popResultSelector(args);\n  return resultSelector\n    ? pipe(combineLatest(...(args as Array<ObservableInput<any>>)), mapOneOrManyArgs(resultSelector))\n    : operate((source, subscriber) => {\n        combineLatestInit([source, ...argsOrArgArray(args)])(subscriber);\n      });\n}\n", "import { ObservableInputTuple, OperatorFunction, Cons } from '../types';\nimport { combineLatest } from './combineLatest';\n\n/**\n * Create an observable that combines the latest values from all passed observables and the source\n * into arrays and emits them.\n *\n * Returns an observable, that when subscribed to, will subscribe to the source observable and all\n * sources provided as arguments. Once all sources emit at least one value, all of the latest values\n * will be emitted as an array. After that, every time any source emits a value, all of the latest values\n * will be emitted as an array.\n *\n * This is a useful operator for eagerly calculating values based off of changed inputs.\n *\n * ## Example\n *\n * Simple concatenation of values from two inputs\n *\n * ```ts\n * import { fromEvent, combineLatestWith, map } from 'rxjs';\n *\n * // Setup: Add two inputs to the page\n * const input1 = document.createElement('input');\n * document.body.appendChild(input1);\n * const input2 = document.createElement('input');\n * document.body.appendChild(input2);\n *\n * // Get streams of changes\n * const input1Changes$ = fromEvent(input1, 'change');\n * const input2Changes$ = fromEvent(input2, 'change');\n *\n * // Combine the changes by adding them together\n * input1Changes$.pipe(\n *   combineLatestWith(input2Changes$),\n *   map(([e1, e2]) => (<HTMLInputElement>e1.target).value + ' - ' + (<HTMLInputElement>e2.target).value)\n * )\n * .subscribe(x => console.log(x));\n * ```\n *\n * @param otherSources the other sources to subscribe to.\n * @return A function that returns an Observable that emits the latest\n * emissions from both source and provided Observables.\n */\nexport function combineLatestWith<T, A extends readonly unknown[]>(\n  ...otherSources: [...ObservableInputTuple<A>]\n): OperatorFunction<T, Cons<T, A>> {\n  return combineLatest(...otherSources);\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { Subscription } from '../Subscription';\nimport { MonoTypeOperatorFunction, SchedulerAction, SchedulerLike } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Emits a notification from the source Observable only after a particular time span\n * has passed without another source emission.\n *\n * <span class=\"informal\">It's like {@link delay}, but passes only the most\n * recent notification from each burst of emissions.</span>\n *\n * ![](debounceTime.png)\n *\n * `debounceTime` delays notifications emitted by the source Observable, but drops\n * previous pending delayed emissions if a new notification arrives on the source\n * Observable. This operator keeps track of the most recent notification from the\n * source Observable, and emits that only when `dueTime` has passed\n * without any other notification appearing on the source Observable. If a new value\n * appears before `dueTime` silence occurs, the previous notification will be dropped\n * and will not be emitted and a new `dueTime` is scheduled.\n * If the completing event happens during `dueTime` the last cached notification\n * is emitted before the completion event is forwarded to the output observable.\n * If the error event happens during `dueTime` or after it only the error event is\n * forwarded to the output observable. The cache notification is not emitted in this case.\n *\n * This is a rate-limiting operator, because it is impossible for more than one\n * notification to be emitted in any time window of duration `dueTime`, but it is also\n * a delay-like operator since output emissions do not occur at the same time as\n * they did on the source Observable. Optionally takes a {@link SchedulerLike} for\n * managing timers.\n *\n * ## Example\n *\n * Emit the most recent click after a burst of clicks\n *\n * ```ts\n * import { fromEvent, debounceTime } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(debounceTime(1000));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link audit}\n * @see {@link auditTime}\n * @see {@link debounce}\n * @see {@link sample}\n * @see {@link sampleTime}\n * @see {@link throttle}\n * @see {@link throttleTime}\n *\n * @param {number} dueTime The timeout duration in milliseconds (or the time\n * unit determined internally by the optional `scheduler`) for the window of\n * time required to wait for emission silence before emitting the most recent\n * source value.\n * @param {SchedulerLike} [scheduler=async] The {@link SchedulerLike} to use for\n * managing the timers that handle the timeout for each value.\n * @return A function that returns an Observable that delays the emissions of\n * the source Observable by the specified `dueTime`, and may drop some values\n * if they occur too frequently.\n */\nexport function debounceTime<T>(dueTime: number, scheduler: SchedulerLike = asyncScheduler): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let activeTask: Subscription | null = null;\n    let lastValue: T | null = null;\n    let lastTime: number | null = null;\n\n    const emit = () => {\n      if (activeTask) {\n        // We have a value! Free up memory first, then emit the value.\n        activeTask.unsubscribe();\n        activeTask = null;\n        const value = lastValue!;\n        lastValue = null;\n        subscriber.next(value);\n      }\n    };\n    function emitWhenIdle(this: SchedulerAction<unknown>) {\n      // This is called `dueTime` after the first value\n      // but we might have received new values during this window!\n\n      const targetTime = lastTime! + dueTime;\n      const now = scheduler.now();\n      if (now < targetTime) {\n        // On that case, re-schedule to the new target\n        activeTask = this.schedule(undefined, targetTime - now);\n        subscriber.add(activeTask);\n        return;\n      }\n\n      emit();\n    }\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value: T) => {\n          lastValue = value;\n          lastTime = scheduler.now();\n\n          // Only set up a task if it's not already up\n          if (!activeTask) {\n            activeTask = scheduler.schedule(emitWhenIdle, dueTime);\n            subscriber.add(activeTask);\n          }\n        },\n        () => {\n          // Source completed.\n          // Emit any pending debounced values then complete\n          emit();\n          subscriber.complete();\n        },\n        // Pass all errors through to consumer.\n        undefined,\n        () => {\n          // Finalization.\n          lastValue = activeTask = null;\n        }\n      )\n    );\n  });\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Emits a given value if the source Observable completes without emitting any\n * `next` value, otherwise mirrors the source Observable.\n *\n * <span class=\"informal\">If the source Observable turns out to be empty, then\n * this operator will emit a default value.</span>\n *\n * ![](defaultIfEmpty.png)\n *\n * `defaultIfEmpty` emits the values emitted by the source Observable or a\n * specified default value if the source Observable is empty (completes without\n * having emitted any `next` value).\n *\n * ## Example\n *\n * If no clicks happen in 5 seconds, then emit 'no clicks'\n *\n * ```ts\n * import { fromEvent, takeUntil, interval, defaultIfEmpty } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const clicksBeforeFive = clicks.pipe(takeUntil(interval(5000)));\n * const result = clicksBeforeFive.pipe(defaultIfEmpty('no clicks'));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link empty}\n * @see {@link last}\n *\n * @param defaultValue The default value used if the source\n * Observable is empty.\n * @return A function that returns an Observable that emits either the\n * specified `defaultValue` if the source Observable emits no items, or the\n * values emitted by the source Observable.\n */\nexport function defaultIfEmpty<T, R>(defaultValue: R): OperatorFunction<T, T | R> {\n  return operate((source, subscriber) => {\n    let hasValue = false;\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          hasValue = true;\n          subscriber.next(value);\n        },\n        () => {\n          if (!hasValue) {\n            subscriber.next(defaultValue!);\n          }\n          subscriber.complete();\n        }\n      )\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { EMPTY } from '../observable/empty';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Emits only the first `count` values emitted by the source Observable.\n *\n * <span class=\"informal\">Takes the first `count` values from the source, then\n * completes.</span>\n *\n * ![](take.png)\n *\n * `take` returns an Observable that emits only the first `count` values emitted\n * by the source Observable. If the source emits fewer than `count` values then\n * all of its values are emitted. After that, it completes, regardless if the\n * source completes.\n *\n * ## Example\n *\n * Take the first 5 seconds of an infinite 1-second interval Observable\n *\n * ```ts\n * import { interval, take } from 'rxjs';\n *\n * const intervalCount = interval(1000);\n * const takeFive = intervalCount.pipe(take(5));\n * takeFive.subscribe(x => console.log(x));\n *\n * // Logs:\n * // 0\n * // 1\n * // 2\n * // 3\n * // 4\n * ```\n *\n * @see {@link takeLast}\n * @see {@link takeUntil}\n * @see {@link takeWhile}\n * @see {@link skip}\n *\n * @param count The maximum number of `next` values to emit.\n * @return A function that returns an Observable that emits only the first\n * `count` values emitted by the source Observable, or all of the values from\n * the source if the source emits fewer than `count` values.\n */\nexport function take<T>(count: number): MonoTypeOperatorFunction<T> {\n  return count <= 0\n    ? // If we are taking no values, that's empty.\n      () => EMPTY\n    : operate((source, subscriber) => {\n        let seen = 0;\n        source.subscribe(\n          createOperatorSubscriber(subscriber, (value) => {\n            // Increment the number of values we have seen,\n            // then check it against the allowed count to see\n            // if we are still letting values through.\n            if (++seen <= count) {\n              subscriber.next(value);\n              // If we have met or passed our allowed count,\n              // we need to complete. We have to do <= here,\n              // because re-entrant code will increment `seen` twice.\n              if (count <= seen) {\n                subscriber.complete();\n              }\n            }\n          })\n        );\n      });\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { noop } from '../util/noop';\n\n/**\n * Ignores all items emitted by the source Observable and only passes calls of `complete` or `error`.\n *\n * ![](ignoreElements.png)\n *\n * The `ignoreElements` operator suppresses all items emitted by the source Observable,\n * but allows its termination notification (either `error` or `complete`) to pass through unchanged.\n *\n * If you do not care about the items being emitted by an Observable, but you do want to be notified\n * when it completes or when it terminates with an error, you can apply the `ignoreElements` operator\n * to the Observable, which will ensure that it will never call its observers\u2019 `next` handlers.\n *\n * ## Example\n *\n * Ignore all `next` emissions from the source\n *\n * ```ts\n * import { of, ignoreElements } from 'rxjs';\n *\n * of('you', 'talking', 'to', 'me')\n *   .pipe(ignoreElements())\n *   .subscribe({\n *     next: word => console.log(word),\n *     error: err => console.log('error:', err),\n *     complete: () => console.log('the end'),\n *   });\n *\n * // result:\n * // 'the end'\n * ```\n *\n * @return A function that returns an empty Observable that only calls\n * `complete` or `error`, based on which one is called by the source\n * Observable.\n */\nexport function ignoreElements(): OperatorFunction<unknown, never> {\n  return operate((source, subscriber) => {\n    source.subscribe(createOperatorSubscriber(subscriber, noop));\n  });\n}\n", "import { OperatorFunction } from '../types';\nimport { map } from './map';\n\n/** @deprecated To be removed in v9. Use {@link map} instead: `map(() => value)`. */\nexport function mapTo<R>(value: R): OperatorFunction<unknown, R>;\n/**\n * @deprecated Do not specify explicit type parameters. Signatures with type parameters\n * that cannot be inferred will be removed in v8. `mapTo` itself will be removed in v9,\n * use {@link map} instead: `map(() => value)`.\n * */\nexport function mapTo<T, R>(value: R): OperatorFunction<T, R>;\n\n/**\n * Emits the given constant value on the output Observable every time the source\n * Observable emits a value.\n *\n * <span class=\"informal\">Like {@link map}, but it maps every source value to\n * the same output value every time.</span>\n *\n * ![](mapTo.png)\n *\n * Takes a constant `value` as argument, and emits that whenever the source\n * Observable emits a value. In other words, ignores the actual source value,\n * and simply uses the emission moment to know when to emit the given `value`.\n *\n * ## Example\n *\n * Map every click to the string `'Hi'`\n *\n * ```ts\n * import { fromEvent, mapTo } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const greetings = clicks.pipe(mapTo('Hi'));\n *\n * greetings.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link map}\n *\n * @param value The value to map each source value to.\n * @return A function that returns an Observable that emits the given `value`\n * every time the source Observable emits.\n * @deprecated To be removed in v9. Use {@link map} instead: `map(() => value)`.\n */\nexport function mapTo<R>(value: R): OperatorFunction<unknown, R> {\n  return map(() => value);\n}\n", "import { Observable } from '../Observable';\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { concat } from '../observable/concat';\nimport { take } from './take';\nimport { ignoreElements } from './ignoreElements';\nimport { mapTo } from './mapTo';\nimport { mergeMap } from './mergeMap';\nimport { innerFrom } from '../observable/innerFrom';\n\n/** @deprecated The `subscriptionDelay` parameter will be removed in v8. */\nexport function delayWhen<T>(\n  delayDurationSelector: (value: T, index: number) => ObservableInput<any>,\n  subscriptionDelay: Observable<any>\n): MonoTypeOperatorFunction<T>;\nexport function delayWhen<T>(delayDurationSelector: (value: T, index: number) => ObservableInput<any>): MonoTypeOperatorFunction<T>;\n\n/**\n * Delays the emission of items from the source Observable by a given time span\n * determined by the emissions of another Observable.\n *\n * <span class=\"informal\">It's like {@link delay}, but the time span of the\n * delay duration is determined by a second Observable.</span>\n *\n * ![](delayWhen.png)\n *\n * `delayWhen` operator shifts each emitted value from the source Observable by\n * a time span determined by another Observable. When the source emits a value,\n * the `delayDurationSelector` function is called with the value emitted from\n * the source Observable as the first argument to the `delayDurationSelector`.\n * The `delayDurationSelector` function should return an {@link ObservableInput},\n * that is internally converted to an Observable that is called the \"duration\"\n * Observable.\n *\n * The source value is emitted on the output Observable only when the \"duration\"\n * Observable emits ({@link guide/glossary-and-semantics#next next}s) any value.\n * Upon that, the \"duration\" Observable gets unsubscribed.\n *\n * Before RxJS V7, the {@link guide/glossary-and-semantics#complete completion}\n * of the \"duration\" Observable would have been triggering the emission of the\n * source value to the output Observable, but with RxJS V7, this is not the case\n * anymore.\n *\n * Only next notifications (from the \"duration\" Observable) trigger values from\n * the source Observable to be passed to the output Observable. If the \"duration\"\n * Observable only emits the complete notification (without next), the value\n * emitted by the source Observable will never get to the output Observable - it\n * will be swallowed. If the \"duration\" Observable errors, the error will be\n * propagated to the output Observable.\n *\n * Optionally, `delayWhen` takes a second argument, `subscriptionDelay`, which\n * is an Observable. When `subscriptionDelay` emits its first value or\n * completes, the source Observable is subscribed to and starts behaving like\n * described in the previous paragraph. If `subscriptionDelay` is not provided,\n * `delayWhen` will subscribe to the source Observable as soon as the output\n * Observable is subscribed.\n *\n * ## Example\n *\n * Delay each click by a random amount of time, between 0 and 5 seconds\n *\n * ```ts\n * import { fromEvent, delayWhen, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const delayedClicks = clicks.pipe(\n *   delayWhen(() => interval(Math.random() * 5000))\n * );\n * delayedClicks.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link delay}\n * @see {@link throttle}\n * @see {@link throttleTime}\n * @see {@link debounce}\n * @see {@link debounceTime}\n * @see {@link sample}\n * @see {@link sampleTime}\n * @see {@link audit}\n * @see {@link auditTime}\n *\n * @param delayDurationSelector A function that returns an `ObservableInput` for\n * each `value` emitted by the source Observable, which is then used to delay the\n * emission of that `value` on the output Observable until the `ObservableInput`\n * returned from this function emits a next value. When called, beside `value`,\n * this function receives a zero-based `index` of the emission order.\n * @param subscriptionDelay An Observable that triggers the subscription to the\n * source Observable once it emits any value.\n * @return A function that returns an Observable that delays the emissions of\n * the source Observable by an amount of time specified by the Observable\n * returned by `delayDurationSelector`.\n */\nexport function delayWhen<T>(\n  delayDurationSelector: (value: T, index: number) => ObservableInput<any>,\n  subscriptionDelay?: Observable<any>\n): MonoTypeOperatorFunction<T> {\n  if (subscriptionDelay) {\n    // DEPRECATED PATH\n    return (source: Observable<T>) =>\n      concat(subscriptionDelay.pipe(take(1), ignoreElements()), source.pipe(delayWhen(delayDurationSelector)));\n  }\n\n  return mergeMap((value, index) => innerFrom(delayDurationSelector(value, index)).pipe(take(1), mapTo(value)));\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { delayWhen } from './delayWhen';\nimport { timer } from '../observable/timer';\n\n/**\n * Delays the emission of items from the source Observable by a given timeout or\n * until a given Date.\n *\n * <span class=\"informal\">Time shifts each item by some specified amount of\n * milliseconds.</span>\n *\n * ![](delay.svg)\n *\n * If the delay argument is a Number, this operator time shifts the source\n * Observable by that amount of time expressed in milliseconds. The relative\n * time intervals between the values are preserved.\n *\n * If the delay argument is a Date, this operator time shifts the start of the\n * Observable execution until the given date occurs.\n *\n * ## Examples\n *\n * Delay each click by one second\n *\n * ```ts\n * import { fromEvent, delay } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const delayedClicks = clicks.pipe(delay(1000)); // each click emitted after 1 second\n * delayedClicks.subscribe(x => console.log(x));\n * ```\n *\n * Delay all clicks until a future date happens\n *\n * ```ts\n * import { fromEvent, delay } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const date = new Date('March 15, 2050 12:00:00'); // in the future\n * const delayedClicks = clicks.pipe(delay(date)); // click emitted only after that date\n * delayedClicks.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link delayWhen}\n * @see {@link throttle}\n * @see {@link throttleTime}\n * @see {@link debounce}\n * @see {@link debounceTime}\n * @see {@link sample}\n * @see {@link sampleTime}\n * @see {@link audit}\n * @see {@link auditTime}\n *\n * @param {number|Date} due The delay duration in milliseconds (a `number`) or\n * a `Date` until which the emission of the source items is delayed.\n * @param {SchedulerLike} [scheduler=async] The {@link SchedulerLike} to use for\n * managing the timers that handle the time-shift for each item.\n * @return A function that returns an Observable that delays the emissions of\n * the source Observable by the specified timeout or Date.\n */\nexport function delay<T>(due: number | Date, scheduler: SchedulerLike = asyncScheduler): MonoTypeOperatorFunction<T> {\n  const duration = timer(due, scheduler);\n  return delayWhen(() => duration);\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { identity } from '../util/identity';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\nexport function distinctUntilChanged<T>(comparator?: (previous: T, current: T) => boolean): MonoTypeOperatorFunction<T>;\nexport function distinctUntilChanged<T, K>(\n  comparator: (previous: K, current: K) => boolean,\n  keySelector: (value: T) => K\n): MonoTypeOperatorFunction<T>;\n\n/**\n * Returns a result {@link Observable} that emits all values pushed by the source observable if they\n * are distinct in comparison to the last value the result observable emitted.\n *\n * When provided without parameters or with the first parameter (`{@link distinctUntilChanged#comparator comparator}`),\n * it behaves like this:\n *\n * 1. It will always emit the first value from the source.\n * 2. For all subsequent values pushed by the source, they will be compared to the previously emitted values\n *    using the provided `comparator` or an `===` equality check.\n * 3. If the value pushed by the source is determined to be unequal by this check, that value is emitted and\n *    becomes the new \"previously emitted value\" internally.\n *\n * When the second parameter (`{@link distinctUntilChanged#keySelector keySelector}`) is provided, the behavior\n * changes:\n *\n * 1. It will always emit the first value from the source.\n * 2. The `keySelector` will be run against all values, including the first value.\n * 3. For all values after the first, the selected key will be compared against the key selected from\n *    the previously emitted value using the `comparator`.\n * 4. If the keys are determined to be unequal by this check, the value (not the key), is emitted\n *    and the selected key from that value is saved for future comparisons against other keys.\n *\n * ## Examples\n *\n * A very basic example with no `{@link distinctUntilChanged#comparator comparator}`. Note that `1` is emitted more than once,\n * because it's distinct in comparison to the _previously emitted_ value,\n * not in comparison to _all other emitted values_.\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * of(1, 1, 1, 2, 2, 2, 1, 1, 3, 3)\n *   .pipe(distinctUntilChanged())\n *   .subscribe(console.log);\n * // Logs: 1, 2, 1, 3\n * ```\n *\n * With a `{@link distinctUntilChanged#comparator comparator}`, you can do custom comparisons. Let's say\n * you only want to emit a value when all of its components have\n * changed:\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * const totallyDifferentBuilds$ = of(\n *   { engineVersion: '1.1.0', transmissionVersion: '1.2.0' },\n *   { engineVersion: '1.1.0', transmissionVersion: '1.4.0' },\n *   { engineVersion: '1.3.0', transmissionVersion: '1.4.0' },\n *   { engineVersion: '1.3.0', transmissionVersion: '1.5.0' },\n *   { engineVersion: '2.0.0', transmissionVersion: '1.5.0' }\n * ).pipe(\n *   distinctUntilChanged((prev, curr) => {\n *     return (\n *       prev.engineVersion === curr.engineVersion ||\n *       prev.transmissionVersion === curr.transmissionVersion\n *     );\n *   })\n * );\n *\n * totallyDifferentBuilds$.subscribe(console.log);\n *\n * // Logs:\n * // { engineVersion: '1.1.0', transmissionVersion: '1.2.0' }\n * // { engineVersion: '1.3.0', transmissionVersion: '1.4.0' }\n * // { engineVersion: '2.0.0', transmissionVersion: '1.5.0' }\n * ```\n *\n * You can also provide a custom `{@link distinctUntilChanged#comparator comparator}` to check that emitted\n * changes are only in one direction. Let's say you only want to get\n * the next record temperature:\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * const temps$ = of(30, 31, 20, 34, 33, 29, 35, 20);\n *\n * const recordHighs$ = temps$.pipe(\n *   distinctUntilChanged((prevHigh, temp) => {\n *     // If the current temp is less than\n *     // or the same as the previous record,\n *     // the record hasn't changed.\n *     return temp <= prevHigh;\n *   })\n * );\n *\n * recordHighs$.subscribe(console.log);\n * // Logs: 30, 31, 34, 35\n * ```\n *\n * Selecting update events only when the `updatedBy` field shows\n * the account changed hands.\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * // A stream of updates to a given account\n * const accountUpdates$ = of(\n *   { updatedBy: 'blesh', data: [] },\n *   { updatedBy: 'blesh', data: [] },\n *   { updatedBy: 'ncjamieson', data: [] },\n *   { updatedBy: 'ncjamieson', data: [] },\n *   { updatedBy: 'blesh', data: [] }\n * );\n *\n * // We only want the events where it changed hands\n * const changedHands$ = accountUpdates$.pipe(\n *   distinctUntilChanged(undefined, update => update.updatedBy)\n * );\n *\n * changedHands$.subscribe(console.log);\n * // Logs:\n * // { updatedBy: 'blesh', data: Array[0] }\n * // { updatedBy: 'ncjamieson', data: Array[0] }\n * // { updatedBy: 'blesh', data: Array[0] }\n * ```\n *\n * @see {@link distinct}\n * @see {@link distinctUntilKeyChanged}\n *\n * @param comparator A function used to compare the previous and current keys for\n * equality. Defaults to a `===` check.\n * @param keySelector Used to select a key value to be passed to the `comparator`.\n *\n * @return A function that returns an Observable that emits items from the\n * source Observable with distinct values.\n */\nexport function distinctUntilChanged<T, K>(\n  comparator?: (previous: K, current: K) => boolean,\n  keySelector: (value: T) => K = identity as (value: T) => K\n): MonoTypeOperatorFunction<T> {\n  // We've been allowing `null` do be passed as the `compare`, so we can't do\n  // a default value for the parameter, because that will only work\n  // for `undefined`.\n  comparator = comparator ?? defaultCompare;\n\n  return operate((source, subscriber) => {\n    // The previous key, used to compare against keys selected\n    // from new arrivals to determine \"distinctiveness\".\n    let previousKey: K;\n    // Whether or not this is the first value we've gotten.\n    let first = true;\n\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value) => {\n        // We always call the key selector.\n        const currentKey = keySelector(value);\n\n        // If it's the first value, we always emit it.\n        // Otherwise, we compare this key to the previous key, and\n        // if the comparer returns false, we emit.\n        if (first || !comparator!(previousKey, currentKey)) {\n          // Update our state *before* we emit the value\n          // as emission can be the source of re-entrant code\n          // in functional libraries like this. We only really\n          // need to do this if it's the first value, or if the\n          // key we're tracking in previous needs to change.\n          first = false;\n          previousKey = currentKey;\n\n          // Emit the value!\n          subscriber.next(value);\n        }\n      })\n    );\n  });\n}\n\nfunction defaultCompare(a: any, b: any) {\n  return a === b;\n}\n", "import { distinctUntilChanged } from './distinctUntilChanged';\nimport { MonoTypeOperatorFunction } from '../types';\n\n/* tslint:disable:max-line-length */\nexport function distinctUntilKeyChanged<T>(key: keyof T): MonoTypeOperatorFunction<T>;\nexport function distinctUntilKeyChanged<T, K extends keyof T>(key: K, compare: (x: T[K], y: T[K]) => boolean): MonoTypeOperatorFunction<T>;\n/* tslint:enable:max-line-length */\n\n/**\n * Returns an Observable that emits all items emitted by the source Observable that are distinct by comparison from the previous item,\n * using a property accessed by using the key provided to check if the two items are distinct.\n *\n * If a comparator function is provided, then it will be called for each item to test for whether or not that value should be emitted.\n *\n * If a comparator function is not provided, an equality check is used by default.\n *\n * ## Examples\n *\n * An example comparing the name of persons\n *\n * ```ts\n * import { of, distinctUntilKeyChanged } from 'rxjs';\n *\n * of(\n *   { age: 4, name: 'Foo' },\n *   { age: 7, name: 'Bar' },\n *   { age: 5, name: 'Foo' },\n *   { age: 6, name: 'Foo' }\n * ).pipe(\n *   distinctUntilKeyChanged('name')\n * )\n * .subscribe(x => console.log(x));\n *\n * // displays:\n * // { age: 4, name: 'Foo' }\n * // { age: 7, name: 'Bar' }\n * // { age: 5, name: 'Foo' }\n * ```\n *\n * An example comparing the first letters of the name\n *\n * ```ts\n * import { of, distinctUntilKeyChanged } from 'rxjs';\n *\n * of(\n *   { age: 4, name: 'Foo1' },\n *   { age: 7, name: 'Bar' },\n *   { age: 5, name: 'Foo2' },\n *   { age: 6, name: 'Foo3' }\n * ).pipe(\n *   distinctUntilKeyChanged('name', (x, y) => x.substring(0, 3) === y.substring(0, 3))\n * )\n * .subscribe(x => console.log(x));\n *\n * // displays:\n * // { age: 4, name: 'Foo1' }\n * // { age: 7, name: 'Bar' }\n * // { age: 5, name: 'Foo2' }\n * ```\n *\n * @see {@link distinct}\n * @see {@link distinctUntilChanged}\n *\n * @param {string} key String key for object property lookup on each item.\n * @param {function} [compare] Optional comparison function called to test if an item is distinct from the previous item in the source.\n * @return A function that returns an Observable that emits items from the\n * source Observable with distinct values based on the key specified.\n */\nexport function distinctUntilKeyChanged<T, K extends keyof T>(key: K, compare?: (x: T[K], y: T[K]) => boolean): MonoTypeOperatorFunction<T> {\n  return distinctUntilChanged((x: T, y: T) => compare ? compare(x[key], y[key]) : x[key] === y[key]);\n}\n", "import { EmptyError } from '../util/EmptyError';\nimport { MonoTypeOperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * If the source observable completes without emitting a value, it will emit\n * an error. The error will be created at that time by the optional\n * `errorFactory` argument, otherwise, the error will be {@link EmptyError}.\n *\n * ![](throwIfEmpty.png)\n *\n * ## Example\n *\n * Throw an error if the document wasn't clicked within 1 second\n *\n * ```ts\n * import { fromEvent, takeUntil, timer, throwIfEmpty } from 'rxjs';\n *\n * const click$ = fromEvent(document, 'click');\n *\n * click$.pipe(\n *   takeUntil(timer(1000)),\n *   throwIfEmpty(() => new Error('The document was not clicked within 1 second'))\n * )\n * .subscribe({\n *   next() {\n *    console.log('The document was clicked');\n *   },\n *   error(err) {\n *     console.error(err.message);\n *   }\n * });\n * ```\n *\n * @param errorFactory A factory function called to produce the\n * error to be thrown when the source observable completes without emitting a\n * value.\n * @return A function that returns an Observable that throws an error if the\n * source Observable completed without emitting.\n */\nexport function throwIfEmpty<T>(errorFactory: () => any = defaultErrorFactory): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let hasValue = false;\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          hasValue = true;\n          subscriber.next(value);\n        },\n        () => (hasValue ? subscriber.complete() : subscriber.error(errorFactory()))\n      )\n    );\n  });\n}\n\nfunction defaultErrorFactory() {\n  return new EmptyError();\n}\n", "/** prettier */\nimport { Observable } from '../Observable';\nimport { concat } from '../observable/concat';\nimport { of } from '../observable/of';\nimport { MonoTypeOperatorFunction, SchedulerLike, OperatorFunction, ValueFromArray } from '../types';\n\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `concatAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function endWith<T>(scheduler: SchedulerLike): MonoTypeOperatorFunction<T>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `concatAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function endWith<T, A extends unknown[] = T[]>(\n  ...valuesAndScheduler: [...A, SchedulerLike]\n): OperatorFunction<T, T | ValueFromArray<A>>;\n\nexport function endWith<T, A extends unknown[] = T[]>(...values: A): OperatorFunction<T, T | ValueFromArray<A>>;\n\n/**\n * Returns an observable that will emit all values from the source, then synchronously emit\n * the provided value(s) immediately after the source completes.\n *\n * NOTE: Passing a last argument of a Scheduler is _deprecated_, and may result in incorrect\n * types in TypeScript.\n *\n * This is useful for knowing when an observable ends. Particularly when paired with an\n * operator like {@link takeUntil}\n *\n * ![](endWith.png)\n *\n * ## Example\n *\n * Emit values to know when an interval starts and stops. The interval will\n * stop when a user clicks anywhere on the document.\n *\n * ```ts\n * import { interval, map, fromEvent, startWith, takeUntil, endWith } from 'rxjs';\n *\n * const ticker$ = interval(5000).pipe(\n *   map(() => 'tick')\n * );\n *\n * const documentClicks$ = fromEvent(document, 'click');\n *\n * ticker$.pipe(\n *   startWith('interval started'),\n *   takeUntil(documentClicks$),\n *   endWith('interval ended by click')\n * )\n * .subscribe(x => console.log(x));\n *\n * // Result (assuming a user clicks after 15 seconds)\n * // 'interval started'\n * // 'tick'\n * // 'tick'\n * // 'tick'\n * // 'interval ended by click'\n * ```\n *\n * @see {@link startWith}\n * @see {@link concat}\n * @see {@link takeUntil}\n *\n * @param values Items you want the modified Observable to emit last.\n * @return A function that returns an Observable that emits all values from the\n * source, then synchronously emits the provided value(s) immediately after the\n * source completes.\n */\nexport function endWith<T>(...values: Array<T | SchedulerLike>): MonoTypeOperatorFunction<T> {\n  return (source: Observable<T>) => concat(source, of(...values)) as Observable<T>;\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { operate } from '../util/lift';\n\n/**\n * Returns an Observable that mirrors the source Observable, but will call a specified function when\n * the source terminates on complete or error.\n * The specified function will also be called when the subscriber explicitly unsubscribes.\n *\n * ## Examples\n *\n * Execute callback function when the observable completes\n *\n * ```ts\n * import { interval, take, finalize } from 'rxjs';\n *\n * // emit value in sequence every 1 second\n * const source = interval(1000);\n * const example = source.pipe(\n *   take(5), //take only the first 5 values\n *   finalize(() => console.log('Sequence complete')) // Execute when the observable completes\n * );\n * const subscribe = example.subscribe(val => console.log(val));\n *\n * // results:\n * // 0\n * // 1\n * // 2\n * // 3\n * // 4\n * // 'Sequence complete'\n * ```\n *\n * Execute callback function when the subscriber explicitly unsubscribes\n *\n * ```ts\n * import { interval, finalize, tap, noop, timer } from 'rxjs';\n *\n * const source = interval(100).pipe(\n *   finalize(() => console.log('[finalize] Called')),\n *   tap({\n *     next: () => console.log('[next] Called'),\n *     error: () => console.log('[error] Not called'),\n *     complete: () => console.log('[tap complete] Not called')\n *   })\n * );\n *\n * const sub = source.subscribe({\n *   next: x => console.log(x),\n *   error: noop,\n *   complete: () => console.log('[complete] Not called')\n * });\n *\n * timer(150).subscribe(() => sub.unsubscribe());\n *\n * // results:\n * // '[next] Called'\n * // 0\n * // '[finalize] Called'\n * ```\n *\n * @param {function} callback Function to be called when source terminates.\n * @return A function that returns an Observable that mirrors the source, but\n * will call the specified function on termination.\n */\nexport function finalize<T>(callback: () => void): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    // TODO: This try/finally was only added for `useDeprecatedSynchronousErrorHandling`.\n    // REMOVE THIS WHEN THAT HOT GARBAGE IS REMOVED IN V8.\n    try {\n      source.subscribe(subscriber);\n    } finally {\n      subscriber.add(callback);\n    }\n  });\n}\n", "import { Observable } from '../Observable';\nimport { EmptyError } from '../util/EmptyError';\nimport { OperatorFunction, TruthyTypesOf } from '../types';\nimport { filter } from './filter';\nimport { take } from './take';\nimport { defaultIfEmpty } from './defaultIfEmpty';\nimport { throwIfEmpty } from './throwIfEmpty';\nimport { identity } from '../util/identity';\n\nexport function first<T, D = T>(predicate?: null, defaultValue?: D): OperatorFunction<T, T | D>;\nexport function first<T>(predicate: BooleanConstructor): OperatorFunction<T, TruthyTypesOf<T>>;\nexport function first<T, D>(predicate: BooleanConstructor, defaultValue: D): OperatorFunction<T, TruthyTypesOf<T> | D>;\nexport function first<T, S extends T>(\n  predicate: (value: T, index: number, source: Observable<T>) => value is S,\n  defaultValue?: S\n): OperatorFunction<T, S>;\nexport function first<T, S extends T, D>(\n  predicate: (value: T, index: number, source: Observable<T>) => value is S,\n  defaultValue: D\n): OperatorFunction<T, S | D>;\nexport function first<T, D = T>(\n  predicate: (value: T, index: number, source: Observable<T>) => boolean,\n  defaultValue?: D\n): OperatorFunction<T, T | D>;\n\n/**\n * Emits only the first value (or the first value that meets some condition)\n * emitted by the source Observable.\n *\n * <span class=\"informal\">Emits only the first value. Or emits only the first\n * value that passes some test.</span>\n *\n * ![](first.png)\n *\n * If called with no arguments, `first` emits the first value of the source\n * Observable, then completes. If called with a `predicate` function, `first`\n * emits the first value of the source that matches the specified condition. Throws an error if\n * `defaultValue` was not provided and a matching element is not found.\n *\n * ## Examples\n *\n * Emit only the first click that happens on the DOM\n *\n * ```ts\n * import { fromEvent, first } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(first());\n * result.subscribe(x => console.log(x));\n * ```\n *\n * Emits the first click that happens on a DIV\n *\n * ```ts\n * import { fromEvent, first } from 'rxjs';\n *\n * const div = document.createElement('div');\n * div.style.cssText = 'width: 200px; height: 200px; background: #09c;';\n * document.body.appendChild(div);\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(first(ev => (<HTMLElement>ev.target).tagName === 'DIV'));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link filter}\n * @see {@link find}\n * @see {@link take}\n *\n * @throws {EmptyError} Delivers an EmptyError to the Observer's `error`\n * callback if the Observable completes before any `next` notification was sent.\n * This is how `first()` is different from {@link take}(1) which completes instead.\n *\n * @param {function(value: T, index: number, source: Observable<T>): boolean} [predicate]\n * An optional function called with each item to test for condition matching.\n * @param {D} [defaultValue] The default value emitted in case no valid value\n * was found on the source.\n * @return A function that returns an Observable that emits the first item that\n * matches the condition.\n */\nexport function first<T, D>(\n  predicate?: ((value: T, index: number, source: Observable<T>) => boolean) | null,\n  defaultValue?: D\n): OperatorFunction<T, T | D> {\n  const hasDefaultValue = arguments.length >= 2;\n  return (source: Observable<T>) =>\n    source.pipe(\n      predicate ? filter((v, i) => predicate(v, i, source)) : identity,\n      take(1),\n      hasDefaultValue ? defaultIfEmpty(defaultValue!) : throwIfEmpty(() => new EmptyError())\n    );\n}\n", "import { ObservableInput, ObservableInputTuple, OperatorFunction, SchedulerLike } from '../types';\nimport { operate } from '../util/lift';\nimport { argsOrArgArray } from '../util/argsOrArgArray';\nimport { mergeAll } from './mergeAll';\nimport { popNumber, popScheduler } from '../util/args';\nimport { from } from '../observable/from';\n\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): OperatorFunction<T, T | A[number]>;\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(\n  ...sourcesAndConcurrency: [...ObservableInputTuple<A>, number]\n): OperatorFunction<T, T | A[number]>;\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(\n  ...sourcesAndScheduler: [...ObservableInputTuple<A>, SchedulerLike]\n): OperatorFunction<T, T | A[number]>;\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(\n  ...sourcesAndConcurrencyAndScheduler: [...ObservableInputTuple<A>, number, SchedulerLike]\n): OperatorFunction<T, T | A[number]>;\n\nexport function merge<T>(...args: unknown[]): OperatorFunction<T, unknown> {\n  const scheduler = popScheduler(args);\n  const concurrent = popNumber(args, Infinity);\n  args = argsOrArgArray(args);\n\n  return operate((source, subscriber) => {\n    mergeAll(concurrent)(from([source, ...(args as ObservableInput<T>[])], scheduler)).subscribe(subscriber);\n  });\n}\n", "import { ObservableInputTuple, OperatorFunction } from '../types';\nimport { merge } from './merge';\n\n/**\n * Merge the values from all observables to a single observable result.\n *\n * Creates an observable, that when subscribed to, subscribes to the source\n * observable, and all other sources provided as arguments. All values from\n * every source are emitted from the resulting subscription.\n *\n * When all sources complete, the resulting observable will complete.\n *\n * When any source errors, the resulting observable will error.\n *\n * ## Example\n *\n * Joining all outputs from multiple user input event streams\n *\n * ```ts\n * import { fromEvent, map, mergeWith } from 'rxjs';\n *\n * const clicks$ = fromEvent(document, 'click').pipe(map(() => 'click'));\n * const mousemoves$ = fromEvent(document, 'mousemove').pipe(map(() => 'mousemove'));\n * const dblclicks$ = fromEvent(document, 'dblclick').pipe(map(() => 'dblclick'));\n *\n * mousemoves$\n *   .pipe(mergeWith(clicks$, dblclicks$))\n *   .subscribe(x => console.log(x));\n *\n * // result (assuming user interactions)\n * // 'mousemove'\n * // 'mousemove'\n * // 'mousemove'\n * // 'click'\n * // 'click'\n * // 'dblclick'\n * ```\n *\n * @see {@link merge}\n *\n * @param otherSources the sources to combine the current source with.\n * @return A function that returns an Observable that merges the values from\n * all given Observables.\n */\nexport function mergeWith<T, A extends readonly unknown[]>(\n  ...otherSources: [...ObservableInputTuple<A>]\n): OperatorFunction<T, T | A[number]> {\n  return merge(...otherSources);\n}\n", "import { Subscription } from '../Subscription';\nimport { EMPTY } from '../observable/empty';\nimport { operate } from '../util/lift';\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { timer } from '../observable/timer';\n\nexport interface RepeatConfig {\n  /**\n   * The number of times to repeat the source. Defaults to `Infinity`.\n   */\n  count?: number;\n\n  /**\n   * If a `number`, will delay the repeat of the source by that number of milliseconds.\n   * If a function, it will provide the number of times the source has been subscribed to,\n   * and the return value should be a valid observable input that will notify when the source\n   * should be repeated. If the notifier observable is empty, the result will complete.\n   */\n  delay?: number | ((count: number) => ObservableInput<any>);\n}\n\n/**\n * Returns an Observable that will resubscribe to the source stream when the source stream completes.\n *\n * <span class=\"informal\">Repeats all values emitted on the source. It's like {@link retry}, but for non error cases.</span>\n *\n * ![](repeat.png)\n *\n * Repeat will output values from a source until the source completes, then it will resubscribe to the\n * source a specified number of times, with a specified delay. Repeat can be particularly useful in\n * combination with closing operators like {@link take}, {@link takeUntil}, {@link first}, or {@link takeWhile},\n * as it can be used to restart a source again from scratch.\n *\n * Repeat is very similar to {@link retry}, where {@link retry} will resubscribe to the source in the error case, but\n * `repeat` will resubscribe if the source completes.\n *\n * Note that `repeat` will _not_ catch errors. Use {@link retry} for that.\n *\n * - `repeat(0)` returns an empty observable\n * - `repeat()` will repeat forever\n * - `repeat({ delay: 200 })` will repeat forever, with a delay of 200ms between repetitions.\n * - `repeat({ count: 2, delay: 400 })` will repeat twice, with a delay of 400ms between repetitions.\n * - `repeat({ delay: (count) => timer(count * 1000) })` will repeat forever, but will have a delay that grows by one second for each repetition.\n *\n * ## Example\n *\n * Repeat a message stream\n *\n * ```ts\n * import { of, repeat } from 'rxjs';\n *\n * const source = of('Repeat message');\n * const result = source.pipe(repeat(3));\n *\n * result.subscribe(x => console.log(x));\n *\n * // Results\n * // 'Repeat message'\n * // 'Repeat message'\n * // 'Repeat message'\n * ```\n *\n * Repeat 3 values, 2 times\n *\n * ```ts\n * import { interval, take, repeat } from 'rxjs';\n *\n * const source = interval(1000);\n * const result = source.pipe(take(3), repeat(2));\n *\n * result.subscribe(x => console.log(x));\n *\n * // Results every second\n * // 0\n * // 1\n * // 2\n * // 0\n * // 1\n * // 2\n * ```\n *\n * Defining two complex repeats with delays on the same source.\n * Note that the second repeat cannot be called until the first\n * repeat as exhausted it's count.\n *\n * ```ts\n * import { defer, of, repeat } from 'rxjs';\n *\n * const source = defer(() => {\n *    return of(`Hello, it is ${new Date()}`)\n * });\n *\n * source.pipe(\n *    // Repeat 3 times with a delay of 1 second between repetitions\n *    repeat({\n *      count: 3,\n *      delay: 1000,\n *    }),\n *\n *    // *Then* repeat forever, but with an exponential step-back\n *    // maxing out at 1 minute.\n *    repeat({\n *      delay: (count) => timer(Math.min(60000, 2 ^ count * 1000))\n *    })\n * )\n * ```\n *\n * @see {@link repeatWhen}\n * @see {@link retry}\n *\n * @param count The number of times the source Observable items are repeated, a count of 0 will yield\n * an empty Observable.\n */\nexport function repeat<T>(countOrConfig?: number | RepeatConfig): MonoTypeOperatorFunction<T> {\n  let count = Infinity;\n  let delay: RepeatConfig['delay'];\n\n  if (countOrConfig != null) {\n    if (typeof countOrConfig === 'object') {\n      ({ count = Infinity, delay } = countOrConfig);\n    } else {\n      count = countOrConfig;\n    }\n  }\n\n  return count <= 0\n    ? () => EMPTY\n    : operate((source, subscriber) => {\n        let soFar = 0;\n        let sourceSub: Subscription | null;\n\n        const resubscribe = () => {\n          sourceSub?.unsubscribe();\n          sourceSub = null;\n          if (delay != null) {\n            const notifier = typeof delay === 'number' ? timer(delay) : innerFrom(delay(soFar));\n            const notifierSubscriber = createOperatorSubscriber(subscriber, () => {\n              notifierSubscriber.unsubscribe();\n              subscribeToSource();\n            });\n            notifier.subscribe(notifierSubscriber);\n          } else {\n            subscribeToSource();\n          }\n        };\n\n        const subscribeToSource = () => {\n          let syncUnsub = false;\n          sourceSub = source.subscribe(\n            createOperatorSubscriber(subscriber, undefined, () => {\n              if (++soFar < count) {\n                if (sourceSub) {\n                  resubscribe();\n                } else {\n                  syncUnsub = true;\n                }\n              } else {\n                subscriber.complete();\n              }\n            })\n          );\n\n          if (syncUnsub) {\n            resubscribe();\n          }\n        };\n\n        subscribeToSource();\n      });\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { scanInternals } from './scanInternals';\n\nexport function scan<V, A = V>(accumulator: (acc: A | V, value: V, index: number) => A): OperatorFunction<V, V | A>;\nexport function scan<V, A>(accumulator: (acc: A, value: V, index: number) => A, seed: A): OperatorFunction<V, A>;\nexport function scan<V, A, S>(accumulator: (acc: A | S, value: V, index: number) => A, seed: S): OperatorFunction<V, A>;\n\n// TODO: link to a \"redux pattern\" section in the guide (location TBD)\n\n/**\n * Useful for encapsulating and managing state. Applies an accumulator (or \"reducer function\")\n * to each value from the source after an initial state is established -- either via\n * a `seed` value (second argument), or from the first value from the source.\n *\n * <span class=\"informal\">It's like {@link reduce}, but emits the current\n * accumulation state after each update</span>\n *\n * ![](scan.png)\n *\n * This operator maintains an internal state and emits it after processing each value as follows:\n *\n * 1. First value arrives\n *   - If a `seed` value was supplied (as the second argument to `scan`), let `state = seed` and `value = firstValue`.\n *   - If NO `seed` value was supplied (no second argument), let `state = firstValue` and go to 3.\n * 2. Let `state = accumulator(state, value)`.\n *   - If an error is thrown by `accumulator`, notify the consumer of an error. The process ends.\n * 3. Emit `state`.\n * 4. Next value arrives, let `value = nextValue`, go to 2.\n *\n * ## Examples\n *\n * An average of previous numbers. This example shows how\n * not providing a `seed` can prime the stream with the\n * first value from the source.\n *\n * ```ts\n * import { of, scan, map } from 'rxjs';\n *\n * const numbers$ = of(1, 2, 3);\n *\n * numbers$\n *   .pipe(\n *     // Get the sum of the numbers coming in.\n *     scan((total, n) => total + n),\n *     // Get the average by dividing the sum by the total number\n *     // received so far (which is 1 more than the zero-based index).\n *     map((sum, index) => sum / (index + 1))\n *   )\n *   .subscribe(console.log);\n * ```\n *\n * The Fibonacci sequence. This example shows how you can use\n * a seed to prime accumulation process. Also... you know... Fibonacci.\n * So important to like, computers and stuff that its whiteboarded\n * in job interviews. Now you can show them the Rx version! (Please don't, haha)\n *\n * ```ts\n * import { interval, scan, map, startWith } from 'rxjs';\n *\n * const firstTwoFibs = [0, 1];\n * // An endless stream of Fibonacci numbers.\n * const fibonacci$ = interval(1000).pipe(\n *   // Scan to get the fibonacci numbers (after 0, 1)\n *   scan(([a, b]) => [b, a + b], firstTwoFibs),\n *   // Get the second number in the tuple, it's the one you calculated\n *   map(([, n]) => n),\n *   // Start with our first two digits :)\n *   startWith(...firstTwoFibs)\n * );\n *\n * fibonacci$.subscribe(console.log);\n * ```\n *\n * @see {@link expand}\n * @see {@link mergeScan}\n * @see {@link reduce}\n * @see {@link switchScan}\n *\n * @param accumulator A \"reducer function\". This will be called for each value after an initial state is\n * acquired.\n * @param seed The initial state. If this is not provided, the first value from the source will\n * be used as the initial state, and emitted without going through the accumulator. All subsequent values\n * will be processed by the accumulator function. If this is provided, all values will go through\n * the accumulator function.\n * @return A function that returns an Observable of the accumulated values.\n */\nexport function scan<V, A, S>(accumulator: (acc: V | A | S, value: V, index: number) => A, seed?: S): OperatorFunction<V, V | A> {\n  // providing a seed of `undefined` *should* be valid and trigger\n  // hasSeed! so don't use `seed !== undefined` checks!\n  // For this reason, we have to check it here at the original call site\n  // otherwise inside Operator/Subscriber we won't know if `undefined`\n  // means they didn't provide anything or if they literally provided `undefined`\n  return operate(scanInternals(accumulator, seed as S, arguments.length >= 2, true));\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { Subject } from '../Subject';\nimport { SafeSubscriber } from '../Subscriber';\nimport { Subscription } from '../Subscription';\nimport { MonoTypeOperatorFunction, SubjectLike, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\n\nexport interface ShareConfig<T> {\n  /**\n   * The factory used to create the subject that will connect the source observable to\n   * multicast consumers.\n   */\n  connector?: () => SubjectLike<T>;\n  /**\n   * If `true`, the resulting observable will reset internal state on error from source and return to a \"cold\" state. This\n   * allows the resulting observable to be \"retried\" in the event of an error.\n   * If `false`, when an error comes from the source it will push the error into the connecting subject, and the subject\n   * will remain the connecting subject, meaning the resulting observable will not go \"cold\" again, and subsequent retries\n   * or resubscriptions will resubscribe to that same subject. In all cases, RxJS subjects will emit the same error again, however\n   * {@link ReplaySubject} will also push its buffered values before pushing the error.\n   * It is also possible to pass a notifier factory returning an `ObservableInput` instead which grants more fine-grained\n   * control over how and when the reset should happen. This allows behaviors like conditional or delayed resets.\n   */\n  resetOnError?: boolean | ((error: any) => ObservableInput<any>);\n  /**\n   * If `true`, the resulting observable will reset internal state on completion from source and return to a \"cold\" state. This\n   * allows the resulting observable to be \"repeated\" after it is done.\n   * If `false`, when the source completes, it will push the completion through the connecting subject, and the subject\n   * will remain the connecting subject, meaning the resulting observable will not go \"cold\" again, and subsequent repeats\n   * or resubscriptions will resubscribe to that same subject.\n   * It is also possible to pass a notifier factory returning an `ObservableInput` instead which grants more fine-grained\n   * control over how and when the reset should happen. This allows behaviors like conditional or delayed resets.\n   */\n  resetOnComplete?: boolean | (() => ObservableInput<any>);\n  /**\n   * If `true`, when the number of subscribers to the resulting observable reaches zero due to those subscribers unsubscribing, the\n   * internal state will be reset and the resulting observable will return to a \"cold\" state. This means that the next\n   * time the resulting observable is subscribed to, a new subject will be created and the source will be subscribed to\n   * again.\n   * If `false`, when the number of subscribers to the resulting observable reaches zero due to unsubscription, the subject\n   * will remain connected to the source, and new subscriptions to the result will be connected through that same subject.\n   * It is also possible to pass a notifier factory returning an `ObservableInput` instead which grants more fine-grained\n   * control over how and when the reset should happen. This allows behaviors like conditional or delayed resets.\n   */\n  resetOnRefCountZero?: boolean | (() => ObservableInput<any>);\n}\n\nexport function share<T>(): MonoTypeOperatorFunction<T>;\n\nexport function share<T>(options: ShareConfig<T>): MonoTypeOperatorFunction<T>;\n\n/**\n * Returns a new Observable that multicasts (shares) the original Observable. As long as there is at least one\n * Subscriber this Observable will be subscribed and emitting data. When all subscribers have unsubscribed it will\n * unsubscribe from the source Observable. Because the Observable is multicasting it makes the stream `hot`.\n * This is an alias for `multicast(() => new Subject()), refCount()`.\n *\n * The subscription to the underlying source Observable can be reset (unsubscribe and resubscribe for new subscribers),\n * if the subscriber count to the shared observable drops to 0, or if the source Observable errors or completes. It is\n * possible to use notifier factories for the resets to allow for behaviors like conditional or delayed resets. Please\n * note that resetting on error or complete of the source Observable does not behave like a transparent retry or restart\n * of the source because the error or complete will be forwarded to all subscribers and their subscription will be\n * closed. Only new subscribers after a reset on error or complete happened will cause a fresh subscription to the\n * source. To achieve transparent retries or restarts pipe the source through appropriate operators before sharing.\n *\n * ![](share.png)\n *\n * ## Example\n *\n * Generate new multicast Observable from the `source` Observable value\n *\n * ```ts\n * import { interval, tap, map, take, share } from 'rxjs';\n *\n * const source = interval(1000).pipe(\n *   tap(x => console.log('Processing: ', x)),\n *   map(x => x * x),\n *   take(6),\n *   share()\n * );\n *\n * source.subscribe(x => console.log('subscription 1: ', x));\n * source.subscribe(x => console.log('subscription 2: ', x));\n *\n * // Logs:\n * // Processing: 0\n * // subscription 1: 0\n * // subscription 2: 0\n * // Processing: 1\n * // subscription 1: 1\n * // subscription 2: 1\n * // Processing: 2\n * // subscription 1: 4\n * // subscription 2: 4\n * // Processing: 3\n * // subscription 1: 9\n * // subscription 2: 9\n * // Processing: 4\n * // subscription 1: 16\n * // subscription 2: 16\n * // Processing: 5\n * // subscription 1: 25\n * // subscription 2: 25\n * ```\n *\n * ## Example with notifier factory: Delayed reset\n *\n * ```ts\n * import { interval, take, share, timer } from 'rxjs';\n *\n * const source = interval(1000).pipe(\n *   take(3),\n *   share({\n *     resetOnRefCountZero: () => timer(1000)\n *   })\n * );\n *\n * const subscriptionOne = source.subscribe(x => console.log('subscription 1: ', x));\n * setTimeout(() => subscriptionOne.unsubscribe(), 1300);\n *\n * setTimeout(() => source.subscribe(x => console.log('subscription 2: ', x)), 1700);\n *\n * setTimeout(() => source.subscribe(x => console.log('subscription 3: ', x)), 5000);\n *\n * // Logs:\n * // subscription 1:  0\n * // (subscription 1 unsubscribes here)\n * // (subscription 2 subscribes here ~400ms later, source was not reset)\n * // subscription 2:  1\n * // subscription 2:  2\n * // (subscription 2 unsubscribes here)\n * // (subscription 3 subscribes here ~2000ms later, source did reset before)\n * // subscription 3:  0\n * // subscription 3:  1\n * // subscription 3:  2\n * ```\n *\n * @see {@link shareReplay}\n *\n * @return A function that returns an Observable that mirrors the source.\n */\nexport function share<T>(options: ShareConfig<T> = {}): MonoTypeOperatorFunction<T> {\n  const { connector = () => new Subject<T>(), resetOnError = true, resetOnComplete = true, resetOnRefCountZero = true } = options;\n  // It's necessary to use a wrapper here, as the _operator_ must be\n  // referentially transparent. Otherwise, it cannot be used in calls to the\n  // static `pipe` function - to create a partial pipeline.\n  //\n  // The _operator function_ - the function returned by the _operator_ - will\n  // not be referentially transparent - as it shares its source - but the\n  // _operator function_ is called when the complete pipeline is composed via a\n  // call to a source observable's `pipe` method - not when the static `pipe`\n  // function is called.\n  return (wrapperSource) => {\n    let connection: SafeSubscriber<T> | undefined;\n    let resetConnection: Subscription | undefined;\n    let subject: SubjectLike<T> | undefined;\n    let refCount = 0;\n    let hasCompleted = false;\n    let hasErrored = false;\n\n    const cancelReset = () => {\n      resetConnection?.unsubscribe();\n      resetConnection = undefined;\n    };\n    // Used to reset the internal state to a \"cold\"\n    // state, as though it had never been subscribed to.\n    const reset = () => {\n      cancelReset();\n      connection = subject = undefined;\n      hasCompleted = hasErrored = false;\n    };\n    const resetAndUnsubscribe = () => {\n      // We need to capture the connection before\n      // we reset (if we need to reset).\n      const conn = connection;\n      reset();\n      conn?.unsubscribe();\n    };\n\n    return operate<T, T>((source, subscriber) => {\n      refCount++;\n      if (!hasErrored && !hasCompleted) {\n        cancelReset();\n      }\n\n      // Create the subject if we don't have one yet. Grab a local reference to\n      // it as well, which avoids non-null assertions when using it and, if we\n      // connect to it now, then error/complete need a reference after it was\n      // reset.\n      const dest = (subject = subject ?? connector());\n\n      // Add the finalization directly to the subscriber - instead of returning it -\n      // so that the handling of the subscriber's unsubscription will be wired\n      // up _before_ the subscription to the source occurs. This is done so that\n      // the assignment to the source connection's `closed` property will be seen\n      // by synchronous firehose sources.\n      subscriber.add(() => {\n        refCount--;\n\n        // If we're resetting on refCount === 0, and it's 0, we only want to do\n        // that on \"unsubscribe\", really. Resetting on error or completion is a different\n        // configuration.\n        if (refCount === 0 && !hasErrored && !hasCompleted) {\n          resetConnection = handleReset(resetAndUnsubscribe, resetOnRefCountZero);\n        }\n      });\n\n      // The following line adds the subscription to the subscriber passed.\n      // Basically, `subscriber === dest.subscribe(subscriber)` is `true`.\n      dest.subscribe(subscriber);\n\n      if (\n        !connection &&\n        // Check this shareReplay is still activate - it can be reset to 0\n        // and be \"unsubscribed\" _before_ it actually subscribes.\n        // If we were to subscribe then, it'd leak and get stuck.\n        refCount > 0\n      ) {\n        // We need to create a subscriber here - rather than pass an observer and\n        // assign the returned subscription to connection - because it's possible\n        // for reentrant subscriptions to the shared observable to occur and in\n        // those situations we want connection to be already-assigned so that we\n        // don't create another connection to the source.\n        connection = new SafeSubscriber({\n          next: (value) => dest.next(value),\n          error: (err) => {\n            hasErrored = true;\n            cancelReset();\n            resetConnection = handleReset(reset, resetOnError, err);\n            dest.error(err);\n          },\n          complete: () => {\n            hasCompleted = true;\n            cancelReset();\n            resetConnection = handleReset(reset, resetOnComplete);\n            dest.complete();\n          },\n        });\n        innerFrom(source).subscribe(connection);\n      }\n    })(wrapperSource);\n  };\n}\n\nfunction handleReset<T extends unknown[] = never[]>(\n  reset: () => void,\n  on: boolean | ((...args: T) => ObservableInput<any>),\n  ...args: T\n): Subscription | undefined {\n  if (on === true) {\n    reset();\n    return;\n  }\n\n  if (on === false) {\n    return;\n  }\n\n  const onSubscriber = new SafeSubscriber({\n    next: () => {\n      onSubscriber.unsubscribe();\n      reset();\n    },\n  });\n\n  return innerFrom(on(...args)).subscribe(onSubscriber);\n}\n", "import { ReplaySubject } from '../ReplaySubject';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { share } from './share';\n\nexport interface ShareReplayConfig {\n  bufferSize?: number;\n  windowTime?: number;\n  refCount: boolean;\n  scheduler?: SchedulerLike;\n}\n\nexport function shareReplay<T>(config: ShareReplayConfig): MonoTypeOperatorFunction<T>;\nexport function shareReplay<T>(bufferSize?: number, windowTime?: number, scheduler?: SchedulerLike): MonoTypeOperatorFunction<T>;\n\n/**\n * Share source and replay specified number of emissions on subscription.\n *\n * This operator is a specialization of `replay` that connects to a source observable\n * and multicasts through a `ReplaySubject` constructed with the specified arguments.\n * A successfully completed source will stay cached in the `shareReplay`ed observable forever,\n * but an errored source can be retried.\n *\n * ## Why use `shareReplay`?\n *\n * You generally want to use `shareReplay` when you have side-effects or taxing computations\n * that you do not wish to be executed amongst multiple subscribers.\n * It may also be valuable in situations where you know you will have late subscribers to\n * a stream that need access to previously emitted values.\n * This ability to replay values on subscription is what differentiates {@link share} and `shareReplay`.\n *\n * ## Reference counting\n *\n * By default `shareReplay` will use `refCount` of false, meaning that it will _not_ unsubscribe the\n * source when the reference counter drops to zero, i.e. the inner `ReplaySubject` will _not_ be unsubscribed\n * (and potentially run for ever).\n * This is the default as it is expected that `shareReplay` is often used to keep around expensive to setup\n * observables which we want to keep running instead of having to do the expensive setup again.\n *\n * As of RXJS version 6.4.0 a new overload signature was added to allow for manual control over what\n * happens when the operators internal reference counter drops to zero.\n * If `refCount` is true, the source will be unsubscribed from once the reference count drops to zero, i.e.\n * the inner `ReplaySubject` will be unsubscribed. All new subscribers will receive value emissions from a\n * new `ReplaySubject` which in turn will cause a new subscription to the source observable.\n *\n * ## Examples\n *\n * Example with a third subscriber coming late to the party\n *\n * ```ts\n * import { interval, take, shareReplay } from 'rxjs';\n *\n * const shared$ = interval(2000).pipe(\n *   take(6),\n *   shareReplay(3)\n * );\n *\n * shared$.subscribe(x => console.log('sub A: ', x));\n * shared$.subscribe(y => console.log('sub B: ', y));\n *\n * setTimeout(() => {\n *   shared$.subscribe(y => console.log('sub C: ', y));\n * }, 11000);\n *\n * // Logs:\n * // (after ~2000 ms)\n * // sub A: 0\n * // sub B: 0\n * // (after ~4000 ms)\n * // sub A: 1\n * // sub B: 1\n * // (after ~6000 ms)\n * // sub A: 2\n * // sub B: 2\n * // (after ~8000 ms)\n * // sub A: 3\n * // sub B: 3\n * // (after ~10000 ms)\n * // sub A: 4\n * // sub B: 4\n * // (after ~11000 ms, sub C gets the last 3 values)\n * // sub C: 2\n * // sub C: 3\n * // sub C: 4\n * // (after ~12000 ms)\n * // sub A: 5\n * // sub B: 5\n * // sub C: 5\n * ```\n *\n * Example for `refCount` usage\n *\n * ```ts\n * import { Observable, tap, interval, shareReplay, take } from 'rxjs';\n *\n * const log = <T>(name: string, source: Observable<T>) => source.pipe(\n *   tap({\n *     subscribe: () => console.log(`${ name }: subscribed`),\n *     next: value => console.log(`${ name }: ${ value }`),\n *     complete: () => console.log(`${ name }: completed`),\n *     finalize: () => console.log(`${ name }: unsubscribed`)\n *   })\n * );\n *\n * const obs$ = log('source', interval(1000));\n *\n * const shared$ = log('shared', obs$.pipe(\n *   shareReplay({ bufferSize: 1, refCount: true }),\n *   take(2)\n * ));\n *\n * shared$.subscribe(x => console.log('sub A: ', x));\n * shared$.subscribe(y => console.log('sub B: ', y));\n *\n * // PRINTS:\n * // shared: subscribed <-- reference count = 1\n * // source: subscribed\n * // shared: subscribed <-- reference count = 2\n * // source: 0\n * // shared: 0\n * // sub A: 0\n * // shared: 0\n * // sub B: 0\n * // source: 1\n * // shared: 1\n * // sub A: 1\n * // shared: completed <-- take(2) completes the subscription for sub A\n * // shared: unsubscribed <-- reference count = 1\n * // shared: 1\n * // sub B: 1\n * // shared: completed <-- take(2) completes the subscription for sub B\n * // shared: unsubscribed <-- reference count = 0\n * // source: unsubscribed <-- replaySubject unsubscribes from source observable because the reference count dropped to 0 and refCount is true\n *\n * // In case of refCount being false, the unsubscribe is never called on the source and the source would keep on emitting, even if no subscribers\n * // are listening.\n * // source: 2\n * // source: 3\n * // source: 4\n * // ...\n * ```\n *\n * @see {@link publish}\n * @see {@link share}\n * @see {@link publishReplay}\n *\n * @param configOrBufferSize Maximum element count of the replay buffer or {@link ShareReplayConfig configuration}\n * object.\n * @param windowTime Maximum time length of the replay buffer in milliseconds.\n * @param scheduler Scheduler where connected observers within the selector function\n * will be invoked on.\n * @return A function that returns an Observable sequence that contains the\n * elements of a sequence produced by multicasting the source sequence within a\n * selector function.\n */\nexport function shareReplay<T>(\n  configOrBufferSize?: ShareReplayConfig | number,\n  windowTime?: number,\n  scheduler?: SchedulerLike\n): MonoTypeOperatorFunction<T> {\n  let bufferSize: number;\n  let refCount = false;\n  if (configOrBufferSize && typeof configOrBufferSize === 'object') {\n    ({ bufferSize = Infinity, windowTime = Infinity, refCount = false, scheduler } = configOrBufferSize);\n  } else {\n    bufferSize = (configOrBufferSize ?? Infinity) as number;\n  }\n  return share<T>({\n    connector: () => new ReplaySubject(bufferSize, windowTime, scheduler),\n    resetOnError: true,\n    resetOnComplete: false,\n    resetOnRefCountZero: refCount,\n  });\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { filter } from './filter';\n\n/**\n * Returns an Observable that skips the first `count` items emitted by the source Observable.\n *\n * ![](skip.png)\n *\n * Skips the values until the sent notifications are equal or less than provided skip count. It raises\n * an error if skip count is equal or more than the actual number of emits and source raises an error.\n *\n * ## Example\n *\n * Skip the values before the emission\n *\n * ```ts\n * import { interval, skip } from 'rxjs';\n *\n * // emit every half second\n * const source = interval(500);\n * // skip the first 10 emitted values\n * const result = source.pipe(skip(10));\n *\n * result.subscribe(value => console.log(value));\n * // output: 10...11...12...13...\n * ```\n *\n * @see {@link last}\n * @see {@link skipWhile}\n * @see {@link skipUntil}\n * @see {@link skipLast}\n *\n * @param {Number} count - The number of times, items emitted by source Observable should be skipped.\n * @return A function that returns an Observable that skips the first `count`\n * values emitted by the source Observable.\n */\nexport function skip<T>(count: number): MonoTypeOperatorFunction<T> {\n  return filter((_, index) => count <= index);\n}\n", "import { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { noop } from '../util/noop';\n\n/**\n * Returns an Observable that skips items emitted by the source Observable until a second Observable emits an item.\n *\n * The `skipUntil` operator causes the observable stream to skip the emission of values until the passed in observable\n * emits the first value. This can be particularly useful in combination with user interactions, responses of HTTP\n * requests or waiting for specific times to pass by.\n *\n * ![](skipUntil.png)\n *\n * Internally, the `skipUntil` operator subscribes to the passed in `notifier` `ObservableInput` (which gets converted\n * to an Observable) in order to recognize the emission of its first value. When `notifier` emits next, the operator\n * unsubscribes from it and starts emitting the values of the *source* observable until it completes or errors. It\n * will never let the *source* observable emit any values if the `notifier` completes or throws an error without\n * emitting a value before.\n *\n * ## Example\n *\n * In the following example, all emitted values of the interval observable are skipped until the user clicks anywhere\n * within the page\n *\n * ```ts\n * import { interval, fromEvent, skipUntil } from 'rxjs';\n *\n * const intervalObservable = interval(1000);\n * const click = fromEvent(document, 'click');\n *\n * const emitAfterClick = intervalObservable.pipe(\n *   skipUntil(click)\n * );\n * // clicked at 4.6s. output: 5...6...7...8........ or\n * // clicked at 7.3s. output: 8...9...10..11.......\n * emitAfterClick.subscribe(value => console.log(value));\n * ```\n *\n * @see {@link last}\n * @see {@link skip}\n * @see {@link skipWhile}\n * @see {@link skipLast}\n *\n * @param notifier An `ObservableInput` that has to emit an item before the source Observable elements begin to\n * be mirrored by the resulting Observable.\n * @return A function that returns an Observable that skips items from the\n * source Observable until the `notifier` Observable emits an item, then emits the\n * remaining items.\n */\nexport function skipUntil<T>(notifier: ObservableInput<any>): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let taking = false;\n\n    const skipSubscriber = createOperatorSubscriber(\n      subscriber,\n      () => {\n        skipSubscriber?.unsubscribe();\n        taking = true;\n      },\n      noop\n    );\n\n    innerFrom(notifier).subscribe(skipSubscriber);\n\n    source.subscribe(createOperatorSubscriber(subscriber, (value) => taking && subscriber.next(value)));\n  });\n}\n", "import { concat } from '../observable/concat';\nimport { OperatorFunction, SchedulerLike, ValueFromArray } from '../types';\nimport { popScheduler } from '../util/args';\nimport { operate } from '../util/lift';\n\n// Devs are more likely to pass null or undefined than they are a scheduler\n// without accompanying values. To make things easier for (naughty) devs who\n// use the `strictNullChecks: false` TypeScript compiler option, these\n// overloads with explicit null and undefined values are included.\n\nexport function startWith<T>(value: null): OperatorFunction<T, T | null>;\nexport function startWith<T>(value: undefined): OperatorFunction<T, T | undefined>;\n\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `concatAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function startWith<T, A extends readonly unknown[] = T[]>(\n  ...valuesAndScheduler: [...A, SchedulerLike]\n): OperatorFunction<T, T | ValueFromArray<A>>;\nexport function startWith<T, A extends readonly unknown[] = T[]>(...values: A): OperatorFunction<T, T | ValueFromArray<A>>;\n\n/**\n * Returns an observable that, at the moment of subscription, will synchronously emit all\n * values provided to this operator, then subscribe to the source and mirror all of its emissions\n * to subscribers.\n *\n * This is a useful way to know when subscription has occurred on an existing observable.\n *\n * <span class=\"informal\">First emits its arguments in order, and then any\n * emissions from the source.</span>\n *\n * ![](startWith.png)\n *\n * ## Examples\n *\n * Emit a value when a timer starts.\n *\n * ```ts\n * import { timer, map, startWith } from 'rxjs';\n *\n * timer(1000)\n *   .pipe(\n *     map(() => 'timer emit'),\n *     startWith('timer start')\n *   )\n *   .subscribe(x => console.log(x));\n *\n * // results:\n * // 'timer start'\n * // 'timer emit'\n * ```\n *\n * @param values Items you want the modified Observable to emit first.\n * @return A function that returns an Observable that synchronously emits\n * provided values before subscribing to the source Observable.\n *\n * @see {@link endWith}\n * @see {@link finalize}\n * @see {@link concat}\n */\nexport function startWith<T, D>(...values: D[]): OperatorFunction<T, T | D> {\n  const scheduler = popScheduler(values);\n  return operate((source, subscriber) => {\n    // Here we can't pass `undefined` as a scheduler, because if we did, the\n    // code inside of `concat` would be confused by the `undefined`, and treat it\n    // like an invalid observable. So we have to split it two different ways.\n    (scheduler ? concat(values, source, scheduler) : concat(values, source)).subscribe(subscriber);\n  });\n}\n", "import { Subscriber } from '../Subscriber';\nimport { ObservableInput, OperatorFunction, ObservedValueOf } from '../types';\nimport { innerFrom } from '../observable/innerFrom';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/* tslint:disable:max-line-length */\nexport function switchMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function switchMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: undefined\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function switchMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: (outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R\n): OperatorFunction<T, R>;\n/* tslint:enable:max-line-length */\n\n/**\n * Projects each source value to an Observable which is merged in the output\n * Observable, emitting values only from the most recently projected Observable.\n *\n * <span class=\"informal\">Maps each value to an Observable, then flattens all of\n * these inner Observables using {@link switchAll}.</span>\n *\n * ![](switchMap.png)\n *\n * Returns an Observable that emits items based on applying a function that you\n * supply to each item emitted by the source Observable, where that function\n * returns an (so-called \"inner\") Observable. Each time it observes one of these\n * inner Observables, the output Observable begins emitting the items emitted by\n * that inner Observable. When a new inner Observable is emitted, `switchMap`\n * stops emitting items from the earlier-emitted inner Observable and begins\n * emitting items from the new one. It continues to behave like this for\n * subsequent inner Observables.\n *\n * ## Example\n *\n * Generate new Observable according to source Observable values\n *\n * ```ts\n * import { of, switchMap } from 'rxjs';\n *\n * const switched = of(1, 2, 3).pipe(switchMap(x => of(x, x ** 2, x ** 3)));\n * switched.subscribe(x => console.log(x));\n * // outputs\n * // 1\n * // 1\n * // 1\n * // 2\n * // 4\n * // 8\n * // 3\n * // 9\n * // 27\n * ```\n *\n * Restart an interval Observable on every click event\n *\n * ```ts\n * import { fromEvent, switchMap, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(switchMap(() => interval(1000)));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link concatMap}\n * @see {@link exhaustMap}\n * @see {@link mergeMap}\n * @see {@link switchAll}\n * @see {@link switchMapTo}\n *\n * @param {function(value: T, index: number): ObservableInput} project A function\n * that, when applied to an item emitted by the source Observable, returns an\n * Observable.\n * @return A function that returns an Observable that emits the result of\n * applying the projection function (and the optional deprecated\n * `resultSelector`) to each item emitted by the source Observable and taking\n * only the values from the most recently projected inner Observable.\n */\nexport function switchMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector?: (outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R\n): OperatorFunction<T, ObservedValueOf<O> | R> {\n  return operate((source, subscriber) => {\n    let innerSubscriber: Subscriber<ObservedValueOf<O>> | null = null;\n    let index = 0;\n    // Whether or not the source subscription has completed\n    let isComplete = false;\n\n    // We only complete the result if the source is complete AND we don't have an active inner subscription.\n    // This is called both when the source completes and when the inners complete.\n    const checkComplete = () => isComplete && !innerSubscriber && subscriber.complete();\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          // Cancel the previous inner subscription if there was one\n          innerSubscriber?.unsubscribe();\n          let innerIndex = 0;\n          const outerIndex = index++;\n          // Start the next inner subscription\n          innerFrom(project(value, outerIndex)).subscribe(\n            (innerSubscriber = createOperatorSubscriber(\n              subscriber,\n              // When we get a new inner value, next it through. Note that this is\n              // handling the deprecate result selector here. This is because with this architecture\n              // it ends up being smaller than using the map operator.\n              (innerValue) => subscriber.next(resultSelector ? resultSelector(value, innerValue, outerIndex, innerIndex++) : innerValue),\n              () => {\n                // The inner has completed. Null out the inner subscriber to\n                // free up memory and to signal that we have no inner subscription\n                // currently.\n                innerSubscriber = null!;\n                checkComplete();\n              }\n            ))\n          );\n        },\n        () => {\n          isComplete = true;\n          checkComplete();\n        }\n      )\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { noop } from '../util/noop';\n\n/**\n * Emits the values emitted by the source Observable until a `notifier`\n * Observable emits a value.\n *\n * <span class=\"informal\">Lets values pass until a second Observable,\n * `notifier`, emits a value. Then, it completes.</span>\n *\n * ![](takeUntil.png)\n *\n * `takeUntil` subscribes and begins mirroring the source Observable. It also\n * monitors a second Observable, `notifier` that you provide. If the `notifier`\n * emits a value, the output Observable stops mirroring the source Observable\n * and completes. If the `notifier` doesn't emit any value and completes\n * then `takeUntil` will pass all values.\n *\n * ## Example\n *\n * Tick every second until the first click happens\n *\n * ```ts\n * import { interval, fromEvent, takeUntil } from 'rxjs';\n *\n * const source = interval(1000);\n * const clicks = fromEvent(document, 'click');\n * const result = source.pipe(takeUntil(clicks));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link take}\n * @see {@link takeLast}\n * @see {@link takeWhile}\n * @see {@link skip}\n *\n * @param {Observable} notifier The Observable whose first emitted value will\n * cause the output Observable of `takeUntil` to stop emitting values from the\n * source Observable.\n * @return A function that returns an Observable that emits the values from the\n * source Observable until `notifier` emits its first value.\n */\nexport function takeUntil<T>(notifier: ObservableInput<any>): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    innerFrom(notifier).subscribe(createOperatorSubscriber(subscriber, () => subscriber.complete(), noop));\n    !subscriber.closed && source.subscribe(subscriber);\n  });\n}\n", "import { OperatorFunction, MonoTypeOperatorFunction, TruthyTypesOf } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\nexport function takeWhile<T>(predicate: BooleanConstructor, inclusive: true): MonoTypeOperatorFunction<T>;\nexport function takeWhile<T>(predicate: BooleanConstructor, inclusive: false): OperatorFunction<T, TruthyTypesOf<T>>;\nexport function takeWhile<T>(predicate: BooleanConstructor): OperatorFunction<T, TruthyTypesOf<T>>;\nexport function takeWhile<T, S extends T>(predicate: (value: T, index: number) => value is S): OperatorFunction<T, S>;\nexport function takeWhile<T, S extends T>(predicate: (value: T, index: number) => value is S, inclusive: false): OperatorFunction<T, S>;\nexport function takeWhile<T>(predicate: (value: T, index: number) => boolean, inclusive?: boolean): MonoTypeOperatorFunction<T>;\n\n/**\n * Emits values emitted by the source Observable so long as each value satisfies\n * the given `predicate`, and then completes as soon as this `predicate` is not\n * satisfied.\n *\n * <span class=\"informal\">Takes values from the source only while they pass the\n * condition given. When the first value does not satisfy, it completes.</span>\n *\n * ![](takeWhile.png)\n *\n * `takeWhile` subscribes and begins mirroring the source Observable. Each value\n * emitted on the source is given to the `predicate` function which returns a\n * boolean, representing a condition to be satisfied by the source values. The\n * output Observable emits the source values until such time as the `predicate`\n * returns false, at which point `takeWhile` stops mirroring the source\n * Observable and completes the output Observable.\n *\n * ## Example\n *\n * Emit click events only while the clientX property is greater than 200\n *\n * ```ts\n * import { fromEvent, takeWhile } from 'rxjs';\n *\n * const clicks = fromEvent<PointerEvent>(document, 'click');\n * const result = clicks.pipe(takeWhile(ev => ev.clientX > 200));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link take}\n * @see {@link takeLast}\n * @see {@link takeUntil}\n * @see {@link skip}\n *\n * @param {function(value: T, index: number): boolean} predicate A function that\n * evaluates a value emitted by the source Observable and returns a boolean.\n * Also takes the (zero-based) index as the second argument.\n * @param {boolean} inclusive When set to `true` the value that caused\n * `predicate` to return `false` will also be emitted.\n * @return A function that returns an Observable that emits values from the\n * source Observable so long as each value satisfies the condition defined by\n * the `predicate`, then completes.\n */\nexport function takeWhile<T>(predicate: (value: T, index: number) => boolean, inclusive = false): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let index = 0;\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value) => {\n        const result = predicate(value, index++);\n        (result || inclusive) && subscriber.next(value);\n        !result && subscriber.complete();\n      })\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction, Observer } from '../types';\nimport { isFunction } from '../util/isFunction';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { identity } from '../util/identity';\n\n/**\n * An extension to the {@link Observer} interface used only by the {@link tap} operator.\n *\n * It provides a useful set of callbacks a user can register to do side-effects in\n * cases other than what the usual {@link Observer} callbacks are\n * ({@link guide/glossary-and-semantics#next next},\n * {@link guide/glossary-and-semantics#error error} and/or\n * {@link guide/glossary-and-semantics#complete complete}).\n *\n * ## Example\n *\n * ```ts\n * import { fromEvent, switchMap, tap, interval, take } from 'rxjs';\n *\n * const source$ = fromEvent(document, 'click');\n * const result$ = source$.pipe(\n *   switchMap((_, i) => i % 2 === 0\n *     ? fromEvent(document, 'mousemove').pipe(\n *         tap({\n *           subscribe: () => console.log('Subscribed to the mouse move events after click #' + i),\n *           unsubscribe: () => console.log('Mouse move events #' + i + ' unsubscribed'),\n *           finalize: () => console.log('Mouse move events #' + i + ' finalized')\n *         })\n *       )\n *     : interval(1_000).pipe(\n *         take(5),\n *         tap({\n *           subscribe: () => console.log('Subscribed to the 1-second interval events after click #' + i),\n *           unsubscribe: () => console.log('1-second interval events #' + i + ' unsubscribed'),\n *           finalize: () => console.log('1-second interval events #' + i + ' finalized')\n *         })\n *       )\n *   )\n * );\n *\n * const subscription = result$.subscribe({\n *   next: console.log\n * });\n *\n * setTimeout(() => {\n *   console.log('Unsubscribe after 60 seconds');\n *   subscription.unsubscribe();\n * }, 60_000);\n * ```\n */\nexport interface TapObserver<T> extends Observer<T> {\n  /**\n   * The callback that `tap` operator invokes at the moment when the source Observable\n   * gets subscribed to.\n   */\n  subscribe: () => void;\n  /**\n   * The callback that `tap` operator invokes when an explicit\n   * {@link guide/glossary-and-semantics#unsubscription unsubscribe} happens. It won't get invoked on\n   * `error` or `complete` events.\n   */\n  unsubscribe: () => void;\n  /**\n   * The callback that `tap` operator invokes when any kind of\n   * {@link guide/glossary-and-semantics#finalization finalization} happens - either when\n   * the source Observable `error`s or `complete`s or when it gets explicitly unsubscribed\n   * by the user. There is no difference in using this callback or the {@link finalize}\n   * operator, but if you're already using `tap` operator, you can use this callback\n   * instead. You'd get the same result in either case.\n   */\n  finalize: () => void;\n}\nexport function tap<T>(observerOrNext?: Partial<TapObserver<T>> | ((value: T) => void)): MonoTypeOperatorFunction<T>;\n/** @deprecated Instead of passing separate callback arguments, use an observer argument. Signatures taking separate callback arguments will be removed in v8. Details: https://rxjs.dev/deprecations/subscribe-arguments */\nexport function tap<T>(\n  next?: ((value: T) => void) | null,\n  error?: ((error: any) => void) | null,\n  complete?: (() => void) | null\n): MonoTypeOperatorFunction<T>;\n\n/**\n * Used to perform side-effects for notifications from the source observable\n *\n * <span class=\"informal\">Used when you want to affect outside state with a notification without altering the notification</span>\n *\n * ![](tap.png)\n *\n * Tap is designed to allow the developer a designated place to perform side effects. While you _could_ perform side-effects\n * inside of a `map` or a `mergeMap`, that would make their mapping functions impure, which isn't always a big deal, but will\n * make it so you can't do things like memoize those functions. The `tap` operator is designed solely for such side-effects to\n * help you remove side-effects from other operations.\n *\n * For any notification, next, error, or complete, `tap` will call the appropriate callback you have provided to it, via a function\n * reference, or a partial observer, then pass that notification down the stream.\n *\n * The observable returned by `tap` is an exact mirror of the source, with one exception: Any error that occurs -- synchronously -- in a handler\n * provided to `tap` will be emitted as an error from the returned observable.\n *\n * > Be careful! You can mutate objects as they pass through the `tap` operator's handlers.\n *\n * The most common use of `tap` is actually for debugging. You can place a `tap(console.log)` anywhere\n * in your observable `pipe`, log out the notifications as they are emitted by the source returned by the previous\n * operation.\n *\n * ## Examples\n *\n * Check a random number before it is handled. Below is an observable that will use a random number between 0 and 1,\n * and emit `'big'` or `'small'` depending on the size of that number. But we wanted to log what the original number\n * was, so we have added a `tap(console.log)`.\n *\n * ```ts\n * import { of, tap, map } from 'rxjs';\n *\n * of(Math.random()).pipe(\n *   tap(console.log),\n *   map(n => n > 0.5 ? 'big' : 'small')\n * ).subscribe(console.log);\n * ```\n *\n * Using `tap` to analyze a value and force an error. Below is an observable where in our system we only\n * want to emit numbers 3 or less we get from another source. We can force our observable to error\n * using `tap`.\n *\n * ```ts\n * import { of, tap } from 'rxjs';\n *\n * const source = of(1, 2, 3, 4, 5);\n *\n * source.pipe(\n *   tap(n => {\n *     if (n > 3) {\n *       throw new TypeError(`Value ${ n } is greater than 3`);\n *     }\n *   })\n * )\n * .subscribe({ next: console.log, error: err => console.log(err.message) });\n * ```\n *\n * We want to know when an observable completes before moving on to the next observable. The system\n * below will emit a random series of `'X'` characters from 3 different observables in sequence. The\n * only way we know when one observable completes and moves to the next one, in this case, is because\n * we have added a `tap` with the side effect of logging to console.\n *\n * ```ts\n * import { of, concatMap, interval, take, map, tap } from 'rxjs';\n *\n * of(1, 2, 3).pipe(\n *   concatMap(n => interval(1000).pipe(\n *     take(Math.round(Math.random() * 10)),\n *     map(() => 'X'),\n *     tap({ complete: () => console.log(`Done with ${ n }`) })\n *   ))\n * )\n * .subscribe(console.log);\n * ```\n *\n * @see {@link finalize}\n * @see {@link TapObserver}\n *\n * @param observerOrNext A next handler or partial observer\n * @param error An error handler\n * @param complete A completion handler\n * @return A function that returns an Observable identical to the source, but\n * runs the specified Observer or callback(s) for each item.\n */\nexport function tap<T>(\n  observerOrNext?: Partial<TapObserver<T>> | ((value: T) => void) | null,\n  error?: ((e: any) => void) | null,\n  complete?: (() => void) | null\n): MonoTypeOperatorFunction<T> {\n  // We have to check to see not only if next is a function,\n  // but if error or complete were passed. This is because someone\n  // could technically call tap like `tap(null, fn)` or `tap(null, null, fn)`.\n  const tapObserver =\n    isFunction(observerOrNext) || error || complete\n      ? // tslint:disable-next-line: no-object-literal-type-assertion\n        ({ next: observerOrNext as Exclude<typeof observerOrNext, Partial<TapObserver<T>>>, error, complete } as Partial<TapObserver<T>>)\n      : observerOrNext;\n\n  return tapObserver\n    ? operate((source, subscriber) => {\n        tapObserver.subscribe?.();\n        let isUnsub = true;\n        source.subscribe(\n          createOperatorSubscriber(\n            subscriber,\n            (value) => {\n              tapObserver.next?.(value);\n              subscriber.next(value);\n            },\n            () => {\n              isUnsub = false;\n              tapObserver.complete?.();\n              subscriber.complete();\n            },\n            (err) => {\n              isUnsub = false;\n              tapObserver.error?.(err);\n              subscriber.error(err);\n            },\n            () => {\n              if (isUnsub) {\n                tapObserver.unsubscribe?.();\n              }\n              tapObserver.finalize?.();\n            }\n          )\n        );\n      })\n    : // Tap was called with no valid tap observer or handler\n      // (e.g. `tap(null, null, null)` or `tap(null)` or `tap()`)\n      // so we're going to just mirror the source.\n      identity;\n}\n", "import { Subscription } from '../Subscription';\n\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\n\n/**\n * An object interface used by {@link throttle} or {@link throttleTime} that ensure\n * configuration options of these operators.\n *\n * @see {@link throttle}\n * @see {@link throttleTime}\n */\nexport interface ThrottleConfig {\n  /**\n   * If `true`, the resulting Observable will emit the first value from the source\n   * Observable at the **start** of the \"throttling\" process (when starting an\n   * internal timer that prevents other emissions from the source to pass through).\n   * If `false`, it will not emit the first value from the source Observable at the\n   * start of the \"throttling\" process.\n   *\n   * If not provided, defaults to: `true`.\n   */\n  leading?: boolean;\n  /**\n   * If `true`, the resulting Observable will emit the last value from the source\n   * Observable at the **end** of the \"throttling\" process (when ending an internal\n   * timer that prevents other emissions from the source to pass through).\n   * If `false`, it will not emit the last value from the source Observable at the\n   * end of the \"throttling\" process.\n   *\n   * If not provided, defaults to: `false`.\n   */\n  trailing?: boolean;\n}\n\n/**\n * Emits a value from the source Observable, then ignores subsequent source\n * values for a duration determined by another Observable, then repeats this\n * process.\n *\n * <span class=\"informal\">It's like {@link throttleTime}, but the silencing\n * duration is determined by a second Observable.</span>\n *\n * ![](throttle.svg)\n *\n * `throttle` emits the source Observable values on the output Observable\n * when its internal timer is disabled, and ignores source values when the timer\n * is enabled. Initially, the timer is disabled. As soon as the first source\n * value arrives, it is forwarded to the output Observable, and then the timer\n * is enabled by calling the `durationSelector` function with the source value,\n * which returns the \"duration\" Observable. When the duration Observable emits a\n * value, the timer is disabled, and this process repeats for the\n * next source value.\n *\n * ## Example\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, throttle, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(throttle(() => interval(1000)));\n *\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link audit}\n * @see {@link debounce}\n * @see {@link delayWhen}\n * @see {@link sample}\n * @see {@link throttleTime}\n *\n * @param durationSelector A function that receives a value from the source\n * Observable, for computing the silencing duration for each source value,\n * returned as an `ObservableInput`.\n * @param config A configuration object to define `leading` and `trailing`\n * behavior. Defaults to `{ leading: true, trailing: false }`.\n * @return A function that returns an Observable that performs the throttle\n * operation to limit the rate of emissions from the source.\n */\nexport function throttle<T>(durationSelector: (value: T) => ObservableInput<any>, config?: ThrottleConfig): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    const { leading = true, trailing = false } = config ?? {};\n    let hasValue = false;\n    let sendValue: T | null = null;\n    let throttled: Subscription | null = null;\n    let isComplete = false;\n\n    const endThrottling = () => {\n      throttled?.unsubscribe();\n      throttled = null;\n      if (trailing) {\n        send();\n        isComplete && subscriber.complete();\n      }\n    };\n\n    const cleanupThrottling = () => {\n      throttled = null;\n      isComplete && subscriber.complete();\n    };\n\n    const startThrottle = (value: T) =>\n      (throttled = innerFrom(durationSelector(value)).subscribe(createOperatorSubscriber(subscriber, endThrottling, cleanupThrottling)));\n\n    const send = () => {\n      if (hasValue) {\n        // Ensure we clear out our value and hasValue flag\n        // before we emit, otherwise reentrant code can cause\n        // issues here.\n        hasValue = false;\n        const value = sendValue!;\n        sendValue = null;\n        // Emit the value.\n        subscriber.next(value);\n        !isComplete && startThrottle(value);\n      }\n    };\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        // Regarding the presence of throttled.closed in the following\n        // conditions, if a synchronous duration selector is specified - weird,\n        // but legal - an already-closed subscription will be assigned to\n        // throttled, so the subscription's closed property needs to be checked,\n        // too.\n        (value) => {\n          hasValue = true;\n          sendValue = value;\n          !(throttled && !throttled.closed) && (leading ? send() : startThrottle(value));\n        },\n        () => {\n          isComplete = true;\n          !(trailing && hasValue && throttled && !throttled.closed) && subscriber.complete();\n        }\n      )\n    );\n  });\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { throttle, ThrottleConfig } from './throttle';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { timer } from '../observable/timer';\n\n/**\n * Emits a value from the source Observable, then ignores subsequent source\n * values for `duration` milliseconds, then repeats this process.\n *\n * <span class=\"informal\">Lets a value pass, then ignores source values for the\n * next `duration` milliseconds.</span>\n *\n * ![](throttleTime.png)\n *\n * `throttleTime` emits the source Observable values on the output Observable\n * when its internal timer is disabled, and ignores source values when the timer\n * is enabled. Initially, the timer is disabled. As soon as the first source\n * value arrives, it is forwarded to the output Observable, and then the timer\n * is enabled. After `duration` milliseconds (or the time unit determined\n * internally by the optional `scheduler`) has passed, the timer is disabled,\n * and this process repeats for the next source value. Optionally takes a\n * {@link SchedulerLike} for managing timers.\n *\n * ## Examples\n *\n * ### Limit click rate\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, throttleTime } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(throttleTime(1000));\n *\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link auditTime}\n * @see {@link debounceTime}\n * @see {@link delay}\n * @see {@link sampleTime}\n * @see {@link throttle}\n *\n * @param duration Time to wait before emitting another value after\n * emitting the last value, measured in milliseconds or the time unit determined\n * internally by the optional `scheduler`.\n * @param scheduler The {@link SchedulerLike} to use for\n * managing the timers that handle the throttling. Defaults to {@link asyncScheduler}.\n * @param config A configuration object to define `leading` and\n * `trailing` behavior. Defaults to `{ leading: true, trailing: false }`.\n * @return A function that returns an Observable that performs the throttle\n * operation to limit the rate of emissions from the source.\n */\nexport function throttleTime<T>(\n  duration: number,\n  scheduler: SchedulerLike = asyncScheduler,\n  config?: ThrottleConfig\n): MonoTypeOperatorFunction<T> {\n  const duration$ = timer(duration, scheduler);\n  return throttle(() => duration$, config);\n}\n", "import { OperatorFunction, ObservableInputTuple } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { identity } from '../util/identity';\nimport { noop } from '../util/noop';\nimport { popResultSelector } from '../util/args';\n\nexport function withLatestFrom<T, O extends unknown[]>(...inputs: [...ObservableInputTuple<O>]): OperatorFunction<T, [T, ...O]>;\n\nexport function withLatestFrom<T, O extends unknown[], R>(\n  ...inputs: [...ObservableInputTuple<O>, (...value: [T, ...O]) => R]\n): OperatorFunction<T, R>;\n\n/**\n * Combines the source Observable with other Observables to create an Observable\n * whose values are calculated from the latest values of each, only when the\n * source emits.\n *\n * <span class=\"informal\">Whenever the source Observable emits a value, it\n * computes a formula using that value plus the latest values from other input\n * Observables, then emits the output of that formula.</span>\n *\n * ![](withLatestFrom.png)\n *\n * `withLatestFrom` combines each value from the source Observable (the\n * instance) with the latest values from the other input Observables only when\n * the source emits a value, optionally using a `project` function to determine\n * the value to be emitted on the output Observable. All input Observables must\n * emit at least one value before the output Observable will emit a value.\n *\n * ## Example\n *\n * On every click event, emit an array with the latest timer event plus the click event\n *\n * ```ts\n * import { fromEvent, interval, withLatestFrom } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const timer = interval(1000);\n * const result = clicks.pipe(withLatestFrom(timer));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link combineLatest}\n *\n * @param {ObservableInput} other An input Observable to combine with the source\n * Observable. More than one input Observables may be given as argument.\n * @param {Function} [project] Projection function for combining values\n * together. Receives all values in order of the Observables passed, where the\n * first parameter is a value from the source Observable. (e.g.\n * `a.pipe(withLatestFrom(b, c), map(([a1, b1, c1]) => a1 + b1 + c1))`). If this is not\n * passed, arrays will be emitted on the output Observable.\n * @return A function that returns an Observable of projected values from the\n * most recent values from each input Observable, or an array of the most\n * recent values from each input Observable.\n */\nexport function withLatestFrom<T, R>(...inputs: any[]): OperatorFunction<T, R | any[]> {\n  const project = popResultSelector(inputs) as ((...args: any[]) => R) | undefined;\n\n  return operate((source, subscriber) => {\n    const len = inputs.length;\n    const otherValues = new Array(len);\n    // An array of whether or not the other sources have emitted. Matched with them by index.\n    // TODO: At somepoint, we should investigate the performance implications here, and look\n    // into using a `Set()` and checking the `size` to see if we're ready.\n    let hasValue = inputs.map(() => false);\n    // Flipped true when we have at least one value from all other sources and\n    // we are ready to start emitting values.\n    let ready = false;\n\n    // Other sources. Note that here we are not checking `subscriber.closed`,\n    // this causes all inputs to be subscribed to, even if nothing can be emitted\n    // from them. This is an important distinction because subscription constitutes\n    // a side-effect.\n    for (let i = 0; i < len; i++) {\n      innerFrom(inputs[i]).subscribe(\n        createOperatorSubscriber(\n          subscriber,\n          (value) => {\n            otherValues[i] = value;\n            if (!ready && !hasValue[i]) {\n              // If we're not ready yet, flag to show this observable has emitted.\n              hasValue[i] = true;\n              // Intentionally terse code.\n              // If all of our other observables have emitted, set `ready` to `true`,\n              // so we know we can start emitting values, then clean up the `hasValue` array,\n              // because we don't need it anymore.\n              (ready = hasValue.every(identity)) && (hasValue = null!);\n            }\n          },\n          // Completing one of the other sources has\n          // no bearing on the completion of our result.\n          noop\n        )\n      );\n    }\n\n    // Source subscription\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value) => {\n        if (ready) {\n          // We have at least one value from the other sources. Go ahead and emit.\n          const values = [value, ...otherValues];\n          subscriber.next(project ? project(...values) : values);\n        }\n      })\n    );\n  });\n}\n", "import { zip as zipStatic } from '../observable/zip';\nimport { ObservableInput, ObservableInputTuple, OperatorFunction, Cons } from '../types';\nimport { operate } from '../util/lift';\n\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[]>(otherInputs: [...ObservableInputTuple<A>]): OperatorFunction<T, Cons<T, A>>;\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[], R>(\n  otherInputsAndProject: [...ObservableInputTuple<A>],\n  project: (...values: Cons<T, A>) => R\n): OperatorFunction<T, R>;\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[]>(...otherInputs: [...ObservableInputTuple<A>]): OperatorFunction<T, Cons<T, A>>;\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[], R>(\n  ...otherInputsAndProject: [...ObservableInputTuple<A>, (...values: Cons<T, A>) => R]\n): OperatorFunction<T, R>;\n\n/**\n * @deprecated Replaced with {@link zipWith}. Will be removed in v8.\n */\nexport function zip<T, R>(...sources: Array<ObservableInput<any> | ((...values: Array<any>) => R)>): OperatorFunction<T, any> {\n  return operate((source, subscriber) => {\n    zipStatic(source as ObservableInput<any>, ...(sources as Array<ObservableInput<any>>)).subscribe(subscriber);\n  });\n}\n", "import { ObservableInputTuple, OperatorFunction, Cons } from '../types';\nimport { zip } from './zip';\n\n/**\n * Subscribes to the source, and the observable inputs provided as arguments, and combines their values, by index, into arrays.\n *\n * What is meant by \"combine by index\": The first value from each will be made into a single array, then emitted,\n * then the second value from each will be combined into a single array and emitted, then the third value\n * from each will be combined into a single array and emitted, and so on.\n *\n * This will continue until it is no longer able to combine values of the same index into an array.\n *\n * After the last value from any one completed source is emitted in an array, the resulting observable will complete,\n * as there is no way to continue \"zipping\" values together by index.\n *\n * Use-cases for this operator are limited. There are memory concerns if one of the streams is emitting\n * values at a much faster rate than the others. Usage should likely be limited to streams that emit\n * at a similar pace, or finite streams of known length.\n *\n * In many cases, authors want `combineLatestWith` and not `zipWith`.\n *\n * @param otherInputs other observable inputs to collate values from.\n * @return A function that returns an Observable that emits items by index\n * combined from the source Observable and provided Observables, in form of an\n * array.\n */\nexport function zipWith<T, A extends readonly unknown[]>(...otherInputs: [...ObservableInputTuple<A>]): OperatorFunction<T, Cons<T, A>> {\n  return zip(...otherInputs);\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  ReplaySubject,\n  Subject,\n  fromEvent\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch document\n *\n * Documents are implemented as subjects, so all downstream observables are\n * automatically updated when a new document is emitted.\n *\n * @returns Document subject\n */\nexport function watchDocument(): Subject<Document> {\n  const document$ = new ReplaySubject<Document>(1)\n  fromEvent(document, \"DOMContentLoaded\", { once: true })\n    .subscribe(() => document$.next(document))\n\n  /* Return document */\n  return document$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve all elements matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getElements<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T][]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T[]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T[] {\n  return Array.from(node.querySelectorAll<T>(selector))\n}\n\n/**\n * Retrieve an element matching a query selector or throw a reference error\n *\n * Note that this function assumes that the element is present. If unsure if an\n * element is existent, use the `getOptionalElement` function instead.\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T]\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T {\n  const el = getOptionalElement<T>(selector, node)\n  if (typeof el === \"undefined\")\n    throw new ReferenceError(\n      `Missing element: expected \"${selector}\" to be present`\n    )\n\n  /* Return element */\n  return el\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Retrieve an optional element matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element or nothing\n */\nexport function getOptionalElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T] | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T | undefined {\n  return node.querySelector<T>(selector) || undefined\n}\n\n/**\n * Retrieve the currently active element\n *\n * @returns Element or nothing\n */\nexport function getActiveElement(): HTMLElement | undefined {\n  return document.activeElement instanceof HTMLElement\n    ? document.activeElement || undefined\n    : undefined\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  debounceTime,\n  distinctUntilChanged,\n  fromEvent,\n  map,\n  merge,\n  shareReplay,\n  startWith\n} from \"rxjs\"\n\nimport { getActiveElement } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Focus observable\n *\n * Previously, this observer used `focus` and `blur` events to determine whether\n * an element is focused, but this doesn't work if there are focusable elements\n * within the elements itself. A better solutions are `focusin` and `focusout`\n * events, which bubble up the tree and allow for more fine-grained control.\n *\n * `debounceTime` is necessary, because when a focus change happens inside an\n * element, the observable would first emit `false` and then `true` again.\n */\nconst observer$ = merge(\n  fromEvent(document.body, \"focusin\"),\n  fromEvent(document.body, \"focusout\")\n)\n  .pipe(\n    debounceTime(1),\n    startWith(undefined),\n    map(() => getActiveElement() || document.body),\n    shareReplay(1)\n  )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch element focus\n *\n * @param el - Element\n *\n * @returns Element focus observable\n */\nexport function watchElementFocus(\n  el: HTMLElement\n): Observable<boolean> {\n  return observer$\n    .pipe(\n      map(active => el.contains(active)),\n      distinctUntilChanged()\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  animationFrameScheduler,\n  auditTime,\n  fromEvent,\n  map,\n  merge,\n  startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Element offset\n */\nexport interface ElementOffset {\n  x: number                            /* Horizontal offset */\n  y: number                            /* Vertical offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element offset\n *\n * @param el - Element\n *\n * @returns Element offset\n */\nexport function getElementOffset(\n  el: HTMLElement\n): ElementOffset {\n  return {\n    x: el.offsetLeft,\n    y: el.offsetTop\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element offset\n *\n * @param el - Element\n *\n * @returns Element offset observable\n */\nexport function watchElementOffset(\n  el: HTMLElement\n): Observable<ElementOffset> {\n  return merge(\n    fromEvent(window, \"load\"),\n    fromEvent(window, \"resize\")\n  )\n    .pipe(\n      auditTime(0, animationFrameScheduler),\n      map(() => getElementOffset(el)),\n      startWith(getElementOffset(el))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  animationFrameScheduler,\n  auditTime,\n  fromEvent,\n  map,\n  merge,\n  startWith\n} from \"rxjs\"\n\nimport { ElementOffset } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element content offset (= scroll offset)\n *\n * @param el - Element\n *\n * @returns Element content offset\n */\nexport function getElementContentOffset(\n  el: HTMLElement\n): ElementOffset {\n  return {\n    x: el.scrollLeft,\n    y: el.scrollTop\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element content offset\n *\n * @param el - Element\n *\n * @returns Element content offset observable\n */\nexport function watchElementContentOffset(\n  el: HTMLElement\n): Observable<ElementOffset> {\n  return merge(\n    fromEvent(el, \"scroll\"),\n    fromEvent(window, \"resize\")\n  )\n    .pipe(\n      auditTime(0, animationFrameScheduler),\n      map(() => getElementContentOffset(el)),\n      startWith(getElementContentOffset(el))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { JSX as JSXInternal } from \"preact\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * HTML attributes\n */\ntype Attributes =\n  & JSXInternal.HTMLAttributes\n  & JSXInternal.SVGAttributes\n  & Record<string, any>\n\n/**\n * Child element\n */\ntype Child =\n  | ChildNode\n  | HTMLElement\n  | Text\n  | string\n  | number\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Append a child node to an element\n *\n * @param el - Element\n * @param child - Child node(s)\n */\nfunction appendChild(el: HTMLElement, child: Child | Child[]): void {\n\n  /* Handle primitive types (including raw HTML) */\n  if (typeof child === \"string\" || typeof child === \"number\") {\n    el.innerHTML += child.toString()\n\n  /* Handle nodes */\n  } else if (child instanceof Node) {\n    el.appendChild(child)\n\n  /* Handle nested children */\n  } else if (Array.isArray(child)) {\n    for (const node of child)\n      appendChild(el, node)\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * JSX factory\n *\n * @template T - Element type\n *\n * @param tag - HTML tag\n * @param attributes - HTML attributes\n * @param children - Child elements\n *\n * @returns Element\n */\nexport function h<T extends keyof HTMLElementTagNameMap>(\n  tag: T, attributes?: Attributes | null, ...children: Child[]\n): HTMLElementTagNameMap[T]\n\nexport function h<T extends h.JSX.Element>(\n  tag: string, attributes?: Attributes | null, ...children: Child[]\n): T\n\nexport function h<T extends h.JSX.Element>(\n  tag: string, attributes?: Attributes | null, ...children: Child[]\n): T {\n  const el = document.createElement(tag)\n\n  /* Set attributes, if any */\n  if (attributes)\n    for (const attr of Object.keys(attributes)) {\n      if (typeof attributes[attr] === \"undefined\")\n        continue\n\n      /* Set default attribute or boolean */\n      if (typeof attributes[attr] !== \"boolean\")\n        el.setAttribute(attr, attributes[attr])\n      else\n        el.setAttribute(attr, \"\")\n    }\n\n  /* Append child nodes */\n  for (const child of children)\n    appendChild(el, child)\n\n  /* Return element */\n  return el as T\n}\n\n/* ----------------------------------------------------------------------------\n * Namespace\n * ------------------------------------------------------------------------- */\n\nexport declare namespace h {\n  namespace JSX {\n    type Element = HTMLElement\n    type IntrinsicElements = JSXInternal.IntrinsicElements\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Round a number for display with repository facts\n *\n * This is a reverse-engineered version of GitHub's weird rounding algorithm\n * for stars, forks and all other numbers. While all numbers below `1,000` are\n * returned as-is, bigger numbers are converted to fixed numbers:\n *\n * - `1,049` => `1k`\n * - `1,050` => `1.1k`\n * - `1,949` => `1.9k`\n * - `1,950` => `2k`\n *\n * @param value - Original value\n *\n * @returns Rounded value\n */\nexport function round(value: number): string {\n  if (value > 999) {\n    const digits = +((value - 950) % 1000 > 99)\n    return `${((value + 0.000001) / 1000).toFixed(digits)}k`\n  } else {\n    return value.toString()\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  defer,\n  finalize,\n  fromEvent,\n  map,\n  merge,\n  switchMap,\n  take,\n  throwError\n} from \"rxjs\"\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create and load a `script` element\n *\n * This function returns an observable that will emit when the script was\n * successfully loaded, or throw an error if it wasn't.\n *\n * @param src - Script URL\n *\n * @returns Script observable\n */\nexport function watchScript(src: string): Observable<void> {\n  const script = h(\"script\", { src })\n  return defer(() => {\n    document.head.appendChild(script)\n    return merge(\n      fromEvent(script, \"load\"),\n      fromEvent(script, \"error\")\n        .pipe(\n          switchMap(() => (\n            throwError(() => new ReferenceError(`Invalid script: ${src}`))\n          ))\n        )\n    )\n      .pipe(\n        map(() => undefined),\n        finalize(() => document.head.removeChild(script)),\n        take(1)\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  NEVER,\n  Observable,\n  Subject,\n  defer,\n  filter,\n  finalize,\n  map,\n  merge,\n  of,\n  shareReplay,\n  startWith,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { watchScript } from \"../../../script\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Element offset\n */\nexport interface ElementSize {\n  width: number                        /* Element width */\n  height: number                       /* Element height */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Resize observer entry subject\n */\nconst entry$ = new Subject<ResizeObserverEntry>()\n\n/**\n * Resize observer observable\n *\n * This observable will create a `ResizeObserver` on the first subscription\n * and will automatically terminate it when there are no more subscribers.\n * It's quite important to centralize observation in a single `ResizeObserver`,\n * as the performance difference can be quite dramatic, as the link shows.\n *\n * If the browser doesn't have a `ResizeObserver` implementation available, a\n * polyfill is automatically downloaded from unpkg.com. This is also compatible\n * with the built-in privacy plugin, which will download the polyfill and put\n * it alongside the built site for self-hosting.\n *\n * @see https://bit.ly/3iIYfEm - Google Groups on performance\n */\nconst observer$ = defer(() => (\n  typeof ResizeObserver === \"undefined\"\n    ? watchScript(\"https://unpkg.com/resize-observer-polyfill\")\n    : of(undefined)\n))\n  .pipe(\n    map(() => new ResizeObserver(entries => {\n      for (const entry of entries)\n        entry$.next(entry)\n    })),\n    switchMap(observer => merge(NEVER, of(observer))\n      .pipe(\n        finalize(() => observer.disconnect())\n      )\n    ),\n    shareReplay(1)\n  )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element size\n *\n * @param el - Element\n *\n * @returns Element size\n */\nexport function getElementSize(\n  el: HTMLElement\n): ElementSize {\n  return {\n    width:  el.offsetWidth,\n    height: el.offsetHeight\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element size\n *\n * This function returns an observable that subscribes to a single internal\n * instance of `ResizeObserver` upon subscription, and emit resize events until\n * termination. Note that this function should not be called with the same\n * element twice, as the first unsubscription will terminate observation.\n *\n * Sadly, we can't use the `DOMRect` objects returned by the observer, because\n * we need the emitted values to be consistent with `getElementSize`, which will\n * return the used values (rounded) and not actual values (unrounded). Thus, we\n * use the `offset*` properties. See the linked GitHub issue.\n *\n * @see https://bit.ly/3m0k3he - GitHub issue\n *\n * @param el - Element\n *\n * @returns Element size observable\n */\nexport function watchElementSize(\n  el: HTMLElement\n): Observable<ElementSize> {\n  return observer$\n    .pipe(\n      tap(observer => observer.observe(el)),\n      switchMap(observer => entry$\n        .pipe(\n          filter(({ target }) => target === el),\n          finalize(() => observer.unobserve(el)),\n          map(() => getElementSize(el))\n        )\n      ),\n      startWith(getElementSize(el))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ElementSize } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element content size (= scroll width and height)\n *\n * @param el - Element\n *\n * @returns Element content size\n */\nexport function getElementContentSize(\n  el: HTMLElement\n): ElementSize {\n  return {\n    width:  el.scrollWidth,\n    height: el.scrollHeight\n  }\n}\n\n/**\n * Retrieve the overflowing container of an element, if any\n *\n * @param el - Element\n *\n * @returns Overflowing container or nothing\n */\nexport function getElementContainer(\n  el: HTMLElement\n): HTMLElement | undefined {\n  let parent = el.parentElement\n  while (parent)\n    if (\n      el.scrollWidth  <= parent.scrollWidth &&\n      el.scrollHeight <= parent.scrollHeight\n    )\n      parent = (el = parent).parentElement\n    else\n      break\n\n  /* Return overflowing container */\n  return parent ? el : undefined\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  NEVER,\n  Observable,\n  Subject,\n  defer,\n  distinctUntilChanged,\n  filter,\n  finalize,\n  map,\n  merge,\n  of,\n  shareReplay,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport {\n  getElementContentSize,\n  getElementSize,\n  watchElementContentOffset\n} from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Intersection observer entry subject\n */\nconst entry$ = new Subject<IntersectionObserverEntry>()\n\n/**\n * Intersection observer observable\n *\n * This observable will create an `IntersectionObserver` on first subscription\n * and will automatically terminate it when there are no more subscribers.\n *\n * @see https://bit.ly/3iIYfEm - Google Groups on performance\n */\nconst observer$ = defer(() => of(\n  new IntersectionObserver(entries => {\n    for (const entry of entries)\n      entry$.next(entry)\n  }, {\n    threshold: 0\n  })\n))\n  .pipe(\n    switchMap(observer => merge(NEVER, of(observer))\n      .pipe(\n        finalize(() => observer.disconnect())\n      )\n    ),\n    shareReplay(1)\n  )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch element visibility\n *\n * @param el - Element\n *\n * @returns Element visibility observable\n */\nexport function watchElementVisibility(\n  el: HTMLElement\n): Observable<boolean> {\n  return observer$\n    .pipe(\n      tap(observer => observer.observe(el)),\n      switchMap(observer => entry$\n        .pipe(\n          filter(({ target }) => target === el),\n          finalize(() => observer.unobserve(el)),\n          map(({ isIntersecting }) => isIntersecting)\n        )\n      )\n    )\n}\n\n/**\n * Watch element boundary\n *\n * This function returns an observable which emits whether the bottom content\n * boundary (= scroll offset) of an element is within a certain threshold.\n *\n * @param el - Element\n * @param threshold - Threshold\n *\n * @returns Element boundary observable\n */\nexport function watchElementBoundary(\n  el: HTMLElement, threshold = 16\n): Observable<boolean> {\n  return watchElementContentOffset(el)\n    .pipe(\n      map(({ y }) => {\n        const visible = getElementSize(el)\n        const content = getElementContentSize(el)\n        return y >= (\n          content.height - visible.height - threshold\n        )\n      }),\n      distinctUntilChanged()\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  startWith\n} from \"rxjs\"\n\nimport { getElement } from \"../element\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Toggle\n */\nexport type Toggle =\n  | \"drawer\"                           /* Toggle for drawer */\n  | \"search\"                           /* Toggle for search */\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Toggle map\n */\nconst toggles: Record<Toggle, HTMLInputElement> = {\n  drawer: getElement(\"[data-md-toggle=drawer]\"),\n  search: getElement(\"[data-md-toggle=search]\")\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve the value of a toggle\n *\n * @param name - Toggle\n *\n * @returns Toggle value\n */\nexport function getToggle(name: Toggle): boolean {\n  return toggles[name].checked\n}\n\n/**\n * Set toggle\n *\n * Simulating a click event seems to be the most cross-browser compatible way\n * of changing the value while also emitting a `change` event. Before, Material\n * used `CustomEvent` to programmatically change the value of a toggle, but this\n * is a much simpler and cleaner solution which doesn't require a polyfill.\n *\n * @param name - Toggle\n * @param value - Toggle value\n */\nexport function setToggle(name: Toggle, value: boolean): void {\n  if (toggles[name].checked !== value)\n    toggles[name].click()\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch toggle\n *\n * @param name - Toggle\n *\n * @returns Toggle value observable\n */\nexport function watchToggle(name: Toggle): Observable<boolean> {\n  const el = toggles[name]\n  return fromEvent(el, \"change\")\n    .pipe(\n      map(() => el.checked),\n      startWith(el.checked)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  filter,\n  fromEvent,\n  map,\n  merge,\n  share,\n  startWith,\n  switchMap\n} from \"rxjs\"\n\nimport { getActiveElement } from \"../element\"\nimport { getToggle } from \"../toggle\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Keyboard mode\n */\nexport type KeyboardMode =\n  | \"global\"                           /* Global */\n  | \"search\"                           /* Search is open */\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Keyboard\n */\nexport interface Keyboard {\n  mode: KeyboardMode                   /* Keyboard mode */\n  type: string                         /* Key type */\n  claim(): void                        /* Key claim */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Check whether an element may receive keyboard input\n *\n * @param el - Element\n * @param type - Key type\n *\n * @returns Test result\n */\nfunction isSusceptibleToKeyboard(\n  el: HTMLElement, type: string\n): boolean {\n  switch (el.constructor) {\n\n    /* Input elements */\n    case HTMLInputElement:\n      /* @ts-expect-error - omit unnecessary type cast */\n      if (el.type === \"radio\")\n        return /^Arrow/.test(type)\n      else\n        return true\n\n    /* Select element and textarea */\n    case HTMLSelectElement:\n    case HTMLTextAreaElement:\n      return true\n\n    /* Everything else */\n    default:\n      return el.isContentEditable\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch composition events\n *\n * @returns Composition observable\n */\nexport function watchComposition(): Observable<boolean> {\n  return merge(\n    fromEvent(window, \"compositionstart\").pipe(map(() => true)),\n    fromEvent(window, \"compositionend\").pipe(map(() => false))\n  )\n    .pipe(\n      startWith(false)\n    )\n}\n\n/**\n * Watch keyboard\n *\n * @returns Keyboard observable\n */\nexport function watchKeyboard(): Observable<Keyboard> {\n  const keyboard$ = fromEvent<KeyboardEvent>(window, \"keydown\")\n    .pipe(\n      filter(ev => !(ev.metaKey || ev.ctrlKey)),\n      map(ev => ({\n        mode: getToggle(\"search\") ? \"search\" : \"global\",\n        type: ev.key,\n        claim() {\n          ev.preventDefault()\n          ev.stopPropagation()\n        }\n      } as Keyboard)),\n      filter(({ mode, type }) => {\n        if (mode === \"global\") {\n          const active = getActiveElement()\n          if (typeof active !== \"undefined\")\n            return !isSusceptibleToKeyboard(active, type)\n        }\n        return true\n      }),\n      share()\n    )\n\n  /* Don't emit during composition events - see https://bit.ly/3te3Wl8 */\n  return watchComposition()\n    .pipe(\n      switchMap(active => !active ? keyboard$ : EMPTY)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Subject } from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve location\n *\n * This function returns a `URL` object (and not `Location`) to normalize the\n * typings across the application. Furthermore, locations need to be tracked\n * without setting them and `Location` is a singleton which represents the\n * current location.\n *\n * @returns URL\n */\nexport function getLocation(): URL {\n  return new URL(location.href)\n}\n\n/**\n * Set location\n *\n * @param url - URL to change to\n */\nexport function setLocation(url: URL | HTMLLinkElement): void {\n  location.href = url.href\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch location\n *\n * @returns Location subject\n */\nexport function watchLocation(): Subject<URL> {\n  return new Subject<URL>()\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  filter,\n  fromEvent,\n  map,\n  merge,\n  shareReplay,\n  startWith\n} from \"rxjs\"\n\nimport { getOptionalElement } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve location hash\n *\n * @returns Location hash\n */\nexport function getLocationHash(): string {\n  return location.hash.slice(1)\n}\n\n/**\n * Set location hash\n *\n * Setting a new fragment identifier via `location.hash` will have no effect\n * if the value doesn't change. When a new fragment identifier is set, we want\n * the browser to target the respective element at all times, which is why we\n * use this dirty little trick.\n *\n * @param hash - Location hash\n */\nexport function setLocationHash(hash: string): void {\n  const el = h(\"a\", { href: hash })\n  el.addEventListener(\"click\", ev => ev.stopPropagation())\n  el.click()\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch location hash\n *\n * @param location$ - Location observable\n *\n * @returns Location hash observable\n */\nexport function watchLocationHash(\n  location$: Observable<URL>\n): Observable<string> {\n  return merge(\n    fromEvent<HashChangeEvent>(window, \"hashchange\"),\n    location$\n  )\n    .pipe(\n      map(getLocationHash),\n      startWith(getLocationHash()),\n      filter(hash => hash.length > 0),\n      shareReplay(1)\n    )\n}\n\n/**\n * Watch location target\n *\n * @param location$ - Location observable\n *\n * @returns Location target observable\n */\nexport function watchLocationTarget(\n  location$: Observable<URL>\n): Observable<HTMLElement> {\n  return watchLocationHash(location$)\n    .pipe(\n      map(id => getOptionalElement(`[id=\"${id}\"]`)!),\n      filter(el => typeof el !== \"undefined\")\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  fromEvent,\n  fromEventPattern,\n  map,\n  merge,\n  startWith,\n  switchMap\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch media query\n *\n * Note that although `MediaQueryList.addListener` is deprecated we have to\n * use it, because it's the only way to ensure proper downward compatibility.\n *\n * @see https://bit.ly/3dUBH2m - GitHub issue\n *\n * @param query - Media query\n *\n * @returns Media observable\n */\nexport function watchMedia(query: string): Observable<boolean> {\n  const media = matchMedia(query)\n  return fromEventPattern<boolean>(next => (\n    media.addListener(() => next(media.matches))\n  ))\n    .pipe(\n      startWith(media.matches)\n    )\n}\n\n/**\n * Watch print mode\n *\n * @returns Print observable\n */\nexport function watchPrint(): Observable<boolean> {\n  const media = matchMedia(\"print\")\n  return merge(\n    fromEvent(window, \"beforeprint\").pipe(map(() => true)),\n    fromEvent(window, \"afterprint\").pipe(map(() => false))\n  )\n    .pipe(\n      startWith(media.matches)\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Toggle an observable with a media observable\n *\n * @template T - Data type\n *\n * @param query$ - Media observable\n * @param factory - Observable factory\n *\n * @returns Toggled observable\n */\nexport function at<T>(\n  query$: Observable<boolean>, factory: () => Observable<T>\n): Observable<T> {\n  return query$\n    .pipe(\n      switchMap(active => active ? factory() : EMPTY)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  catchError,\n  from,\n  map,\n  of,\n  shareReplay,\n  switchMap,\n  throwError\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch the given URL\n *\n * If the request fails (e.g. when dispatched from `file://` locations), the\n * observable will complete without emitting a value.\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Response observable\n */\nexport function request(\n  url: URL | string, options: RequestInit = { credentials: \"same-origin\" }\n): Observable<Response> {\n  return from(fetch(`${url}`, options))\n    .pipe(\n      catchError(() => EMPTY),\n      switchMap(res => res.status !== 200\n        ? throwError(() => new Error(res.statusText))\n        : of(res)\n      )\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Fetch JSON from the given URL\n *\n * @template T - Data type\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Data observable\n */\nexport function requestJSON<T>(\n  url: URL | string, options?: RequestInit\n): Observable<T> {\n  return request(url, options)\n    .pipe(\n      switchMap(res => res.json()),\n      shareReplay(1)\n    )\n}\n\n/**\n * Fetch XML from the given URL\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Data observable\n */\nexport function requestXML(\n  url: URL | string, options?: RequestInit\n): Observable<Document> {\n  const dom = new DOMParser()\n  return request(url, options)\n    .pipe(\n      switchMap(res => res.text()),\n      map(res => dom.parseFromString(res, \"text/xml\")),\n      shareReplay(1)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  merge,\n  startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport offset\n */\nexport interface ViewportOffset {\n  x: number                            /* Horizontal offset */\n  y: number                            /* Vertical offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve viewport offset\n *\n * On iOS Safari, viewport offset can be negative due to overflow scrolling.\n * As this may induce strange behaviors downstream, we'll just limit it to 0.\n *\n * @returns Viewport offset\n */\nexport function getViewportOffset(): ViewportOffset {\n  return {\n    x: Math.max(0, scrollX),\n    y: Math.max(0, scrollY)\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport offset\n *\n * @returns Viewport offset observable\n */\nexport function watchViewportOffset(): Observable<ViewportOffset> {\n  return merge(\n    fromEvent(window, \"scroll\", { passive: true }),\n    fromEvent(window, \"resize\", { passive: true })\n  )\n    .pipe(\n      map(getViewportOffset),\n      startWith(getViewportOffset())\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport size\n */\nexport interface ViewportSize {\n  width: number                        /* Viewport width */\n  height: number                       /* Viewport height */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve viewport size\n *\n * @returns Viewport size\n */\nexport function getViewportSize(): ViewportSize {\n  return {\n    width:  innerWidth,\n    height: innerHeight\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport size\n *\n * @returns Viewport size observable\n */\nexport function watchViewportSize(): Observable<ViewportSize> {\n  return fromEvent(window, \"resize\", { passive: true })\n    .pipe(\n      map(getViewportSize),\n      startWith(getViewportSize())\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  map,\n  shareReplay\n} from \"rxjs\"\n\nimport {\n  ViewportOffset,\n  watchViewportOffset\n} from \"../offset\"\nimport {\n  ViewportSize,\n  watchViewportSize\n} from \"../size\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport\n */\nexport interface Viewport {\n  offset: ViewportOffset               /* Viewport offset */\n  size: ViewportSize                   /* Viewport size */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport\n *\n * @returns Viewport observable\n */\nexport function watchViewport(): Observable<Viewport> {\n  return combineLatest([\n    watchViewportOffset(),\n    watchViewportSize()\n  ])\n    .pipe(\n      map(([offset, size]) => ({ offset, size })),\n      shareReplay(1)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  distinctUntilKeyChanged,\n  map\n} from \"rxjs\"\n\nimport { Header } from \"~/components\"\n\nimport { getElementOffset } from \"../../element\"\nimport { Viewport } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport relative to element\n *\n * @param el - Element\n * @param options - Options\n *\n * @returns Viewport observable\n */\nexport function watchViewportAt(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<Viewport> {\n  const size$ = viewport$\n    .pipe(\n      distinctUntilKeyChanged(\"size\")\n    )\n\n  /* Compute element offset */\n  const offset$ = combineLatest([size$, header$])\n    .pipe(\n      map(() => getElementOffset(el))\n    )\n\n  /* Compute relative viewport, return hot observable */\n  return combineLatest([header$, viewport$, offset$])\n    .pipe(\n      map(([{ height }, { offset, size }, { x, y }]) => ({\n        offset: {\n          x: offset.x - x,\n          y: offset.y - y + height\n        },\n        size\n      }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  endWith,\n  fromEvent,\n  ignoreElements,\n  mergeWith,\n  share,\n  takeUntil\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Worker message\n */\nexport interface WorkerMessage {\n  type: unknown                        /* Message type */\n  data?: unknown                       /* Message data */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create an observable for receiving from a web worker\n *\n * @template T - Data type\n *\n * @param worker - Web worker\n *\n * @returns Message observable\n */\nfunction recv<T>(worker: Worker): Observable<T> {\n  return fromEvent<MessageEvent<T>, T>(worker, \"message\", ev => ev.data)\n}\n\n/**\n * Create a subject for sending to a web worker\n *\n * @template T - Data type\n *\n * @param worker - Web worker\n *\n * @returns Message subject\n */\nfunction send<T>(worker: Worker): Subject<T> {\n  const send$ = new Subject<T>()\n  send$.subscribe(data => worker.postMessage(data))\n\n  /* Return message subject */\n  return send$\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a bidirectional communication channel to a web worker\n *\n * @template T - Data type\n *\n * @param url - Worker URL\n * @param worker - Worker\n *\n * @returns Worker subject\n */\nexport function watchWorker<T extends WorkerMessage>(\n  url: string, worker = new Worker(url)\n): Subject<T> {\n  const recv$ = recv<T>(worker)\n  const send$ = send<T>(worker)\n\n  /* Create worker subject and forward messages */\n  const worker$ = new Subject<T>()\n  worker$.subscribe(send$)\n\n  /* Return worker subject */\n  const done$ = send$.pipe(ignoreElements(), endWith(true))\n  return worker$\n    .pipe(\n      ignoreElements(),\n      mergeWith(recv$.pipe(takeUntil(done$))),\n      share()\n    ) as Subject<T>\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { getElement, getLocation } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Feature flag\n */\nexport type Flag =\n  | \"announce.dismiss\"                 /* Dismissable announcement bar */\n  | \"content.code.annotate\"            /* Code annotations */\n  | \"content.code.copy\"                /* Code copy button */\n  | \"content.lazy\"                     /* Lazy content elements */\n  | \"content.tabs.link\"                /* Link content tabs */\n  | \"header.autohide\"                  /* Hide header */\n  | \"navigation.expand\"                /* Automatic expansion */\n  | \"navigation.indexes\"               /* Section pages */\n  | \"navigation.instant\"               /* Instant loading */\n  | \"navigation.sections\"              /* Section navigation */\n  | \"navigation.tabs\"                  /* Tabs navigation */\n  | \"navigation.tabs.sticky\"           /* Tabs navigation (sticky) */\n  | \"navigation.top\"                   /* Back-to-top button */\n  | \"navigation.tracking\"              /* Anchor tracking */\n  | \"search.highlight\"                 /* Search highlighting */\n  | \"search.share\"                     /* Search sharing */\n  | \"search.suggest\"                   /* Search suggestions */\n  | \"toc.follow\"                       /* Following table of contents */\n  | \"toc.integrate\"                    /* Integrated table of contents */\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Translation\n */\nexport type Translation =\n  | \"clipboard.copy\"                   /* Copy to clipboard */\n  | \"clipboard.copied\"                 /* Copied to clipboard */\n  | \"search.result.placeholder\"        /* Type to start searching */\n  | \"search.result.none\"               /* No matching documents */\n  | \"search.result.one\"                /* 1 matching document */\n  | \"search.result.other\"              /* # matching documents */\n  | \"search.result.more.one\"           /* 1 more on this page */\n  | \"search.result.more.other\"         /* # more on this page */\n  | \"search.result.term.missing\"       /* Missing */\n  | \"select.version\"                   /* Version selector */\n\n/**\n * Translations\n */\nexport type Translations =\n  Record<Translation, string>\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Versioning\n */\nexport interface Versioning {\n  provider: \"mike\"                     /* Version provider */\n  default?: string | string[]          /* Default version */\n}\n\n/**\n * Configuration\n */\nexport interface Config {\n  base: string                         /* Base URL */\n  features: Flag[]                     /* Feature flags */\n  translations: Translations           /* Translations */\n  search: string                       /* Search worker URL */\n  tags?: Record<string, string>        /* Tags mapping */\n  version?: Versioning                 /* Versioning */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve global configuration and make base URL absolute\n */\nconst script = getElement(\"#__config\")\nconst config: Config = JSON.parse(script.textContent!)\nconfig.base = `${new URL(config.base, getLocation())}`\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve global configuration\n *\n * @returns Global configuration\n */\nexport function configuration(): Config {\n  return config\n}\n\n/**\n * Check whether a feature flag is enabled\n *\n * @param flag - Feature flag\n *\n * @returns Test result\n */\nexport function feature(flag: Flag): boolean {\n  return config.features.includes(flag)\n}\n\n/**\n * Retrieve the translation for the given key\n *\n * @param key - Key to be translated\n * @param value - Positional value, if any\n *\n * @returns Translation\n */\nexport function translation(\n  key: Translation, value?: string | number\n): string {\n  return typeof value !== \"undefined\"\n    ? config.translations[key].replace(\"#\", value.toString())\n    : config.translations[key]\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { getElement, getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Component type\n */\nexport type ComponentType =\n  | \"announce\"                         /* Announcement bar */\n  | \"container\"                        /* Container */\n  | \"consent\"                          /* Consent */\n  | \"content\"                          /* Content */\n  | \"dialog\"                           /* Dialog */\n  | \"header\"                           /* Header */\n  | \"header-title\"                     /* Header title */\n  | \"header-topic\"                     /* Header topic */\n  | \"main\"                             /* Main area */\n  | \"outdated\"                         /* Version warning */\n  | \"palette\"                          /* Color palette */\n  | \"search\"                           /* Search */\n  | \"search-query\"                     /* Search input */\n  | \"search-result\"                    /* Search results */\n  | \"search-share\"                     /* Search sharing */\n  | \"search-suggest\"                   /* Search suggestions */\n  | \"sidebar\"                          /* Sidebar */\n  | \"skip\"                             /* Skip link */\n  | \"source\"                           /* Repository information */\n  | \"tabs\"                             /* Navigation tabs */\n  | \"toc\"                              /* Table of contents */\n  | \"top\"                              /* Back-to-top button */\n\n/**\n * Component\n *\n * @template T - Component type\n * @template U - Reference type\n */\nexport type Component<\n  T extends {} = {},\n  U extends HTMLElement = HTMLElement\n> =\n  T & {\n    ref: U                             /* Component reference */\n  }\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Component type map\n */\ninterface ComponentTypeMap {\n  \"announce\": HTMLElement              /* Announcement bar */\n  \"container\": HTMLElement             /* Container */\n  \"consent\": HTMLElement               /* Consent */\n  \"content\": HTMLElement               /* Content */\n  \"dialog\": HTMLElement                /* Dialog */\n  \"header\": HTMLElement                /* Header */\n  \"header-title\": HTMLElement          /* Header title */\n  \"header-topic\": HTMLElement          /* Header topic */\n  \"main\": HTMLElement                  /* Main area */\n  \"outdated\": HTMLElement              /* Version warning */\n  \"palette\": HTMLElement               /* Color palette */\n  \"search\": HTMLElement                /* Search */\n  \"search-query\": HTMLInputElement     /* Search input */\n  \"search-result\": HTMLElement         /* Search results */\n  \"search-share\": HTMLAnchorElement    /* Search sharing */\n  \"search-suggest\": HTMLElement        /* Search suggestions */\n  \"sidebar\": HTMLElement               /* Sidebar */\n  \"skip\": HTMLAnchorElement            /* Skip link */\n  \"source\": HTMLAnchorElement          /* Repository information */\n  \"tabs\": HTMLElement                  /* Navigation tabs */\n  \"toc\": HTMLElement                   /* Table of contents */\n  \"top\": HTMLAnchorElement             /* Back-to-top button */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve the element for a given component or throw a reference error\n *\n * @template T - Component type\n *\n * @param type - Component type\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getComponentElement<T extends ComponentType>(\n  type: T, node: ParentNode = document\n): ComponentTypeMap[T] {\n  return getElement(`[data-md-component=${type}]`, node)\n}\n\n/**\n * Retrieve all elements for a given component\n *\n * @template T - Component type\n *\n * @param type - Component type\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getComponentElements<T extends ComponentType>(\n  type: T, node: ParentNode = document\n): ComponentTypeMap[T][] {\n  return getElements(`[data-md-component=${type}]`, node)\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  finalize,\n  fromEvent,\n  map,\n  tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport { getElement } from \"~/browser\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Announcement bar\n */\nexport interface Announce {\n  hash: number                        /* Content hash */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch announcement bar\n *\n * @param el - Announcement bar element\n *\n * @returns Announcement bar observable\n */\nexport function watchAnnounce(\n  el: HTMLElement\n): Observable<Announce> {\n  const button = getElement(\".md-typeset > :first-child\", el)\n  return fromEvent(button, \"click\", { once: true })\n    .pipe(\n      map(() => getElement(\".md-typeset\", el)),\n      map(content => ({ hash: __md_hash(content.innerHTML) }))\n    )\n}\n\n/**\n * Mount announcement bar\n *\n * @param el - Announcement bar element\n *\n * @returns Announcement bar component observable\n */\nexport function mountAnnounce(\n  el: HTMLElement\n): Observable<Component<Announce>> {\n  if (!feature(\"announce.dismiss\") || !el.childElementCount)\n    return EMPTY\n\n  /* Support instant loading - see https://t.ly/3FTme */\n  if (!el.hidden) {\n    const content = getElement(\".md-typeset\", el)\n    if (__md_hash(content.innerHTML) === __md_get(\"__announce\"))\n      el.hidden = true\n  }\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject<Announce>()\n    push$.subscribe(({ hash }) => {\n      el.hidden = true\n\n      /* Persist preference in local storage */\n      __md_set<number>(\"__announce\", hash)\n    })\n\n    /* Create and return component */\n    return watchAnnounce(el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  finalize,\n  map,\n  tap\n} from \"rxjs\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Consent\n */\nexport interface Consent {\n  hidden: boolean                      /* Consent is hidden */\n}\n\n/**\n * Consent defaults\n */\nexport interface ConsentDefaults {\n  analytics?: boolean                  /* Consent for Analytics */\n  github?: boolean                     /* Consent for GitHub */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  target$: Observable<HTMLElement>     /* Target observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch consent\n *\n * @param el - Consent element\n * @param options - Options\n *\n * @returns Consent observable\n */\nexport function watchConsent(\n  el: HTMLElement, { target$ }: WatchOptions\n): Observable<Consent> {\n  return target$\n    .pipe(\n      map(target => ({ hidden: target !== el }))\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount consent\n *\n * @param el - Consent element\n * @param options - Options\n *\n * @returns Consent component observable\n */\nexport function mountConsent(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Consent>> {\n  const internal$ = new Subject<Consent>()\n  internal$.subscribe(({ hidden }) => {\n    el.hidden = hidden\n  })\n\n  /* Create and return component */\n  return watchConsent(el, options)\n    .pipe(\n      tap(state => internal$.next(state)),\n      finalize(() => internal$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  animationFrameScheduler,\n  auditTime,\n  combineLatest,\n  debounceTime,\n  defer,\n  delay,\n  endWith,\n  filter,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  merge,\n  switchMap,\n  take,\n  takeUntil,\n  tap,\n  throttleTime,\n  withLatestFrom\n} from \"rxjs\"\n\nimport {\n  ElementOffset,\n  getActiveElement,\n  getElementSize,\n  watchElementContentOffset,\n  watchElementFocus,\n  watchElementOffset,\n  watchElementVisibility\n} from \"~/browser\"\n\nimport { Component } from \"../../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Annotation\n */\nexport interface Annotation {\n  active: boolean                      /* Annotation is active */\n  offset: ElementOffset                /* Annotation offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch annotation\n *\n * @param el - Annotation element\n * @param container - Containing element\n *\n * @returns Annotation observable\n */\nexport function watchAnnotation(\n  el: HTMLElement, container: HTMLElement\n): Observable<Annotation> {\n  const offset$ = defer(() => combineLatest([\n    watchElementOffset(el),\n    watchElementContentOffset(container)\n  ]))\n    .pipe(\n      map(([{ x, y }, scroll]): ElementOffset => {\n        const { width, height } = getElementSize(el)\n        return ({\n          x: x - scroll.x + width  / 2,\n          y: y - scroll.y + height / 2\n        })\n      })\n    )\n\n  /* Actively watch annotation on focus */\n  return watchElementFocus(el)\n    .pipe(\n      switchMap(active => offset$\n        .pipe(\n          map(offset => ({ active, offset })),\n          take(+!active || Infinity)\n        )\n      )\n    )\n}\n\n/**\n * Mount annotation\n *\n * @param el - Annotation element\n * @param container - Containing element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotation(\n  el: HTMLElement, container: HTMLElement, { target$ }: MountOptions\n): Observable<Component<Annotation>> {\n  const [tooltip, index] = Array.from(el.children)\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject<Annotation>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    push$.subscribe({\n\n      /* Handle emission */\n      next({ offset }) {\n        el.style.setProperty(\"--md-tooltip-x\", `${offset.x}px`)\n        el.style.setProperty(\"--md-tooltip-y\", `${offset.y}px`)\n      },\n\n      /* Handle complete */\n      complete() {\n        el.style.removeProperty(\"--md-tooltip-x\")\n        el.style.removeProperty(\"--md-tooltip-y\")\n      }\n    })\n\n    /* Start animation only when annotation is visible */\n    watchElementVisibility(el)\n      .pipe(\n        takeUntil(done$)\n      )\n        .subscribe(visible => {\n          el.toggleAttribute(\"data-md-visible\", visible)\n        })\n\n    /* Toggle tooltip presence to mitigate empty lines when copying */\n    merge(\n      push$.pipe(filter(({ active }) => active)),\n      push$.pipe(debounceTime(250), filter(({ active }) => !active))\n    )\n      .subscribe({\n\n        /* Handle emission */\n        next({ active }) {\n          if (active)\n            el.prepend(tooltip)\n          else\n            tooltip.remove()\n        },\n\n        /* Handle complete */\n        complete() {\n          el.prepend(tooltip)\n        }\n      })\n\n    /* Toggle tooltip visibility */\n    push$\n      .pipe(\n        auditTime(16, animationFrameScheduler)\n      )\n        .subscribe(({ active }) => {\n          tooltip.classList.toggle(\"md-tooltip--active\", active)\n        })\n\n    /* Track relative origin of tooltip */\n    push$\n      .pipe(\n        throttleTime(125, animationFrameScheduler),\n        filter(() => !!el.offsetParent),\n        map(() => el.offsetParent!.getBoundingClientRect()),\n        map(({ x }) => x)\n      )\n        .subscribe({\n\n          /* Handle emission */\n          next(origin) {\n            if (origin)\n              el.style.setProperty(\"--md-tooltip-0\", `${-origin}px`)\n            else\n              el.style.removeProperty(\"--md-tooltip-0\")\n          },\n\n          /* Handle complete */\n          complete() {\n            el.style.removeProperty(\"--md-tooltip-0\")\n          }\n        })\n\n    /* Allow to copy link without scrolling to anchor */\n    fromEvent<MouseEvent>(index, \"click\")\n      .pipe(\n        takeUntil(done$),\n        filter(ev => !(ev.metaKey || ev.ctrlKey))\n      )\n        .subscribe(ev => {\n          ev.stopPropagation()\n          ev.preventDefault()\n        })\n\n    /* Allow to open link in new tab or blur on close */\n    fromEvent<MouseEvent>(index, \"mousedown\")\n      .pipe(\n        takeUntil(done$),\n        withLatestFrom(push$)\n      )\n        .subscribe(([ev, { active }]) => {\n\n          /* Open in new tab */\n          if (ev.button !== 0 || ev.metaKey || ev.ctrlKey) {\n            ev.preventDefault()\n\n          /* Close annotation */\n          } else if (active) {\n            ev.preventDefault()\n\n            /* Focus parent annotation, if any */\n            const parent = el.parentElement!.closest(\".md-annotation\")\n            if (parent instanceof HTMLElement)\n              parent.focus()\n            else\n              getActiveElement()?.blur()\n          }\n        })\n\n    /* Open and focus annotation on location target */\n    target$\n      .pipe(\n        takeUntil(done$),\n        filter(target => target === tooltip),\n        delay(125)\n      )\n        .subscribe(() => el.focus())\n\n    /* Create and return component */\n    return watchAnnotation(el, container)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a tooltip\n *\n * @param id - Tooltip identifier\n *\n * @returns Element\n */\nexport function renderTooltip(id?: string): HTMLElement {\n  return (\n    <div class=\"md-tooltip\" id={id}>\n      <div class=\"md-tooltip__inner md-typeset\"></div>\n    </div>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\nimport { renderTooltip } from \"../tooltip\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render an annotation\n *\n * @param id - Annotation identifier\n * @param prefix - Tooltip identifier prefix\n *\n * @returns Element\n */\nexport function renderAnnotation(\n  id: string | number, prefix?: string\n): HTMLElement {\n  prefix = prefix ? `${prefix}_annotation_${id}` : undefined\n\n  /* Render tooltip with anchor, if given */\n  if (prefix) {\n    const anchor = prefix ? `#${prefix}` : undefined\n    return (\n      <aside class=\"md-annotation\" tabIndex={0}>\n        {renderTooltip(prefix)}\n        <a href={anchor} class=\"md-annotation__index\" tabIndex={-1}>\n          <span data-md-annotation-id={id}></span>\n        </a>\n      </aside>\n    )\n  } else {\n    return (\n      <aside class=\"md-annotation\" tabIndex={0}>\n        {renderTooltip(prefix)}\n        <span class=\"md-annotation__index\" tabIndex={-1}>\n          <span data-md-annotation-id={id}></span>\n        </span>\n      </aside>\n    )\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { translation } from \"~/_\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a 'copy-to-clipboard' button\n *\n * @param id - Unique identifier\n *\n * @returns Element\n */\nexport function renderClipboardButton(id: string): HTMLElement {\n  return (\n    <button\n      class=\"md-clipboard md-icon\"\n      title={translation(\"clipboard.copy\")}\n      data-clipboard-target={`#${id} > code`}\n    ></button>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ComponentChild } from \"preact\"\n\nimport { configuration, feature, translation } from \"~/_\"\nimport { SearchItem } from \"~/integrations/search\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Render flag\n */\nconst enum Flag {\n  TEASER = 1,                          /* Render teaser */\n  PARENT = 2                           /* Render as parent */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper function\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a search document\n *\n * @param document - Search document\n * @param flag - Render flags\n *\n * @returns Element\n */\nfunction renderSearchDocument(\n  document: SearchItem, flag: Flag\n): HTMLElement {\n  const parent = flag & Flag.PARENT\n  const teaser = flag & Flag.TEASER\n\n  /* Render missing query terms */\n  const missing = Object.keys(document.terms)\n    .filter(key => !document.terms[key])\n    .reduce<ComponentChild[]>((list, key) => [\n      ...list, <del>{key}</del>, \" \"\n    ], [])\n    .slice(0, -1)\n\n  /* Assemble query string for highlighting */\n  const config = configuration()\n  const url = new URL(document.location, config.base)\n  if (feature(\"search.highlight\"))\n    url.searchParams.set(\"h\", Object.entries(document.terms)\n      .filter(([, match]) => match)\n      .reduce((highlight, [value]) => `${highlight} ${value}`.trim(), \"\")\n    )\n\n  /* Render article or section, depending on flags */\n  const { tags } = configuration()\n  return (\n    <a href={`${url}`} class=\"md-search-result__link\" tabIndex={-1}>\n      <article\n        class=\"md-search-result__article md-typeset\"\n        data-md-score={document.score.toFixed(2)}\n      >\n        {parent > 0 && <div class=\"md-search-result__icon md-icon\"></div>}\n        {parent > 0 && <h1>{document.title}</h1>}\n        {parent <= 0 && <h2>{document.title}</h2>}\n        {teaser > 0 && document.text.length > 0 &&\n          document.text\n        }\n        {document.tags && document.tags.map(tag => {\n          const type = tags\n            ? tag in tags\n              ? `md-tag-icon md-tag--${tags[tag]}`\n              : \"md-tag-icon\"\n            : \"\"\n          return (\n            <span class={`md-tag ${type}`}>{tag}</span>\n          )\n        })}\n        {teaser > 0 && missing.length > 0 &&\n          <p class=\"md-search-result__terms\">\n            {translation(\"search.result.term.missing\")}: {...missing}\n          </p>\n        }\n      </article>\n    </a>\n  )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a search result\n *\n * @param result - Search result\n *\n * @returns Element\n */\nexport function renderSearchResultItem(\n  result: SearchItem[]\n): HTMLElement {\n  const threshold = result[0].score\n  const docs = [...result]\n\n  const config = configuration()\n\n  /* Find and extract parent article */\n  const parent = docs.findIndex(doc => {\n    const l = `${new URL(doc.location, config.base)}` // @todo hacky\n    return !l.includes(\"#\")\n  })\n  const [article] = docs.splice(parent, 1)\n\n  /* Determine last index above threshold */\n  let index = docs.findIndex(doc => doc.score < threshold)\n  if (index === -1)\n    index = docs.length\n\n  /* Partition sections */\n  const best = docs.slice(0, index)\n  const more = docs.slice(index)\n\n  /* Render children */\n  const children = [\n    renderSearchDocument(article, Flag.PARENT | +(!parent && index === 0)),\n    ...best.map(section => renderSearchDocument(section, Flag.TEASER)),\n    ...more.length ? [\n      <details class=\"md-search-result__more\">\n        <summary tabIndex={-1}>\n          <div>\n            {more.length > 0 && more.length === 1\n              ? translation(\"search.result.more.one\")\n              : translation(\"search.result.more.other\", more.length)\n            }\n          </div>\n        </summary>\n        {...more.map(section => renderSearchDocument(section, Flag.TEASER))}\n      </details>\n    ] : []\n  ]\n\n  /* Render search result */\n  return (\n    <li class=\"md-search-result__item\">\n      {children}\n    </li>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { SourceFacts } from \"~/components\"\nimport { h, round } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render repository facts\n *\n * @param facts - Repository facts\n *\n * @returns Element\n */\nexport function renderSourceFacts(facts: SourceFacts): HTMLElement {\n  return (\n    <ul class=\"md-source__facts\">\n      {Object.entries(facts).map(([key, value]) => (\n        <li class={`md-source__fact md-source__fact--${key}`}>\n          {typeof value === \"number\" ? round(value) : value}\n        </li>\n      ))}\n    </ul>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Tabbed control type\n */\ntype TabbedControlType =\n  | \"prev\"\n  | \"next\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render control for content tabs\n *\n * @param type - Control type\n *\n * @returns Element\n */\nexport function renderTabbedControl(\n  type: TabbedControlType\n): HTMLElement {\n  const classes = `tabbed-control tabbed-control--${type}`\n  return (\n    <div class={classes} hidden>\n      <button class=\"tabbed-button\" tabIndex={-1} aria-hidden=\"true\"></button>\n    </div>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a table inside a wrapper to improve scrolling on mobile\n *\n * @param table - Table element\n *\n * @returns Element\n */\nexport function renderTable(table: HTMLElement): HTMLElement {\n  return (\n    <div class=\"md-typeset__scrollwrap\">\n      <div class=\"md-typeset__table\">\n        {table}\n      </div>\n    </div>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { configuration, translation } from \"~/_\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Version\n */\nexport interface Version {\n  version: string                      /* Version identifier */\n  title: string                        /* Version title */\n  aliases: string[]                    /* Version aliases */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a version\n *\n * @param version - Version\n *\n * @returns Element\n */\nfunction renderVersion(version: Version): HTMLElement {\n  const config = configuration()\n\n  /* Ensure trailing slash - see https://bit.ly/3rL5u3f */\n  const url = new URL(`../${version.version}/`, config.base)\n  return (\n    <li class=\"md-version__item\">\n      <a href={`${url}`} class=\"md-version__link\">\n        {version.title}\n      </a>\n    </li>\n  )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a version selector\n *\n * @param versions - Versions\n * @param active - Active version\n *\n * @returns Element\n */\nexport function renderVersionSelector(\n  versions: Version[], active: Version\n): HTMLElement {\n  return (\n    <div class=\"md-version\">\n      <button\n        class=\"md-version__current\"\n        aria-label={translation(\"select.version\")}\n      >\n        {active.title}\n      </button>\n      <ul class=\"md-version__list\">\n        {versions.map(renderVersion)}\n      </ul>\n    </div>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  endWith,\n  finalize,\n  ignoreElements,\n  merge,\n  share,\n  takeUntil\n} from \"rxjs\"\n\nimport {\n  getElement,\n  getElements,\n  getOptionalElement\n} from \"~/browser\"\nimport { renderAnnotation } from \"~/templates\"\n\nimport { Component } from \"../../../_\"\nimport {\n  Annotation,\n  mountAnnotation\n} from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find all annotation hosts in the containing element\n *\n * @param container - Containing element\n *\n * @returns Annotation hosts\n */\nfunction findHosts(container: HTMLElement): HTMLElement[] {\n  return container.tagName === \"CODE\"\n    ? getElements(\".c, .c1, .cm\", container)\n    : [container]\n}\n\n/**\n * Find all annotation markers in the containing element\n *\n * @param container - Containing element\n *\n * @returns Annotation markers\n */\nfunction findMarkers(container: HTMLElement): Text[] {\n  const markers: Text[] = []\n  for (const el of findHosts(container)) {\n    const nodes: Text[] = []\n\n    /* Find all text nodes in current element */\n    const it = document.createNodeIterator(el, NodeFilter.SHOW_TEXT)\n    for (let node = it.nextNode(); node; node = it.nextNode())\n      nodes.push(node as Text)\n\n    /* Find all markers in each text node */\n    for (let text of nodes) {\n      let match: RegExpExecArray | null\n\n      /* Split text at marker and add to list */\n      while ((match = /(\\(\\d+\\))(!)?/.exec(text.textContent!))) {\n        const [, id, force] = match\n        if (typeof force === \"undefined\") {\n          const marker = text.splitText(match.index)\n          text = marker.splitText(id.length)\n          markers.push(marker)\n\n        /* Replace entire text with marker */\n        } else {\n          text.textContent = id\n          markers.push(text)\n          break\n        }\n      }\n    }\n  }\n  return markers\n}\n\n/**\n * Swap the child nodes of two elements\n *\n * @param source - Source element\n * @param target - Target element\n */\nfunction swap(source: HTMLElement, target: HTMLElement): void {\n  target.append(...Array.from(source.childNodes))\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount annotation list\n *\n * This function analyzes the containing code block and checks for markers\n * referring to elements in the given annotation list. If no markers are found,\n * the list is left untouched. Otherwise, list elements are rendered as\n * annotations inside the code block.\n *\n * @param el - Annotation list element\n * @param container - Containing element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotationList(\n  el: HTMLElement, container: HTMLElement, { target$, print$ }: MountOptions\n): Observable<Component<Annotation>> {\n\n  /* Compute prefix for tooltip anchors */\n  const parent = container.closest(\"[id]\")\n  const prefix = parent?.id\n\n  /* Find and replace all markers with empty annotations */\n  const annotations = new Map<string, HTMLElement>()\n  for (const marker of findMarkers(container)) {\n    const [, id] = marker.textContent!.match(/\\((\\d+)\\)/)!\n    if (getOptionalElement(`:scope > li:nth-child(${id})`, el)) {\n      annotations.set(id, renderAnnotation(id, prefix))\n      marker.replaceWith(annotations.get(id)!)\n    }\n  }\n\n  /* Keep list if there are no annotations to render */\n  if (annotations.size === 0)\n    return EMPTY\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n\n    /* Retrieve container pairs for swapping */\n    const pairs: [HTMLElement, HTMLElement][] = []\n    for (const [id, annotation] of annotations)\n      pairs.push([\n        getElement(\".md-typeset\", annotation),\n        getElement(`:scope > li:nth-child(${id})`, el)\n      ])\n\n    /* Handle print mode - see https://bit.ly/3rgPdpt */\n    print$.pipe(takeUntil(done$))\n      .subscribe(active => {\n        el.hidden = !active\n\n        /* Add class to discern list element */\n        el.classList.toggle(\"md-annotation-list\", active)\n\n        /* Show annotations in code block or list (print) */\n        for (const [inner, child] of pairs)\n          if (!active)\n            swap(child, inner)\n          else\n            swap(inner, child)\n      })\n\n    /* Create and return component */\n    return merge(...[...annotations]\n      .map(([, annotation]) => (\n        mountAnnotation(annotation, container, { target$ })\n      ))\n    )\n      .pipe(\n        finalize(() => push$.complete()),\n        share()\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { EMPTY, Observable, defer } from \"rxjs\"\n\nimport { Component } from \"../../../_\"\nimport { Annotation } from \"../_\"\nimport { mountAnnotationList } from \"../list\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find list element directly following a block\n *\n * @param el - Annotation block element\n *\n * @returns List element or nothing\n */\nfunction findList(el: HTMLElement): HTMLElement | undefined {\n  if (el.nextElementSibling) {\n    const sibling = el.nextElementSibling as HTMLElement\n    if (sibling.tagName === \"OL\")\n      return sibling\n\n    /* Skip empty paragraphs - see https://bit.ly/3r4ZJ2O */\n    else if (sibling.tagName === \"P\" && !sibling.children.length)\n      return findList(sibling)\n  }\n\n  /* Everything else */\n  return undefined\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount annotation block\n *\n * @param el - Annotation block element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotationBlock(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Annotation>> {\n  return defer(() => {\n    const list = findList(el)\n    return typeof list !== \"undefined\"\n      ? mountAnnotationList(list, el, options)\n      : EMPTY\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport ClipboardJS from \"clipboard\"\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  filter,\n  finalize,\n  map,\n  mergeWith,\n  switchMap,\n  take,\n  tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  getElementContentSize,\n  watchElementSize,\n  watchElementVisibility\n} from \"~/browser\"\nimport { renderClipboardButton } from \"~/templates\"\n\nimport { Component } from \"../../../_\"\nimport {\n  Annotation,\n  mountAnnotationList\n} from \"../../annotation\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Code block\n */\nexport interface CodeBlock {\n  scrollable: boolean                  /* Code block overflows */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Global sequence number for code blocks\n */\nlet sequence = 0\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find candidate list element directly following a code block\n *\n * @param el - Code block element\n *\n * @returns List element or nothing\n */\nfunction findCandidateList(el: HTMLElement): HTMLElement | undefined {\n  if (el.nextElementSibling) {\n    const sibling = el.nextElementSibling as HTMLElement\n    if (sibling.tagName === \"OL\")\n      return sibling\n\n    /* Skip empty paragraphs - see https://bit.ly/3r4ZJ2O */\n    else if (sibling.tagName === \"P\" && !sibling.children.length)\n      return findCandidateList(sibling)\n  }\n\n  /* Everything else */\n  return undefined\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch code block\n *\n * This function monitors size changes of the viewport, as well as switches of\n * content tabs with embedded code blocks, as both may trigger overflow.\n *\n * @param el - Code block element\n *\n * @returns Code block observable\n */\nexport function watchCodeBlock(\n  el: HTMLElement\n): Observable<CodeBlock> {\n  return watchElementSize(el)\n    .pipe(\n      map(({ width }) => {\n        const content = getElementContentSize(el)\n        return {\n          scrollable: content.width > width\n        }\n      }),\n      distinctUntilKeyChanged(\"scrollable\")\n    )\n}\n\n/**\n * Mount code block\n *\n * This function ensures that an overflowing code block is focusable through\n * keyboard, so it can be scrolled without a mouse to improve on accessibility.\n * Furthermore, if code annotations are enabled, they are mounted if and only\n * if the code block is currently visible, e.g., not in a hidden content tab.\n *\n * Note that code blocks may be mounted eagerly or lazily. If they're mounted\n * lazily (on first visibility), code annotation anchor links will not work,\n * as they are evaluated on initial page load, and code annotations in general\n * might feel a little bumpier.\n *\n * @param el - Code block element\n * @param options - Options\n *\n * @returns Code block and annotation component observable\n */\nexport function mountCodeBlock(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<CodeBlock | Annotation>> {\n  const { matches: hover } = matchMedia(\"(hover)\")\n\n  /* Defer mounting of code block - see https://bit.ly/3vHVoVD */\n  const factory$ = defer(() => {\n    const push$ = new Subject<CodeBlock>()\n    push$.subscribe(({ scrollable }) => {\n      if (scrollable && hover)\n        el.setAttribute(\"tabindex\", \"0\")\n      else\n        el.removeAttribute(\"tabindex\")\n    })\n\n    /* Render button for Clipboard.js integration */\n    if (ClipboardJS.isSupported()) {\n      if (el.closest(\".copy\") || (\n        feature(\"content.code.copy\") && !el.closest(\".no-copy\")\n      )) {\n        const parent = el.closest(\"pre\")!\n        parent.id = `__code_${sequence++}`\n        parent.insertBefore(\n          renderClipboardButton(parent.id),\n          el\n        )\n      }\n    }\n\n    /* Handle code annotations */\n    const container = el.closest(\".highlight\")\n    if (container instanceof HTMLElement) {\n      const list = findCandidateList(container)\n\n      /* Mount code annotations, if enabled */\n      if (typeof list !== \"undefined\" && (\n        container.classList.contains(\"annotate\") ||\n        feature(\"content.code.annotate\")\n      )) {\n        const annotations$ = mountAnnotationList(list, el, options)\n\n        /* Create and return component */\n        return watchCodeBlock(el)\n          .pipe(\n            tap(state => push$.next(state)),\n            finalize(() => push$.complete()),\n            map(state => ({ ref: el, ...state })),\n            mergeWith(\n              watchElementSize(container)\n                .pipe(\n                  map(({ width, height }) => width && height),\n                  distinctUntilChanged(),\n                  switchMap(active => active ? annotations$ : EMPTY)\n                )\n            )\n          )\n      }\n    }\n\n    /* Create and return component */\n    return watchCodeBlock(el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n\n  /* Mount code block lazily */\n  if (feature(\"content.lazy\"))\n    return watchElementVisibility(el)\n      .pipe(\n        filter(visible => visible),\n        take(1),\n        switchMap(() => factory$)\n      )\n\n  /* Mount code block */\n  return factory$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  defer,\n  filter,\n  finalize,\n  map,\n  merge,\n  tap\n} from \"rxjs\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Details\n */\nexport interface Details {\n  action: \"open\" | \"close\"             /* Details state */\n  reveal?: boolean                     /* Details is revealed */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch details\n *\n * @param el - Details element\n * @param options - Options\n *\n * @returns Details observable\n */\nexport function watchDetails(\n  el: HTMLDetailsElement, { target$, print$ }: WatchOptions\n): Observable<Details> {\n  let open = true\n  return merge(\n\n    /* Open and focus details on location target */\n    target$\n      .pipe(\n        map(target => target.closest(\"details:not([open])\")!),\n        filter(details => el === details),\n        map(() => ({\n          action: \"open\", reveal: true\n        }) as Details)\n      ),\n\n    /* Open details on print and close afterwards */\n    print$\n      .pipe(\n        filter(active => active || !open),\n        tap(() => open = el.open),\n        map(active => ({\n          action: active ? \"open\" : \"close\"\n        }) as Details)\n      )\n  )\n}\n\n/**\n * Mount details\n *\n * This function ensures that `details` tags are opened on anchor jumps and\n * prior to printing, so the whole content of the page is visible.\n *\n * @param el - Details element\n * @param options - Options\n *\n * @returns Details component observable\n */\nexport function mountDetails(\n  el: HTMLDetailsElement, options: MountOptions\n): Observable<Component<Details>> {\n  return defer(() => {\n    const push$ = new Subject<Details>()\n    push$.subscribe(({ action, reveal }) => {\n      el.toggleAttribute(\"open\", action === \"open\")\n      if (reveal)\n        el.scrollIntoView()\n    })\n\n    /* Create and return component */\n    return watchDetails(el, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", ".node circle,.node ellipse,.node path,.node polygon,.node rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}marker{fill:var(--md-mermaid-edge-color)!important}.edgeLabel .label rect{fill:#0000}.label{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.label foreignObject{line-height:normal;overflow:visible}.label div .edgeLabel{color:var(--md-mermaid-label-fg-color)}.edgeLabel,.edgeLabel rect,.label div .edgeLabel{background-color:var(--md-mermaid-label-bg-color)}.edgeLabel,.edgeLabel rect{fill:var(--md-mermaid-label-bg-color);color:var(--md-mermaid-edge-color)}.edgePath .path,.flowchart-link{stroke:var(--md-mermaid-edge-color);stroke-width:.05rem}.edgePath .arrowheadPath{fill:var(--md-mermaid-edge-color);stroke:none}.cluster rect{fill:var(--md-default-fg-color--lightest);stroke:var(--md-default-fg-color--lighter)}.cluster span{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}g #flowchart-circleEnd,g #flowchart-circleStart,g #flowchart-crossEnd,g #flowchart-crossStart,g #flowchart-pointEnd,g #flowchart-pointStart{stroke:none}g.classGroup line,g.classGroup rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}g.classGroup text{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.classLabel .box{fill:var(--md-mermaid-label-bg-color);background-color:var(--md-mermaid-label-bg-color);opacity:1}.classLabel .label{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.node .divider{stroke:var(--md-mermaid-node-fg-color)}.relation{stroke:var(--md-mermaid-edge-color)}.cardinality{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.cardinality text{fill:inherit!important}defs #classDiagram-compositionEnd,defs #classDiagram-compositionStart,defs #classDiagram-dependencyEnd,defs #classDiagram-dependencyStart,defs #classDiagram-extensionEnd,defs #classDiagram-extensionStart{fill:var(--md-mermaid-edge-color)!important;stroke:var(--md-mermaid-edge-color)!important}defs #classDiagram-aggregationEnd,defs #classDiagram-aggregationStart{fill:var(--md-mermaid-label-bg-color)!important;stroke:var(--md-mermaid-edge-color)!important}g.stateGroup rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}g.stateGroup .state-title{fill:var(--md-mermaid-label-fg-color)!important;font-family:var(--md-mermaid-font-family)}g.stateGroup .composit{fill:var(--md-mermaid-label-bg-color)}.nodeLabel{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.node circle.state-end,.node circle.state-start,.start-state{fill:var(--md-mermaid-edge-color);stroke:none}.end-state-inner,.end-state-outer{fill:var(--md-mermaid-edge-color)}.end-state-inner,.node circle.state-end{stroke:var(--md-mermaid-label-bg-color)}.transition{stroke:var(--md-mermaid-edge-color)}[id^=state-fork] rect,[id^=state-join] rect{fill:var(--md-mermaid-edge-color)!important;stroke:none!important}.statediagram-cluster.statediagram-cluster .inner{fill:var(--md-default-bg-color)}.statediagram-cluster rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}.statediagram-state rect.divider{fill:var(--md-default-fg-color--lightest);stroke:var(--md-default-fg-color--lighter)}defs #statediagram-barbEnd{stroke:var(--md-mermaid-edge-color)}.attributeBoxEven,.attributeBoxOdd{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}.entityBox{fill:var(--md-mermaid-label-bg-color);stroke:var(--md-mermaid-node-fg-color)}.entityLabel{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.relationshipLabelBox{fill:var(--md-mermaid-label-bg-color);fill-opacity:1;background-color:var(--md-mermaid-label-bg-color);opacity:1}.relationshipLabel{fill:var(--md-mermaid-label-fg-color)}.relationshipLine{stroke:var(--md-mermaid-edge-color)}defs #ONE_OR_MORE_END *,defs #ONE_OR_MORE_START *,defs #ONLY_ONE_END *,defs #ONLY_ONE_START *,defs #ZERO_OR_MORE_END *,defs #ZERO_OR_MORE_START *,defs #ZERO_OR_ONE_END *,defs #ZERO_OR_ONE_START *{stroke:var(--md-mermaid-edge-color)!important}defs #ZERO_OR_MORE_END circle,defs #ZERO_OR_MORE_START circle{fill:var(--md-mermaid-label-bg-color)}.actor{fill:var(--md-mermaid-sequence-actor-bg-color);stroke:var(--md-mermaid-sequence-actor-border-color)}text.actor>tspan{fill:var(--md-mermaid-sequence-actor-fg-color);font-family:var(--md-mermaid-font-family)}line{stroke:var(--md-mermaid-sequence-actor-line-color)}.actor-man circle,.actor-man line{fill:var(--md-mermaid-sequence-actorman-bg-color);stroke:var(--md-mermaid-sequence-actorman-line-color)}.messageLine0,.messageLine1{stroke:var(--md-mermaid-sequence-message-line-color)}.note{fill:var(--md-mermaid-sequence-note-bg-color);stroke:var(--md-mermaid-sequence-note-border-color)}.loopText,.loopText>tspan,.messageText,.noteText>tspan{stroke:none;font-family:var(--md-mermaid-font-family)!important}.messageText{fill:var(--md-mermaid-sequence-message-fg-color)}.loopText,.loopText>tspan{fill:var(--md-mermaid-sequence-loop-fg-color)}.noteText>tspan{fill:var(--md-mermaid-sequence-note-fg-color)}#arrowhead path{fill:var(--md-mermaid-sequence-message-line-color);stroke:none}.loopLine{fill:var(--md-mermaid-sequence-loop-bg-color);stroke:var(--md-mermaid-sequence-loop-border-color)}.labelBox{fill:var(--md-mermaid-sequence-label-bg-color);stroke:none}.labelText,.labelText>span{fill:var(--md-mermaid-sequence-label-fg-color);font-family:var(--md-mermaid-font-family)}.sequenceNumber{fill:var(--md-mermaid-sequence-number-fg-color)}rect.rect{fill:var(--md-mermaid-sequence-box-bg-color);stroke:none}rect.rect+text.text{fill:var(--md-mermaid-sequence-box-fg-color)}defs #sequencenumber{fill:var(--md-mermaid-sequence-number-bg-color)!important}", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  map,\n  of,\n  shareReplay,\n  tap\n} from \"rxjs\"\n\nimport { watchScript } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\nimport themeCSS from \"./index.css\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mermaid diagram\n */\nexport interface Mermaid {}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Mermaid instance observable\n */\nlet mermaid$: Observable<void>\n\n/**\n * Global sequence number for diagrams\n */\nlet sequence = 0\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch Mermaid script\n *\n * @returns Mermaid scripts observable\n */\nfunction fetchScripts(): Observable<void> {\n  return typeof mermaid === \"undefined\" || mermaid instanceof Element\n    ? watchScript(\"https://unpkg.com/mermaid@9.4.3/dist/mermaid.min.js\")\n    : of(undefined)\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount Mermaid diagram\n *\n * @param el - Code block element\n *\n * @returns Mermaid diagram component observable\n */\nexport function mountMermaid(\n  el: HTMLElement\n): Observable<Component<Mermaid>> {\n  el.classList.remove(\"mermaid\") // Hack: mitigate https://bit.ly/3CiN6Du\n  mermaid$ ||= fetchScripts()\n    .pipe(\n      tap(() => mermaid.initialize({\n        startOnLoad: false,\n        themeCSS,\n        sequence: {\n          actorFontSize: \"16px\", // Hack: mitigate https://bit.ly/3y0NEi3\n          messageFontSize: \"16px\",\n          noteFontSize: \"16px\"\n        }\n      })),\n      map(() => undefined),\n      shareReplay(1)\n    )\n\n  /* Render diagram */\n  mermaid$.subscribe(() => {\n    el.classList.add(\"mermaid\") // Hack: mitigate https://bit.ly/3CiN6Du\n    const id = `__mermaid_${sequence++}`\n\n    /* Create host element to replace code block */\n    const host = h(\"div\", { class: \"mermaid\" })\n    const text = el.textContent\n\n    /* Render and inject diagram */\n    mermaid.mermaidAPI.render(id, text, (svg: string, fn: Function) => {\n\n      /* Create a shadow root and inject diagram */\n      const shadow = host.attachShadow({ mode: \"closed\" })\n      shadow.innerHTML = svg\n\n      /* Replace code block with diagram and bind functions */\n      el.replaceWith(host)\n      fn?.(shadow)\n    })\n  })\n\n  /* Create and return component */\n  return mermaid$\n    .pipe(\n      map(() => ({ ref: el }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Observable, of } from \"rxjs\"\n\nimport { renderTable } from \"~/templates\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Data table\n */\nexport interface DataTable {}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Sentinel for replacement\n */\nconst sentinel = h(\"table\")\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount data table\n *\n * This function wraps a data table in another scrollable container, so it can\n * be smoothly scrolled on smaller screen sizes and won't break the layout.\n *\n * @param el - Data table element\n *\n * @returns Data table component observable\n */\nexport function mountDataTable(\n  el: HTMLElement\n): Observable<Component<DataTable>> {\n  el.replaceWith(sentinel)\n  sentinel.replaceWith(renderTable(el))\n\n  /* Create and return component */\n  return of({ ref: el })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  animationFrameScheduler,\n  asyncScheduler,\n  auditTime,\n  combineLatest,\n  defer,\n  endWith,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  merge,\n  skip,\n  startWith,\n  subscribeOn,\n  takeUntil,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  getElement,\n  getElementContentOffset,\n  getElementContentSize,\n  getElementOffset,\n  getElementSize,\n  getElements,\n  watchElementContentOffset,\n  watchElementSize\n} from \"~/browser\"\nimport { renderTabbedControl } from \"~/templates\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Content tabs\n */\nexport interface ContentTabs {\n  active: HTMLLabelElement             /* Active tab label */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch content tabs\n *\n * @param el - Content tabs element\n *\n * @returns Content tabs observable\n */\nexport function watchContentTabs(\n  el: HTMLElement\n): Observable<ContentTabs> {\n  const inputs = getElements<HTMLInputElement>(\":scope > input\", el)\n  const initial = inputs.find(input => input.checked) || inputs[0]\n  return merge(...inputs.map(input => fromEvent(input, \"change\")\n    .pipe(\n      map(() => getElement<HTMLLabelElement>(`label[for=\"${input.id}\"]`))\n    )\n  ))\n    .pipe(\n      startWith(getElement<HTMLLabelElement>(`label[for=\"${initial.id}\"]`)),\n      map(active => ({ active }))\n    )\n}\n\n/**\n * Mount content tabs\n *\n * This function scrolls the active tab into view. While this functionality is\n * provided by browsers as part of `scrollInfoView`, browsers will always also\n * scroll the vertical axis, which we do not want. Thus, we decided to provide\n * this functionality ourselves.\n *\n * @param el - Content tabs element\n * @param options - Options\n *\n * @returns Content tabs component observable\n */\nexport function mountContentTabs(\n  el: HTMLElement, { viewport$ }: MountOptions\n): Observable<Component<ContentTabs>> {\n\n  /* Render content tab previous button for pagination */\n  const prev = renderTabbedControl(\"prev\")\n  el.append(prev)\n\n  /* Render content tab next button for pagination */\n  const next = renderTabbedControl(\"next\")\n  el.append(next)\n\n  /* Mount component on subscription */\n  const container = getElement(\".tabbed-labels\", el)\n  return defer(() => {\n    const push$ = new Subject<ContentTabs>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    combineLatest([push$, watchElementSize(el)])\n      .pipe(\n        auditTime(1, animationFrameScheduler),\n        takeUntil(done$)\n      )\n        .subscribe({\n\n          /* Handle emission */\n          next([{ active }, size]) {\n            const offset = getElementOffset(active)\n            const { width } = getElementSize(active)\n\n            /* Set tab indicator offset and width */\n            el.style.setProperty(\"--md-indicator-x\", `${offset.x}px`)\n            el.style.setProperty(\"--md-indicator-width\", `${width}px`)\n\n            /* Scroll container to active content tab */\n            const content = getElementContentOffset(container)\n            if (\n              offset.x         < content.x              ||\n              offset.x + width > content.x + size.width\n            )\n              container.scrollTo({\n                left: Math.max(0, offset.x - 16),\n                behavior: \"smooth\"\n              })\n          },\n\n          /* Handle complete */\n          complete() {\n            el.style.removeProperty(\"--md-indicator-x\")\n            el.style.removeProperty(\"--md-indicator-width\")\n          }\n        })\n\n    /* Hide content tab buttons on borders */\n    combineLatest([\n      watchElementContentOffset(container),\n      watchElementSize(container)\n    ])\n      .pipe(\n        takeUntil(done$)\n      )\n        .subscribe(([offset, size]) => {\n          const content = getElementContentSize(container)\n          prev.hidden = offset.x < 16\n          next.hidden = offset.x > content.width - size.width - 16\n        })\n\n    /* Paginate content tab container on click */\n    merge(\n      fromEvent(prev, \"click\").pipe(map(() => -1)),\n      fromEvent(next, \"click\").pipe(map(() => +1))\n    )\n      .pipe(\n        takeUntil(done$)\n      )\n        .subscribe(direction => {\n          const { width } = getElementSize(container)\n          container.scrollBy({\n            left: width * direction,\n            behavior: \"smooth\"\n          })\n        })\n\n    /* Set up linking of content tabs, if enabled */\n    if (feature(\"content.tabs.link\"))\n      push$.pipe(\n        skip(1),\n        withLatestFrom(viewport$)\n      )\n        .subscribe(([{ active }, { offset }]) => {\n          const tab = active.innerText.trim()\n          if (active.hasAttribute(\"data-md-switching\")) {\n            active.removeAttribute(\"data-md-switching\")\n\n          /* Determine viewport offset of active tab */\n          } else {\n            const y = el.offsetTop - offset.y\n\n            /* Passively activate other tabs */\n            for (const set of getElements(\"[data-tabs]\"))\n              for (const input of getElements<HTMLInputElement>(\n                \":scope > input\", set\n              )) {\n                const label = getElement(`label[for=\"${input.id}\"]`)\n                if (\n                  label !== active &&\n                  label.innerText.trim() === tab\n                ) {\n                  label.setAttribute(\"data-md-switching\", \"\")\n                  input.click()\n                  break\n                }\n              }\n\n            /* Bring active tab into view */\n            window.scrollTo({\n              top: el.offsetTop - y\n            })\n\n            /* Persist active tabs in local storage */\n            const tabs = __md_get<string[]>(\"__tabs\") || []\n            __md_set(\"__tabs\", [...new Set([tab, ...tabs])])\n          }\n        })\n\n    /* Pause media (audio, video) on switch - see https://bit.ly/3Bk6cel */\n    push$.pipe(takeUntil(done$))\n      .subscribe(() => {\n        for (const media of getElements<HTMLAudioElement>(\"audio, video\", el))\n          media.pause()\n      })\n\n    /* Create and return component */\n    return watchContentTabs(el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n    .pipe(\n      subscribeOn(asyncScheduler)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Observable, merge } from \"rxjs\"\n\nimport { Viewport, getElements } from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport {\n  Annotation,\n  mountAnnotationBlock\n} from \"../annotation\"\nimport {\n  CodeBlock,\n  mountCodeBlock\n} from \"../code\"\nimport {\n  Details,\n  mountDetails\n} from \"../details\"\nimport {\n  Mermaid,\n  mountMermaid\n} from \"../mermaid\"\nimport {\n  DataTable,\n  mountDataTable\n} from \"../table\"\nimport {\n  ContentTabs,\n  mountContentTabs\n} from \"../tabs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Content\n */\nexport type Content =\n  | Annotation\n  | CodeBlock\n  | ContentTabs\n  | DataTable\n  | Details\n  | Mermaid\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount content\n *\n * This function mounts all components that are found in the content of the\n * actual article, including code blocks, data tables and details.\n *\n * @param el - Content element\n * @param options - Options\n *\n * @returns Content component observable\n */\nexport function mountContent(\n  el: HTMLElement, { viewport$, target$, print$ }: MountOptions\n): Observable<Component<Content>> {\n  return merge(\n\n    /* Annotations */\n    ...getElements(\".annotate:not(.highlight)\", el)\n      .map(child => mountAnnotationBlock(child, { target$, print$ })),\n\n    /* Code blocks */\n    ...getElements(\"pre:not(.mermaid) > code\", el)\n      .map(child => mountCodeBlock(child, { target$, print$ })),\n\n    /* Mermaid diagrams */\n    ...getElements(\"pre.mermaid\", el)\n      .map(child => mountMermaid(child)),\n\n    /* Data tables */\n    ...getElements(\"table:not([class])\", el)\n      .map(child => mountDataTable(child)),\n\n    /* Details */\n    ...getElements(\"details\", el)\n      .map(child => mountDetails(child, { target$, print$ })),\n\n    /* Content tabs */\n    ...getElements(\"[data-tabs]\", el)\n      .map(child => mountContentTabs(child, { viewport$ }))\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  defer,\n  delay,\n  finalize,\n  map,\n  merge,\n  of,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { getElement } from \"~/browser\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Dialog\n */\nexport interface Dialog {\n  message: string                      /* Dialog message */\n  active: boolean                      /* Dialog is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  alert$: Subject<string>              /* Alert subject */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  alert$: Subject<string>              /* Alert subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch dialog\n *\n * @param _el - Dialog element\n * @param options - Options\n *\n * @returns Dialog observable\n */\nexport function watchDialog(\n  _el: HTMLElement, { alert$ }: WatchOptions\n): Observable<Dialog> {\n  return alert$\n    .pipe(\n      switchMap(message => merge(\n        of(true),\n        of(false).pipe(delay(2000))\n      )\n        .pipe(\n          map(active => ({ message, active }))\n        )\n      )\n    )\n}\n\n/**\n * Mount dialog\n *\n * This function reveals the dialog in the right corner when a new alert is\n * emitted through the subject that is passed as part of the options.\n *\n * @param el - Dialog element\n * @param options - Options\n *\n * @returns Dialog component observable\n */\nexport function mountDialog(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Dialog>> {\n  const inner = getElement(\".md-typeset\", el)\n  return defer(() => {\n    const push$ = new Subject<Dialog>()\n    push$.subscribe(({ message, active }) => {\n      el.classList.toggle(\"md-dialog--active\", active)\n      inner.textContent = message\n    })\n\n    /* Create and return component */\n    return watchDialog(el, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  bufferCount,\n  combineLatest,\n  combineLatestWith,\n  defer,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  filter,\n  ignoreElements,\n  map,\n  of,\n  shareReplay,\n  startWith,\n  switchMap,\n  takeUntil\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  watchElementSize,\n  watchToggle\n} from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { Main } from \"../../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Header\n */\nexport interface Header {\n  height: number                       /* Header visible height */\n  hidden: boolean                      /* Header is hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Compute whether the header is hidden\n *\n * If the user scrolls past a certain threshold, the header can be hidden when\n * scrolling down, and shown when scrolling up.\n *\n * @param options - Options\n *\n * @returns Toggle observable\n */\nfunction isHidden({ viewport$ }: WatchOptions): Observable<boolean> {\n  if (!feature(\"header.autohide\"))\n    return of(false)\n\n  /* Compute direction and turning point */\n  const direction$ = viewport$\n    .pipe(\n      map(({ offset: { y } }) => y),\n      bufferCount(2, 1),\n      map(([a, b]) => [a < b, b] as const),\n      distinctUntilKeyChanged(0)\n    )\n\n  /* Compute whether header should be hidden */\n  const hidden$ = combineLatest([viewport$, direction$])\n    .pipe(\n      filter(([{ offset }, [, y]]) => Math.abs(y - offset.y) > 100),\n      map(([, [direction]]) => direction),\n      distinctUntilChanged()\n    )\n\n  /* Compute threshold for hiding */\n  const search$ = watchToggle(\"search\")\n  return combineLatest([viewport$, search$])\n    .pipe(\n      map(([{ offset }, search]) => offset.y > 400 && !search),\n      distinctUntilChanged(),\n      switchMap(active => active ? hidden$ : of(false)),\n      startWith(false)\n    )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch header\n *\n * @param el - Header element\n * @param options - Options\n *\n * @returns Header observable\n */\nexport function watchHeader(\n  el: HTMLElement, options: WatchOptions\n): Observable<Header> {\n  return defer(() => combineLatest([\n    watchElementSize(el),\n    isHidden(options)\n  ]))\n    .pipe(\n      map(([{ height }, hidden]) => ({\n        height,\n        hidden\n      })),\n      distinctUntilChanged((a, b) => (\n        a.height === b.height &&\n        a.hidden === b.hidden\n      )),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount header\n *\n * This function manages the different states of the header, i.e. whether it's\n * hidden or rendered with a shadow. This depends heavily on the main area.\n *\n * @param el - Header element\n * @param options - Options\n *\n * @returns Header component observable\n */\nexport function mountHeader(\n  el: HTMLElement, { header$, main$ }: MountOptions\n): Observable<Component<Header>> {\n  return defer(() => {\n    const push$ = new Subject<Main>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    push$\n      .pipe(\n        distinctUntilKeyChanged(\"active\"),\n        combineLatestWith(header$)\n      )\n        .subscribe(([{ active }, { hidden }]) => {\n          el.classList.toggle(\"md-header--shadow\", active && !hidden)\n          el.hidden = hidden\n        })\n\n    /* Link to main area */\n    main$.subscribe(push$)\n\n    /* Create and return component */\n    return header$\n      .pipe(\n        takeUntil(done$),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  distinctUntilKeyChanged,\n  finalize,\n  map,\n  tap\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  getElementSize,\n  getOptionalElement,\n  watchViewportAt\n} from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { Header } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Header\n */\nexport interface HeaderTitle {\n  active: boolean                      /* Header title is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch header title\n *\n * @param el - Heading element\n * @param options - Options\n *\n * @returns Header title observable\n */\nexport function watchHeaderTitle(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<HeaderTitle> {\n  return watchViewportAt(el, { viewport$, header$ })\n    .pipe(\n      map(({ offset: { y } }) => {\n        const { height } = getElementSize(el)\n        return {\n          active: y >= height\n        }\n      }),\n      distinctUntilKeyChanged(\"active\")\n    )\n}\n\n/**\n * Mount header title\n *\n * This function swaps the header title from the site title to the title of the\n * current page when the user scrolls past the first headline.\n *\n * @param el - Header title element\n * @param options - Options\n *\n * @returns Header title component observable\n */\nexport function mountHeaderTitle(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<HeaderTitle>> {\n  return defer(() => {\n    const push$ = new Subject<HeaderTitle>()\n    push$.subscribe({\n\n      /* Handle emission */\n      next({ active }) {\n        el.classList.toggle(\"md-header__title--active\", active)\n      },\n\n      /* Handle complete */\n      complete() {\n        el.classList.remove(\"md-header__title--active\")\n      }\n    })\n\n    /* Obtain headline, if any */\n    const heading = getOptionalElement(\".md-content h1\")\n    if (typeof heading === \"undefined\")\n      return EMPTY\n\n    /* Create and return component */\n    return watchHeaderTitle(heading, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  map,\n  switchMap\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  watchElementSize\n} from \"~/browser\"\n\nimport { Header } from \"../header\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Main area\n */\nexport interface Main {\n  offset: number                       /* Main area top offset */\n  height: number                       /* Main area visible height */\n  active: boolean                      /* Main area is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch main area\n *\n * This function returns an observable that computes the visual parameters of\n * the main area which depends on the viewport vertical offset and height, as\n * well as the height of the header element, if the header is fixed.\n *\n * @param el - Main area element\n * @param options - Options\n *\n * @returns Main area observable\n */\nexport function watchMain(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<Main> {\n\n  /* Compute necessary adjustment for header */\n  const adjust$ = header$\n    .pipe(\n      map(({ height }) => height),\n      distinctUntilChanged()\n    )\n\n  /* Compute the main area's top and bottom borders */\n  const border$ = adjust$\n    .pipe(\n      switchMap(() => watchElementSize(el)\n        .pipe(\n          map(({ height }) => ({\n            top:    el.offsetTop,\n            bottom: el.offsetTop + height\n          })),\n          distinctUntilKeyChanged(\"bottom\")\n        )\n      )\n    )\n\n  /* Compute the main area's offset, visible height and if we scrolled past */\n  return combineLatest([adjust$, border$, viewport$])\n    .pipe(\n      map(([header, { top, bottom }, { offset: { y }, size: { height } }]) => {\n        height = Math.max(0, height\n          - Math.max(0, top    - y,  header)\n          - Math.max(0, height + y - bottom)\n        )\n        return {\n          offset: top - header,\n          height,\n          active: top - header <= y\n        }\n      }),\n      distinctUntilChanged((a, b) => (\n        a.offset === b.offset &&\n        a.height === b.height &&\n        a.active === b.active\n      ))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  asyncScheduler,\n  defer,\n  finalize,\n  fromEvent,\n  map,\n  mergeMap,\n  observeOn,\n  of,\n  shareReplay,\n  startWith,\n  tap\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\nimport {\n  Component,\n  getComponentElement\n} from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Palette colors\n */\nexport interface PaletteColor {\n  scheme?: string                      /* Color scheme */\n  primary?: string                     /* Primary color */\n  accent?: string                      /* Accent color */\n}\n\n/**\n * Palette\n */\nexport interface Palette {\n  index: number                        /* Palette index */\n  color: PaletteColor                  /* Palette colors */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch color palette\n *\n * @param inputs - Color palette element\n *\n * @returns Color palette observable\n */\nexport function watchPalette(\n  inputs: HTMLInputElement[]\n): Observable<Palette> {\n  const current = __md_get<Palette>(\"__palette\") || {\n    index: inputs.findIndex(input => matchMedia(\n      input.getAttribute(\"data-md-color-media\")!\n    ).matches)\n  }\n\n  /* Emit changes in color palette */\n  return of(...inputs)\n    .pipe(\n      mergeMap(input => fromEvent(input, \"change\")\n        .pipe(\n          map(() => input)\n        )\n      ),\n      startWith(inputs[Math.max(0, current.index)]),\n      map(input => ({\n        index: inputs.indexOf(input),\n        color: {\n          scheme:  input.getAttribute(\"data-md-color-scheme\"),\n          primary: input.getAttribute(\"data-md-color-primary\"),\n          accent:  input.getAttribute(\"data-md-color-accent\")\n        }\n      } as Palette)),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount color palette\n *\n * @param el - Color palette element\n *\n * @returns Color palette component observable\n */\nexport function mountPalette(\n  el: HTMLElement\n): Observable<Component<Palette>> {\n  const meta = h(\"meta\", { name: \"theme-color\" })\n  document.head.appendChild(meta)\n\n  // Add color scheme meta tag\n  const scheme = h(\"meta\", { name: \"color-scheme\" })\n  document.head.appendChild(scheme)\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject<Palette>()\n    push$.subscribe(palette => {\n      document.body.setAttribute(\"data-md-color-switching\", \"\")\n\n      /* Set color palette */\n      for (const [key, value] of Object.entries(palette.color))\n        document.body.setAttribute(`data-md-color-${key}`, value)\n\n      /* Toggle visibility */\n      for (let index = 0; index < inputs.length; index++) {\n        const label = inputs[index].nextElementSibling\n        if (label instanceof HTMLElement)\n          label.hidden = palette.index !== index\n      }\n\n      /* Persist preference in local storage */\n      __md_set(\"__palette\", palette)\n    })\n\n    /* Update theme-color meta tag */\n    push$\n      .pipe(\n        map(() => {\n          const header = getComponentElement(\"header\")\n          const style  = window.getComputedStyle(header)\n\n          // Set color scheme\n          scheme.content = style.colorScheme\n\n          /* Return color in hexadecimal format */\n          return style.backgroundColor.match(/\\d+/g)!\n            .map(value => (+value).toString(16).padStart(2, \"0\"))\n            .join(\"\")\n        })\n      )\n        .subscribe(color => meta.content = `#${color}`)\n\n    /* Revert transition durations after color switch */\n    push$.pipe(observeOn(asyncScheduler))\n      .subscribe(() => {\n        document.body.removeAttribute(\"data-md-color-switching\")\n      })\n\n    /* Create and return component */\n    const inputs = getElements<HTMLInputElement>(\"input\", el)\n    return watchPalette(inputs)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport ClipboardJS from \"clipboard\"\nimport {\n  Observable,\n  Subject,\n  map,\n  tap\n} from \"rxjs\"\n\nimport { translation } from \"~/_\"\nimport { getElement } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n  alert$: Subject<string>              /* Alert subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Extract text to copy\n *\n * @param el - HTML element\n *\n * @returns Extracted text\n */\nfunction extract(el: HTMLElement): string {\n  el.setAttribute(\"data-md-copying\", \"\")\n  const text = el.innerText\n  el.removeAttribute(\"data-md-copying\")\n  return text\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up Clipboard.js integration\n *\n * @param options - Options\n */\nexport function setupClipboardJS(\n  { alert$ }: SetupOptions\n): void {\n  if (ClipboardJS.isSupported()) {\n    new Observable<ClipboardJS.Event>(subscriber => {\n      new ClipboardJS(\"[data-clipboard-target], [data-clipboard-text]\", {\n        text: el => (\n          el.getAttribute(\"data-clipboard-text\")! ||\n          extract(getElement(\n            el.getAttribute(\"data-clipboard-target\")!\n          ))\n        )\n      })\n        .on(\"success\", ev => subscriber.next(ev))\n    })\n      .pipe(\n        tap(ev => {\n          const trigger = ev.trigger as HTMLElement\n          trigger.focus()\n        }),\n        map(() => translation(\"clipboard.copied\"))\n      )\n        .subscribe(alert$)\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  catchError,\n  defaultIfEmpty,\n  map,\n  of,\n  tap\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport { getElements, requestXML } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Sitemap, i.e. a list of URLs\n */\nexport type Sitemap = string[]\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Preprocess a list of URLs\n *\n * This function replaces the `site_url` in the sitemap with the actual base\n * URL, to allow instant loading to work in occasions like Netlify previews.\n *\n * @param urls - URLs\n *\n * @returns URL path parts\n */\nfunction preprocess(urls: Sitemap): Sitemap {\n  if (urls.length < 2)\n    return [\"\"]\n\n  /* Take the first two URLs and remove everything after the last slash */\n  const [root, next] = [...urls]\n    .sort((a, b) => a.length - b.length)\n    .map(url => url.replace(/[^/]+$/, \"\"))\n\n  /* Compute common prefix */\n  let index = 0\n  if (root === next)\n    index = root.length\n  else\n    while (root.charCodeAt(index) === next.charCodeAt(index))\n      index++\n\n  /* Remove common prefix and return in original order */\n  return urls.map(url => url.replace(root.slice(0, index), \"\"))\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch the sitemap for the given base URL\n *\n * @param base - Base URL\n *\n * @returns Sitemap observable\n */\nexport function fetchSitemap(base?: URL): Observable<Sitemap> {\n  const cached = __md_get<Sitemap>(\"__sitemap\", sessionStorage, base)\n  if (cached) {\n    return of(cached)\n  } else {\n    const config = configuration()\n    return requestXML(new URL(\"sitemap.xml\", base || config.base))\n      .pipe(\n        map(sitemap => preprocess(getElements(\"loc\", sitemap)\n          .map(node => node.textContent!)\n        )),\n        catchError(() => EMPTY), // @todo refactor instant loading\n        defaultIfEmpty([]),\n        tap(sitemap => __md_set(\"__sitemap\", sitemap, sessionStorage, base))\n      )\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  bufferCount,\n  catchError,\n  concat,\n  debounceTime,\n  distinctUntilKeyChanged,\n  endWith,\n  fromEvent,\n  ignoreElements,\n  map,\n  of,\n  share,\n  skip,\n  startWith,\n  switchMap,\n  take,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { configuration, feature } from \"~/_\"\nimport {\n  Viewport,\n  getElements,\n  getLocation,\n  getOptionalElement,\n  request,\n  setLocation,\n  setLocationHash\n} from \"~/browser\"\nimport { getComponentElement } from \"~/components\"\n\nimport { fetchSitemap } from \"../sitemap\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n  location$: Subject<URL>              /* Location subject */\n  viewport$: Observable<Viewport>      /* Viewport observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up instant navigation\n *\n * This is a heavily orchestrated operation - see inline comments to learn how\n * this works with Material for MkDocs, and how you can hook into it.\n *\n * @param options - Options\n *\n * @returns Document observable\n */\nexport function setupInstantLoading(\n  { location$, viewport$ }: SetupOptions\n): Observable<Document> {\n  const config = configuration()\n  if (location.protocol === \"file:\")\n    return EMPTY\n\n  // Load sitemap immediately, so we have it available when the user initiates\n  // the first instant navigation request, and canonicalize URLs to the current\n  // base URL. The base URL will remain stable in between loads, as it's only\n  // read at the first initialization of the application.\n  const sitemap$ = fetchSitemap()\n    .pipe(\n      map(paths => paths.map(path => `${new URL(path, config.base)}`))\n    )\n\n  // Intercept inter-site navigation - to keep the number of event listeners\n  // low we use the fact that uncaptured events bubble up to the body. This also\n  // has the nice property that we don't need to detach and then again attach\n  // event listeners when instant navigation occurs.\n  const instant$ = fromEvent<MouseEvent>(document.body, \"click\")\n    .pipe(\n      withLatestFrom(sitemap$),\n      switchMap(([ev, sitemap]) => {\n        if (!(ev.target instanceof Element))\n          return EMPTY\n\n        // Skip, as target is not within a link - clicks on non-link elements\n        // are also captured, which we need to exclude from processing\n        const el = ev.target.closest(\"a\")\n        if (el === null)\n          return EMPTY\n\n        // Skip, as link opens in new window - we now know we have captured a\n        // click on a link, but the link either has a `target` property defined,\n        // or the user pressed the `meta` or `ctrl` key to open it in a new\n        // window. Thus, we need to filter those events, too.\n        if (el.target || ev.metaKey || ev.ctrlKey)\n          return EMPTY\n\n        // Next, we must check if the URL is relevant for us, i.e., if it's an\n        // internal link to a page that is managed by MkDocs. Only then we can\n        // be sure that the structure of the page to be loaded adheres to the\n        // current document structure and can subsequently be injected into it\n        // without doing a full reload. For this reason, we must canonicalize\n        // the URL by removing all search parameters and hash fragments.\n        const url = new URL(el.href)\n        url.search = url.hash = \"\"\n\n        // Skip, if URL is not included in the sitemap - this could be the case\n        // when linking between versions or languages, or to another page that\n        // the author included as part of the build, but that is not managed by\n        // MkDocs. In that case we must not continue with instant navigation.\n        if (!sitemap.includes(`${url}`))\n          return EMPTY\n\n        // We now know that we have a link to an internal page, so we prevent\n        // the browser from navigation and emit the URL for instant navigation.\n        // Note that this also includes anchor links, which means we need to\n        // implement anchor positioning ourselves. The reason for this is that\n        // if we wouldn't manage anchor links as well, scroll restoration will\n        // not work correctly (e.g. following an anchor link and scrolling).\n        ev.preventDefault()\n        return of(new URL(el.href))\n      }),\n      share()\n    )\n\n  // Before fetching for the first time, resolve the absolute favicon position,\n  // as the browser will try to fetch the icon immediately\n  instant$.pipe(take(1))\n    .subscribe(() => {\n      const favicon = getOptionalElement<HTMLLinkElement>(\"link[rel=icon]\")\n      if (typeof favicon !== \"undefined\")\n        favicon.href = favicon.href\n    })\n\n  // Enable scroll restoration before window unloads - this is essential to\n  // ensure that full reloads (F5) restore the viewport offset correctly. If\n  // only popstate events wouldn't reset the scroll position prior to their\n  // emission, we could just reset this in popstate. Meh.\n  fromEvent(window, \"beforeunload\")\n    .subscribe(() => {\n      history.scrollRestoration = \"auto\"\n    })\n\n  // When an instant navigation event occurs, disable scroll restoration, since\n  // we must normalize and synchronize the behavior across all browsers. For\n  // instance, when the user clicks the back or forward button, the browser\n  // would immediately jump to the position of the previous document.\n  instant$.pipe(withLatestFrom(viewport$))\n    .subscribe(([url, { offset }]) => {\n      history.scrollRestoration = \"manual\"\n\n      // While it would be better UX to defer the history state change until the\n      // document was fully fetched and parsed, we must schedule it here, since\n      // popstate events are emitted when history state changes happen. Moreover\n      // we need to back up the current viewport offset, so we can restore it\n      // when popstate events occur, e.g., when the browser's back and forward\n      // buttons are used for navigation.\n      history.replaceState(offset, \"\")\n      history.pushState(null, \"\", url)\n    })\n\n  // Emit URL that should be fetched via instant loading on location subject,\n  // which was passed into this function. The idea is that instant loading can\n  // be intercepted by other parts of the application, which can synchronously\n  // back up or restore state before instant loading happens.\n  instant$.subscribe(location$)\n\n  // Fetch document - when fetching, we could use `responseType: document`, but\n  // since all MkDocs links are relative, we need to make sure that the current\n  // location matches the document we just loaded. Otherwise any relative links\n  // in the document might use the old location. If the request fails for some\n  // reason, we fall back to regular navigation and set the location explicitly,\n  // which will force-load the page. Furthermore, we must pre-warm the buffer\n  // for the duplicate check, or the first click on an anchor link will also\n  // trigger an instant loading event, which doesn't make sense.\n  const response$ = location$\n    .pipe(\n      startWith(getLocation()),\n      distinctUntilKeyChanged(\"pathname\"),\n      skip(1),\n      switchMap(url => request(url)\n        .pipe(\n          catchError(() => {\n            setLocation(url)\n            return EMPTY\n          })\n        )\n      )\n    )\n\n  // Initialize the DOM parser, parse the returned HTML, and replace selected\n  // meta tags and components before handing control down to the application\n  const dom = new DOMParser()\n  const document$ = response$\n    .pipe(\n      switchMap(res => res.text()),\n      switchMap(res => {\n        const document = dom.parseFromString(res, \"text/html\")\n        for (const selector of [\n\n          // Meta tags\n          \"title\",\n          \"link[rel=canonical]\",\n          \"meta[name=author]\",\n          \"meta[name=description]\",\n\n          // Components\n          \"[data-md-component=announce]\",\n          \"[data-md-component=container]\",\n          \"[data-md-component=header-topic]\",\n          \"[data-md-component=outdated]\",\n          \"[data-md-component=logo]\",\n          \"[data-md-component=skip]\",\n          ...feature(\"navigation.tabs.sticky\")\n            ? [\"[data-md-component=tabs]\"]\n            : []\n        ]) {\n          const source = getOptionalElement(selector)\n          const target = getOptionalElement(selector, document)\n          if (\n            typeof source !== \"undefined\" &&\n            typeof target !== \"undefined\"\n          ) {\n            source.replaceWith(target)\n          }\n        }\n\n        // After meta tags and components were replaced, re-evaluate scripts\n        // that were provided by the author as part of Markdown files\n        const container = getComponentElement(\"container\")\n        return concat(getElements(\"script\", container))\n          .pipe(\n            switchMap(el => {\n              const script = document.createElement(\"script\")\n              if (el.src) {\n                for (const name of el.getAttributeNames())\n                  script.setAttribute(name, el.getAttribute(name)!)\n                el.replaceWith(script)\n\n                // Complete when script is loaded\n                return new Observable(observer => {\n                  script.onload = () => observer.complete()\n                })\n\n              // Complete immediately\n              } else {\n                script.textContent = el.textContent\n                el.replaceWith(script)\n                return EMPTY\n              }\n            }),\n            ignoreElements(),\n            endWith(document)\n          )\n      }),\n      share()\n    )\n\n  // Intercept popstate events, e.g. when using the browser's back and forward\n  // buttons, and emit new location for fetching and parsing\n  const popstate$ = fromEvent<PopStateEvent>(window, \"popstate\")\n  popstate$.pipe(map(getLocation))\n    .subscribe(location$)\n\n  // Intercept clicks on anchor links, and scroll document into position. As\n  // we disabled scroll restoration, we need to do this manually here.\n  location$\n    .pipe(\n      startWith(getLocation()),\n      bufferCount(2, 1),\n      switchMap(([prev, next]) => (\n        prev.pathname === next.pathname &&\n        prev.hash     !== next.hash\n      )\n        ? of(next)\n        : EMPTY\n      )\n    )\n      .subscribe(url => {\n        if (history.state !== null || !url.hash) {\n          window.scrollTo(0, history.state?.y ?? 0)\n        } else {\n          history.scrollRestoration = \"auto\"\n          setLocationHash(url.hash)\n          history.scrollRestoration = \"manual\"\n        }\n      })\n\n  // After parsing the document, check if the current history entry has a state.\n  // This may happen when users press the back or forward button to visit a page\n  // that was already seen. If there's no state, it means a new page was visited\n  // and we should scroll to the top, unless an anchor is given.\n  document$.pipe(withLatestFrom(location$))\n    .subscribe(([, url]) => {\n      if (history.state !== null || !url.hash) {\n        window.scrollTo(0, history.state?.y ?? 0)\n      } else {\n        setLocationHash(url.hash)\n      }\n    })\n\n  // If the current history is not empty, register an event listener updating\n  // the current history state whenever the scroll position changes. This must\n  // be debounced and cannot be done in popstate, as popstate has already\n  // removed the entry from the history.\n  document$\n    .pipe(\n      switchMap(() => viewport$),\n      distinctUntilKeyChanged(\"offset\"),\n      debounceTime(100)\n    )\n      .subscribe(({ offset }) => {\n        history.replaceState(offset, \"\")\n      })\n\n  // Return document\n  return document$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport escapeHTML from \"escape-html\"\n\nimport { SearchConfig } from \"../config\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search highlight function\n *\n * @param value - Value\n *\n * @returns Highlighted value\n */\nexport type SearchHighlightFn = (value: string) => string\n\n/**\n * Search highlight factory function\n *\n * @param query - Query value\n *\n * @returns Search highlight function\n */\nexport type SearchHighlightFactoryFn = (query: string) => SearchHighlightFn\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search highlighter\n *\n * @param config - Search configuration\n *\n * @returns Search highlight factory function\n */\nexport function setupSearchHighlighter(\n  config: SearchConfig\n): SearchHighlightFactoryFn {\n  // Hack: temporarily remove pure lookaheads and lookbehinds\n  const regex = config.separator.split(\"|\").map(term => {\n    const temp = term.replace(/(\\(\\?[!=<][^)]+\\))/g, \"\")\n    return temp.length === 0 ? \"\uFFFD\" : term\n  })\n    .join(\"|\")\n\n  const separator = new RegExp(regex, \"img\")\n  const highlight = (_: unknown, data: string, term: string) => {\n    return `${data}<mark data-md-highlight>${term}</mark>`\n  }\n\n  /* Return factory function */\n  return (query: string) => {\n    query = query\n      .replace(/[\\s*+\\-:~^]+/g, \" \")\n      .trim()\n\n    /* Create search term match expression */\n    const match = new RegExp(`(^|${config.separator}|)(${\n      query\n        .replace(/[|\\\\{}()[\\]^$+*?.-]/g, \"\\\\$&\")\n        .replace(separator, \"|\")\n    })`, \"img\")\n\n    /* Highlight string value */\n    return value => escapeHTML(value)\n      .replace(match, highlight)\n      .replace(/<\\/mark>(\\s+)<mark[^>]*>/img, \"$1\")\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { SearchResult } from \"../../_\"\nimport { SearchIndex } from \"../../config\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search message type\n */\nexport const enum SearchMessageType {\n  SETUP,                               /* Search index setup */\n  READY,                               /* Search index ready */\n  QUERY,                               /* Search query */\n  RESULT                               /* Search results */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Message containing the data necessary to setup the search index\n */\nexport interface SearchSetupMessage {\n  type: SearchMessageType.SETUP        /* Message type */\n  data: SearchIndex                    /* Message data */\n}\n\n/**\n * Message indicating the search index is ready\n */\nexport interface SearchReadyMessage {\n  type: SearchMessageType.READY        /* Message type */\n}\n\n/**\n * Message containing a search query\n */\nexport interface SearchQueryMessage {\n  type: SearchMessageType.QUERY        /* Message type */\n  data: string                         /* Message data */\n}\n\n/**\n * Message containing results for a search query\n */\nexport interface SearchResultMessage {\n  type: SearchMessageType.RESULT       /* Message type */\n  data: SearchResult                   /* Message data */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Message exchanged with the search worker\n */\nexport type SearchMessage =\n  | SearchSetupMessage\n  | SearchReadyMessage\n  | SearchQueryMessage\n  | SearchResultMessage\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Type guard for search ready messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchReadyMessage(\n  message: SearchMessage\n): message is SearchReadyMessage {\n  return message.type === SearchMessageType.READY\n}\n\n/**\n * Type guard for search result messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchResultMessage(\n  message: SearchMessage\n): message is SearchResultMessage {\n  return message.type === SearchMessageType.RESULT\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  ObservableInput,\n  Subject,\n  first,\n  merge,\n  of,\n  switchMap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport { watchToggle, watchWorker } from \"~/browser\"\n\nimport { SearchIndex } from \"../../config\"\nimport {\n  SearchMessage,\n  SearchMessageType\n} from \"../message\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up search worker\n *\n * This function creates and initializes a web worker that is used for search,\n * so that the user interface doesn't freeze. In general, the application does\n * not care how search is implemented, as long as the web worker conforms to\n * the format expected by the application as defined in `SearchMessage`. This\n * allows the author to implement custom search functionality, by providing a\n * custom web worker via configuration.\n *\n * Material for MkDocs' built-in search implementation makes use of Lunr.js, an\n * efficient and fast implementation for client-side search. Leveraging a tiny\n * iframe-based web worker shim, search is even supported for the `file://`\n * protocol, enabling search for local non-hosted builds.\n *\n * If the protocol is `file://`, search initialization is deferred to mitigate\n * freezing, as it's now synchronous by design - see https://bit.ly/3C521EO\n *\n * @see https://bit.ly/3igvtQv - How to implement custom search\n *\n * @param url - Worker URL\n * @param index$ - Search index observable input\n *\n * @returns Search worker\n */\nexport function setupSearchWorker(\n  url: string, index$: ObservableInput<SearchIndex>\n): Subject<SearchMessage> {\n  const worker$ = watchWorker<SearchMessage>(url)\n  merge(\n    of(location.protocol !== \"file:\"),\n    watchToggle(\"search\")\n  )\n    .pipe(\n      first(active => active),\n      switchMap(() => index$)\n    )\n      .subscribe(({ config, docs }) => worker$.next({\n        type: SearchMessageType.SETUP,\n        data: {\n          config,\n          docs,\n          options: {\n            suggest: feature(\"search.suggest\")\n          }\n        }\n      }))\n\n  /* Return search worker */\n  return worker$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Subject,\n  catchError,\n  combineLatest,\n  filter,\n  fromEvent,\n  map,\n  of,\n  switchMap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport {\n  getElement,\n  getLocation,\n  requestJSON,\n  setLocation\n} from \"~/browser\"\nimport { getComponentElements } from \"~/components\"\nimport {\n  Version,\n  renderVersionSelector\n} from \"~/templates\"\n\nimport { fetchSitemap } from \"../sitemap\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n  document$: Subject<Document>         /* Document subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up version selector\n *\n * @param options - Options\n */\nexport function setupVersionSelector(\n  { document$ }: SetupOptions\n): void {\n  const config = configuration()\n  const versions$ = requestJSON<Version[]>(\n    new URL(\"../versions.json\", config.base)\n  )\n    .pipe(\n      catchError(() => EMPTY) // @todo refactor instant loading\n    )\n\n  /* Determine current version */\n  const current$ = versions$\n    .pipe(\n      map(versions => {\n        const [, current] = config.base.match(/([^/]+)\\/?$/)!\n        return versions.find(({ version, aliases }) => (\n          version === current || aliases.includes(current)\n        )) || versions[0]\n      })\n    )\n\n  /* Intercept inter-version navigation */\n  versions$\n    .pipe(\n      map(versions => new Map(versions.map(version => [\n        `${new URL(`../${version.version}/`, config.base)}`,\n        version\n      ]))),\n      switchMap(urls => fromEvent<MouseEvent>(document.body, \"click\")\n        .pipe(\n          filter(ev => !ev.metaKey && !ev.ctrlKey),\n          withLatestFrom(current$),\n          switchMap(([ev, current]) => {\n            if (ev.target instanceof Element) {\n              const el = ev.target.closest(\"a\")\n              if (el && !el.target && urls.has(el.href)) {\n                const url = el.href\n                // This is a temporary hack to detect if a version inside the\n                // version selector or on another part of the site was clicked.\n                // If we're inside the version selector, we definitely want to\n                // find the same page, as we might have different deployments\n                // due to aliases. However, if we're outside the version\n                // selector, we must abort here, because we might otherwise\n                // interfere with instant loading. We need to refactor this\n                // at some point together with instant loading.\n                //\n                // See https://github.com/squidfunk/mkdocs-material/issues/4012\n                if (!ev.target.closest(\".md-version\")) {\n                  const version = urls.get(url)!\n                  if (version === current)\n                    return EMPTY\n                }\n                ev.preventDefault()\n                return of(url)\n              }\n            }\n            return EMPTY\n          }),\n          switchMap(url => {\n            const { version } = urls.get(url)!\n            return fetchSitemap(new URL(url))\n              .pipe(\n                map(sitemap => {\n                  const location = getLocation()\n                  const path = location.href.replace(config.base, \"\")\n                  return sitemap.includes(path.split(\"#\")[0])\n                    ? new URL(`../${version}/${path}`, config.base)\n                    : new URL(url)\n                })\n              )\n          })\n        )\n      )\n    )\n      .subscribe(url => setLocation(url))\n\n  /* Render version selector and warning */\n  combineLatest([versions$, current$])\n    .subscribe(([versions, current]) => {\n      const topic = getElement(\".md-header__topic\")\n      topic.appendChild(renderVersionSelector(versions, current))\n    })\n\n  /* Integrate outdated version banner with instant loading */\n  document$.pipe(switchMap(() => current$))\n    .subscribe(current => {\n\n      /* Check if version state was already determined */\n      let outdated = __md_get(\"__outdated\", sessionStorage)\n      if (outdated === null) {\n        outdated = true\n\n        /* Obtain and normalize default versions */\n        let ignored = config.version?.default || \"latest\"\n        if (!Array.isArray(ignored))\n          ignored = [ignored]\n\n        /* Check if version is considered a default */\n        main: for (const ignore of ignored)\n          for (const alias of current.aliases)\n            if (new RegExp(ignore, \"i\").test(alias)) {\n              outdated = false\n              break main\n            }\n\n        /* Persist version state in session storage */\n        __md_set(\"__outdated\", outdated, sessionStorage)\n      }\n\n      /* Unhide outdated version banner */\n      if (outdated)\n        for (const warning of getComponentElements(\"outdated\"))\n          warning.hidden = false\n    })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  combineLatest,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  finalize,\n  first,\n  fromEvent,\n  ignoreElements,\n  map,\n  merge,\n  shareReplay,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport {\n  getElement,\n  getLocation,\n  setToggle,\n  watchElementFocus,\n  watchToggle\n} from \"~/browser\"\nimport {\n  SearchMessage,\n  SearchMessageType,\n  isSearchReadyMessage\n} from \"~/integrations\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search query\n */\nexport interface SearchQuery {\n  value: string                        /* Query value */\n  focus: boolean                       /* Query focus */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch search query\n *\n * Note that the focus event which triggers re-reading the current query value\n * is delayed by `1ms` so the input's empty state is allowed to propagate.\n *\n * @param el - Search query element\n * @param options - Options\n *\n * @returns Search query observable\n */\nexport function watchSearchQuery(\n  el: HTMLInputElement, { worker$ }: WatchOptions\n): Observable<SearchQuery> {\n\n  /* Support search deep linking */\n  const { searchParams } = getLocation()\n  if (searchParams.has(\"q\")) {\n    setToggle(\"search\", true)\n\n    /* Set query from parameter */\n    el.value = searchParams.get(\"q\")!\n    el.focus()\n\n    /* Remove query parameter on close */\n    watchToggle(\"search\")\n      .pipe(\n        first(active => !active)\n      )\n        .subscribe(() => {\n          const url = new URL(location.href)\n          url.searchParams.delete(\"q\")\n          history.replaceState({}, \"\", `${url}`)\n        })\n  }\n\n  /* Intercept focus and input events */\n  const focus$ = watchElementFocus(el)\n  const value$ = merge(\n    worker$.pipe(first(isSearchReadyMessage)),\n    fromEvent(el, \"keyup\"),\n    focus$\n  )\n    .pipe(\n      map(() => el.value),\n      distinctUntilChanged()\n    )\n\n  /* Combine into single observable */\n  return combineLatest([value$, focus$])\n    .pipe(\n      map(([value, focus]) => ({ value, focus })),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount search query\n *\n * @param el - Search query element\n * @param options - Options\n *\n * @returns Search query component observable\n */\nexport function mountSearchQuery(\n  el: HTMLInputElement, { worker$ }: MountOptions\n): Observable<Component<SearchQuery, HTMLInputElement>> {\n  const push$ = new Subject<SearchQuery>()\n  const done$ = push$.pipe(ignoreElements(), endWith(true))\n\n  /* Handle value change */\n  combineLatest([\n    worker$.pipe(first(isSearchReadyMessage)),\n    push$\n  ], (_, query) => query)\n    .pipe(\n      distinctUntilKeyChanged(\"value\")\n    )\n      .subscribe(({ value }) => worker$.next({\n        type: SearchMessageType.QUERY,\n        data: value\n      }))\n\n  /* Handle focus change */\n  push$\n    .pipe(\n      distinctUntilKeyChanged(\"focus\")\n    )\n      .subscribe(({ focus }) => {\n        if (focus)\n          setToggle(\"search\", focus)\n      })\n\n  /* Handle reset */\n  fromEvent(el.form!, \"reset\")\n    .pipe(\n      takeUntil(done$)\n    )\n      .subscribe(() => el.focus())\n\n  // Focus search query on label click - note that this is necessary to bring\n  // up the keyboard on iOS and other mobile platforms, as the search dialog is\n  // not visible at first, and programatically focusing an input element must\n  // be triggered by a user interaction - see https://t.ly/Cb30n\n  const label = getElement(\"header [for=__search]\")\n  fromEvent(label, \"click\")\n    .subscribe(() => el.focus())\n\n  /* Create and return component */\n  return watchSearchQuery(el, { worker$ })\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state })),\n      shareReplay(1)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  bufferCount,\n  filter,\n  finalize,\n  first,\n  fromEvent,\n  map,\n  merge,\n  mergeMap,\n  of,\n  share,\n  skipUntil,\n  switchMap,\n  takeUntil,\n  tap,\n  withLatestFrom,\n  zipWith\n} from \"rxjs\"\n\nimport { translation } from \"~/_\"\nimport {\n  getElement,\n  getOptionalElement,\n  watchElementBoundary,\n  watchToggle\n} from \"~/browser\"\nimport {\n  SearchMessage,\n  SearchResult,\n  isSearchReadyMessage,\n  isSearchResultMessage\n} from \"~/integrations\"\nimport { renderSearchResultItem } from \"~/templates\"\nimport { round } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\nimport { SearchQuery } from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  query$: Observable<SearchQuery>      /* Search query observable */\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search result list\n *\n * This function performs a lazy rendering of the search results, depending on\n * the vertical offset of the search result container.\n *\n * @param el - Search result list element\n * @param options - Options\n *\n * @returns Search result list component observable\n */\nexport function mountSearchResult(\n  el: HTMLElement, { worker$, query$ }: MountOptions\n): Observable<Component<SearchResult>> {\n  const push$ = new Subject<SearchResult>()\n  const boundary$ = watchElementBoundary(el.parentElement!)\n    .pipe(\n      filter(Boolean)\n    )\n\n  /* Retrieve container */\n  const container = el.parentElement!\n\n  /* Retrieve nested components */\n  const meta = getElement(\":scope > :first-child\", el)\n  const list = getElement(\":scope > :last-child\", el)\n\n  /* Reveal to accessibility tree \u2013 see https://bit.ly/3iAA7t8 */\n  watchToggle(\"search\")\n    .subscribe(active => list.setAttribute(\n      \"role\", active ? \"list\" : \"presentation\"\n    ))\n\n  /* Update search result metadata */\n  push$\n    .pipe(\n      withLatestFrom(query$),\n      skipUntil(worker$.pipe(first(isSearchReadyMessage)))\n    )\n      .subscribe(([{ items }, { value }]) => {\n        switch (items.length) {\n\n          /* No results */\n          case 0:\n            meta.textContent = value.length\n              ? translation(\"search.result.none\")\n              : translation(\"search.result.placeholder\")\n            break\n\n          /* One result */\n          case 1:\n            meta.textContent = translation(\"search.result.one\")\n            break\n\n          /* Multiple result */\n          default:\n            const count = round(items.length)\n            meta.textContent = translation(\"search.result.other\", count)\n        }\n      })\n\n  /* Render search result item */\n  const render$ = push$\n    .pipe(\n      tap(() => list.innerHTML = \"\"),\n      switchMap(({ items }) => merge(\n        of(...items.slice(0, 10)),\n        of(...items.slice(10))\n          .pipe(\n            bufferCount(4),\n            zipWith(boundary$),\n            switchMap(([chunk]) => chunk)\n          )\n      )),\n      map(renderSearchResultItem),\n      share()\n    )\n\n  /* Update search result list */\n  render$.subscribe(item => list.appendChild(item))\n  render$\n    .pipe(\n      mergeMap(item => {\n        const details = getOptionalElement(\"details\", item)\n        if (typeof details === \"undefined\")\n          return EMPTY\n\n        /* Keep position of details element stable */\n        return fromEvent(details, \"toggle\")\n          .pipe(\n            takeUntil(push$),\n            map(() => details)\n          )\n      })\n    )\n      .subscribe(details => {\n        if (\n          details.open === false &&\n          details.offsetTop <= container.scrollTop\n        )\n          container.scrollTo({ top: details.offsetTop })\n      })\n\n  /* Filter search result message */\n  const result$ = worker$\n    .pipe(\n      filter(isSearchResultMessage),\n      map(({ data }) => data)\n    )\n\n  /* Create and return component */\n  return result$\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  endWith,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport { getLocation } from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { SearchQuery } from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search sharing\n */\nexport interface SearchShare {\n  url: URL                             /* Deep link for sharing */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  query$: Observable<SearchQuery>      /* Search query observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  query$: Observable<SearchQuery>      /* Search query observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search sharing\n *\n * @param _el - Search sharing element\n * @param options - Options\n *\n * @returns Search sharing observable\n */\nexport function watchSearchShare(\n  _el: HTMLElement, { query$ }: WatchOptions\n): Observable<SearchShare> {\n  return query$\n    .pipe(\n      map(({ value }) => {\n        const url = getLocation()\n        url.hash = \"\"\n\n        /* Compute readable query strings */\n        value = value\n          .replace(/\\s+/g, \"+\")        /* Collapse whitespace */\n          .replace(/&/g, \"%26\")        /* Escape '&' character */\n          .replace(/=/g, \"%3D\")        /* Escape '=' character */\n\n        /* Replace query string */\n        url.search = `q=${value}`\n        return { url }\n      })\n    )\n}\n\n/**\n * Mount search sharing\n *\n * @param el - Search sharing element\n * @param options - Options\n *\n * @returns Search sharing component observable\n */\nexport function mountSearchShare(\n  el: HTMLAnchorElement, options: MountOptions\n): Observable<Component<SearchShare>> {\n  const push$ = new Subject<SearchShare>()\n  const done$ = push$.pipe(ignoreElements(), endWith(true))\n  push$.subscribe(({ url }) => {\n    el.setAttribute(\"data-clipboard-text\", el.href)\n    el.href = `${url}`\n  })\n\n  /* Prevent following of link */\n  fromEvent(el, \"click\")\n    .pipe(\n      takeUntil(done$)\n    )\n      .subscribe(ev => ev.preventDefault())\n\n  /* Create and return component */\n  return watchSearchShare(el, options)\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  asyncScheduler,\n  combineLatestWith,\n  distinctUntilChanged,\n  filter,\n  finalize,\n  fromEvent,\n  map,\n  merge,\n  observeOn,\n  tap\n} from \"rxjs\"\n\nimport { Keyboard } from \"~/browser\"\nimport {\n  SearchMessage,\n  SearchResult,\n  isSearchResultMessage\n} from \"~/integrations\"\n\nimport { Component, getComponentElement } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search suggestions\n */\nexport interface SearchSuggest {}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  keyboard$: Observable<Keyboard>      /* Keyboard observable */\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search suggestions\n *\n * This function will perform a lazy rendering of the search results, depending\n * on the vertical offset of the search result container.\n *\n * @param el - Search result list element\n * @param options - Options\n *\n * @returns Search result list component observable\n */\nexport function mountSearchSuggest(\n  el: HTMLElement, { worker$, keyboard$ }: MountOptions\n): Observable<Component<SearchSuggest>> {\n  const push$ = new Subject<SearchResult>()\n\n  /* Retrieve query component and track all changes */\n  const query  = getComponentElement(\"search-query\")\n  const query$ = merge(\n    fromEvent(query, \"keydown\"),\n    fromEvent(query, \"focus\")\n  )\n    .pipe(\n      observeOn(asyncScheduler),\n      map(() => query.value),\n      distinctUntilChanged(),\n    )\n\n  /* Update search suggestions */\n  push$\n    .pipe(\n      combineLatestWith(query$),\n      map(([{ suggest }, value]) => {\n        const words = value.split(/([\\s-]+)/)\n        if (suggest?.length && words[words.length - 1]) {\n          const last = suggest[suggest.length - 1]\n          if (last.startsWith(words[words.length - 1]))\n            words[words.length - 1] = last\n        } else {\n          words.length = 0\n        }\n        return words\n      })\n    )\n      .subscribe(words => el.innerHTML = words\n        .join(\"\")\n        .replace(/\\s/g, \"&nbsp;\")\n      )\n\n  /* Set up search keyboard handlers */\n  keyboard$\n    .pipe(\n      filter(({ mode }) => mode === \"search\")\n    )\n      .subscribe(key => {\n        switch (key.type) {\n\n          /* Right arrow: accept current suggestion */\n          case \"ArrowRight\":\n            if (\n              el.innerText.length &&\n              query.selectionStart === query.value.length\n            )\n              query.value = el.innerText\n            break\n        }\n      })\n\n  /* Filter search result message */\n  const result$ = worker$\n    .pipe(\n      filter(isSearchResultMessage),\n      map(({ data }) => data)\n    )\n\n  /* Create and return component */\n  return result$\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(() => ({ ref: el }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  NEVER,\n  Observable,\n  ObservableInput,\n  filter,\n  fromEvent,\n  merge,\n  mergeWith\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport {\n  Keyboard,\n  getActiveElement,\n  getElements,\n  setToggle\n} from \"~/browser\"\nimport {\n  SearchIndex,\n  SearchResult,\n  setupSearchWorker\n} from \"~/integrations\"\n\nimport {\n  Component,\n  getComponentElement,\n  getComponentElements\n} from \"../../_\"\nimport {\n  SearchQuery,\n  mountSearchQuery\n} from \"../query\"\nimport { mountSearchResult } from \"../result\"\nimport {\n  SearchShare,\n  mountSearchShare\n} from \"../share\"\nimport {\n  SearchSuggest,\n  mountSearchSuggest\n} from \"../suggest\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search\n */\nexport type Search =\n  | SearchQuery\n  | SearchResult\n  | SearchShare\n  | SearchSuggest\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  index$: ObservableInput<SearchIndex> /* Search index observable */\n  keyboard$: Observable<Keyboard>      /* Keyboard observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search\n *\n * This function sets up the search functionality, including the underlying\n * web worker and all keyboard bindings.\n *\n * @param el - Search element\n * @param options - Options\n *\n * @returns Search component observable\n */\nexport function mountSearch(\n  el: HTMLElement, { index$, keyboard$ }: MountOptions\n): Observable<Component<Search>> {\n  const config = configuration()\n  try {\n    const worker$ = setupSearchWorker(config.search, index$)\n\n    /* Retrieve query and result components */\n    const query  = getComponentElement(\"search-query\", el)\n    const result = getComponentElement(\"search-result\", el)\n\n    /* Always close search on result selection */\n    fromEvent<PointerEvent>(el, \"click\")\n      .pipe(\n        filter(({ target }) => (\n          target instanceof Element && !!target.closest(\"a\")\n        ))\n      )\n        .subscribe(() => setToggle(\"search\", false))\n\n    /* Set up search keyboard handlers */\n    keyboard$\n      .pipe(\n        filter(({ mode }) => mode === \"search\")\n      )\n        .subscribe(key => {\n          const active = getActiveElement()\n          switch (key.type) {\n\n            /* Enter: go to first (best) result */\n            case \"Enter\":\n              if (active === query) {\n                const anchors = new Map<HTMLAnchorElement, number>()\n                for (const anchor of getElements<HTMLAnchorElement>(\n                  \":first-child [href]\", result\n                )) {\n                  const article = anchor.firstElementChild!\n                  anchors.set(anchor, parseFloat(\n                    article.getAttribute(\"data-md-score\")!\n                  ))\n                }\n\n                /* Go to result with highest score, if any */\n                if (anchors.size) {\n                  const [[best]] = [...anchors].sort(([, a], [, b]) => b - a)\n                  best.click()\n                }\n\n                /* Otherwise omit form submission */\n                key.claim()\n              }\n              break\n\n            /* Escape or Tab: close search */\n            case \"Escape\":\n            case \"Tab\":\n              setToggle(\"search\", false)\n              query.blur()\n              break\n\n            /* Vertical arrows: select previous or next search result */\n            case \"ArrowUp\":\n            case \"ArrowDown\":\n              if (typeof active === \"undefined\") {\n                query.focus()\n              } else {\n                const els = [query, ...getElements(\n                  \":not(details) > [href], summary, details[open] [href]\",\n                  result\n                )]\n                const i = Math.max(0, (\n                  Math.max(0, els.indexOf(active)) + els.length + (\n                    key.type === \"ArrowUp\" ? -1 : +1\n                  )\n                ) % els.length)\n                els[i].focus()\n              }\n\n              /* Prevent scrolling of page */\n              key.claim()\n              break\n\n            /* All other keys: hand to search query */\n            default:\n              if (query !== getActiveElement())\n                query.focus()\n          }\n        })\n\n    /* Set up global keyboard handlers */\n    keyboard$\n      .pipe(\n        filter(({ mode }) => mode === \"global\")\n      )\n        .subscribe(key => {\n          switch (key.type) {\n\n            /* Open search and select query */\n            case \"f\":\n            case \"s\":\n            case \"/\":\n              query.focus()\n              query.select()\n\n              /* Prevent scrolling of page */\n              key.claim()\n              break\n          }\n        })\n\n    /* Create and return component */\n    const query$ = mountSearchQuery(query, { worker$ })\n    return merge(\n      query$,\n      mountSearchResult(result, { worker$, query$ })\n    )\n      .pipe(\n        mergeWith(\n\n          /* Search sharing */\n          ...getComponentElements(\"search-share\", el)\n            .map(child => mountSearchShare(child, { query$ })),\n\n          /* Search suggestions */\n          ...getComponentElements(\"search-suggest\", el)\n            .map(child => mountSearchSuggest(child, { worker$, keyboard$ }))\n        )\n      )\n\n  /* Gracefully handle broken search */\n  } catch (err) {\n    el.hidden = true\n    return NEVER\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  ObservableInput,\n  combineLatest,\n  filter,\n  map,\n  startWith\n} from \"rxjs\"\n\nimport { getLocation } from \"~/browser\"\nimport {\n  SearchIndex,\n  setupSearchHighlighter\n} from \"~/integrations\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search highlighting\n */\nexport interface SearchHighlight {\n  nodes: Map<ChildNode, string>        /* Map of replacements */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  index$: ObservableInput<SearchIndex> /* Search index observable */\n  location$: Observable<URL>           /* Location observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search highlighting\n *\n * @param el - Content element\n * @param options - Options\n *\n * @returns Search highlighting component observable\n */\nexport function mountSearchHiglight(\n  el: HTMLElement, { index$, location$ }: MountOptions\n): Observable<Component<SearchHighlight>> {\n  return combineLatest([\n    index$,\n    location$\n      .pipe(\n        startWith(getLocation()),\n        filter(url => !!url.searchParams.get(\"h\"))\n      )\n  ])\n    .pipe(\n      map(([index, url]) => setupSearchHighlighter(index.config)(\n        url.searchParams.get(\"h\")!\n      )),\n      map(fn => {\n        const nodes = new Map<ChildNode, string>()\n\n        /* Traverse text nodes and collect matches */\n        const it = document.createNodeIterator(el, NodeFilter.SHOW_TEXT)\n        for (let node = it.nextNode(); node; node = it.nextNode()) {\n          if (node.parentElement?.offsetHeight) {\n            const original = node.textContent!\n            const replaced = fn(original)\n            if (replaced.length > original.length)\n              nodes.set(node as ChildNode, replaced)\n          }\n        }\n\n        /* Replace original nodes with matches */\n        for (const [node, text] of nodes) {\n          const { childNodes } = h(\"span\", null, text)\n          node.replaceWith(...Array.from(childNodes))\n        }\n\n        /* Return component */\n        return { ref: el, nodes }\n      })\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  animationFrameScheduler,\n  auditTime,\n  combineLatest,\n  defer,\n  distinctUntilChanged,\n  endWith,\n  finalize,\n  first,\n  from,\n  fromEvent,\n  ignoreElements,\n  map,\n  mergeMap,\n  takeUntil,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  getElement,\n  getElementContainer,\n  getElementOffset,\n  getElementSize,\n  getElements\n} from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Sidebar\n */\nexport interface Sidebar {\n  height: number                       /* Sidebar height */\n  locked: boolean                      /* Sidebar is locked */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  main$: Observable<Main>              /* Main area observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch sidebar\n *\n * This function returns an observable that computes the visual parameters of\n * the sidebar which depends on the vertical viewport offset, as well as the\n * height of the main area. When the page is scrolled beyond the header, the\n * sidebar is locked and fills the remaining space.\n *\n * @param el - Sidebar element\n * @param options - Options\n *\n * @returns Sidebar observable\n */\nexport function watchSidebar(\n  el: HTMLElement, { viewport$, main$ }: WatchOptions\n): Observable<Sidebar> {\n  const parent = el.closest<HTMLElement>(\".md-grid\")!\n  const adjust =\n    parent.offsetTop -\n    parent.parentElement!.offsetTop\n\n  /* Compute the sidebar's available height and if it should be locked */\n  return combineLatest([main$, viewport$])\n    .pipe(\n      map(([{ offset, height }, { offset: { y } }]) => {\n        height = height\n          + Math.min(adjust, Math.max(0, y - offset))\n          - adjust\n        return {\n          height,\n          locked: y >= offset + adjust\n        }\n      }),\n      distinctUntilChanged((a, b) => (\n        a.height === b.height &&\n        a.locked === b.locked\n      ))\n    )\n}\n\n/**\n * Mount sidebar\n *\n * This function doesn't set the height of the actual sidebar, but of its first\n * child \u2013 the `.md-sidebar__scrollwrap` element in order to mitigiate jittery\n * sidebars when the footer is scrolled into view. At some point we switched\n * from `absolute` / `fixed` positioning to `sticky` positioning, significantly\n * reducing jitter in some browsers (respectively Firefox and Safari) when\n * scrolling from the top. However, top-aligned sticky positioning means that\n * the sidebar snaps to the bottom when the end of the container is reached.\n * This is what leads to the mentioned jitter, as the sidebar's height may be\n * updated too slowly.\n *\n * This behaviour can be mitigiated by setting the height of the sidebar to `0`\n * while preserving the padding, and the height on its first element.\n *\n * @param el - Sidebar element\n * @param options - Options\n *\n * @returns Sidebar component observable\n */\nexport function mountSidebar(\n  el: HTMLElement, { header$, ...options }: MountOptions\n): Observable<Component<Sidebar>> {\n  const inner = getElement(\".md-sidebar__scrollwrap\", el)\n  const { y } = getElementOffset(inner)\n  return defer(() => {\n    const push$ = new Subject<Sidebar>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    const next$ = push$\n      .pipe(\n        auditTime(0, animationFrameScheduler)\n      )\n\n    /* Update sidebar height and offset */\n    next$.pipe(withLatestFrom(header$))\n      .subscribe({\n\n        /* Handle emission */\n        next([{ height }, { height: offset }]) {\n          inner.style.height = `${height - 2 * y}px`\n          el.style.top       = `${offset}px`\n        },\n\n        /* Handle complete */\n        complete() {\n          inner.style.height = \"\"\n          el.style.top       = \"\"\n        }\n      })\n\n    /* Bring active item into view on initial load */\n    next$.pipe(first())\n      .subscribe(() => {\n        for (const item of getElements(\".md-nav__link--active[href]\", el)) {\n          const container = getElementContainer(item)\n          if (typeof container !== \"undefined\") {\n            const offset = item.offsetTop - container.offsetTop\n            const { height } = getElementSize(container)\n            container.scrollTo({\n              top: offset - height / 2\n            })\n          }\n        }\n      })\n\n    /* Handle accessibility for expandable items, see https://bit.ly/3jaod9p */\n    from(getElements<HTMLLabelElement>(\"label[tabindex]\", el))\n      .pipe(\n        mergeMap(label => fromEvent(label, \"click\")\n          .pipe(\n            map(() => label),\n            takeUntil(done$)\n          )\n        )\n      )\n        .subscribe(label => {\n          const input = getElement<HTMLInputElement>(`[id=\"${label.htmlFor}\"]`)\n          const nav = getElement(`[aria-labelledby=\"${label.id}\"]`)\n          nav.setAttribute(\"aria-expanded\", `${input.checked}`)\n        })\n\n    /* Create and return component */\n    return watchSidebar(el, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Repo, User } from \"github-types\"\nimport {\n  EMPTY,\n  Observable,\n  catchError,\n  defaultIfEmpty,\n  map,\n  zip\n} from \"rxjs\"\n\nimport { requestJSON } from \"~/browser\"\n\nimport { SourceFacts } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * GitHub release (partial)\n */\ninterface Release {\n  tag_name: string                     /* Tag name */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch GitHub repository facts\n *\n * @param user - GitHub user or organization\n * @param repo - GitHub repository\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFactsFromGitHub(\n  user: string, repo?: string\n): Observable<SourceFacts> {\n  if (typeof repo !== \"undefined\") {\n    const url = `https://api.github.com/repos/${user}/${repo}`\n    return zip(\n\n      /* Fetch version */\n      requestJSON<Release>(`${url}/releases/latest`)\n        .pipe(\n          catchError(() => EMPTY), // @todo refactor instant loading\n          map(release => ({\n            version: release.tag_name\n          })),\n          defaultIfEmpty({})\n        ),\n\n      /* Fetch stars and forks */\n      requestJSON<Repo>(url)\n        .pipe(\n          catchError(() => EMPTY), // @todo refactor instant loading\n          map(info => ({\n            stars: info.stargazers_count,\n            forks: info.forks_count\n          })),\n          defaultIfEmpty({})\n        )\n    )\n      .pipe(\n        map(([release, info]) => ({ ...release, ...info }))\n      )\n\n  /* User or organization */\n  } else {\n    const url = `https://api.github.com/users/${user}`\n    return requestJSON<User>(url)\n      .pipe(\n        map(info => ({\n          repositories: info.public_repos\n        })),\n        defaultIfEmpty({})\n      )\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ProjectSchema } from \"gitlab\"\nimport {\n  EMPTY,\n  Observable,\n  catchError,\n  defaultIfEmpty,\n  map\n} from \"rxjs\"\n\nimport { requestJSON } from \"~/browser\"\n\nimport { SourceFacts } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch GitLab repository facts\n *\n * @param base - GitLab base\n * @param project - GitLab project\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFactsFromGitLab(\n  base: string, project: string\n): Observable<SourceFacts> {\n  const url = `https://${base}/api/v4/projects/${encodeURIComponent(project)}`\n  return requestJSON<ProjectSchema>(url)\n    .pipe(\n      catchError(() => EMPTY), // @todo refactor instant loading\n      map(({ star_count, forks_count }) => ({\n        stars: star_count,\n        forks: forks_count\n      })),\n      defaultIfEmpty({})\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { EMPTY, Observable } from \"rxjs\"\n\nimport { fetchSourceFactsFromGitHub } from \"../github\"\nimport { fetchSourceFactsFromGitLab } from \"../gitlab\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository facts for repositories\n */\nexport interface RepositoryFacts {\n  stars?: number                       /* Number of stars */\n  forks?: number                       /* Number of forks */\n  version?: string                     /* Latest version */\n}\n\n/**\n * Repository facts for organizations\n */\nexport interface OrganizationFacts {\n  repositories?: number                /* Number of repositories */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Repository facts\n */\nexport type SourceFacts =\n  | RepositoryFacts\n  | OrganizationFacts\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch repository facts\n *\n * @param url - Repository URL\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFacts(\n  url: string\n): Observable<SourceFacts> {\n\n  /* Try to match GitHub repository */\n  let match = url.match(/^.+github\\.com\\/([^/]+)\\/?([^/]+)?/i)\n  if (match) {\n    const [, user, repo] = match\n    return fetchSourceFactsFromGitHub(user, repo)\n  }\n\n  /* Try to match GitLab repository */\n  match = url.match(/^.+?([^/]*gitlab[^/]+)\\/(.+?)\\/?$/i)\n  if (match) {\n    const [, base, slug] = match\n    return fetchSourceFactsFromGitLab(base, slug)\n  }\n\n  /* Fallback */\n  return EMPTY\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  catchError,\n  defer,\n  filter,\n  finalize,\n  map,\n  of,\n  shareReplay,\n  tap\n} from \"rxjs\"\n\nimport { getElement } from \"~/browser\"\nimport { ConsentDefaults } from \"~/components/consent\"\nimport { renderSourceFacts } from \"~/templates\"\n\nimport {\n  Component,\n  getComponentElements\n} from \"../../_\"\nimport {\n  SourceFacts,\n  fetchSourceFacts\n} from \"../facts\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository information\n */\nexport interface Source {\n  facts: SourceFacts                   /* Repository facts */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository information observable\n */\nlet fetch$: Observable<Source>\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch repository information\n *\n * This function tries to read the repository facts from session storage, and\n * if unsuccessful, fetches them from the underlying provider.\n *\n * @param el - Repository information element\n *\n * @returns Repository information observable\n */\nexport function watchSource(\n  el: HTMLAnchorElement\n): Observable<Source> {\n  return fetch$ ||= defer(() => {\n    const cached = __md_get<SourceFacts>(\"__source\", sessionStorage)\n    if (cached) {\n      return of(cached)\n    } else {\n\n      /* Check if consent is configured and was given */\n      const els = getComponentElements(\"consent\")\n      if (els.length) {\n        const consent = __md_get<ConsentDefaults>(\"__consent\")\n        if (!(consent && consent.github))\n          return EMPTY\n      }\n\n      /* Fetch repository facts */\n      return fetchSourceFacts(el.href)\n        .pipe(\n          tap(facts => __md_set(\"__source\", facts, sessionStorage))\n        )\n    }\n  })\n    .pipe(\n      catchError(() => EMPTY),\n      filter(facts => Object.keys(facts).length > 0),\n      map(facts => ({ facts })),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount repository information\n *\n * @param el - Repository information element\n *\n * @returns Repository information component observable\n */\nexport function mountSource(\n  el: HTMLAnchorElement\n): Observable<Component<Source>> {\n  const inner = getElement(\":scope > :last-child\", el)\n  return defer(() => {\n    const push$ = new Subject<Source>()\n    push$.subscribe(({ facts }) => {\n      inner.appendChild(renderSourceFacts(facts))\n      inner.classList.add(\"md-source__repository--active\")\n    })\n\n    /* Create and return component */\n    return watchSource(el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  defer,\n  distinctUntilKeyChanged,\n  finalize,\n  map,\n  of,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  watchElementSize,\n  watchViewportAt\n} from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Navigation tabs\n */\nexport interface Tabs {\n  hidden: boolean                      /* Navigation tabs are hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch navigation tabs\n *\n * @param el - Navigation tabs element\n * @param options - Options\n *\n * @returns Navigation tabs observable\n */\nexport function watchTabs(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<Tabs> {\n  return watchElementSize(document.body)\n    .pipe(\n      switchMap(() => watchViewportAt(el, { header$, viewport$ })),\n      map(({ offset: { y } }) => {\n        return {\n          hidden: y >= 10\n        }\n      }),\n      distinctUntilKeyChanged(\"hidden\")\n    )\n}\n\n/**\n * Mount navigation tabs\n *\n * This function hides the navigation tabs when scrolling past the threshold\n * and makes them reappear in a nice CSS animation when scrolling back up.\n *\n * @param el - Navigation tabs element\n * @param options - Options\n *\n * @returns Navigation tabs component observable\n */\nexport function mountTabs(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Tabs>> {\n  return defer(() => {\n    const push$ = new Subject<Tabs>()\n    push$.subscribe({\n\n      /* Handle emission */\n      next({ hidden }) {\n        el.hidden = hidden\n      },\n\n      /* Handle complete */\n      complete() {\n        el.hidden = false\n      }\n    })\n\n    /* Create and return component */\n    return (\n      feature(\"navigation.tabs.sticky\")\n        ? of({ hidden: false })\n        : watchTabs(el, options)\n    )\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  asyncScheduler,\n  bufferCount,\n  combineLatestWith,\n  debounceTime,\n  defer,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  filter,\n  finalize,\n  ignoreElements,\n  map,\n  merge,\n  observeOn,\n  of,\n  repeat,\n  scan,\n  share,\n  skip,\n  startWith,\n  switchMap,\n  takeUntil,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  getElement,\n  getElementContainer,\n  getElementSize,\n  getElements,\n  getLocation,\n  getOptionalElement,\n  watchElementSize\n} from \"~/browser\"\n\nimport {\n  Component,\n  getComponentElement\n} from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Table of contents\n */\nexport interface TableOfContents {\n  prev: HTMLAnchorElement[][]          /* Anchors (previous) */\n  next: HTMLAnchorElement[][]          /* Anchors (next) */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch table of contents\n *\n * This is effectively a scroll spy implementation which will account for the\n * fixed header and automatically re-calculate anchor offsets when the viewport\n * is resized. The returned observable will only emit if the table of contents\n * needs to be repainted.\n *\n * This implementation tracks an anchor element's entire path starting from its\n * level up to the top-most anchor element, e.g. `[h3, h2, h1]`. Although the\n * Material theme currently doesn't make use of this information, it enables\n * the styling of the entire hierarchy through customization.\n *\n * Note that the current anchor is the last item of the `prev` anchor list.\n *\n * @param el - Table of contents element\n * @param options - Options\n *\n * @returns Table of contents observable\n */\nexport function watchTableOfContents(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<TableOfContents> {\n  const table = new Map<HTMLAnchorElement, HTMLElement>()\n\n  /* Compute anchor-to-target mapping */\n  const anchors = getElements<HTMLAnchorElement>(\"[href^=\\\\#]\", el)\n  for (const anchor of anchors) {\n    const id = decodeURIComponent(anchor.hash.substring(1))\n    const target = getOptionalElement(`[id=\"${id}\"]`)\n    if (typeof target !== \"undefined\")\n      table.set(anchor, target)\n  }\n\n  /* Compute necessary adjustment for header */\n  const adjust$ = header$\n    .pipe(\n      distinctUntilKeyChanged(\"height\"),\n      map(({ height }) => {\n        const main = getComponentElement(\"main\")\n        const grid = getElement(\":scope > :first-child\", main)\n        return height + 0.8 * (\n          grid.offsetTop -\n          main.offsetTop\n        )\n      }),\n      share()\n    )\n\n  /* Compute partition of previous and next anchors */\n  const partition$ = watchElementSize(document.body)\n    .pipe(\n      distinctUntilKeyChanged(\"height\"),\n\n      /* Build index to map anchor paths to vertical offsets */\n      switchMap(body => defer(() => {\n        let path: HTMLAnchorElement[] = []\n        return of([...table].reduce((index, [anchor, target]) => {\n          while (path.length) {\n            const last = table.get(path[path.length - 1])!\n            if (last.tagName >= target.tagName) {\n              path.pop()\n            } else {\n              break\n            }\n          }\n\n          /* If the current anchor is hidden, continue with its parent */\n          let offset = target.offsetTop\n          while (!offset && target.parentElement) {\n            target = target.parentElement\n            offset = target.offsetTop\n          }\n\n          /* Fix anchor offsets in tables - see https://bit.ly/3CUFOcn */\n          let parent = target.offsetParent as HTMLElement\n          for (; parent; parent = parent.offsetParent as HTMLElement)\n            offset += parent.offsetTop\n\n          /* Map reversed anchor path to vertical offset */\n          return index.set(\n            [...path = [...path, anchor]].reverse(),\n            offset\n          )\n        }, new Map<HTMLAnchorElement[], number>()))\n      })\n        .pipe(\n\n          /* Sort index by vertical offset (see https://bit.ly/30z6QSO) */\n          map(index => new Map([...index].sort(([, a], [, b]) => a - b))),\n          combineLatestWith(adjust$),\n\n          /* Re-compute partition when viewport offset changes */\n          switchMap(([index, adjust]) => viewport$\n            .pipe(\n              scan(([prev, next], { offset: { y }, size }) => {\n                const last = y + size.height >= Math.floor(body.height)\n\n                /* Look forward */\n                while (next.length) {\n                  const [, offset] = next[0]\n                  if (offset - adjust < y || last) {\n                    prev = [...prev, next.shift()!]\n                  } else {\n                    break\n                  }\n                }\n\n                /* Look backward */\n                while (prev.length) {\n                  const [, offset] = prev[prev.length - 1]\n                  if (offset - adjust >= y && !last) {\n                    next = [prev.pop()!, ...next]\n                  } else {\n                    break\n                  }\n                }\n\n                /* Return partition */\n                return [prev, next]\n              }, [[], [...index]]),\n              distinctUntilChanged((a, b) => (\n                a[0] === b[0] &&\n                a[1] === b[1]\n              ))\n            )\n          )\n        )\n      )\n    )\n\n  /* Compute and return anchor list migrations */\n  return partition$\n    .pipe(\n      map(([prev, next]) => ({\n        prev: prev.map(([path]) => path),\n        next: next.map(([path]) => path)\n      })),\n\n      /* Extract anchor list migrations */\n      startWith({ prev: [], next: [] }),\n      bufferCount(2, 1),\n      map(([a, b]) => {\n\n        /* Moving down */\n        if (a.prev.length < b.prev.length) {\n          return {\n            prev: b.prev.slice(Math.max(0, a.prev.length - 1), b.prev.length),\n            next: []\n          }\n\n        /* Moving up */\n        } else {\n          return {\n            prev: b.prev.slice(-1),\n            next: b.next.slice(0, b.next.length - a.next.length)\n          }\n        }\n      })\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount table of contents\n *\n * @param el - Table of contents element\n * @param options - Options\n *\n * @returns Table of contents component observable\n */\nexport function mountTableOfContents(\n  el: HTMLElement, { viewport$, header$, main$, target$ }: MountOptions\n): Observable<Component<TableOfContents>> {\n  return defer(() => {\n    const push$ = new Subject<TableOfContents>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    push$.subscribe(({ prev, next }) => {\n\n      /* Look forward */\n      for (const [anchor] of next) {\n        anchor.classList.remove(\"md-nav__link--passed\")\n        anchor.classList.remove(\"md-nav__link--active\")\n      }\n\n      /* Look backward */\n      for (const [index, [anchor]] of prev.entries()) {\n        anchor.classList.add(\"md-nav__link--passed\")\n        anchor.classList.toggle(\n          \"md-nav__link--active\",\n          index === prev.length - 1\n        )\n      }\n    })\n\n    /* Set up following, if enabled */\n    if (feature(\"toc.follow\")) {\n\n      /* Toggle smooth scrolling only for anchor clicks */\n      const smooth$ = merge(\n        viewport$.pipe(debounceTime(1), map(() => undefined)),\n        viewport$.pipe(debounceTime(250), map(() => \"smooth\" as const))\n      )\n\n      /* Bring active anchor into view */ // @todo: refactor\n      push$\n        .pipe(\n          filter(({ prev }) => prev.length > 0),\n          combineLatestWith(main$.pipe(observeOn(asyncScheduler))),\n          withLatestFrom(smooth$)\n        )\n          .subscribe(([[{ prev }], behavior]) => {\n            const [anchor] = prev[prev.length - 1]\n            if (anchor.offsetHeight) {\n\n              /* Retrieve overflowing container and scroll */\n              const container = getElementContainer(anchor)\n              if (typeof container !== \"undefined\") {\n                const offset = anchor.offsetTop - container.offsetTop\n                const { height } = getElementSize(container)\n                container.scrollTo({\n                  top: offset - height / 2,\n                  behavior\n                })\n              }\n            }\n          })\n    }\n\n    /* Set up anchor tracking, if enabled */\n    if (feature(\"navigation.tracking\"))\n      viewport$\n        .pipe(\n          takeUntil(done$),\n          distinctUntilKeyChanged(\"offset\"),\n          debounceTime(250),\n          skip(1),\n          takeUntil(target$.pipe(skip(1))),\n          repeat({ delay: 250 }),\n          withLatestFrom(push$)\n        )\n          .subscribe(([, { prev }]) => {\n            const url = getLocation()\n\n            /* Set hash fragment to active anchor */\n            const anchor = prev[prev.length - 1]\n            if (anchor && anchor.length) {\n              const [active] = anchor\n              const { hash } = new URL(active.href)\n              if (url.hash !== hash) {\n                url.hash = hash\n                history.replaceState({}, \"\", `${url}`)\n              }\n\n            /* Reset anchor when at the top */\n            } else {\n              url.hash = \"\"\n              history.replaceState({}, \"\", `${url}`)\n            }\n          })\n\n    /* Create and return component */\n    return watchTableOfContents(el, { viewport$, header$ })\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  bufferCount,\n  combineLatest,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  repeat,\n  skip,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport { Viewport } from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Back-to-top button\n */\nexport interface BackToTop {\n  hidden: boolean                      /* Back-to-top button is hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  main$: Observable<Main>              /* Main area observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch back-to-top\n *\n * @param _el - Back-to-top element\n * @param options - Options\n *\n * @returns Back-to-top observable\n */\nexport function watchBackToTop(\n  _el: HTMLElement, { viewport$, main$, target$ }: WatchOptions\n): Observable<BackToTop> {\n\n  /* Compute direction */\n  const direction$ = viewport$\n    .pipe(\n      map(({ offset: { y } }) => y),\n      bufferCount(2, 1),\n      map(([a, b]) => a > b && b > 0),\n      distinctUntilChanged()\n    )\n\n  /* Compute whether main area is active */\n  const active$ = main$\n    .pipe(\n      map(({ active }) => active)\n    )\n\n  /* Compute threshold for hiding */\n  return combineLatest([active$, direction$])\n    .pipe(\n      map(([active, direction]) => !(active && direction)),\n      distinctUntilChanged(),\n      takeUntil(target$.pipe(skip(1))),\n      endWith(true),\n      repeat({ delay: 250 }),\n      map(hidden => ({ hidden }))\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount back-to-top\n *\n * @param el - Back-to-top element\n * @param options - Options\n *\n * @returns Back-to-top component observable\n */\nexport function mountBackToTop(\n  el: HTMLElement, { viewport$, header$, main$, target$ }: MountOptions\n): Observable<Component<BackToTop>> {\n  const push$ = new Subject<BackToTop>()\n  const done$ = push$.pipe(ignoreElements(), endWith(true))\n  push$.subscribe({\n\n    /* Handle emission */\n    next({ hidden }) {\n      el.hidden = hidden\n      if (hidden) {\n        el.setAttribute(\"tabindex\", \"-1\")\n        el.blur()\n      } else {\n        el.removeAttribute(\"tabindex\")\n      }\n    },\n\n    /* Handle complete */\n    complete() {\n      el.style.top = \"\"\n      el.hidden = true\n      el.removeAttribute(\"tabindex\")\n    }\n  })\n\n  /* Watch header height */\n  header$\n    .pipe(\n      takeUntil(done$),\n      distinctUntilKeyChanged(\"height\")\n    )\n      .subscribe(({ height }) => {\n        el.style.top = `${height + 16}px`\n      })\n\n  /* Go back to top */\n  fromEvent(el, \"click\")\n    .subscribe(ev => {\n      ev.preventDefault()\n      window.scrollTo({ top: 0 })\n    })\n\n  /* Create and return component */\n  return watchBackToTop(el, { viewport$, main$, target$ })\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  mergeMap,\n  switchMap,\n  takeWhile,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n  document$: Observable<Document>      /* Document observable */\n  tablet$: Observable<boolean>         /* Media tablet observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch indeterminate checkboxes\n *\n * This function replaces the indeterminate \"pseudo state\" with the actual\n * indeterminate state, which is used to keep navigation always expanded.\n *\n * @param options - Options\n */\nexport function patchIndeterminate(\n  { document$, tablet$ }: PatchOptions\n): void {\n  document$\n    .pipe(\n      switchMap(() => getElements<HTMLInputElement>(\n        \".md-toggle--indeterminate\"\n      )),\n      tap(el => {\n        el.indeterminate = true\n        el.checked = false\n      }),\n      mergeMap(el => fromEvent(el, \"change\")\n        .pipe(\n          takeWhile(() => el.classList.contains(\"md-toggle--indeterminate\")),\n          map(() => el)\n        )\n      ),\n      withLatestFrom(tablet$)\n    )\n      .subscribe(([el, tablet]) => {\n        el.classList.remove(\"md-toggle--indeterminate\")\n        if (tablet)\n          el.checked = false\n      })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  filter,\n  fromEvent,\n  map,\n  mergeMap,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n  document$: Observable<Document>      /* Document observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Check whether the given device is an Apple device\n *\n * @returns Test result\n */\nfunction isAppleDevice(): boolean {\n  return /(iPad|iPhone|iPod)/.test(navigator.userAgent)\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch all elements with `data-md-scrollfix` attributes\n *\n * This is a year-old patch which ensures that overflow scrolling works at the\n * top and bottom of containers on iOS by ensuring a `1px` scroll offset upon\n * the start of a touch event.\n *\n * @see https://bit.ly/2SCtAOO - Original source\n *\n * @param options - Options\n */\nexport function patchScrollfix(\n  { document$ }: PatchOptions\n): void {\n  document$\n    .pipe(\n      switchMap(() => getElements(\"[data-md-scrollfix]\")),\n      tap(el => el.removeAttribute(\"data-md-scrollfix\")),\n      filter(isAppleDevice),\n      mergeMap(el => fromEvent(el, \"touchstart\")\n        .pipe(\n          map(() => el)\n        )\n      )\n    )\n      .subscribe(el => {\n        const top = el.scrollTop\n\n        /* We're at the top of the container */\n        if (top === 0) {\n          el.scrollTop = 1\n\n        /* We're at the bottom of the container */\n        } else if (top + el.offsetHeight === el.scrollHeight) {\n          el.scrollTop = top - 1\n        }\n      })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  delay,\n  map,\n  of,\n  switchMap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  watchToggle\n} from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  tablet$: Observable<boolean>         /* Media tablet observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch the document body to lock when search is open\n *\n * For mobile and tablet viewports, the search is rendered full screen, which\n * leads to scroll leaking when at the top or bottom of the search result. This\n * function locks the body when the search is in full screen mode, and restores\n * the scroll position when leaving.\n *\n * @param options - Options\n */\nexport function patchScrolllock(\n  { viewport$, tablet$ }: PatchOptions\n): void {\n  combineLatest([watchToggle(\"search\"), tablet$])\n    .pipe(\n      map(([active, tablet]) => active && !tablet),\n      switchMap(active => of(active)\n        .pipe(\n          delay(active ? 400 : 100)\n        )\n      ),\n      withLatestFrom(viewport$)\n    )\n      .subscribe(([active, { offset: { y }}]) => {\n        if (active) {\n          document.body.setAttribute(\"data-md-scrolllock\", \"\")\n          document.body.style.top = `-${y}px`\n        } else {\n          const value = -1 * parseInt(document.body.style.top, 10)\n          document.body.removeAttribute(\"data-md-scrolllock\")\n          document.body.style.top = \"\"\n          if (value)\n            window.scrollTo(0, value)\n        }\n      })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Polyfills\n * ------------------------------------------------------------------------- */\n\n/* Polyfill `Object.entries` */\nif (!Object.entries)\n  Object.entries = function (obj: object) {\n    const data: [string, string][] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push([key, obj[key]])\n\n    /* Return entries */\n    return data\n  }\n\n/* Polyfill `Object.values` */\nif (!Object.values)\n  Object.values = function (obj: object) {\n    const data: string[] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push(obj[key])\n\n    /* Return values */\n    return data\n  }\n\n/* ------------------------------------------------------------------------- */\n\n/* Polyfills for `Element` */\nif (typeof Element !== \"undefined\") {\n\n  /* Polyfill `Element.scrollTo` */\n  if (!Element.prototype.scrollTo)\n    Element.prototype.scrollTo = function (\n      x?: ScrollToOptions | number, y?: number\n    ): void {\n      if (typeof x === \"object\") {\n        this.scrollLeft = x.left!\n        this.scrollTop = x.top!\n      } else {\n        this.scrollLeft = x!\n        this.scrollTop = y!\n      }\n    }\n\n  /* Polyfill `Element.replaceWith` */\n  if (!Element.prototype.replaceWith)\n    Element.prototype.replaceWith = function (\n      ...nodes: Array<string | Node>\n    ): void {\n      const parent = this.parentNode\n      if (parent) {\n        if (nodes.length === 0)\n          parent.removeChild(this)\n\n        /* Replace children and create text nodes */\n        for (let i = nodes.length - 1; i >= 0; i--) {\n          let node = nodes[i]\n          if (typeof node === \"string\")\n            node = document.createTextNode(node)\n          else if (node.parentNode)\n            node.parentNode.removeChild(node)\n\n          /* Replace child or insert before previous sibling */\n          if (!i)\n            parent.replaceChild(node, this)\n          else\n            parent.insertBefore(this.previousSibling!, node)\n        }\n      }\n    }\n}\n"],
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r=new x,o=r.pipe(ee(),oe(!0));B([t.pipe(Pe(Ht)),r],(i,s)=>s).pipe(te("value")).subscribe(({value:i})=>t.next({type:2,data:i})),r.pipe(te("focus")).subscribe(({focus:i})=>{i&&Ye("search",i)}),h(e.form,"reset").pipe(j(o)).subscribe(()=>e.focus());let n=U("header [for=__search]");return h(n,"click").subscribe(()=>e.focus()),Da(e,{worker$:t}).pipe(T(i=>r.next(i)),A(()=>r.complete()),m(i=>P({ref:e},i)),Z(1))}function Gn(e,{worker$:t,query$:r}){let o=new x,n=Go(e.parentElement).pipe(v(Boolean)),i=e.parentElement,s=U(":scope > :first-child",e),a=U(":scope > :last-child",e);Ne("search").subscribe(l=>a.setAttribute("role",l?"list":"presentation")),o.pipe(ae(r),Wr(t.pipe(Pe(Ht)))).subscribe(([{items:l},{value:f}])=>{switch(l.length){case 0:s.textContent=f.length?we("search.result.none"):we("search.result.placeholder");break;case 1:s.textContent=we("search.result.one");break;default:let u=ar(l.length);s.textContent=we("search.result.other",u)}});let c=o.pipe(T(()=>a.innerHTML=""),w(({items:l})=>L(R(...l.slice(0,10)),R(...l.slice(10)).pipe(Ce(4),Nr(n),w(([f])=>f)))),m(hn),de());return c.subscribe(l=>a.appendChild(l)),c.pipe(re(l=>{let f=ce("details",l);return typeof f=="undefined"?M:h(f,"toggle").pipe(j(o),m(()=>f))})).subscribe(l=>{l.open===!1&&l.offsetTop<=i.scrollTop&&i.scrollTo({top:l.offsetTop})}),t.pipe(v(dr),m(({data:l})=>l)).pipe(T(l=>o.next(l)),A(()=>o.complete()),m(l=>P({ref:e},l)))}function Va(e,{query$:t}){return t.pipe(m(({value:r})=>{let o=me();return o.hash="",r=r.replace(/\s+/g,"+").replace(/&/g,"%26").replace(/=/g,"%3D"),o.search=`q=${r}`,{url:o}}))}function Jn(e,t){let r=new x,o=r.pipe(ee(),oe(!0));return r.subscribe(({url:n})=>{e.setAttribute("data-clipboard-text",e.href),e.href=`${n}`}),h(e,"click").pipe(j(o)).subscribe(n=>n.preventDefault()),Va(e,t).pipe(T(n=>r.next(n)),A(()=>r.complete()),m(n=>P({ref:e},n)))}function Xn(e,{worker$:t,keyboard$:r}){let o=new 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r=`https://api.github.com/users/${e}`;return De(r).pipe(m(o=>({repositories:o.public_repos})),$e({}))}}function ri(e,t){let r=`https://${e}/api/v4/projects/${encodeURIComponent(t)}`;return De(r).pipe(xe(()=>M),m(({star_count:o,forks_count:n})=>({stars:o,forks:n})),$e({}))}function oi(e){let t=e.match(/^.+github\.com\/([^/]+)\/?([^/]+)?/i);if(t){let[,r,o]=t;return ti(r,o)}if(t=e.match(/^.+?([^/]*gitlab[^/]+)\/(.+?)\/?$/i),t){let[,r,o]=t;return ri(r,o)}return M}var qa;function Ka(e){return qa||(qa=H(()=>{let t=__md_get("__source",sessionStorage);if(t)return R(t);if(ne("consent").length){let o=__md_get("__consent");if(!(o&&o.github))return M}return oi(e.href).pipe(T(o=>__md_set("__source",o,sessionStorage)))}).pipe(xe(()=>M),v(t=>Object.keys(t).length>0),m(t=>({facts:t})),Z(1)))}function ni(e){let t=U(":scope > :last-child",e);return H(()=>{let r=new x;return 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i=new x,s=i.pipe(ee(),oe(!0));if(i.subscribe(({prev:a,next:c})=>{for(let[p]of c)p.classList.remove("md-nav__link--passed"),p.classList.remove("md-nav__link--active");for(let[p,[l]]of a.entries())l.classList.add("md-nav__link--passed"),l.classList.toggle("md-nav__link--active",p===a.length-1)}),G("toc.follow")){let a=L(t.pipe(ye(1),m(()=>{})),t.pipe(ye(250),m(()=>"smooth")));i.pipe(v(({prev:c})=>c.length>0),Ze(o.pipe(Me(ie))),ae(a)).subscribe(([[{prev:c}],p])=>{let[l]=c[c.length-1];if(l.offsetHeight){let f=sr(l);if(typeof f!="undefined"){let u=l.offsetTop-f.offsetTop,{height:d}=le(f);f.scrollTo({top:u-d/2,behavior:p})}}})}return G("navigation.tracking")&&t.pipe(j(s),te("offset"),ye(250),Ee(1),j(n.pipe(Ee(1))),at({delay:250}),ae(i)).subscribe(([,{prev:a}])=>{let c=me(),p=a[a.length-1];if(p&&p.length){let[l]=p,{hash:f}=new URL(l.href);c.hash!==f&&(c.hash=f,history.replaceState({},"",`${c}`))}else 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t!="undefined"&&st(t);break;case"n":case".":let r=ce("link[rel=next]");typeof r!="undefined"&&st(r);break;case"Enter":let o=Ie();o instanceof HTMLLabelElement&&o.click()}});ci({document$:rt});pi({document$:rt,tablet$:hr});li({document$:rt});mi({viewport$:_e,tablet$:hr});var tt=Pn(Oe("header"),{viewport$:_e}),$t=rt.pipe(m(()=>Oe("main")),w(e=>Fn(e,{viewport$:_e,header$:tt})),Z(1)),Xa=L(...ne("consent").map(e=>fn(e,{target$:wt})),...ne("dialog").map(e=>$n(e,{alert$:Xr})),...ne("header").map(e=>Rn(e,{viewport$:_e,header$:tt,main$:$t})),...ne("palette").map(e=>jn(e)),...ne("progress").map(e=>Wn(e,{progress$:Zr})),...ne("search").map(e=>Zn(e,{index$:hi,keyboard$:Jr})),...ne("source").map(e=>ni(e))),Za=H(()=>L(...ne("announce").map(e=>mn(e)),...ne("content").map(e=>Hn(e,{viewport$:_e,target$:wt,print$:di})),...ne("content").map(e=>G("search.highlight")?ei(e,{index$:hi,location$:Pt}):M),...ne("header-title").map(e=>In(e,{viewport$:_e,header$:tt})),...ne("sidebar").map(e=>e.getAttribute("data-md-type")==="navigation"?Dr(ui,()=>Br(e,{viewport$:_e,header$:tt,main$:$t})):Dr(hr,()=>Br(e,{viewport$:_e,header$:tt,main$:$t}))),...ne("tabs").map(e=>ii(e,{viewport$:_e,header$:tt})),...ne("toc").map(e=>ai(e,{viewport$:_e,header$:tt,main$:$t,target$:wt})),...ne("top").map(e=>si(e,{viewport$:_e,header$:tt,main$:$t,target$:wt})))),bi=rt.pipe(w(()=>Za),Re(Xa),Z(1));bi.subscribe();window.document$=rt;window.location$=Pt;window.target$=wt;window.keyboard$=Jr;window.viewport$=_e;window.tablet$=hr;window.screen$=ui;window.print$=di;window.alert$=Xr;window.progress$=Zr;window.component$=bi;})();
+//# sourceMappingURL=bundle.d7c377c4.min.js.map
+
diff --git a/assets/javascripts/bundle.d7c377c4.min.js.map b/assets/javascripts/bundle.d7c377c4.min.js.map
new file mode 100644
index 000000000..a57d388af
--- /dev/null
+++ b/assets/javascripts/bundle.d7c377c4.min.js.map
@@ -0,0 +1,7 @@
+{
+  "version": 3,
+  "sources": ["node_modules/focus-visible/dist/focus-visible.js", "node_modules/clipboard/dist/clipboard.js", "node_modules/escape-html/index.js", "src/templates/assets/javascripts/bundle.ts", "node_modules/rxjs/node_modules/tslib/tslib.es6.js", "node_modules/rxjs/src/internal/util/isFunction.ts", "node_modules/rxjs/src/internal/util/createErrorClass.ts", "node_modules/rxjs/src/internal/util/UnsubscriptionError.ts", "node_modules/rxjs/src/internal/util/arrRemove.ts", "node_modules/rxjs/src/internal/Subscription.ts", "node_modules/rxjs/src/internal/config.ts", "node_modules/rxjs/src/internal/scheduler/timeoutProvider.ts", "node_modules/rxjs/src/internal/util/reportUnhandledError.ts", "node_modules/rxjs/src/internal/util/noop.ts", "node_modules/rxjs/src/internal/NotificationFactories.ts", "node_modules/rxjs/src/internal/util/errorContext.ts", "node_modules/rxjs/src/internal/Subscriber.ts", "node_modules/rxjs/src/internal/symbol/observable.ts", "node_modules/rxjs/src/internal/util/identity.ts", "node_modules/rxjs/src/internal/util/pipe.ts", "node_modules/rxjs/src/internal/Observable.ts", "node_modules/rxjs/src/internal/util/lift.ts", "node_modules/rxjs/src/internal/operators/OperatorSubscriber.ts", "node_modules/rxjs/src/internal/scheduler/animationFrameProvider.ts", "node_modules/rxjs/src/internal/util/ObjectUnsubscribedError.ts", "node_modules/rxjs/src/internal/Subject.ts", "node_modules/rxjs/src/internal/scheduler/dateTimestampProvider.ts", "node_modules/rxjs/src/internal/ReplaySubject.ts", "node_modules/rxjs/src/internal/scheduler/Action.ts", "node_modules/rxjs/src/internal/scheduler/intervalProvider.ts", "node_modules/rxjs/src/internal/scheduler/AsyncAction.ts", "node_modules/rxjs/src/internal/Scheduler.ts", "node_modules/rxjs/src/internal/scheduler/AsyncScheduler.ts", "node_modules/rxjs/src/internal/scheduler/async.ts", "node_modules/rxjs/src/internal/scheduler/AnimationFrameAction.ts", "node_modules/rxjs/src/internal/scheduler/AnimationFrameScheduler.ts", "node_modules/rxjs/src/internal/scheduler/animationFrame.ts", "node_modules/rxjs/src/internal/observable/empty.ts", "node_modules/rxjs/src/internal/util/isScheduler.ts", "node_modules/rxjs/src/internal/util/args.ts", "node_modules/rxjs/src/internal/util/isArrayLike.ts", "node_modules/rxjs/src/internal/util/isPromise.ts", "node_modules/rxjs/src/internal/util/isInteropObservable.ts", "node_modules/rxjs/src/internal/util/isAsyncIterable.ts", "node_modules/rxjs/src/internal/util/throwUnobservableError.ts", "node_modules/rxjs/src/internal/symbol/iterator.ts", "node_modules/rxjs/src/internal/util/isIterable.ts", "node_modules/rxjs/src/internal/util/isReadableStreamLike.ts", "node_modules/rxjs/src/internal/observable/innerFrom.ts", "node_modules/rxjs/src/internal/util/executeSchedule.ts", "node_modules/rxjs/src/internal/operators/observeOn.ts", "node_modules/rxjs/src/internal/operators/subscribeOn.ts", "node_modules/rxjs/src/internal/scheduled/scheduleObservable.ts", "node_modules/rxjs/src/internal/scheduled/schedulePromise.ts", "node_modules/rxjs/src/internal/scheduled/scheduleArray.ts", "node_modules/rxjs/src/internal/scheduled/scheduleIterable.ts", "node_modules/rxjs/src/internal/scheduled/scheduleAsyncIterable.ts", "node_modules/rxjs/src/internal/scheduled/scheduleReadableStreamLike.ts", "node_modules/rxjs/src/internal/scheduled/scheduled.ts", "node_modules/rxjs/src/internal/observable/from.ts", "node_modules/rxjs/src/internal/observable/of.ts", "node_modules/rxjs/src/internal/observable/throwError.ts", "node_modules/rxjs/src/internal/util/EmptyError.ts", "node_modules/rxjs/src/internal/util/isDate.ts", "node_modules/rxjs/src/internal/operators/map.ts", "node_modules/rxjs/src/internal/util/mapOneOrManyArgs.ts", "node_modules/rxjs/src/internal/util/argsArgArrayOrObject.ts", "node_modules/rxjs/src/internal/util/createObject.ts", "node_modules/rxjs/src/internal/observable/combineLatest.ts", "node_modules/rxjs/src/internal/operators/mergeInternals.ts", "node_modules/rxjs/src/internal/operators/mergeMap.ts", "node_modules/rxjs/src/internal/operators/mergeAll.ts", "node_modules/rxjs/src/internal/operators/concatAll.ts", "node_modules/rxjs/src/internal/observable/concat.ts", "node_modules/rxjs/src/internal/observable/defer.ts", "node_modules/rxjs/src/internal/observable/fromEvent.ts", "node_modules/rxjs/src/internal/observable/fromEventPattern.ts", "node_modules/rxjs/src/internal/observable/timer.ts", "node_modules/rxjs/src/internal/observable/merge.ts", "node_modules/rxjs/src/internal/observable/never.ts", "node_modules/rxjs/src/internal/util/argsOrArgArray.ts", "node_modules/rxjs/src/internal/operators/filter.ts", "node_modules/rxjs/src/internal/observable/zip.ts", "node_modules/rxjs/src/internal/operators/audit.ts", "node_modules/rxjs/src/internal/operators/auditTime.ts", "node_modules/rxjs/src/internal/operators/bufferCount.ts", "node_modules/rxjs/src/internal/operators/catchError.ts", "node_modules/rxjs/src/internal/operators/scanInternals.ts", "node_modules/rxjs/src/internal/operators/combineLatest.ts", "node_modules/rxjs/src/internal/operators/combineLatestWith.ts", "node_modules/rxjs/src/internal/operators/debounceTime.ts", "node_modules/rxjs/src/internal/operators/defaultIfEmpty.ts", "node_modules/rxjs/src/internal/operators/take.ts", "node_modules/rxjs/src/internal/operators/ignoreElements.ts", "node_modules/rxjs/src/internal/operators/mapTo.ts", "node_modules/rxjs/src/internal/operators/delayWhen.ts", "node_modules/rxjs/src/internal/operators/delay.ts", "node_modules/rxjs/src/internal/operators/distinctUntilChanged.ts", "node_modules/rxjs/src/internal/operators/distinctUntilKeyChanged.ts", "node_modules/rxjs/src/internal/operators/throwIfEmpty.ts", "node_modules/rxjs/src/internal/operators/endWith.ts", "node_modules/rxjs/src/internal/operators/finalize.ts", "node_modules/rxjs/src/internal/operators/first.ts", "node_modules/rxjs/src/internal/operators/takeLast.ts", "node_modules/rxjs/src/internal/operators/merge.ts", "node_modules/rxjs/src/internal/operators/mergeWith.ts", "node_modules/rxjs/src/internal/operators/repeat.ts", "node_modules/rxjs/src/internal/operators/sample.ts", "node_modules/rxjs/src/internal/operators/scan.ts", "node_modules/rxjs/src/internal/operators/share.ts", "node_modules/rxjs/src/internal/operators/shareReplay.ts", "node_modules/rxjs/src/internal/operators/skip.ts", "node_modules/rxjs/src/internal/operators/skipUntil.ts", "node_modules/rxjs/src/internal/operators/startWith.ts", "node_modules/rxjs/src/internal/operators/switchMap.ts", "node_modules/rxjs/src/internal/operators/takeUntil.ts", "node_modules/rxjs/src/internal/operators/takeWhile.ts", "node_modules/rxjs/src/internal/operators/tap.ts", "node_modules/rxjs/src/internal/operators/throttle.ts", 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+  "sourcesContent": ["(function (global, factory) {\n  typeof exports === 'object' && typeof module !== 'undefined' ? factory() :\n  typeof define === 'function' && define.amd ? define(factory) :\n  (factory());\n}(this, (function () { 'use strict';\n\n  /**\n   * Applies the :focus-visible polyfill at the given scope.\n   * A scope in this case is either the top-level Document or a Shadow Root.\n   *\n   * @param {(Document|ShadowRoot)} scope\n   * @see https://github.com/WICG/focus-visible\n   */\n  function applyFocusVisiblePolyfill(scope) {\n    var hadKeyboardEvent = true;\n    var hadFocusVisibleRecently = false;\n    var hadFocusVisibleRecentlyTimeout = null;\n\n    var inputTypesAllowlist = {\n      text: true,\n      search: true,\n      url: true,\n      tel: true,\n      email: true,\n      password: true,\n      number: true,\n      date: true,\n      month: true,\n      week: true,\n      time: true,\n      datetime: true,\n      'datetime-local': true\n    };\n\n    /**\n     * Helper function for legacy browsers and iframes which sometimes focus\n     * elements like document, body, and non-interactive SVG.\n     * @param {Element} el\n     */\n    function isValidFocusTarget(el) {\n      if (\n        el &&\n        el !== document &&\n        el.nodeName !== 'HTML' &&\n        el.nodeName !== 'BODY' &&\n        'classList' in el &&\n        'contains' in el.classList\n      ) {\n        return true;\n      }\n      return false;\n    }\n\n    /**\n     * Computes whether the given element should automatically trigger the\n     * `focus-visible` class being added, i.e. whether it should always match\n     * `:focus-visible` when focused.\n     * @param {Element} el\n     * @return {boolean}\n     */\n    function focusTriggersKeyboardModality(el) {\n      var type = el.type;\n      var tagName = el.tagName;\n\n      if (tagName === 'INPUT' && inputTypesAllowlist[type] && !el.readOnly) {\n        return true;\n      }\n\n      if (tagName === 'TEXTAREA' && !el.readOnly) {\n        return true;\n      }\n\n      if (el.isContentEditable) {\n        return true;\n      }\n\n      return false;\n    }\n\n    /**\n     * Add the `focus-visible` class to the given element if it was not added by\n     * the author.\n     * @param {Element} el\n     */\n    function addFocusVisibleClass(el) {\n      if (el.classList.contains('focus-visible')) {\n        return;\n      }\n      el.classList.add('focus-visible');\n      el.setAttribute('data-focus-visible-added', '');\n    }\n\n    /**\n     * Remove the `focus-visible` class from the given element if it was not\n     * originally added by the author.\n     * @param {Element} el\n     */\n    function removeFocusVisibleClass(el) {\n      if (!el.hasAttribute('data-focus-visible-added')) {\n        return;\n      }\n      el.classList.remove('focus-visible');\n      el.removeAttribute('data-focus-visible-added');\n    }\n\n    /**\n     * If the most recent user interaction was via the keyboard;\n     * and the key press did not include a meta, alt/option, or control key;\n     * then the modality is keyboard. Otherwise, the modality is not keyboard.\n     * Apply `focus-visible` to any current active element and keep track\n     * of our keyboard modality state with `hadKeyboardEvent`.\n     * @param {KeyboardEvent} e\n     */\n    function onKeyDown(e) {\n      if (e.metaKey || e.altKey || e.ctrlKey) {\n        return;\n      }\n\n      if (isValidFocusTarget(scope.activeElement)) {\n        addFocusVisibleClass(scope.activeElement);\n      }\n\n      hadKeyboardEvent = true;\n    }\n\n    /**\n     * If at any point a user clicks with a pointing device, ensure that we change\n     * the modality away from keyboard.\n     * This avoids the situation where a user presses a key on an already focused\n     * element, and then clicks on a different element, focusing it with a\n     * pointing device, while we still think we're in keyboard modality.\n     * @param {Event} e\n     */\n    function onPointerDown(e) {\n      hadKeyboardEvent = false;\n    }\n\n    /**\n     * On `focus`, add the `focus-visible` class to the target if:\n     * - the target received focus as a result of keyboard navigation, or\n     * - the event target is an element that will likely require interaction\n     *   via the keyboard (e.g. a text box)\n     * @param {Event} e\n     */\n    function onFocus(e) {\n      // Prevent IE from focusing the document or HTML element.\n      if (!isValidFocusTarget(e.target)) {\n        return;\n      }\n\n      if (hadKeyboardEvent || focusTriggersKeyboardModality(e.target)) {\n        addFocusVisibleClass(e.target);\n      }\n    }\n\n    /**\n     * On `blur`, remove the `focus-visible` class from the target.\n     * @param {Event} e\n     */\n    function onBlur(e) {\n      if (!isValidFocusTarget(e.target)) {\n        return;\n      }\n\n      if (\n        e.target.classList.contains('focus-visible') ||\n        e.target.hasAttribute('data-focus-visible-added')\n      ) {\n        // To detect a tab/window switch, we look for a blur event followed\n        // rapidly by a visibility change.\n        // If we don't see a visibility change within 100ms, it's probably a\n        // regular focus change.\n        hadFocusVisibleRecently = true;\n        window.clearTimeout(hadFocusVisibleRecentlyTimeout);\n        hadFocusVisibleRecentlyTimeout = window.setTimeout(function() {\n          hadFocusVisibleRecently = false;\n        }, 100);\n        removeFocusVisibleClass(e.target);\n      }\n    }\n\n    /**\n     * If the user changes tabs, keep track of whether or not the previously\n     * focused element had .focus-visible.\n     * @param {Event} e\n     */\n    function onVisibilityChange(e) {\n      if (document.visibilityState === 'hidden') {\n        // If the tab becomes active again, the browser will handle calling focus\n        // on the element (Safari actually calls it twice).\n        // If this tab change caused a blur on an element with focus-visible,\n        // re-apply the class when the user switches back to the tab.\n        if (hadFocusVisibleRecently) {\n          hadKeyboardEvent = true;\n        }\n        addInitialPointerMoveListeners();\n      }\n    }\n\n    /**\n     * Add a group of listeners to detect usage of any pointing devices.\n     * These listeners will be added when the polyfill first loads, and anytime\n     * the window is blurred, so that they are active when the window regains\n     * focus.\n     */\n    function addInitialPointerMoveListeners() {\n      document.addEventListener('mousemove', onInitialPointerMove);\n      document.addEventListener('mousedown', onInitialPointerMove);\n      document.addEventListener('mouseup', onInitialPointerMove);\n      document.addEventListener('pointermove', onInitialPointerMove);\n      document.addEventListener('pointerdown', onInitialPointerMove);\n      document.addEventListener('pointerup', onInitialPointerMove);\n      document.addEventListener('touchmove', onInitialPointerMove);\n      document.addEventListener('touchstart', onInitialPointerMove);\n      document.addEventListener('touchend', onInitialPointerMove);\n    }\n\n    function removeInitialPointerMoveListeners() {\n      document.removeEventListener('mousemove', onInitialPointerMove);\n      document.removeEventListener('mousedown', onInitialPointerMove);\n      document.removeEventListener('mouseup', onInitialPointerMove);\n      document.removeEventListener('pointermove', onInitialPointerMove);\n      document.removeEventListener('pointerdown', onInitialPointerMove);\n      document.removeEventListener('pointerup', onInitialPointerMove);\n      document.removeEventListener('touchmove', onInitialPointerMove);\n      document.removeEventListener('touchstart', onInitialPointerMove);\n      document.removeEventListener('touchend', onInitialPointerMove);\n    }\n\n    /**\n     * When the polfyill first loads, assume the user is in keyboard modality.\n     * If any event is received from a pointing device (e.g. mouse, pointer,\n     * touch), turn off keyboard modality.\n     * This accounts for situations where focus enters the page from the URL bar.\n     * @param {Event} e\n     */\n    function onInitialPointerMove(e) {\n      // Work around a Safari quirk that fires a mousemove on <html> whenever the\n      // window blurs, even if you're tabbing out of the page. \u00AF\\_(\u30C4)_/\u00AF\n      if (e.target.nodeName && e.target.nodeName.toLowerCase() === 'html') {\n        return;\n      }\n\n      hadKeyboardEvent = false;\n      removeInitialPointerMoveListeners();\n    }\n\n    // For some kinds of state, we are interested in changes at the global scope\n    // only. For example, global pointer input, global key presses and global\n    // visibility change should affect the state at every scope:\n    document.addEventListener('keydown', onKeyDown, true);\n    document.addEventListener('mousedown', onPointerDown, true);\n    document.addEventListener('pointerdown', onPointerDown, true);\n    document.addEventListener('touchstart', onPointerDown, true);\n    document.addEventListener('visibilitychange', onVisibilityChange, true);\n\n    addInitialPointerMoveListeners();\n\n    // For focus and blur, we specifically care about state changes in the local\n    // scope. This is because focus / blur events that originate from within a\n    // shadow root are not re-dispatched from the host element if it was already\n    // the active element in its own scope:\n    scope.addEventListener('focus', onFocus, true);\n    scope.addEventListener('blur', onBlur, true);\n\n    // We detect that a node is a ShadowRoot by ensuring that it is a\n    // DocumentFragment and also has a host property. This check covers native\n    // implementation and polyfill implementation transparently. If we only cared\n    // about the native implementation, we could just check if the scope was\n    // an instance of a ShadowRoot.\n    if (scope.nodeType === Node.DOCUMENT_FRAGMENT_NODE && scope.host) {\n      // Since a ShadowRoot is a special kind of DocumentFragment, it does not\n      // have a root element to add a class to. So, we add this attribute to the\n      // host element instead:\n      scope.host.setAttribute('data-js-focus-visible', '');\n    } else if (scope.nodeType === Node.DOCUMENT_NODE) {\n      document.documentElement.classList.add('js-focus-visible');\n      document.documentElement.setAttribute('data-js-focus-visible', '');\n    }\n  }\n\n  // It is important to wrap all references to global window and document in\n  // these checks to support server-side rendering use cases\n  // @see https://github.com/WICG/focus-visible/issues/199\n  if (typeof window !== 'undefined' && typeof document !== 'undefined') {\n    // Make the polyfill helper globally available. This can be used as a signal\n    // to interested libraries that wish to coordinate with the polyfill for e.g.,\n    // applying the polyfill to a shadow root:\n    window.applyFocusVisiblePolyfill = applyFocusVisiblePolyfill;\n\n    // Notify interested libraries of the polyfill's presence, in case the\n    // polyfill was loaded lazily:\n    var event;\n\n    try {\n      event = new CustomEvent('focus-visible-polyfill-ready');\n    } catch (error) {\n      // IE11 does not support using CustomEvent as a constructor directly:\n      event = document.createEvent('CustomEvent');\n      event.initCustomEvent('focus-visible-polyfill-ready', false, false, {});\n    }\n\n    window.dispatchEvent(event);\n  }\n\n  if (typeof document !== 'undefined') {\n    // Apply the polyfill to the global document, so that no JavaScript\n    // coordination is required to use the polyfill in the top-level document:\n    applyFocusVisiblePolyfill(document);\n  }\n\n})));\n", "/*!\n * clipboard.js v2.0.11\n * https://clipboardjs.com/\n *\n * Licensed MIT \u00A9 Zeno Rocha\n */\n(function webpackUniversalModuleDefinition(root, factory) {\n\tif(typeof exports === 'object' && typeof module === 'object')\n\t\tmodule.exports = factory();\n\telse if(typeof define === 'function' && define.amd)\n\t\tdefine([], factory);\n\telse if(typeof exports === 'object')\n\t\texports[\"ClipboardJS\"] = factory();\n\telse\n\t\troot[\"ClipboardJS\"] = factory();\n})(this, function() {\nreturn /******/ (function() { // webpackBootstrap\n/******/ \tvar __webpack_modules__ = ({\n\n/***/ 686:\n/***/ (function(__unused_webpack_module, __webpack_exports__, __webpack_require__) {\n\n\"use strict\";\n\n// EXPORTS\n__webpack_require__.d(__webpack_exports__, {\n  \"default\": function() { return /* binding */ clipboard; }\n});\n\n// EXTERNAL MODULE: ./node_modules/tiny-emitter/index.js\nvar tiny_emitter = __webpack_require__(279);\nvar tiny_emitter_default = /*#__PURE__*/__webpack_require__.n(tiny_emitter);\n// EXTERNAL MODULE: ./node_modules/good-listener/src/listen.js\nvar listen = __webpack_require__(370);\nvar listen_default = /*#__PURE__*/__webpack_require__.n(listen);\n// EXTERNAL MODULE: ./node_modules/select/src/select.js\nvar src_select = __webpack_require__(817);\nvar select_default = /*#__PURE__*/__webpack_require__.n(src_select);\n;// CONCATENATED MODULE: ./src/common/command.js\n/**\n * Executes a given operation type.\n * @param {String} type\n * @return {Boolean}\n */\nfunction command(type) {\n  try {\n    return document.execCommand(type);\n  } catch (err) {\n    return false;\n  }\n}\n;// CONCATENATED MODULE: ./src/actions/cut.js\n\n\n/**\n * Cut action wrapper.\n * @param {String|HTMLElement} target\n * @return {String}\n */\n\nvar ClipboardActionCut = function ClipboardActionCut(target) {\n  var selectedText = select_default()(target);\n  command('cut');\n  return selectedText;\n};\n\n/* harmony default export */ var actions_cut = (ClipboardActionCut);\n;// CONCATENATED MODULE: ./src/common/create-fake-element.js\n/**\n * Creates a fake textarea element with a value.\n * @param {String} value\n * @return {HTMLElement}\n */\nfunction createFakeElement(value) {\n  var isRTL = document.documentElement.getAttribute('dir') === 'rtl';\n  var fakeElement = document.createElement('textarea'); // Prevent zooming on iOS\n\n  fakeElement.style.fontSize = '12pt'; // Reset box model\n\n  fakeElement.style.border = '0';\n  fakeElement.style.padding = '0';\n  fakeElement.style.margin = '0'; // Move element out of screen horizontally\n\n  fakeElement.style.position = 'absolute';\n  fakeElement.style[isRTL ? 'right' : 'left'] = '-9999px'; // Move element to the same position vertically\n\n  var yPosition = window.pageYOffset || document.documentElement.scrollTop;\n  fakeElement.style.top = \"\".concat(yPosition, \"px\");\n  fakeElement.setAttribute('readonly', '');\n  fakeElement.value = value;\n  return fakeElement;\n}\n;// CONCATENATED MODULE: ./src/actions/copy.js\n\n\n\n/**\n * Create fake copy action wrapper using a fake element.\n * @param {String} target\n * @param {Object} options\n * @return {String}\n */\n\nvar fakeCopyAction = function fakeCopyAction(value, options) {\n  var fakeElement = createFakeElement(value);\n  options.container.appendChild(fakeElement);\n  var selectedText = select_default()(fakeElement);\n  command('copy');\n  fakeElement.remove();\n  return selectedText;\n};\n/**\n * Copy action wrapper.\n * @param {String|HTMLElement} target\n * @param {Object} options\n * @return {String}\n */\n\n\nvar ClipboardActionCopy = function ClipboardActionCopy(target) {\n  var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n    container: document.body\n  };\n  var selectedText = '';\n\n  if (typeof target === 'string') {\n    selectedText = fakeCopyAction(target, options);\n  } else if (target instanceof HTMLInputElement && !['text', 'search', 'url', 'tel', 'password'].includes(target === null || target === void 0 ? void 0 : target.type)) {\n    // If input type doesn't support `setSelectionRange`. Simulate it. https://developer.mozilla.org/en-US/docs/Web/API/HTMLInputElement/setSelectionRange\n    selectedText = fakeCopyAction(target.value, options);\n  } else {\n    selectedText = select_default()(target);\n    command('copy');\n  }\n\n  return selectedText;\n};\n\n/* harmony default export */ var actions_copy = (ClipboardActionCopy);\n;// CONCATENATED MODULE: ./src/actions/default.js\nfunction _typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { _typeof = function _typeof(obj) { return typeof obj; }; } else { _typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return _typeof(obj); }\n\n\n\n/**\n * Inner function which performs selection from either `text` or `target`\n * properties and then executes copy or cut operations.\n * @param {Object} options\n */\n\nvar ClipboardActionDefault = function ClipboardActionDefault() {\n  var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n  // Defines base properties passed from constructor.\n  var _options$action = options.action,\n      action = _options$action === void 0 ? 'copy' : _options$action,\n      container = options.container,\n      target = options.target,\n      text = options.text; // Sets the `action` to be performed which can be either 'copy' or 'cut'.\n\n  if (action !== 'copy' && action !== 'cut') {\n    throw new Error('Invalid \"action\" value, use either \"copy\" or \"cut\"');\n  } // Sets the `target` property using an element that will be have its content copied.\n\n\n  if (target !== undefined) {\n    if (target && _typeof(target) === 'object' && target.nodeType === 1) {\n      if (action === 'copy' && target.hasAttribute('disabled')) {\n        throw new Error('Invalid \"target\" attribute. Please use \"readonly\" instead of \"disabled\" attribute');\n      }\n\n      if (action === 'cut' && (target.hasAttribute('readonly') || target.hasAttribute('disabled'))) {\n        throw new Error('Invalid \"target\" attribute. You can\\'t cut text from elements with \"readonly\" or \"disabled\" attributes');\n      }\n    } else {\n      throw new Error('Invalid \"target\" value, use a valid Element');\n    }\n  } // Define selection strategy based on `text` property.\n\n\n  if (text) {\n    return actions_copy(text, {\n      container: container\n    });\n  } // Defines which selection strategy based on `target` property.\n\n\n  if (target) {\n    return action === 'cut' ? actions_cut(target) : actions_copy(target, {\n      container: container\n    });\n  }\n};\n\n/* harmony default export */ var actions_default = (ClipboardActionDefault);\n;// CONCATENATED MODULE: ./src/clipboard.js\nfunction clipboard_typeof(obj) { \"@babel/helpers - typeof\"; if (typeof Symbol === \"function\" && typeof Symbol.iterator === \"symbol\") { clipboard_typeof = function _typeof(obj) { return typeof obj; }; } else { clipboard_typeof = function _typeof(obj) { return obj && typeof Symbol === \"function\" && obj.constructor === Symbol && obj !== Symbol.prototype ? \"symbol\" : typeof obj; }; } return clipboard_typeof(obj); }\n\nfunction _classCallCheck(instance, Constructor) { if (!(instance instanceof Constructor)) { throw new TypeError(\"Cannot call a class as a function\"); } }\n\nfunction _defineProperties(target, props) { for (var i = 0; i < props.length; i++) { var descriptor = props[i]; descriptor.enumerable = descriptor.enumerable || false; descriptor.configurable = true; if (\"value\" in descriptor) descriptor.writable = true; Object.defineProperty(target, descriptor.key, descriptor); } }\n\nfunction _createClass(Constructor, protoProps, staticProps) { if (protoProps) _defineProperties(Constructor.prototype, protoProps); if (staticProps) _defineProperties(Constructor, staticProps); return Constructor; }\n\nfunction _inherits(subClass, superClass) { if (typeof superClass !== \"function\" && superClass !== null) { throw new TypeError(\"Super expression must either be null or a function\"); } subClass.prototype = Object.create(superClass && superClass.prototype, { constructor: { value: subClass, writable: true, configurable: true } }); if (superClass) _setPrototypeOf(subClass, superClass); }\n\nfunction _setPrototypeOf(o, p) { _setPrototypeOf = Object.setPrototypeOf || function _setPrototypeOf(o, p) { o.__proto__ = p; return o; }; return _setPrototypeOf(o, p); }\n\nfunction _createSuper(Derived) { var hasNativeReflectConstruct = _isNativeReflectConstruct(); return function _createSuperInternal() { var Super = _getPrototypeOf(Derived), result; if (hasNativeReflectConstruct) { var NewTarget = _getPrototypeOf(this).constructor; result = Reflect.construct(Super, arguments, NewTarget); } else { result = Super.apply(this, arguments); } return _possibleConstructorReturn(this, result); }; }\n\nfunction _possibleConstructorReturn(self, call) { if (call && (clipboard_typeof(call) === \"object\" || typeof call === \"function\")) { return call; } return _assertThisInitialized(self); }\n\nfunction _assertThisInitialized(self) { if (self === void 0) { throw new ReferenceError(\"this hasn't been initialised - super() hasn't been called\"); } return self; }\n\nfunction _isNativeReflectConstruct() { if (typeof Reflect === \"undefined\" || !Reflect.construct) return false; if (Reflect.construct.sham) return false; if (typeof Proxy === \"function\") return true; try { Date.prototype.toString.call(Reflect.construct(Date, [], function () {})); return true; } catch (e) { return false; } }\n\nfunction _getPrototypeOf(o) { _getPrototypeOf = Object.setPrototypeOf ? Object.getPrototypeOf : function _getPrototypeOf(o) { return o.__proto__ || Object.getPrototypeOf(o); }; return _getPrototypeOf(o); }\n\n\n\n\n\n\n/**\n * Helper function to retrieve attribute value.\n * @param {String} suffix\n * @param {Element} element\n */\n\nfunction getAttributeValue(suffix, element) {\n  var attribute = \"data-clipboard-\".concat(suffix);\n\n  if (!element.hasAttribute(attribute)) {\n    return;\n  }\n\n  return element.getAttribute(attribute);\n}\n/**\n * Base class which takes one or more elements, adds event listeners to them,\n * and instantiates a new `ClipboardAction` on each click.\n */\n\n\nvar Clipboard = /*#__PURE__*/function (_Emitter) {\n  _inherits(Clipboard, _Emitter);\n\n  var _super = _createSuper(Clipboard);\n\n  /**\n   * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n   * @param {Object} options\n   */\n  function Clipboard(trigger, options) {\n    var _this;\n\n    _classCallCheck(this, Clipboard);\n\n    _this = _super.call(this);\n\n    _this.resolveOptions(options);\n\n    _this.listenClick(trigger);\n\n    return _this;\n  }\n  /**\n   * Defines if attributes would be resolved using internal setter functions\n   * or custom functions that were passed in the constructor.\n   * @param {Object} options\n   */\n\n\n  _createClass(Clipboard, [{\n    key: \"resolveOptions\",\n    value: function resolveOptions() {\n      var options = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : {};\n      this.action = typeof options.action === 'function' ? options.action : this.defaultAction;\n      this.target = typeof options.target === 'function' ? options.target : this.defaultTarget;\n      this.text = typeof options.text === 'function' ? options.text : this.defaultText;\n      this.container = clipboard_typeof(options.container) === 'object' ? options.container : document.body;\n    }\n    /**\n     * Adds a click event listener to the passed trigger.\n     * @param {String|HTMLElement|HTMLCollection|NodeList} trigger\n     */\n\n  }, {\n    key: \"listenClick\",\n    value: function listenClick(trigger) {\n      var _this2 = this;\n\n      this.listener = listen_default()(trigger, 'click', function (e) {\n        return _this2.onClick(e);\n      });\n    }\n    /**\n     * Defines a new `ClipboardAction` on each click event.\n     * @param {Event} e\n     */\n\n  }, {\n    key: \"onClick\",\n    value: function onClick(e) {\n      var trigger = e.delegateTarget || e.currentTarget;\n      var action = this.action(trigger) || 'copy';\n      var text = actions_default({\n        action: action,\n        container: this.container,\n        target: this.target(trigger),\n        text: this.text(trigger)\n      }); // Fires an event based on the copy operation result.\n\n      this.emit(text ? 'success' : 'error', {\n        action: action,\n        text: text,\n        trigger: trigger,\n        clearSelection: function clearSelection() {\n          if (trigger) {\n            trigger.focus();\n          }\n\n          window.getSelection().removeAllRanges();\n        }\n      });\n    }\n    /**\n     * Default `action` lookup function.\n     * @param {Element} trigger\n     */\n\n  }, {\n    key: \"defaultAction\",\n    value: function defaultAction(trigger) {\n      return getAttributeValue('action', trigger);\n    }\n    /**\n     * Default `target` lookup function.\n     * @param {Element} trigger\n     */\n\n  }, {\n    key: \"defaultTarget\",\n    value: function defaultTarget(trigger) {\n      var selector = getAttributeValue('target', trigger);\n\n      if (selector) {\n        return document.querySelector(selector);\n      }\n    }\n    /**\n     * Allow fire programmatically a copy action\n     * @param {String|HTMLElement} target\n     * @param {Object} options\n     * @returns Text copied.\n     */\n\n  }, {\n    key: \"defaultText\",\n\n    /**\n     * Default `text` lookup function.\n     * @param {Element} trigger\n     */\n    value: function defaultText(trigger) {\n      return getAttributeValue('text', trigger);\n    }\n    /**\n     * Destroy lifecycle.\n     */\n\n  }, {\n    key: \"destroy\",\n    value: function destroy() {\n      this.listener.destroy();\n    }\n  }], [{\n    key: \"copy\",\n    value: function copy(target) {\n      var options = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : {\n        container: document.body\n      };\n      return actions_copy(target, options);\n    }\n    /**\n     * Allow fire programmatically a cut action\n     * @param {String|HTMLElement} target\n     * @returns Text cutted.\n     */\n\n  }, {\n    key: \"cut\",\n    value: function cut(target) {\n      return actions_cut(target);\n    }\n    /**\n     * Returns the support of the given action, or all actions if no action is\n     * given.\n     * @param {String} [action]\n     */\n\n  }, {\n    key: \"isSupported\",\n    value: function isSupported() {\n      var action = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : ['copy', 'cut'];\n      var actions = typeof action === 'string' ? [action] : action;\n      var support = !!document.queryCommandSupported;\n      actions.forEach(function (action) {\n        support = support && !!document.queryCommandSupported(action);\n      });\n      return support;\n    }\n  }]);\n\n  return Clipboard;\n}((tiny_emitter_default()));\n\n/* harmony default export */ var clipboard = (Clipboard);\n\n/***/ }),\n\n/***/ 828:\n/***/ (function(module) {\n\nvar DOCUMENT_NODE_TYPE = 9;\n\n/**\n * A polyfill for Element.matches()\n */\nif (typeof Element !== 'undefined' && !Element.prototype.matches) {\n    var proto = Element.prototype;\n\n    proto.matches = proto.matchesSelector ||\n                    proto.mozMatchesSelector ||\n                    proto.msMatchesSelector ||\n                    proto.oMatchesSelector ||\n                    proto.webkitMatchesSelector;\n}\n\n/**\n * Finds the closest parent that matches a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @return {Function}\n */\nfunction closest (element, selector) {\n    while (element && element.nodeType !== DOCUMENT_NODE_TYPE) {\n        if (typeof element.matches === 'function' &&\n            element.matches(selector)) {\n          return element;\n        }\n        element = element.parentNode;\n    }\n}\n\nmodule.exports = closest;\n\n\n/***/ }),\n\n/***/ 438:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar closest = __webpack_require__(828);\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction _delegate(element, selector, type, callback, useCapture) {\n    var listenerFn = listener.apply(this, arguments);\n\n    element.addEventListener(type, listenerFn, useCapture);\n\n    return {\n        destroy: function() {\n            element.removeEventListener(type, listenerFn, useCapture);\n        }\n    }\n}\n\n/**\n * Delegates event to a selector.\n *\n * @param {Element|String|Array} [elements]\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @param {Boolean} useCapture\n * @return {Object}\n */\nfunction delegate(elements, selector, type, callback, useCapture) {\n    // Handle the regular Element usage\n    if (typeof elements.addEventListener === 'function') {\n        return _delegate.apply(null, arguments);\n    }\n\n    // Handle Element-less usage, it defaults to global delegation\n    if (typeof type === 'function') {\n        // Use `document` as the first parameter, then apply arguments\n        // This is a short way to .unshift `arguments` without running into deoptimizations\n        return _delegate.bind(null, document).apply(null, arguments);\n    }\n\n    // Handle Selector-based usage\n    if (typeof elements === 'string') {\n        elements = document.querySelectorAll(elements);\n    }\n\n    // Handle Array-like based usage\n    return Array.prototype.map.call(elements, function (element) {\n        return _delegate(element, selector, type, callback, useCapture);\n    });\n}\n\n/**\n * Finds closest match and invokes callback.\n *\n * @param {Element} element\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Function}\n */\nfunction listener(element, selector, type, callback) {\n    return function(e) {\n        e.delegateTarget = closest(e.target, selector);\n\n        if (e.delegateTarget) {\n            callback.call(element, e);\n        }\n    }\n}\n\nmodule.exports = delegate;\n\n\n/***/ }),\n\n/***/ 879:\n/***/ (function(__unused_webpack_module, exports) {\n\n/**\n * Check if argument is a HTML element.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.node = function(value) {\n    return value !== undefined\n        && value instanceof HTMLElement\n        && value.nodeType === 1;\n};\n\n/**\n * Check if argument is a list of HTML elements.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.nodeList = function(value) {\n    var type = Object.prototype.toString.call(value);\n\n    return value !== undefined\n        && (type === '[object NodeList]' || type === '[object HTMLCollection]')\n        && ('length' in value)\n        && (value.length === 0 || exports.node(value[0]));\n};\n\n/**\n * Check if argument is a string.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.string = function(value) {\n    return typeof value === 'string'\n        || value instanceof String;\n};\n\n/**\n * Check if argument is a function.\n *\n * @param {Object} value\n * @return {Boolean}\n */\nexports.fn = function(value) {\n    var type = Object.prototype.toString.call(value);\n\n    return type === '[object Function]';\n};\n\n\n/***/ }),\n\n/***/ 370:\n/***/ (function(module, __unused_webpack_exports, __webpack_require__) {\n\nvar is = __webpack_require__(879);\nvar delegate = __webpack_require__(438);\n\n/**\n * Validates all params and calls the right\n * listener function based on its target type.\n *\n * @param {String|HTMLElement|HTMLCollection|NodeList} target\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listen(target, type, callback) {\n    if (!target && !type && !callback) {\n        throw new Error('Missing required arguments');\n    }\n\n    if (!is.string(type)) {\n        throw new TypeError('Second argument must be a String');\n    }\n\n    if (!is.fn(callback)) {\n        throw new TypeError('Third argument must be a Function');\n    }\n\n    if (is.node(target)) {\n        return listenNode(target, type, callback);\n    }\n    else if (is.nodeList(target)) {\n        return listenNodeList(target, type, callback);\n    }\n    else if (is.string(target)) {\n        return listenSelector(target, type, callback);\n    }\n    else {\n        throw new TypeError('First argument must be a String, HTMLElement, HTMLCollection, or NodeList');\n    }\n}\n\n/**\n * Adds an event listener to a HTML element\n * and returns a remove listener function.\n *\n * @param {HTMLElement} node\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNode(node, type, callback) {\n    node.addEventListener(type, callback);\n\n    return {\n        destroy: function() {\n            node.removeEventListener(type, callback);\n        }\n    }\n}\n\n/**\n * Add an event listener to a list of HTML elements\n * and returns a remove listener function.\n *\n * @param {NodeList|HTMLCollection} nodeList\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenNodeList(nodeList, type, callback) {\n    Array.prototype.forEach.call(nodeList, function(node) {\n        node.addEventListener(type, callback);\n    });\n\n    return {\n        destroy: function() {\n            Array.prototype.forEach.call(nodeList, function(node) {\n                node.removeEventListener(type, callback);\n            });\n        }\n    }\n}\n\n/**\n * Add an event listener to a selector\n * and returns a remove listener function.\n *\n * @param {String} selector\n * @param {String} type\n * @param {Function} callback\n * @return {Object}\n */\nfunction listenSelector(selector, type, callback) {\n    return delegate(document.body, selector, type, callback);\n}\n\nmodule.exports = listen;\n\n\n/***/ }),\n\n/***/ 817:\n/***/ (function(module) {\n\nfunction select(element) {\n    var selectedText;\n\n    if (element.nodeName === 'SELECT') {\n        element.focus();\n\n        selectedText = element.value;\n    }\n    else if (element.nodeName === 'INPUT' || element.nodeName === 'TEXTAREA') {\n        var isReadOnly = element.hasAttribute('readonly');\n\n        if (!isReadOnly) {\n            element.setAttribute('readonly', '');\n        }\n\n        element.select();\n        element.setSelectionRange(0, element.value.length);\n\n        if (!isReadOnly) {\n            element.removeAttribute('readonly');\n        }\n\n        selectedText = element.value;\n    }\n    else {\n        if (element.hasAttribute('contenteditable')) {\n            element.focus();\n        }\n\n        var selection = window.getSelection();\n        var range = document.createRange();\n\n        range.selectNodeContents(element);\n        selection.removeAllRanges();\n        selection.addRange(range);\n\n        selectedText = selection.toString();\n    }\n\n    return selectedText;\n}\n\nmodule.exports = select;\n\n\n/***/ }),\n\n/***/ 279:\n/***/ (function(module) {\n\nfunction E () {\n  // Keep this empty so it's easier to inherit from\n  // (via https://github.com/lipsmack from https://github.com/scottcorgan/tiny-emitter/issues/3)\n}\n\nE.prototype = {\n  on: function (name, callback, ctx) {\n    var e = this.e || (this.e = {});\n\n    (e[name] || (e[name] = [])).push({\n      fn: callback,\n      ctx: ctx\n    });\n\n    return this;\n  },\n\n  once: function (name, callback, ctx) {\n    var self = this;\n    function listener () {\n      self.off(name, listener);\n      callback.apply(ctx, arguments);\n    };\n\n    listener._ = callback\n    return this.on(name, listener, ctx);\n  },\n\n  emit: function (name) {\n    var data = [].slice.call(arguments, 1);\n    var evtArr = ((this.e || (this.e = {}))[name] || []).slice();\n    var i = 0;\n    var len = evtArr.length;\n\n    for (i; i < len; i++) {\n      evtArr[i].fn.apply(evtArr[i].ctx, data);\n    }\n\n    return this;\n  },\n\n  off: function (name, callback) {\n    var e = this.e || (this.e = {});\n    var evts = e[name];\n    var liveEvents = [];\n\n    if (evts && callback) {\n      for (var i = 0, len = evts.length; i < len; i++) {\n        if (evts[i].fn !== callback && evts[i].fn._ !== callback)\n          liveEvents.push(evts[i]);\n      }\n    }\n\n    // Remove event from queue to prevent memory leak\n    // Suggested by https://github.com/lazd\n    // Ref: https://github.com/scottcorgan/tiny-emitter/commit/c6ebfaa9bc973b33d110a84a307742b7cf94c953#commitcomment-5024910\n\n    (liveEvents.length)\n      ? e[name] = liveEvents\n      : delete e[name];\n\n    return this;\n  }\n};\n\nmodule.exports = E;\nmodule.exports.TinyEmitter = E;\n\n\n/***/ })\n\n/******/ \t});\n/************************************************************************/\n/******/ \t// The module cache\n/******/ \tvar __webpack_module_cache__ = {};\n/******/ \t\n/******/ \t// The require function\n/******/ \tfunction __webpack_require__(moduleId) {\n/******/ \t\t// Check if module is in cache\n/******/ \t\tif(__webpack_module_cache__[moduleId]) {\n/******/ \t\t\treturn __webpack_module_cache__[moduleId].exports;\n/******/ \t\t}\n/******/ \t\t// Create a new module (and put it into the cache)\n/******/ \t\tvar module = __webpack_module_cache__[moduleId] = {\n/******/ \t\t\t// no module.id needed\n/******/ \t\t\t// no module.loaded needed\n/******/ \t\t\texports: {}\n/******/ \t\t};\n/******/ \t\n/******/ \t\t// Execute the module function\n/******/ \t\t__webpack_modules__[moduleId](module, module.exports, __webpack_require__);\n/******/ \t\n/******/ \t\t// Return the exports of the module\n/******/ \t\treturn module.exports;\n/******/ \t}\n/******/ \t\n/************************************************************************/\n/******/ \t/* webpack/runtime/compat get default export */\n/******/ \t!function() {\n/******/ \t\t// getDefaultExport function for compatibility with non-harmony modules\n/******/ \t\t__webpack_require__.n = function(module) {\n/******/ \t\t\tvar getter = module && module.__esModule ?\n/******/ \t\t\t\tfunction() { return module['default']; } :\n/******/ \t\t\t\tfunction() { return module; };\n/******/ \t\t\t__webpack_require__.d(getter, { a: getter });\n/******/ \t\t\treturn getter;\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/define property getters */\n/******/ \t!function() {\n/******/ \t\t// define getter functions for harmony exports\n/******/ \t\t__webpack_require__.d = function(exports, definition) {\n/******/ \t\t\tfor(var key in definition) {\n/******/ \t\t\t\tif(__webpack_require__.o(definition, key) && !__webpack_require__.o(exports, key)) {\n/******/ \t\t\t\t\tObject.defineProperty(exports, key, { enumerable: true, get: definition[key] });\n/******/ \t\t\t\t}\n/******/ \t\t\t}\n/******/ \t\t};\n/******/ \t}();\n/******/ \t\n/******/ \t/* webpack/runtime/hasOwnProperty shorthand */\n/******/ \t!function() {\n/******/ \t\t__webpack_require__.o = function(obj, prop) { return Object.prototype.hasOwnProperty.call(obj, prop); }\n/******/ \t}();\n/******/ \t\n/************************************************************************/\n/******/ \t// module exports must be returned from runtime so entry inlining is disabled\n/******/ \t// startup\n/******/ \t// Load entry module and return exports\n/******/ \treturn __webpack_require__(686);\n/******/ })()\n.default;\n});", "/*!\n * escape-html\n * Copyright(c) 2012-2013 TJ Holowaychuk\n * Copyright(c) 2015 Andreas Lubbe\n * Copyright(c) 2015 Tiancheng \"Timothy\" Gu\n * MIT Licensed\n */\n\n'use strict';\n\n/**\n * Module variables.\n * @private\n */\n\nvar matchHtmlRegExp = /[\"'&<>]/;\n\n/**\n * Module exports.\n * @public\n */\n\nmodule.exports = escapeHtml;\n\n/**\n * Escape special characters in the given string of html.\n *\n * @param  {string} string The string to escape for inserting into HTML\n * @return {string}\n * @public\n */\n\nfunction escapeHtml(string) {\n  var str = '' + string;\n  var match = matchHtmlRegExp.exec(str);\n\n  if (!match) {\n    return str;\n  }\n\n  var escape;\n  var html = '';\n  var index = 0;\n  var lastIndex = 0;\n\n  for (index = match.index; index < str.length; index++) {\n    switch (str.charCodeAt(index)) {\n      case 34: // \"\n        escape = '&quot;';\n        break;\n      case 38: // &\n        escape = '&amp;';\n        break;\n      case 39: // '\n        escape = '&#39;';\n        break;\n      case 60: // <\n        escape = '&lt;';\n        break;\n      case 62: // >\n        escape = '&gt;';\n        break;\n      default:\n        continue;\n    }\n\n    if (lastIndex !== index) {\n      html += str.substring(lastIndex, index);\n    }\n\n    lastIndex = index + 1;\n    html += escape;\n  }\n\n  return lastIndex !== index\n    ? html + str.substring(lastIndex, index)\n    : html;\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport \"focus-visible\"\n\nimport {\n  EMPTY,\n  NEVER,\n  Observable,\n  Subject,\n  defer,\n  delay,\n  filter,\n  map,\n  merge,\n  mergeWith,\n  shareReplay,\n  switchMap\n} from \"rxjs\"\n\nimport { configuration, feature } from \"./_\"\nimport {\n  at,\n  getActiveElement,\n  getOptionalElement,\n  requestJSON,\n  setLocation,\n  setToggle,\n  watchDocument,\n  watchKeyboard,\n  watchLocation,\n  watchLocationTarget,\n  watchMedia,\n  watchPrint,\n  watchScript,\n  watchViewport\n} from \"./browser\"\nimport {\n  getComponentElement,\n  getComponentElements,\n  mountAnnounce,\n  mountBackToTop,\n  mountConsent,\n  mountContent,\n  mountDialog,\n  mountHeader,\n  mountHeaderTitle,\n  mountPalette,\n  mountProgress,\n  mountSearch,\n  mountSearchHiglight,\n  mountSidebar,\n  mountSource,\n  mountTableOfContents,\n  mountTabs,\n  watchHeader,\n  watchMain\n} from \"./components\"\nimport {\n  SearchIndex,\n  setupClipboardJS,\n  setupInstantNavigation,\n  setupVersionSelector\n} from \"./integrations\"\nimport {\n  patchEllipsis,\n  patchIndeterminate,\n  patchScrollfix,\n  patchScrolllock\n} from \"./patches\"\nimport \"./polyfills\"\n\n/* ----------------------------------------------------------------------------\n * Functions - @todo refactor\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch search index\n *\n * @returns Search index observable\n */\nfunction fetchSearchIndex(): Observable<SearchIndex> {\n  if (location.protocol === \"file:\") {\n    return watchScript(\n      `${new URL(\"search/search_index.js\", config.base)}`\n    )\n      .pipe(\n        // @ts-ignore - @todo fix typings\n        map(() => __index),\n        shareReplay(1)\n      )\n  } else {\n    return requestJSON<SearchIndex>(\n      new URL(\"search/search_index.json\", config.base)\n    )\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Application\n * ------------------------------------------------------------------------- */\n\n/* Yay, JavaScript is available */\ndocument.documentElement.classList.remove(\"no-js\")\ndocument.documentElement.classList.add(\"js\")\n\n/* Set up navigation observables and subjects */\nconst document$ = watchDocument()\nconst location$ = watchLocation()\nconst target$   = watchLocationTarget(location$)\nconst keyboard$ = watchKeyboard()\n\n/* Set up media observables */\nconst viewport$ = watchViewport()\nconst tablet$   = watchMedia(\"(min-width: 960px)\")\nconst screen$   = watchMedia(\"(min-width: 1220px)\")\nconst print$    = watchPrint()\n\n/* Retrieve search index, if search is enabled */\nconst config = configuration()\nconst index$ = document.forms.namedItem(\"search\")\n  ? fetchSearchIndex()\n  : NEVER\n\n/* Set up Clipboard.js integration */\nconst alert$ = new Subject<string>()\nsetupClipboardJS({ alert$ })\n\n/* Set up progress indicator */\nconst progress$ = new Subject<number>()\n\n/* Set up instant navigation, if enabled */\nif (feature(\"navigation.instant\"))\n  setupInstantNavigation({ location$, viewport$, progress$ })\n    .subscribe(document$)\n\n/* Set up version selector */\nif (config.version?.provider === \"mike\")\n  setupVersionSelector({ document$ })\n\n/* Always close drawer and search on navigation */\nmerge(location$, target$)\n  .pipe(\n    delay(125)\n  )\n    .subscribe(() => {\n      setToggle(\"drawer\", false)\n      setToggle(\"search\", false)\n    })\n\n/* Set up global keyboard handlers */\nkeyboard$\n  .pipe(\n    filter(({ mode }) => mode === \"global\")\n  )\n    .subscribe(key => {\n      switch (key.type) {\n\n        /* Go to previous page */\n        case \"p\":\n        case \",\":\n          const prev = getOptionalElement<HTMLLinkElement>(\"link[rel=prev]\")\n          if (typeof prev !== \"undefined\")\n            setLocation(prev)\n          break\n\n        /* Go to next page */\n        case \"n\":\n        case \".\":\n          const next = getOptionalElement<HTMLLinkElement>(\"link[rel=next]\")\n          if (typeof next !== \"undefined\")\n            setLocation(next)\n          break\n\n        /* Expand navigation, see https://bit.ly/3ZjG5io */\n        case \"Enter\":\n          const active = getActiveElement()\n          if (active instanceof HTMLLabelElement)\n            active.click()\n      }\n    })\n\n/* Set up patches */\npatchEllipsis({ document$ })\npatchIndeterminate({ document$, tablet$ })\npatchScrollfix({ document$ })\npatchScrolllock({ viewport$, tablet$ })\n\n/* Set up header and main area observable */\nconst header$ = watchHeader(getComponentElement(\"header\"), { viewport$ })\nconst main$ = document$\n  .pipe(\n    map(() => getComponentElement(\"main\")),\n    switchMap(el => watchMain(el, { viewport$, header$ })),\n    shareReplay(1)\n  )\n\n/* Set up control component observables */\nconst control$ = merge(\n\n  /* Consent */\n  ...getComponentElements(\"consent\")\n    .map(el => mountConsent(el, { target$ })),\n\n  /* Dialog */\n  ...getComponentElements(\"dialog\")\n    .map(el => mountDialog(el, { alert$ })),\n\n  /* Header */\n  ...getComponentElements(\"header\")\n    .map(el => mountHeader(el, { viewport$, header$, main$ })),\n\n  /* Color palette */\n  ...getComponentElements(\"palette\")\n    .map(el => mountPalette(el)),\n\n  /* Progress bar */\n  ...getComponentElements(\"progress\")\n    .map(el => mountProgress(el, { progress$ })),\n\n  /* Search */\n  ...getComponentElements(\"search\")\n    .map(el => mountSearch(el, { index$, keyboard$ })),\n\n  /* Repository information */\n  ...getComponentElements(\"source\")\n    .map(el => mountSource(el))\n)\n\n/* Set up content component observables */\nconst content$ = defer(() => merge(\n\n  /* Announcement bar */\n  ...getComponentElements(\"announce\")\n    .map(el => mountAnnounce(el)),\n\n  /* Content */\n  ...getComponentElements(\"content\")\n    .map(el => mountContent(el, { viewport$, target$, print$ })),\n\n  /* Search highlighting */\n  ...getComponentElements(\"content\")\n    .map(el => feature(\"search.highlight\")\n      ? mountSearchHiglight(el, { index$, location$ })\n      : EMPTY\n    ),\n\n  /* Header title */\n  ...getComponentElements(\"header-title\")\n    .map(el => mountHeaderTitle(el, { viewport$, header$ })),\n\n  /* Sidebar */\n  ...getComponentElements(\"sidebar\")\n    .map(el => el.getAttribute(\"data-md-type\") === \"navigation\"\n      ? at(screen$, () => mountSidebar(el, { viewport$, header$, main$ }))\n      : at(tablet$, () => mountSidebar(el, { viewport$, header$, main$ }))\n    ),\n\n  /* Navigation tabs */\n  ...getComponentElements(\"tabs\")\n    .map(el => mountTabs(el, { viewport$, header$ })),\n\n  /* Table of contents */\n  ...getComponentElements(\"toc\")\n    .map(el => mountTableOfContents(el, {\n      viewport$, header$, main$, target$\n    })),\n\n  /* Back-to-top button */\n  ...getComponentElements(\"top\")\n    .map(el => mountBackToTop(el, { viewport$, header$, main$, target$ }))\n))\n\n/* Set up component observables */\nconst component$ = document$\n  .pipe(\n    switchMap(() => content$),\n    mergeWith(control$),\n    shareReplay(1)\n  )\n\n/* Subscribe to all components */\ncomponent$.subscribe()\n\n/* ----------------------------------------------------------------------------\n * Exports\n * ------------------------------------------------------------------------- */\n\nwindow.document$  = document$          /* Document observable */\nwindow.location$  = location$          /* Location subject */\nwindow.target$    = target$            /* Location target observable */\nwindow.keyboard$  = keyboard$          /* Keyboard observable */\nwindow.viewport$  = viewport$          /* Viewport observable */\nwindow.tablet$    = tablet$            /* Media tablet observable */\nwindow.screen$    = screen$            /* Media screen observable */\nwindow.print$     = print$             /* Media print observable */\nwindow.alert$     = alert$             /* Alert subject */\nwindow.progress$  = progress$          /* Progress indicator subject */\nwindow.component$ = component$         /* Component observable */\n", "/*! *****************************************************************************\r\nCopyright (c) Microsoft Corporation.\r\n\r\nPermission to use, copy, modify, and/or distribute this software for any\r\npurpose with or without fee is hereby granted.\r\n\r\nTHE SOFTWARE IS PROVIDED \"AS IS\" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH\r\nREGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY\r\nAND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT,\r\nINDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM\r\nLOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR\r\nOTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR\r\nPERFORMANCE OF THIS SOFTWARE.\r\n***************************************************************************** */\r\n/* global Reflect, Promise */\r\n\r\nvar extendStatics = function(d, b) {\r\n    extendStatics = Object.setPrototypeOf ||\r\n        ({ __proto__: [] } instanceof Array && function (d, b) { d.__proto__ = b; }) ||\r\n        function (d, b) { for (var p in b) if (Object.prototype.hasOwnProperty.call(b, p)) d[p] = b[p]; };\r\n    return extendStatics(d, b);\r\n};\r\n\r\nexport function __extends(d, b) {\r\n    if (typeof b !== \"function\" && b !== null)\r\n        throw new TypeError(\"Class extends value \" + String(b) + \" is not a constructor or null\");\r\n    extendStatics(d, b);\r\n    function __() { this.constructor = d; }\r\n    d.prototype = b === null ? Object.create(b) : (__.prototype = b.prototype, new __());\r\n}\r\n\r\nexport var __assign = function() {\r\n    __assign = Object.assign || function __assign(t) {\r\n        for (var s, i = 1, n = arguments.length; i < n; i++) {\r\n            s = arguments[i];\r\n            for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p)) t[p] = s[p];\r\n        }\r\n        return t;\r\n    }\r\n    return __assign.apply(this, arguments);\r\n}\r\n\r\nexport function __rest(s, e) {\r\n    var t = {};\r\n    for (var p in s) if (Object.prototype.hasOwnProperty.call(s, p) && e.indexOf(p) < 0)\r\n        t[p] = s[p];\r\n    if (s != null && typeof Object.getOwnPropertySymbols === \"function\")\r\n        for (var i = 0, p = Object.getOwnPropertySymbols(s); i < p.length; i++) {\r\n            if (e.indexOf(p[i]) < 0 && Object.prototype.propertyIsEnumerable.call(s, p[i]))\r\n                t[p[i]] = s[p[i]];\r\n        }\r\n    return t;\r\n}\r\n\r\nexport function __decorate(decorators, target, key, desc) {\r\n    var c = arguments.length, r = c < 3 ? target : desc === null ? desc = Object.getOwnPropertyDescriptor(target, key) : desc, d;\r\n    if (typeof Reflect === \"object\" && typeof Reflect.decorate === \"function\") r = Reflect.decorate(decorators, target, key, desc);\r\n    else for (var i = decorators.length - 1; i >= 0; i--) if (d = decorators[i]) r = (c < 3 ? d(r) : c > 3 ? d(target, key, r) : d(target, key)) || r;\r\n    return c > 3 && r && Object.defineProperty(target, key, r), r;\r\n}\r\n\r\nexport function __param(paramIndex, decorator) {\r\n    return function (target, key) { decorator(target, key, paramIndex); }\r\n}\r\n\r\nexport function __metadata(metadataKey, metadataValue) {\r\n    if (typeof Reflect === \"object\" && typeof Reflect.metadata === \"function\") return Reflect.metadata(metadataKey, metadataValue);\r\n}\r\n\r\nexport function __awaiter(thisArg, _arguments, P, generator) {\r\n    function adopt(value) { return value instanceof P ? value : new P(function (resolve) { resolve(value); }); }\r\n    return new (P || (P = Promise))(function (resolve, reject) {\r\n        function fulfilled(value) { try { step(generator.next(value)); } catch (e) { reject(e); } }\r\n        function rejected(value) { try { step(generator[\"throw\"](value)); } catch (e) { reject(e); } }\r\n        function step(result) { result.done ? resolve(result.value) : adopt(result.value).then(fulfilled, rejected); }\r\n        step((generator = generator.apply(thisArg, _arguments || [])).next());\r\n    });\r\n}\r\n\r\nexport function __generator(thisArg, body) {\r\n    var _ = { label: 0, sent: function() { if (t[0] & 1) throw t[1]; return t[1]; }, trys: [], ops: [] }, f, y, t, g;\r\n    return g = { next: verb(0), \"throw\": verb(1), \"return\": verb(2) }, typeof Symbol === \"function\" && (g[Symbol.iterator] = function() { return this; }), g;\r\n    function verb(n) { return function (v) { return step([n, v]); }; }\r\n    function step(op) {\r\n        if (f) throw new TypeError(\"Generator is already executing.\");\r\n        while (_) try {\r\n            if (f = 1, y && (t = op[0] & 2 ? y[\"return\"] : op[0] ? y[\"throw\"] || ((t = y[\"return\"]) && t.call(y), 0) : y.next) && !(t = t.call(y, op[1])).done) return t;\r\n            if (y = 0, t) op = [op[0] & 2, t.value];\r\n            switch (op[0]) {\r\n                case 0: case 1: t = op; break;\r\n                case 4: _.label++; return { value: op[1], done: false };\r\n                case 5: _.label++; y = op[1]; op = [0]; continue;\r\n                case 7: op = _.ops.pop(); _.trys.pop(); continue;\r\n                default:\r\n                    if (!(t = _.trys, t = t.length > 0 && t[t.length - 1]) && (op[0] === 6 || op[0] === 2)) { _ = 0; continue; }\r\n                    if (op[0] === 3 && (!t || (op[1] > t[0] && op[1] < t[3]))) { _.label = op[1]; break; }\r\n                    if (op[0] === 6 && _.label < t[1]) { _.label = t[1]; t = op; break; }\r\n                    if (t && _.label < t[2]) { _.label = t[2]; _.ops.push(op); break; }\r\n                    if (t[2]) _.ops.pop();\r\n                    _.trys.pop(); continue;\r\n            }\r\n            op = body.call(thisArg, _);\r\n        } catch (e) { op = [6, e]; y = 0; } finally { f = t = 0; }\r\n        if (op[0] & 5) throw op[1]; return { value: op[0] ? op[1] : void 0, done: true };\r\n    }\r\n}\r\n\r\nexport var __createBinding = Object.create ? (function(o, m, k, k2) {\r\n    if (k2 === undefined) k2 = k;\r\n    Object.defineProperty(o, k2, { enumerable: true, get: function() { return m[k]; } });\r\n}) : (function(o, m, k, k2) {\r\n    if (k2 === undefined) k2 = k;\r\n    o[k2] = m[k];\r\n});\r\n\r\nexport function __exportStar(m, o) {\r\n    for (var p in m) if (p !== \"default\" && !Object.prototype.hasOwnProperty.call(o, p)) __createBinding(o, m, p);\r\n}\r\n\r\nexport function __values(o) {\r\n    var s = typeof Symbol === \"function\" && Symbol.iterator, m = s && o[s], i = 0;\r\n    if (m) return m.call(o);\r\n    if (o && typeof o.length === \"number\") return {\r\n        next: function () {\r\n            if (o && i >= o.length) o = void 0;\r\n            return { value: o && o[i++], done: !o };\r\n        }\r\n    };\r\n    throw new TypeError(s ? \"Object is not iterable.\" : \"Symbol.iterator is not defined.\");\r\n}\r\n\r\nexport function __read(o, n) {\r\n    var m = typeof Symbol === \"function\" && o[Symbol.iterator];\r\n    if (!m) return o;\r\n    var i = m.call(o), r, ar = [], e;\r\n    try {\r\n        while ((n === void 0 || n-- > 0) && !(r = i.next()).done) ar.push(r.value);\r\n    }\r\n    catch (error) { e = { error: error }; }\r\n    finally {\r\n        try {\r\n            if (r && !r.done && (m = i[\"return\"])) m.call(i);\r\n        }\r\n        finally { if (e) throw e.error; }\r\n    }\r\n    return ar;\r\n}\r\n\r\n/** @deprecated */\r\nexport function __spread() {\r\n    for (var ar = [], i = 0; i < arguments.length; i++)\r\n        ar = ar.concat(__read(arguments[i]));\r\n    return ar;\r\n}\r\n\r\n/** @deprecated */\r\nexport function __spreadArrays() {\r\n    for (var s = 0, i = 0, il = arguments.length; i < il; i++) s += arguments[i].length;\r\n    for (var r = Array(s), k = 0, i = 0; i < il; i++)\r\n        for (var a = arguments[i], j = 0, jl = a.length; j < jl; j++, k++)\r\n            r[k] = a[j];\r\n    return r;\r\n}\r\n\r\nexport function __spreadArray(to, from, pack) {\r\n    if (pack || arguments.length === 2) for (var i = 0, l = from.length, ar; i < l; i++) {\r\n        if (ar || !(i in from)) {\r\n            if (!ar) ar = Array.prototype.slice.call(from, 0, i);\r\n            ar[i] = from[i];\r\n        }\r\n    }\r\n    return to.concat(ar || Array.prototype.slice.call(from));\r\n}\r\n\r\nexport function __await(v) {\r\n    return this instanceof __await ? (this.v = v, this) : new __await(v);\r\n}\r\n\r\nexport function __asyncGenerator(thisArg, _arguments, generator) {\r\n    if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\r\n    var g = generator.apply(thisArg, _arguments || []), i, q = [];\r\n    return i = {}, verb(\"next\"), verb(\"throw\"), verb(\"return\"), i[Symbol.asyncIterator] = function () { return this; }, i;\r\n    function verb(n) { if (g[n]) i[n] = function (v) { return new Promise(function (a, b) { q.push([n, v, a, b]) > 1 || resume(n, v); }); }; }\r\n    function resume(n, v) { try { step(g[n](v)); } catch (e) { settle(q[0][3], e); } }\r\n    function step(r) { r.value instanceof __await ? Promise.resolve(r.value.v).then(fulfill, reject) : settle(q[0][2], r); }\r\n    function fulfill(value) { resume(\"next\", value); }\r\n    function reject(value) { resume(\"throw\", value); }\r\n    function settle(f, v) { if (f(v), q.shift(), q.length) resume(q[0][0], q[0][1]); }\r\n}\r\n\r\nexport function __asyncDelegator(o) {\r\n    var i, p;\r\n    return i = {}, verb(\"next\"), verb(\"throw\", function (e) { throw e; }), verb(\"return\"), i[Symbol.iterator] = function () { return this; }, i;\r\n    function verb(n, f) { i[n] = o[n] ? function (v) { return (p = !p) ? { value: __await(o[n](v)), done: n === \"return\" } : f ? f(v) : v; } : f; }\r\n}\r\n\r\nexport function __asyncValues(o) {\r\n    if (!Symbol.asyncIterator) throw new TypeError(\"Symbol.asyncIterator is not defined.\");\r\n    var m = o[Symbol.asyncIterator], i;\r\n    return m ? m.call(o) : (o = typeof __values === \"function\" ? __values(o) : o[Symbol.iterator](), i = {}, verb(\"next\"), verb(\"throw\"), verb(\"return\"), i[Symbol.asyncIterator] = function () { return this; }, i);\r\n    function verb(n) { i[n] = o[n] && function (v) { return new Promise(function (resolve, reject) { v = o[n](v), settle(resolve, reject, v.done, v.value); }); }; }\r\n    function settle(resolve, reject, d, v) { Promise.resolve(v).then(function(v) { resolve({ value: v, done: d }); }, reject); }\r\n}\r\n\r\nexport function __makeTemplateObject(cooked, raw) {\r\n    if (Object.defineProperty) { Object.defineProperty(cooked, \"raw\", { value: raw }); } else { cooked.raw = raw; }\r\n    return cooked;\r\n};\r\n\r\nvar __setModuleDefault = Object.create ? (function(o, v) {\r\n    Object.defineProperty(o, \"default\", { enumerable: true, value: v });\r\n}) : function(o, v) {\r\n    o[\"default\"] = v;\r\n};\r\n\r\nexport function __importStar(mod) {\r\n    if (mod && mod.__esModule) return mod;\r\n    var result = {};\r\n    if (mod != null) for (var k in mod) if (k !== \"default\" && Object.prototype.hasOwnProperty.call(mod, k)) __createBinding(result, mod, k);\r\n    __setModuleDefault(result, mod);\r\n    return result;\r\n}\r\n\r\nexport function __importDefault(mod) {\r\n    return (mod && mod.__esModule) ? mod : { default: mod };\r\n}\r\n\r\nexport function __classPrivateFieldGet(receiver, state, kind, f) {\r\n    if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a getter\");\r\n    if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot read private member from an object whose class did not declare it\");\r\n    return kind === \"m\" ? f : kind === \"a\" ? f.call(receiver) : f ? f.value : state.get(receiver);\r\n}\r\n\r\nexport function __classPrivateFieldSet(receiver, state, value, kind, f) {\r\n    if (kind === \"m\") throw new TypeError(\"Private method is not writable\");\r\n    if (kind === \"a\" && !f) throw new TypeError(\"Private accessor was defined without a setter\");\r\n    if (typeof state === \"function\" ? receiver !== state || !f : !state.has(receiver)) throw new TypeError(\"Cannot write private member to an object whose class did not declare it\");\r\n    return (kind === \"a\" ? f.call(receiver, value) : f ? f.value = value : state.set(receiver, value)), value;\r\n}\r\n", "/**\n * Returns true if the object is a function.\n * @param value The value to check\n */\nexport function isFunction(value: any): value is (...args: any[]) => any {\n  return typeof value === 'function';\n}\n", "/**\n * Used to create Error subclasses until the community moves away from ES5.\n *\n * This is because compiling from TypeScript down to ES5 has issues with subclassing Errors\n * as well as other built-in types: https://github.com/Microsoft/TypeScript/issues/12123\n *\n * @param createImpl A factory function to create the actual constructor implementation. The returned\n * function should be a named function that calls `_super` internally.\n */\nexport function createErrorClass<T>(createImpl: (_super: any) => any): T {\n  const _super = (instance: any) => {\n    Error.call(instance);\n    instance.stack = new Error().stack;\n  };\n\n  const ctorFunc = createImpl(_super);\n  ctorFunc.prototype = Object.create(Error.prototype);\n  ctorFunc.prototype.constructor = ctorFunc;\n  return ctorFunc;\n}\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface UnsubscriptionError extends Error {\n  readonly errors: any[];\n}\n\nexport interface UnsubscriptionErrorCtor {\n  /**\n   * @deprecated Internal implementation detail. Do not construct error instances.\n   * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n   */\n  new (errors: any[]): UnsubscriptionError;\n}\n\n/**\n * An error thrown when one or more errors have occurred during the\n * `unsubscribe` of a {@link Subscription}.\n */\nexport const UnsubscriptionError: UnsubscriptionErrorCtor = createErrorClass(\n  (_super) =>\n    function UnsubscriptionErrorImpl(this: any, errors: (Error | string)[]) {\n      _super(this);\n      this.message = errors\n        ? `${errors.length} errors occurred during unsubscription:\n${errors.map((err, i) => `${i + 1}) ${err.toString()}`).join('\\n  ')}`\n        : '';\n      this.name = 'UnsubscriptionError';\n      this.errors = errors;\n    }\n);\n", "/**\n * Removes an item from an array, mutating it.\n * @param arr The array to remove the item from\n * @param item The item to remove\n */\nexport function arrRemove<T>(arr: T[] | undefined | null, item: T) {\n  if (arr) {\n    const index = arr.indexOf(item);\n    0 <= index && arr.splice(index, 1);\n  }\n}\n", "import { isFunction } from './util/isFunction';\nimport { UnsubscriptionError } from './util/UnsubscriptionError';\nimport { SubscriptionLike, TeardownLogic, Unsubscribable } from './types';\nimport { arrRemove } from './util/arrRemove';\n\n/**\n * Represents a disposable resource, such as the execution of an Observable. A\n * Subscription has one important method, `unsubscribe`, that takes no argument\n * and just disposes the resource held by the subscription.\n *\n * Additionally, subscriptions may be grouped together through the `add()`\n * method, which will attach a child Subscription to the current Subscription.\n * When a Subscription is unsubscribed, all its children (and its grandchildren)\n * will be unsubscribed as well.\n *\n * @class Subscription\n */\nexport class Subscription implements SubscriptionLike {\n  /** @nocollapse */\n  public static EMPTY = (() => {\n    const empty = new Subscription();\n    empty.closed = true;\n    return empty;\n  })();\n\n  /**\n   * A flag to indicate whether this Subscription has already been unsubscribed.\n   */\n  public closed = false;\n\n  private _parentage: Subscription[] | Subscription | null = null;\n\n  /**\n   * The list of registered finalizers to execute upon unsubscription. Adding and removing from this\n   * list occurs in the {@link #add} and {@link #remove} methods.\n   */\n  private _finalizers: Exclude<TeardownLogic, void>[] | null = null;\n\n  /**\n   * @param initialTeardown A function executed first as part of the finalization\n   * process that is kicked off when {@link #unsubscribe} is called.\n   */\n  constructor(private initialTeardown?: () => void) {}\n\n  /**\n   * Disposes the resources held by the subscription. May, for instance, cancel\n   * an ongoing Observable execution or cancel any other type of work that\n   * started when the Subscription was created.\n   * @return {void}\n   */\n  unsubscribe(): void {\n    let errors: any[] | undefined;\n\n    if (!this.closed) {\n      this.closed = true;\n\n      // Remove this from it's parents.\n      const { _parentage } = this;\n      if (_parentage) {\n        this._parentage = null;\n        if (Array.isArray(_parentage)) {\n          for (const parent of _parentage) {\n            parent.remove(this);\n          }\n        } else {\n          _parentage.remove(this);\n        }\n      }\n\n      const { initialTeardown: initialFinalizer } = this;\n      if (isFunction(initialFinalizer)) {\n        try {\n          initialFinalizer();\n        } catch (e) {\n          errors = e instanceof UnsubscriptionError ? e.errors : [e];\n        }\n      }\n\n      const { _finalizers } = this;\n      if (_finalizers) {\n        this._finalizers = null;\n        for (const finalizer of _finalizers) {\n          try {\n            execFinalizer(finalizer);\n          } catch (err) {\n            errors = errors ?? [];\n            if (err instanceof UnsubscriptionError) {\n              errors = [...errors, ...err.errors];\n            } else {\n              errors.push(err);\n            }\n          }\n        }\n      }\n\n      if (errors) {\n        throw new UnsubscriptionError(errors);\n      }\n    }\n  }\n\n  /**\n   * Adds a finalizer to this subscription, so that finalization will be unsubscribed/called\n   * when this subscription is unsubscribed. If this subscription is already {@link #closed},\n   * because it has already been unsubscribed, then whatever finalizer is passed to it\n   * will automatically be executed (unless the finalizer itself is also a closed subscription).\n   *\n   * Closed Subscriptions cannot be added as finalizers to any subscription. Adding a closed\n   * subscription to a any subscription will result in no operation. (A noop).\n   *\n   * Adding a subscription to itself, or adding `null` or `undefined` will not perform any\n   * operation at all. (A noop).\n   *\n   * `Subscription` instances that are added to this instance will automatically remove themselves\n   * if they are unsubscribed. Functions and {@link Unsubscribable} objects that you wish to remove\n   * will need to be removed manually with {@link #remove}\n   *\n   * @param teardown The finalization logic to add to this subscription.\n   */\n  add(teardown: TeardownLogic): void {\n    // Only add the finalizer if it's not undefined\n    // and don't add a subscription to itself.\n    if (teardown && teardown !== this) {\n      if (this.closed) {\n        // If this subscription is already closed,\n        // execute whatever finalizer is handed to it automatically.\n        execFinalizer(teardown);\n      } else {\n        if (teardown instanceof Subscription) {\n          // We don't add closed subscriptions, and we don't add the same subscription\n          // twice. Subscription unsubscribe is idempotent.\n          if (teardown.closed || teardown._hasParent(this)) {\n            return;\n          }\n          teardown._addParent(this);\n        }\n        (this._finalizers = this._finalizers ?? []).push(teardown);\n      }\n    }\n  }\n\n  /**\n   * Checks to see if a this subscription already has a particular parent.\n   * This will signal that this subscription has already been added to the parent in question.\n   * @param parent the parent to check for\n   */\n  private _hasParent(parent: Subscription) {\n    const { _parentage } = this;\n    return _parentage === parent || (Array.isArray(_parentage) && _parentage.includes(parent));\n  }\n\n  /**\n   * Adds a parent to this subscription so it can be removed from the parent if it\n   * unsubscribes on it's own.\n   *\n   * NOTE: THIS ASSUMES THAT {@link _hasParent} HAS ALREADY BEEN CHECKED.\n   * @param parent The parent subscription to add\n   */\n  private _addParent(parent: Subscription) {\n    const { _parentage } = this;\n    this._parentage = Array.isArray(_parentage) ? (_parentage.push(parent), _parentage) : _parentage ? [_parentage, parent] : parent;\n  }\n\n  /**\n   * Called on a child when it is removed via {@link #remove}.\n   * @param parent The parent to remove\n   */\n  private _removeParent(parent: Subscription) {\n    const { _parentage } = this;\n    if (_parentage === parent) {\n      this._parentage = null;\n    } else if (Array.isArray(_parentage)) {\n      arrRemove(_parentage, parent);\n    }\n  }\n\n  /**\n   * Removes a finalizer from this subscription that was previously added with the {@link #add} method.\n   *\n   * Note that `Subscription` instances, when unsubscribed, will automatically remove themselves\n   * from every other `Subscription` they have been added to. This means that using the `remove` method\n   * is not a common thing and should be used thoughtfully.\n   *\n   * If you add the same finalizer instance of a function or an unsubscribable object to a `Subscription` instance\n   * more than once, you will need to call `remove` the same number of times to remove all instances.\n   *\n   * All finalizer instances are removed to free up memory upon unsubscription.\n   *\n   * @param teardown The finalizer to remove from this subscription\n   */\n  remove(teardown: Exclude<TeardownLogic, void>): void {\n    const { _finalizers } = this;\n    _finalizers && arrRemove(_finalizers, teardown);\n\n    if (teardown instanceof Subscription) {\n      teardown._removeParent(this);\n    }\n  }\n}\n\nexport const EMPTY_SUBSCRIPTION = Subscription.EMPTY;\n\nexport function isSubscription(value: any): value is Subscription {\n  return (\n    value instanceof Subscription ||\n    (value && 'closed' in value && isFunction(value.remove) && isFunction(value.add) && isFunction(value.unsubscribe))\n  );\n}\n\nfunction execFinalizer(finalizer: Unsubscribable | (() => void)) {\n  if (isFunction(finalizer)) {\n    finalizer();\n  } else {\n    finalizer.unsubscribe();\n  }\n}\n", "import { Subscriber } from './Subscriber';\nimport { ObservableNotification } from './types';\n\n/**\n * The {@link GlobalConfig} object for RxJS. It is used to configure things\n * like how to react on unhandled errors.\n */\nexport const config: GlobalConfig = {\n  onUnhandledError: null,\n  onStoppedNotification: null,\n  Promise: undefined,\n  useDeprecatedSynchronousErrorHandling: false,\n  useDeprecatedNextContext: false,\n};\n\n/**\n * The global configuration object for RxJS, used to configure things\n * like how to react on unhandled errors. Accessible via {@link config}\n * object.\n */\nexport interface GlobalConfig {\n  /**\n   * A registration point for unhandled errors from RxJS. These are errors that\n   * cannot were not handled by consuming code in the usual subscription path. For\n   * example, if you have this configured, and you subscribe to an observable without\n   * providing an error handler, errors from that subscription will end up here. This\n   * will _always_ be called asynchronously on another job in the runtime. This is because\n   * we do not want errors thrown in this user-configured handler to interfere with the\n   * behavior of the library.\n   */\n  onUnhandledError: ((err: any) => void) | null;\n\n  /**\n   * A registration point for notifications that cannot be sent to subscribers because they\n   * have completed, errored or have been explicitly unsubscribed. By default, next, complete\n   * and error notifications sent to stopped subscribers are noops. However, sometimes callers\n   * might want a different behavior. For example, with sources that attempt to report errors\n   * to stopped subscribers, a caller can configure RxJS to throw an unhandled error instead.\n   * This will _always_ be called asynchronously on another job in the runtime. This is because\n   * we do not want errors thrown in this user-configured handler to interfere with the\n   * behavior of the library.\n   */\n  onStoppedNotification: ((notification: ObservableNotification<any>, subscriber: Subscriber<any>) => void) | null;\n\n  /**\n   * The promise constructor used by default for {@link Observable#toPromise toPromise} and {@link Observable#forEach forEach}\n   * methods.\n   *\n   * @deprecated As of version 8, RxJS will no longer support this sort of injection of a\n   * Promise constructor. If you need a Promise implementation other than native promises,\n   * please polyfill/patch Promise as you see appropriate. Will be removed in v8.\n   */\n  Promise?: PromiseConstructorLike;\n\n  /**\n   * If true, turns on synchronous error rethrowing, which is a deprecated behavior\n   * in v6 and higher. This behavior enables bad patterns like wrapping a subscribe\n   * call in a try/catch block. It also enables producer interference, a nasty bug\n   * where a multicast can be broken for all observers by a downstream consumer with\n   * an unhandled error. DO NOT USE THIS FLAG UNLESS IT'S NEEDED TO BUY TIME\n   * FOR MIGRATION REASONS.\n   *\n   * @deprecated As of version 8, RxJS will no longer support synchronous throwing\n   * of unhandled errors. All errors will be thrown on a separate call stack to prevent bad\n   * behaviors described above. Will be removed in v8.\n   */\n  useDeprecatedSynchronousErrorHandling: boolean;\n\n  /**\n   * If true, enables an as-of-yet undocumented feature from v5: The ability to access\n   * `unsubscribe()` via `this` context in `next` functions created in observers passed\n   * to `subscribe`.\n   *\n   * This is being removed because the performance was severely problematic, and it could also cause\n   * issues when types other than POJOs are passed to subscribe as subscribers, as they will likely have\n   * their `this` context overwritten.\n   *\n   * @deprecated As of version 8, RxJS will no longer support altering the\n   * context of next functions provided as part of an observer to Subscribe. Instead,\n   * you will have access to a subscription or a signal or token that will allow you to do things like\n   * unsubscribe and test closed status. Will be removed in v8.\n   */\n  useDeprecatedNextContext: boolean;\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetTimeoutFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearTimeoutFunction = (handle: TimerHandle) => void;\n\ninterface TimeoutProvider {\n  setTimeout: SetTimeoutFunction;\n  clearTimeout: ClearTimeoutFunction;\n  delegate:\n    | {\n        setTimeout: SetTimeoutFunction;\n        clearTimeout: ClearTimeoutFunction;\n      }\n    | undefined;\n}\n\nexport const timeoutProvider: TimeoutProvider = {\n  // When accessing the delegate, use the variable rather than `this` so that\n  // the functions can be called without being bound to the provider.\n  setTimeout(handler: () => void, timeout?: number, ...args) {\n    const { delegate } = timeoutProvider;\n    if (delegate?.setTimeout) {\n      return delegate.setTimeout(handler, timeout, ...args);\n    }\n    return setTimeout(handler, timeout, ...args);\n  },\n  clearTimeout(handle) {\n    const { delegate } = timeoutProvider;\n    return (delegate?.clearTimeout || clearTimeout)(handle as any);\n  },\n  delegate: undefined,\n};\n", "import { config } from '../config';\nimport { timeoutProvider } from '../scheduler/timeoutProvider';\n\n/**\n * Handles an error on another job either with the user-configured {@link onUnhandledError},\n * or by throwing it on that new job so it can be picked up by `window.onerror`, `process.on('error')`, etc.\n *\n * This should be called whenever there is an error that is out-of-band with the subscription\n * or when an error hits a terminal boundary of the subscription and no error handler was provided.\n *\n * @param err the error to report\n */\nexport function reportUnhandledError(err: any) {\n  timeoutProvider.setTimeout(() => {\n    const { onUnhandledError } = config;\n    if (onUnhandledError) {\n      // Execute the user-configured error handler.\n      onUnhandledError(err);\n    } else {\n      // Throw so it is picked up by the runtime's uncaught error mechanism.\n      throw err;\n    }\n  });\n}\n", "/* tslint:disable:no-empty */\nexport function noop() { }\n", "import { CompleteNotification, NextNotification, ErrorNotification } from './types';\n\n/**\n * A completion object optimized for memory use and created to be the\n * same \"shape\" as other notifications in v8.\n * @internal\n */\nexport const COMPLETE_NOTIFICATION = (() => createNotification('C', undefined, undefined) as CompleteNotification)();\n\n/**\n * Internal use only. Creates an optimized error notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function errorNotification(error: any): ErrorNotification {\n  return createNotification('E', undefined, error) as any;\n}\n\n/**\n * Internal use only. Creates an optimized next notification that is the same \"shape\"\n * as other notifications.\n * @internal\n */\nexport function nextNotification<T>(value: T) {\n  return createNotification('N', value, undefined) as NextNotification<T>;\n}\n\n/**\n * Ensures that all notifications created internally have the same \"shape\" in v8.\n *\n * TODO: This is only exported to support a crazy legacy test in `groupBy`.\n * @internal\n */\nexport function createNotification(kind: 'N' | 'E' | 'C', value: any, error: any) {\n  return {\n    kind,\n    value,\n    error,\n  };\n}\n", "import { config } from '../config';\n\nlet context: { errorThrown: boolean; error: any } | null = null;\n\n/**\n * Handles dealing with errors for super-gross mode. Creates a context, in which\n * any synchronously thrown errors will be passed to {@link captureError}. Which\n * will record the error such that it will be rethrown after the call back is complete.\n * TODO: Remove in v8\n * @param cb An immediately executed function.\n */\nexport function errorContext(cb: () => void) {\n  if (config.useDeprecatedSynchronousErrorHandling) {\n    const isRoot = !context;\n    if (isRoot) {\n      context = { errorThrown: false, error: null };\n    }\n    cb();\n    if (isRoot) {\n      const { errorThrown, error } = context!;\n      context = null;\n      if (errorThrown) {\n        throw error;\n      }\n    }\n  } else {\n    // This is the general non-deprecated path for everyone that\n    // isn't crazy enough to use super-gross mode (useDeprecatedSynchronousErrorHandling)\n    cb();\n  }\n}\n\n/**\n * Captures errors only in super-gross mode.\n * @param err the error to capture\n */\nexport function captureError(err: any) {\n  if (config.useDeprecatedSynchronousErrorHandling && context) {\n    context.errorThrown = true;\n    context.error = err;\n  }\n}\n", "import { isFunction } from './util/isFunction';\nimport { Observer, ObservableNotification } from './types';\nimport { isSubscription, Subscription } from './Subscription';\nimport { config } from './config';\nimport { reportUnhandledError } from './util/reportUnhandledError';\nimport { noop } from './util/noop';\nimport { nextNotification, errorNotification, COMPLETE_NOTIFICATION } from './NotificationFactories';\nimport { timeoutProvider } from './scheduler/timeoutProvider';\nimport { captureError } from './util/errorContext';\n\n/**\n * Implements the {@link Observer} interface and extends the\n * {@link Subscription} class. While the {@link Observer} is the public API for\n * consuming the values of an {@link Observable}, all Observers get converted to\n * a Subscriber, in order to provide Subscription-like capabilities such as\n * `unsubscribe`. Subscriber is a common type in RxJS, and crucial for\n * implementing operators, but it is rarely used as a public API.\n *\n * @class Subscriber<T>\n */\nexport class Subscriber<T> extends Subscription implements Observer<T> {\n  /**\n   * A static factory for a Subscriber, given a (potentially partial) definition\n   * of an Observer.\n   * @param next The `next` callback of an Observer.\n   * @param error The `error` callback of an\n   * Observer.\n   * @param complete The `complete` callback of an\n   * Observer.\n   * @return A Subscriber wrapping the (partially defined)\n   * Observer represented by the given arguments.\n   * @nocollapse\n   * @deprecated Do not use. Will be removed in v8. There is no replacement for this\n   * method, and there is no reason to be creating instances of `Subscriber` directly.\n   * If you have a specific use case, please file an issue.\n   */\n  static create<T>(next?: (x?: T) => void, error?: (e?: any) => void, complete?: () => void): Subscriber<T> {\n    return new SafeSubscriber(next, error, complete);\n  }\n\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  protected isStopped: boolean = false;\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  protected destination: Subscriber<any> | Observer<any>; // this `any` is the escape hatch to erase extra type param (e.g. R)\n\n  /**\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   * There is no reason to directly create an instance of Subscriber. This type is exported for typings reasons.\n   */\n  constructor(destination?: Subscriber<any> | Observer<any>) {\n    super();\n    if (destination) {\n      this.destination = destination;\n      // Automatically chain subscriptions together here.\n      // if destination is a Subscription, then it is a Subscriber.\n      if (isSubscription(destination)) {\n        destination.add(this);\n      }\n    } else {\n      this.destination = EMPTY_OBSERVER;\n    }\n  }\n\n  /**\n   * The {@link Observer} callback to receive notifications of type `next` from\n   * the Observable, with a value. The Observable may call this method 0 or more\n   * times.\n   * @param {T} [value] The `next` value.\n   * @return {void}\n   */\n  next(value?: T): void {\n    if (this.isStopped) {\n      handleStoppedNotification(nextNotification(value), this);\n    } else {\n      this._next(value!);\n    }\n  }\n\n  /**\n   * The {@link Observer} callback to receive notifications of type `error` from\n   * the Observable, with an attached `Error`. Notifies the Observer that\n   * the Observable has experienced an error condition.\n   * @param {any} [err] The `error` exception.\n   * @return {void}\n   */\n  error(err?: any): void {\n    if (this.isStopped) {\n      handleStoppedNotification(errorNotification(err), this);\n    } else {\n      this.isStopped = true;\n      this._error(err);\n    }\n  }\n\n  /**\n   * The {@link Observer} callback to receive a valueless notification of type\n   * `complete` from the Observable. Notifies the Observer that the Observable\n   * has finished sending push-based notifications.\n   * @return {void}\n   */\n  complete(): void {\n    if (this.isStopped) {\n      handleStoppedNotification(COMPLETE_NOTIFICATION, this);\n    } else {\n      this.isStopped = true;\n      this._complete();\n    }\n  }\n\n  unsubscribe(): void {\n    if (!this.closed) {\n      this.isStopped = true;\n      super.unsubscribe();\n      this.destination = null!;\n    }\n  }\n\n  protected _next(value: T): void {\n    this.destination.next(value);\n  }\n\n  protected _error(err: any): void {\n    try {\n      this.destination.error(err);\n    } finally {\n      this.unsubscribe();\n    }\n  }\n\n  protected _complete(): void {\n    try {\n      this.destination.complete();\n    } finally {\n      this.unsubscribe();\n    }\n  }\n}\n\n/**\n * This bind is captured here because we want to be able to have\n * compatibility with monoid libraries that tend to use a method named\n * `bind`. In particular, a library called Monio requires this.\n */\nconst _bind = Function.prototype.bind;\n\nfunction bind<Fn extends (...args: any[]) => any>(fn: Fn, thisArg: any): Fn {\n  return _bind.call(fn, thisArg);\n}\n\n/**\n * Internal optimization only, DO NOT EXPOSE.\n * @internal\n */\nclass ConsumerObserver<T> implements Observer<T> {\n  constructor(private partialObserver: Partial<Observer<T>>) {}\n\n  next(value: T): void {\n    const { partialObserver } = this;\n    if (partialObserver.next) {\n      try {\n        partialObserver.next(value);\n      } catch (error) {\n        handleUnhandledError(error);\n      }\n    }\n  }\n\n  error(err: any): void {\n    const { partialObserver } = this;\n    if (partialObserver.error) {\n      try {\n        partialObserver.error(err);\n      } catch (error) {\n        handleUnhandledError(error);\n      }\n    } else {\n      handleUnhandledError(err);\n    }\n  }\n\n  complete(): void {\n    const { partialObserver } = this;\n    if (partialObserver.complete) {\n      try {\n        partialObserver.complete();\n      } catch (error) {\n        handleUnhandledError(error);\n      }\n    }\n  }\n}\n\nexport class SafeSubscriber<T> extends Subscriber<T> {\n  constructor(\n    observerOrNext?: Partial<Observer<T>> | ((value: T) => void) | null,\n    error?: ((e?: any) => void) | null,\n    complete?: (() => void) | null\n  ) {\n    super();\n\n    let partialObserver: Partial<Observer<T>>;\n    if (isFunction(observerOrNext) || !observerOrNext) {\n      // The first argument is a function, not an observer. The next\n      // two arguments *could* be observers, or they could be empty.\n      partialObserver = {\n        next: (observerOrNext ?? undefined) as (((value: T) => void) | undefined),\n        error: error ?? undefined,\n        complete: complete ?? undefined,\n      };\n    } else {\n      // The first argument is a partial observer.\n      let context: any;\n      if (this && config.useDeprecatedNextContext) {\n        // This is a deprecated path that made `this.unsubscribe()` available in\n        // next handler functions passed to subscribe. This only exists behind a flag\n        // now, as it is *very* slow.\n        context = Object.create(observerOrNext);\n        context.unsubscribe = () => this.unsubscribe();\n        partialObserver = {\n          next: observerOrNext.next && bind(observerOrNext.next, context),\n          error: observerOrNext.error && bind(observerOrNext.error, context),\n          complete: observerOrNext.complete && bind(observerOrNext.complete, context),\n        };\n      } else {\n        // The \"normal\" path. Just use the partial observer directly.\n        partialObserver = observerOrNext;\n      }\n    }\n\n    // Wrap the partial observer to ensure it's a full observer, and\n    // make sure proper error handling is accounted for.\n    this.destination = new ConsumerObserver(partialObserver);\n  }\n}\n\nfunction handleUnhandledError(error: any) {\n  if (config.useDeprecatedSynchronousErrorHandling) {\n    captureError(error);\n  } else {\n    // Ideal path, we report this as an unhandled error,\n    // which is thrown on a new call stack.\n    reportUnhandledError(error);\n  }\n}\n\n/**\n * An error handler used when no error handler was supplied\n * to the SafeSubscriber -- meaning no error handler was supplied\n * do the `subscribe` call on our observable.\n * @param err The error to handle\n */\nfunction defaultErrorHandler(err: any) {\n  throw err;\n}\n\n/**\n * A handler for notifications that cannot be sent to a stopped subscriber.\n * @param notification The notification being sent\n * @param subscriber The stopped subscriber\n */\nfunction handleStoppedNotification(notification: ObservableNotification<any>, subscriber: Subscriber<any>) {\n  const { onStoppedNotification } = config;\n  onStoppedNotification && timeoutProvider.setTimeout(() => onStoppedNotification(notification, subscriber));\n}\n\n/**\n * The observer used as a stub for subscriptions where the user did not\n * pass any arguments to `subscribe`. Comes with the default error handling\n * behavior.\n */\nexport const EMPTY_OBSERVER: Readonly<Observer<any>> & { closed: true } = {\n  closed: true,\n  next: noop,\n  error: defaultErrorHandler,\n  complete: noop,\n};\n", "/**\n * Symbol.observable or a string \"@@observable\". Used for interop\n *\n * @deprecated We will no longer be exporting this symbol in upcoming versions of RxJS.\n * Instead polyfill and use Symbol.observable directly *or* use https://www.npmjs.com/package/symbol-observable\n */\nexport const observable: string | symbol = (() => (typeof Symbol === 'function' && Symbol.observable) || '@@observable')();\n", "/**\n * This function takes one parameter and just returns it. Simply put,\n * this is like `<T>(x: T): T => x`.\n *\n * ## Examples\n *\n * This is useful in some cases when using things like `mergeMap`\n *\n * ```ts\n * import { interval, take, map, range, mergeMap, identity } from 'rxjs';\n *\n * const source$ = interval(1000).pipe(take(5));\n *\n * const result$ = source$.pipe(\n *   map(i => range(i)),\n *   mergeMap(identity) // same as mergeMap(x => x)\n * );\n *\n * result$.subscribe({\n *   next: console.log\n * });\n * ```\n *\n * Or when you want to selectively apply an operator\n *\n * ```ts\n * import { interval, take, identity } from 'rxjs';\n *\n * const shouldLimit = () => Math.random() < 0.5;\n *\n * const source$ = interval(1000);\n *\n * const result$ = source$.pipe(shouldLimit() ? take(5) : identity);\n *\n * result$.subscribe({\n *   next: console.log\n * });\n * ```\n *\n * @param x Any value that is returned by this function\n * @returns The value passed as the first parameter to this function\n */\nexport function identity<T>(x: T): T {\n  return x;\n}\n", "import { identity } from './identity';\nimport { UnaryFunction } from '../types';\n\nexport function pipe(): typeof identity;\nexport function pipe<T, A>(fn1: UnaryFunction<T, A>): UnaryFunction<T, A>;\nexport function pipe<T, A, B>(fn1: UnaryFunction<T, A>, fn2: UnaryFunction<A, B>): UnaryFunction<T, B>;\nexport function pipe<T, A, B, C>(fn1: UnaryFunction<T, A>, fn2: UnaryFunction<A, B>, fn3: UnaryFunction<B, C>): UnaryFunction<T, C>;\nexport function pipe<T, A, B, C, D>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>\n): UnaryFunction<T, D>;\nexport function pipe<T, A, B, C, D, E>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>\n): UnaryFunction<T, E>;\nexport function pipe<T, A, B, C, D, E, F>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>\n): UnaryFunction<T, F>;\nexport function pipe<T, A, B, C, D, E, F, G>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>\n): UnaryFunction<T, G>;\nexport function pipe<T, A, B, C, D, E, F, G, H>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>,\n  fn8: UnaryFunction<G, H>\n): UnaryFunction<T, H>;\nexport function pipe<T, A, B, C, D, E, F, G, H, I>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>,\n  fn8: UnaryFunction<G, H>,\n  fn9: UnaryFunction<H, I>\n): UnaryFunction<T, I>;\nexport function pipe<T, A, B, C, D, E, F, G, H, I>(\n  fn1: UnaryFunction<T, A>,\n  fn2: UnaryFunction<A, B>,\n  fn3: UnaryFunction<B, C>,\n  fn4: UnaryFunction<C, D>,\n  fn5: UnaryFunction<D, E>,\n  fn6: UnaryFunction<E, F>,\n  fn7: UnaryFunction<F, G>,\n  fn8: UnaryFunction<G, H>,\n  fn9: UnaryFunction<H, I>,\n  ...fns: UnaryFunction<any, any>[]\n): UnaryFunction<T, unknown>;\n\n/**\n * pipe() can be called on one or more functions, each of which can take one argument (\"UnaryFunction\")\n * and uses it to return a value.\n * It returns a function that takes one argument, passes it to the first UnaryFunction, and then\n * passes the result to the next one, passes that result to the next one, and so on.  \n */\nexport function pipe(...fns: Array<UnaryFunction<any, any>>): UnaryFunction<any, any> {\n  return pipeFromArray(fns);\n}\n\n/** @internal */\nexport function pipeFromArray<T, R>(fns: Array<UnaryFunction<T, R>>): UnaryFunction<T, R> {\n  if (fns.length === 0) {\n    return identity as UnaryFunction<any, any>;\n  }\n\n  if (fns.length === 1) {\n    return fns[0];\n  }\n\n  return function piped(input: T): R {\n    return fns.reduce((prev: any, fn: UnaryFunction<T, R>) => fn(prev), input as any);\n  };\n}\n", "import { Operator } from './Operator';\nimport { SafeSubscriber, Subscriber } from './Subscriber';\nimport { isSubscription, Subscription } from './Subscription';\nimport { TeardownLogic, OperatorFunction, Subscribable, Observer } from './types';\nimport { observable as Symbol_observable } from './symbol/observable';\nimport { pipeFromArray } from './util/pipe';\nimport { config } from './config';\nimport { isFunction } from './util/isFunction';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A representation of any set of values over any amount of time. This is the most basic building block\n * of RxJS.\n *\n * @class Observable<T>\n */\nexport class Observable<T> implements Subscribable<T> {\n  /**\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   */\n  source: Observable<any> | undefined;\n\n  /**\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   */\n  operator: Operator<any, T> | undefined;\n\n  /**\n   * @constructor\n   * @param {Function} subscribe the function that is called when the Observable is\n   * initially subscribed to. This function is given a Subscriber, to which new values\n   * can be `next`ed, or an `error` method can be called to raise an error, or\n   * `complete` can be called to notify of a successful completion.\n   */\n  constructor(subscribe?: (this: Observable<T>, subscriber: Subscriber<T>) => TeardownLogic) {\n    if (subscribe) {\n      this._subscribe = subscribe;\n    }\n  }\n\n  // HACK: Since TypeScript inherits static properties too, we have to\n  // fight against TypeScript here so Subject can have a different static create signature\n  /**\n   * Creates a new Observable by calling the Observable constructor\n   * @owner Observable\n   * @method create\n   * @param {Function} subscribe? the subscriber function to be passed to the Observable constructor\n   * @return {Observable} a new observable\n   * @nocollapse\n   * @deprecated Use `new Observable()` instead. Will be removed in v8.\n   */\n  static create: (...args: any[]) => any = <T>(subscribe?: (subscriber: Subscriber<T>) => TeardownLogic) => {\n    return new Observable<T>(subscribe);\n  };\n\n  /**\n   * Creates a new Observable, with this Observable instance as the source, and the passed\n   * operator defined as the new observable's operator.\n   * @method lift\n   * @param operator the operator defining the operation to take on the observable\n   * @return a new observable with the Operator applied\n   * @deprecated Internal implementation detail, do not use directly. Will be made internal in v8.\n   * If you have implemented an operator using `lift`, it is recommended that you create an\n   * operator by simply returning `new Observable()` directly. See \"Creating new operators from\n   * scratch\" section here: https://rxjs.dev/guide/operators\n   */\n  lift<R>(operator?: Operator<T, R>): Observable<R> {\n    const observable = new Observable<R>();\n    observable.source = this;\n    observable.operator = operator;\n    return observable;\n  }\n\n  subscribe(observerOrNext?: Partial<Observer<T>> | ((value: T) => void)): Subscription;\n  /** @deprecated Instead of passing separate callback arguments, use an observer argument. Signatures taking separate callback arguments will be removed in v8. Details: https://rxjs.dev/deprecations/subscribe-arguments */\n  subscribe(next?: ((value: T) => void) | null, error?: ((error: any) => void) | null, complete?: (() => void) | null): Subscription;\n  /**\n   * Invokes an execution of an Observable and registers Observer handlers for notifications it will emit.\n   *\n   * <span class=\"informal\">Use it when you have all these Observables, but still nothing is happening.</span>\n   *\n   * `subscribe` is not a regular operator, but a method that calls Observable's internal `subscribe` function. It\n   * might be for example a function that you passed to Observable's constructor, but most of the time it is\n   * a library implementation, which defines what will be emitted by an Observable, and when it be will emitted. This means\n   * that calling `subscribe` is actually the moment when Observable starts its work, not when it is created, as it is often\n   * the thought.\n   *\n   * Apart from starting the execution of an Observable, this method allows you to listen for values\n   * that an Observable emits, as well as for when it completes or errors. You can achieve this in two\n   * of the following ways.\n   *\n   * The first way is creating an object that implements {@link Observer} interface. It should have methods\n   * defined by that interface, but note that it should be just a regular JavaScript object, which you can create\n   * yourself in any way you want (ES6 class, classic function constructor, object literal etc.). In particular, do\n   * not attempt to use any RxJS implementation details to create Observers - you don't need them. Remember also\n   * that your object does not have to implement all methods. If you find yourself creating a method that doesn't\n   * do anything, you can simply omit it. Note however, if the `error` method is not provided and an error happens,\n   * it will be thrown asynchronously. Errors thrown asynchronously cannot be caught using `try`/`catch`. Instead,\n   * use the {@link onUnhandledError} configuration option or use a runtime handler (like `window.onerror` or\n   * `process.on('error)`) to be notified of unhandled errors. Because of this, it's recommended that you provide\n   * an `error` method to avoid missing thrown errors.\n   *\n   * The second way is to give up on Observer object altogether and simply provide callback functions in place of its methods.\n   * This means you can provide three functions as arguments to `subscribe`, where the first function is equivalent\n   * of a `next` method, the second of an `error` method and the third of a `complete` method. Just as in case of an Observer,\n   * if you do not need to listen for something, you can omit a function by passing `undefined` or `null`,\n   * since `subscribe` recognizes these functions by where they were placed in function call. When it comes\n   * to the `error` function, as with an Observer, if not provided, errors emitted by an Observable will be thrown asynchronously.\n   *\n   * You can, however, subscribe with no parameters at all. This may be the case where you're not interested in terminal events\n   * and you also handled emissions internally by using operators (e.g. using `tap`).\n   *\n   * Whichever style of calling `subscribe` you use, in both cases it returns a Subscription object.\n   * This object allows you to call `unsubscribe` on it, which in turn will stop the work that an Observable does and will clean\n   * up all resources that an Observable used. Note that cancelling a subscription will not call `complete` callback\n   * provided to `subscribe` function, which is reserved for a regular completion signal that comes from an Observable.\n   *\n   * Remember that callbacks provided to `subscribe` are not guaranteed to be called asynchronously.\n   * It is an Observable itself that decides when these functions will be called. For example {@link of}\n   * by default emits all its values synchronously. Always check documentation for how given Observable\n   * will behave when subscribed and if its default behavior can be modified with a `scheduler`.\n   *\n   * #### Examples\n   *\n   * Subscribe with an {@link guide/observer Observer}\n   *\n   * ```ts\n   * import { of } from 'rxjs';\n   *\n   * const sumObserver = {\n   *   sum: 0,\n   *   next(value) {\n   *     console.log('Adding: ' + value);\n   *     this.sum = this.sum + value;\n   *   },\n   *   error() {\n   *     // We actually could just remove this method,\n   *     // since we do not really care about errors right now.\n   *   },\n   *   complete() {\n   *     console.log('Sum equals: ' + this.sum);\n   *   }\n   * };\n   *\n   * of(1, 2, 3) // Synchronously emits 1, 2, 3 and then completes.\n   *   .subscribe(sumObserver);\n   *\n   * // Logs:\n   * // 'Adding: 1'\n   * // 'Adding: 2'\n   * // 'Adding: 3'\n   * // 'Sum equals: 6'\n   * ```\n   *\n   * Subscribe with functions ({@link deprecations/subscribe-arguments deprecated})\n   *\n   * ```ts\n   * import { of } from 'rxjs'\n   *\n   * let sum = 0;\n   *\n   * of(1, 2, 3).subscribe(\n   *   value => {\n   *     console.log('Adding: ' + value);\n   *     sum = sum + value;\n   *   },\n   *   undefined,\n   *   () => console.log('Sum equals: ' + sum)\n   * );\n   *\n   * // Logs:\n   * // 'Adding: 1'\n   * // 'Adding: 2'\n   * // 'Adding: 3'\n   * // 'Sum equals: 6'\n   * ```\n   *\n   * Cancel a subscription\n   *\n   * ```ts\n   * import { interval } from 'rxjs';\n   *\n   * const subscription = interval(1000).subscribe({\n   *   next(num) {\n   *     console.log(num)\n   *   },\n   *   complete() {\n   *     // Will not be called, even when cancelling subscription.\n   *     console.log('completed!');\n   *   }\n   * });\n   *\n   * setTimeout(() => {\n   *   subscription.unsubscribe();\n   *   console.log('unsubscribed!');\n   * }, 2500);\n   *\n   * // Logs:\n   * // 0 after 1s\n   * // 1 after 2s\n   * // 'unsubscribed!' after 2.5s\n   * ```\n   *\n   * @param {Observer|Function} observerOrNext (optional) Either an observer with methods to be called,\n   * or the first of three possible handlers, which is the handler for each value emitted from the subscribed\n   * Observable.\n   * @param {Function} error (optional) A handler for a terminal event resulting from an error. If no error handler is provided,\n   * the error will be thrown asynchronously as unhandled.\n   * @param {Function} complete (optional) A handler for a terminal event resulting from successful completion.\n   * @return {Subscription} a subscription reference to the registered handlers\n   * @method subscribe\n   */\n  subscribe(\n    observerOrNext?: Partial<Observer<T>> | ((value: T) => void) | null,\n    error?: ((error: any) => void) | null,\n    complete?: (() => void) | null\n  ): Subscription {\n    const subscriber = isSubscriber(observerOrNext) ? observerOrNext : new SafeSubscriber(observerOrNext, error, complete);\n\n    errorContext(() => {\n      const { operator, source } = this;\n      subscriber.add(\n        operator\n          ? // We're dealing with a subscription in the\n            // operator chain to one of our lifted operators.\n            operator.call(subscriber, source)\n          : source\n          ? // If `source` has a value, but `operator` does not, something that\n            // had intimate knowledge of our API, like our `Subject`, must have\n            // set it. We're going to just call `_subscribe` directly.\n            this._subscribe(subscriber)\n          : // In all other cases, we're likely wrapping a user-provided initializer\n            // function, so we need to catch errors and handle them appropriately.\n            this._trySubscribe(subscriber)\n      );\n    });\n\n    return subscriber;\n  }\n\n  /** @internal */\n  protected _trySubscribe(sink: Subscriber<T>): TeardownLogic {\n    try {\n      return this._subscribe(sink);\n    } catch (err) {\n      // We don't need to return anything in this case,\n      // because it's just going to try to `add()` to a subscription\n      // above.\n      sink.error(err);\n    }\n  }\n\n  /**\n   * Used as a NON-CANCELLABLE means of subscribing to an observable, for use with\n   * APIs that expect promises, like `async/await`. You cannot unsubscribe from this.\n   *\n   * **WARNING**: Only use this with observables you *know* will complete. If the source\n   * observable does not complete, you will end up with a promise that is hung up, and\n   * potentially all of the state of an async function hanging out in memory. To avoid\n   * this situation, look into adding something like {@link timeout}, {@link take},\n   * {@link takeWhile}, or {@link takeUntil} amongst others.\n   *\n   * #### Example\n   *\n   * ```ts\n   * import { interval, take } from 'rxjs';\n   *\n   * const source$ = interval(1000).pipe(take(4));\n   *\n   * async function getTotal() {\n   *   let total = 0;\n   *\n   *   await source$.forEach(value => {\n   *     total += value;\n   *     console.log('observable -> ' + value);\n   *   });\n   *\n   *   return total;\n   * }\n   *\n   * getTotal().then(\n   *   total => console.log('Total: ' + total)\n   * );\n   *\n   * // Expected:\n   * // 'observable -> 0'\n   * // 'observable -> 1'\n   * // 'observable -> 2'\n   * // 'observable -> 3'\n   * // 'Total: 6'\n   * ```\n   *\n   * @param next a handler for each value emitted by the observable\n   * @return a promise that either resolves on observable completion or\n   *  rejects with the handled error\n   */\n  forEach(next: (value: T) => void): Promise<void>;\n\n  /**\n   * @param next a handler for each value emitted by the observable\n   * @param promiseCtor a constructor function used to instantiate the Promise\n   * @return a promise that either resolves on observable completion or\n   *  rejects with the handled error\n   * @deprecated Passing a Promise constructor will no longer be available\n   * in upcoming versions of RxJS. This is because it adds weight to the library, for very\n   * little benefit. If you need this functionality, it is recommended that you either\n   * polyfill Promise, or you create an adapter to convert the returned native promise\n   * to whatever promise implementation you wanted. Will be removed in v8.\n   */\n  forEach(next: (value: T) => void, promiseCtor: PromiseConstructorLike): Promise<void>;\n\n  forEach(next: (value: T) => void, promiseCtor?: PromiseConstructorLike): Promise<void> {\n    promiseCtor = getPromiseCtor(promiseCtor);\n\n    return new promiseCtor<void>((resolve, reject) => {\n      const subscriber = new SafeSubscriber<T>({\n        next: (value) => {\n          try {\n            next(value);\n          } catch (err) {\n            reject(err);\n            subscriber.unsubscribe();\n          }\n        },\n        error: reject,\n        complete: resolve,\n      });\n      this.subscribe(subscriber);\n    }) as Promise<void>;\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<any>): TeardownLogic {\n    return this.source?.subscribe(subscriber);\n  }\n\n  /**\n   * An interop point defined by the es7-observable spec https://github.com/zenparsing/es-observable\n   * @method Symbol.observable\n   * @return {Observable} this instance of the observable\n   */\n  [Symbol_observable]() {\n    return this;\n  }\n\n  /* tslint:disable:max-line-length */\n  pipe(): Observable<T>;\n  pipe<A>(op1: OperatorFunction<T, A>): Observable<A>;\n  pipe<A, B>(op1: OperatorFunction<T, A>, op2: OperatorFunction<A, B>): Observable<B>;\n  pipe<A, B, C>(op1: OperatorFunction<T, A>, op2: OperatorFunction<A, B>, op3: OperatorFunction<B, C>): Observable<C>;\n  pipe<A, B, C, D>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>\n  ): Observable<D>;\n  pipe<A, B, C, D, E>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>\n  ): Observable<E>;\n  pipe<A, B, C, D, E, F>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>\n  ): Observable<F>;\n  pipe<A, B, C, D, E, F, G>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>\n  ): Observable<G>;\n  pipe<A, B, C, D, E, F, G, H>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>,\n    op8: OperatorFunction<G, H>\n  ): Observable<H>;\n  pipe<A, B, C, D, E, F, G, H, I>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>,\n    op8: OperatorFunction<G, H>,\n    op9: OperatorFunction<H, I>\n  ): Observable<I>;\n  pipe<A, B, C, D, E, F, G, H, I>(\n    op1: OperatorFunction<T, A>,\n    op2: OperatorFunction<A, B>,\n    op3: OperatorFunction<B, C>,\n    op4: OperatorFunction<C, D>,\n    op5: OperatorFunction<D, E>,\n    op6: OperatorFunction<E, F>,\n    op7: OperatorFunction<F, G>,\n    op8: OperatorFunction<G, H>,\n    op9: OperatorFunction<H, I>,\n    ...operations: OperatorFunction<any, any>[]\n  ): Observable<unknown>;\n  /* tslint:enable:max-line-length */\n\n  /**\n   * Used to stitch together functional operators into a chain.\n   * @method pipe\n   * @return {Observable} the Observable result of all of the operators having\n   * been called in the order they were passed in.\n   *\n   * ## Example\n   *\n   * ```ts\n   * import { interval, filter, map, scan } from 'rxjs';\n   *\n   * interval(1000)\n   *   .pipe(\n   *     filter(x => x % 2 === 0),\n   *     map(x => x + x),\n   *     scan((acc, x) => acc + x)\n   *   )\n   *   .subscribe(x => console.log(x));\n   * ```\n   */\n  pipe(...operations: OperatorFunction<any, any>[]): Observable<any> {\n    return pipeFromArray(operations)(this);\n  }\n\n  /* tslint:disable:max-line-length */\n  /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n  toPromise(): Promise<T | undefined>;\n  /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n  toPromise(PromiseCtor: typeof Promise): Promise<T | undefined>;\n  /** @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise */\n  toPromise(PromiseCtor: PromiseConstructorLike): Promise<T | undefined>;\n  /* tslint:enable:max-line-length */\n\n  /**\n   * Subscribe to this Observable and get a Promise resolving on\n   * `complete` with the last emission (if any).\n   *\n   * **WARNING**: Only use this with observables you *know* will complete. If the source\n   * observable does not complete, you will end up with a promise that is hung up, and\n   * potentially all of the state of an async function hanging out in memory. To avoid\n   * this situation, look into adding something like {@link timeout}, {@link take},\n   * {@link takeWhile}, or {@link takeUntil} amongst others.\n   *\n   * @method toPromise\n   * @param [promiseCtor] a constructor function used to instantiate\n   * the Promise\n   * @return A Promise that resolves with the last value emit, or\n   * rejects on an error. If there were no emissions, Promise\n   * resolves with undefined.\n   * @deprecated Replaced with {@link firstValueFrom} and {@link lastValueFrom}. Will be removed in v8. Details: https://rxjs.dev/deprecations/to-promise\n   */\n  toPromise(promiseCtor?: PromiseConstructorLike): Promise<T | undefined> {\n    promiseCtor = getPromiseCtor(promiseCtor);\n\n    return new promiseCtor((resolve, reject) => {\n      let value: T | undefined;\n      this.subscribe(\n        (x: T) => (value = x),\n        (err: any) => reject(err),\n        () => resolve(value)\n      );\n    }) as Promise<T | undefined>;\n  }\n}\n\n/**\n * Decides between a passed promise constructor from consuming code,\n * A default configured promise constructor, and the native promise\n * constructor and returns it. If nothing can be found, it will throw\n * an error.\n * @param promiseCtor The optional promise constructor to passed by consuming code\n */\nfunction getPromiseCtor(promiseCtor: PromiseConstructorLike | undefined) {\n  return promiseCtor ?? config.Promise ?? Promise;\n}\n\nfunction isObserver<T>(value: any): value is Observer<T> {\n  return value && isFunction(value.next) && isFunction(value.error) && isFunction(value.complete);\n}\n\nfunction isSubscriber<T>(value: any): value is Subscriber<T> {\n  return (value && value instanceof Subscriber) || (isObserver(value) && isSubscription(value));\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { OperatorFunction } from '../types';\nimport { isFunction } from './isFunction';\n\n/**\n * Used to determine if an object is an Observable with a lift function.\n */\nexport function hasLift(source: any): source is { lift: InstanceType<typeof Observable>['lift'] } {\n  return isFunction(source?.lift);\n}\n\n/**\n * Creates an `OperatorFunction`. Used to define operators throughout the library in a concise way.\n * @param init The logic to connect the liftedSource to the subscriber at the moment of subscription.\n */\nexport function operate<T, R>(\n  init: (liftedSource: Observable<T>, subscriber: Subscriber<R>) => (() => void) | void\n): OperatorFunction<T, R> {\n  return (source: Observable<T>) => {\n    if (hasLift(source)) {\n      return source.lift(function (this: Subscriber<R>, liftedSource: Observable<T>) {\n        try {\n          return init(liftedSource, this);\n        } catch (err) {\n          this.error(err);\n        }\n      });\n    }\n    throw new TypeError('Unable to lift unknown Observable type');\n  };\n}\n", "import { Subscriber } from '../Subscriber';\n\n/**\n * Creates an instance of an `OperatorSubscriber`.\n * @param destination The downstream subscriber.\n * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n * and send to the `destination` error handler.\n * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n * this handler are sent to the `destination` error handler.\n * @param onFinalize Additional teardown logic here. This will only be called on teardown if the\n * subscriber itself is not already closed. This is called after all other teardown logic is executed.\n */\nexport function createOperatorSubscriber<T>(\n  destination: Subscriber<any>,\n  onNext?: (value: T) => void,\n  onComplete?: () => void,\n  onError?: (err: any) => void,\n  onFinalize?: () => void\n): Subscriber<T> {\n  return new OperatorSubscriber(destination, onNext, onComplete, onError, onFinalize);\n}\n\n/**\n * A generic helper for allowing operators to be created with a Subscriber and\n * use closures to capture necessary state from the operator function itself.\n */\nexport class OperatorSubscriber<T> extends Subscriber<T> {\n  /**\n   * Creates an instance of an `OperatorSubscriber`.\n   * @param destination The downstream subscriber.\n   * @param onNext Handles next values, only called if this subscriber is not stopped or closed. Any\n   * error that occurs in this function is caught and sent to the `error` method of this subscriber.\n   * @param onError Handles errors from the subscription, any errors that occur in this handler are caught\n   * and send to the `destination` error handler.\n   * @param onComplete Handles completion notification from the subscription. Any errors that occur in\n   * this handler are sent to the `destination` error handler.\n   * @param onFinalize Additional finalization logic here. This will only be called on finalization if the\n   * subscriber itself is not already closed. This is called after all other finalization logic is executed.\n   * @param shouldUnsubscribe An optional check to see if an unsubscribe call should truly unsubscribe.\n   * NOTE: This currently **ONLY** exists to support the strange behavior of {@link groupBy}, where unsubscription\n   * to the resulting observable does not actually disconnect from the source if there are active subscriptions\n   * to any grouped observable. (DO NOT EXPOSE OR USE EXTERNALLY!!!)\n   */\n  constructor(\n    destination: Subscriber<any>,\n    onNext?: (value: T) => void,\n    onComplete?: () => void,\n    onError?: (err: any) => void,\n    private onFinalize?: () => void,\n    private shouldUnsubscribe?: () => boolean\n  ) {\n    // It's important - for performance reasons - that all of this class's\n    // members are initialized and that they are always initialized in the same\n    // order. This will ensure that all OperatorSubscriber instances have the\n    // same hidden class in V8. This, in turn, will help keep the number of\n    // hidden classes involved in property accesses within the base class as\n    // low as possible. If the number of hidden classes involved exceeds four,\n    // the property accesses will become megamorphic and performance penalties\n    // will be incurred - i.e. inline caches won't be used.\n    //\n    // The reasons for ensuring all instances have the same hidden class are\n    // further discussed in this blog post from Benedikt Meurer:\n    // https://benediktmeurer.de/2018/03/23/impact-of-polymorphism-on-component-based-frameworks-like-react/\n    super(destination);\n    this._next = onNext\n      ? function (this: OperatorSubscriber<T>, value: T) {\n          try {\n            onNext(value);\n          } catch (err) {\n            destination.error(err);\n          }\n        }\n      : super._next;\n    this._error = onError\n      ? function (this: OperatorSubscriber<T>, err: any) {\n          try {\n            onError(err);\n          } catch (err) {\n            // Send any errors that occur down stream.\n            destination.error(err);\n          } finally {\n            // Ensure finalization.\n            this.unsubscribe();\n          }\n        }\n      : super._error;\n    this._complete = onComplete\n      ? function (this: OperatorSubscriber<T>) {\n          try {\n            onComplete();\n          } catch (err) {\n            // Send any errors that occur down stream.\n            destination.error(err);\n          } finally {\n            // Ensure finalization.\n            this.unsubscribe();\n          }\n        }\n      : super._complete;\n  }\n\n  unsubscribe() {\n    if (!this.shouldUnsubscribe || this.shouldUnsubscribe()) {\n      const { closed } = this;\n      super.unsubscribe();\n      // Execute additional teardown if we have any and we didn't already do so.\n      !closed && this.onFinalize?.();\n    }\n  }\n}\n", "import { Subscription } from '../Subscription';\n\ninterface AnimationFrameProvider {\n  schedule(callback: FrameRequestCallback): Subscription;\n  requestAnimationFrame: typeof requestAnimationFrame;\n  cancelAnimationFrame: typeof cancelAnimationFrame;\n  delegate:\n    | {\n        requestAnimationFrame: typeof requestAnimationFrame;\n        cancelAnimationFrame: typeof cancelAnimationFrame;\n      }\n    | undefined;\n}\n\nexport const animationFrameProvider: AnimationFrameProvider = {\n  // When accessing the delegate, use the variable rather than `this` so that\n  // the functions can be called without being bound to the provider.\n  schedule(callback) {\n    let request = requestAnimationFrame;\n    let cancel: typeof cancelAnimationFrame | undefined = cancelAnimationFrame;\n    const { delegate } = animationFrameProvider;\n    if (delegate) {\n      request = delegate.requestAnimationFrame;\n      cancel = delegate.cancelAnimationFrame;\n    }\n    const handle = request((timestamp) => {\n      // Clear the cancel function. The request has been fulfilled, so\n      // attempting to cancel the request upon unsubscription would be\n      // pointless.\n      cancel = undefined;\n      callback(timestamp);\n    });\n    return new Subscription(() => cancel?.(handle));\n  },\n  requestAnimationFrame(...args) {\n    const { delegate } = animationFrameProvider;\n    return (delegate?.requestAnimationFrame || requestAnimationFrame)(...args);\n  },\n  cancelAnimationFrame(...args) {\n    const { delegate } = animationFrameProvider;\n    return (delegate?.cancelAnimationFrame || cancelAnimationFrame)(...args);\n  },\n  delegate: undefined,\n};\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface ObjectUnsubscribedError extends Error {}\n\nexport interface ObjectUnsubscribedErrorCtor {\n  /**\n   * @deprecated Internal implementation detail. Do not construct error instances.\n   * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n   */\n  new (): ObjectUnsubscribedError;\n}\n\n/**\n * An error thrown when an action is invalid because the object has been\n * unsubscribed.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n *\n * @class ObjectUnsubscribedError\n */\nexport const ObjectUnsubscribedError: ObjectUnsubscribedErrorCtor = createErrorClass(\n  (_super) =>\n    function ObjectUnsubscribedErrorImpl(this: any) {\n      _super(this);\n      this.name = 'ObjectUnsubscribedError';\n      this.message = 'object unsubscribed';\n    }\n);\n", "import { Operator } from './Operator';\nimport { Observable } from './Observable';\nimport { Subscriber } from './Subscriber';\nimport { Subscription, EMPTY_SUBSCRIPTION } from './Subscription';\nimport { Observer, SubscriptionLike, TeardownLogic } from './types';\nimport { ObjectUnsubscribedError } from './util/ObjectUnsubscribedError';\nimport { arrRemove } from './util/arrRemove';\nimport { errorContext } from './util/errorContext';\n\n/**\n * A Subject is a special type of Observable that allows values to be\n * multicasted to many Observers. Subjects are like EventEmitters.\n *\n * Every Subject is an Observable and an Observer. You can subscribe to a\n * Subject, and you can call next to feed values as well as error and complete.\n */\nexport class Subject<T> extends Observable<T> implements SubscriptionLike {\n  closed = false;\n\n  private currentObservers: Observer<T>[] | null = null;\n\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  observers: Observer<T>[] = [];\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  isStopped = false;\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  hasError = false;\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  thrownError: any = null;\n\n  /**\n   * Creates a \"subject\" by basically gluing an observer to an observable.\n   *\n   * @nocollapse\n   * @deprecated Recommended you do not use. Will be removed at some point in the future. Plans for replacement still under discussion.\n   */\n  static create: (...args: any[]) => any = <T>(destination: Observer<T>, source: Observable<T>): AnonymousSubject<T> => {\n    return new AnonymousSubject<T>(destination, source);\n  };\n\n  constructor() {\n    // NOTE: This must be here to obscure Observable's constructor.\n    super();\n  }\n\n  /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n  lift<R>(operator: Operator<T, R>): Observable<R> {\n    const subject = new AnonymousSubject(this, this);\n    subject.operator = operator as any;\n    return subject as any;\n  }\n\n  /** @internal */\n  protected _throwIfClosed() {\n    if (this.closed) {\n      throw new ObjectUnsubscribedError();\n    }\n  }\n\n  next(value: T) {\n    errorContext(() => {\n      this._throwIfClosed();\n      if (!this.isStopped) {\n        if (!this.currentObservers) {\n          this.currentObservers = Array.from(this.observers);\n        }\n        for (const observer of this.currentObservers) {\n          observer.next(value);\n        }\n      }\n    });\n  }\n\n  error(err: any) {\n    errorContext(() => {\n      this._throwIfClosed();\n      if (!this.isStopped) {\n        this.hasError = this.isStopped = true;\n        this.thrownError = err;\n        const { observers } = this;\n        while (observers.length) {\n          observers.shift()!.error(err);\n        }\n      }\n    });\n  }\n\n  complete() {\n    errorContext(() => {\n      this._throwIfClosed();\n      if (!this.isStopped) {\n        this.isStopped = true;\n        const { observers } = this;\n        while (observers.length) {\n          observers.shift()!.complete();\n        }\n      }\n    });\n  }\n\n  unsubscribe() {\n    this.isStopped = this.closed = true;\n    this.observers = this.currentObservers = null!;\n  }\n\n  get observed() {\n    return this.observers?.length > 0;\n  }\n\n  /** @internal */\n  protected _trySubscribe(subscriber: Subscriber<T>): TeardownLogic {\n    this._throwIfClosed();\n    return super._trySubscribe(subscriber);\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<T>): Subscription {\n    this._throwIfClosed();\n    this._checkFinalizedStatuses(subscriber);\n    return this._innerSubscribe(subscriber);\n  }\n\n  /** @internal */\n  protected _innerSubscribe(subscriber: Subscriber<any>) {\n    const { hasError, isStopped, observers } = this;\n    if (hasError || isStopped) {\n      return EMPTY_SUBSCRIPTION;\n    }\n    this.currentObservers = null;\n    observers.push(subscriber);\n    return new Subscription(() => {\n      this.currentObservers = null;\n      arrRemove(observers, subscriber);\n    });\n  }\n\n  /** @internal */\n  protected _checkFinalizedStatuses(subscriber: Subscriber<any>) {\n    const { hasError, thrownError, isStopped } = this;\n    if (hasError) {\n      subscriber.error(thrownError);\n    } else if (isStopped) {\n      subscriber.complete();\n    }\n  }\n\n  /**\n   * Creates a new Observable with this Subject as the source. You can do this\n   * to create custom Observer-side logic of the Subject and conceal it from\n   * code that uses the Observable.\n   * @return {Observable} Observable that the Subject casts to\n   */\n  asObservable(): Observable<T> {\n    const observable: any = new Observable<T>();\n    observable.source = this;\n    return observable;\n  }\n}\n\n/**\n * @class AnonymousSubject<T>\n */\nexport class AnonymousSubject<T> extends Subject<T> {\n  constructor(\n    /** @deprecated Internal implementation detail, do not use directly. Will be made internal in v8. */\n    public destination?: Observer<T>,\n    source?: Observable<T>\n  ) {\n    super();\n    this.source = source;\n  }\n\n  next(value: T) {\n    this.destination?.next?.(value);\n  }\n\n  error(err: any) {\n    this.destination?.error?.(err);\n  }\n\n  complete() {\n    this.destination?.complete?.();\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<T>): Subscription {\n    return this.source?.subscribe(subscriber) ?? EMPTY_SUBSCRIPTION;\n  }\n}\n", "import { TimestampProvider } from '../types';\n\ninterface DateTimestampProvider extends TimestampProvider {\n  delegate: TimestampProvider | undefined;\n}\n\nexport const dateTimestampProvider: DateTimestampProvider = {\n  now() {\n    // Use the variable rather than `this` so that the function can be called\n    // without being bound to the provider.\n    return (dateTimestampProvider.delegate || Date).now();\n  },\n  delegate: undefined,\n};\n", "import { Subject } from './Subject';\nimport { TimestampProvider } from './types';\nimport { Subscriber } from './Subscriber';\nimport { Subscription } from './Subscription';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * A variant of {@link Subject} that \"replays\" old values to new subscribers by emitting them when they first subscribe.\n *\n * `ReplaySubject` has an internal buffer that will store a specified number of values that it has observed. Like `Subject`,\n * `ReplaySubject` \"observes\" values by having them passed to its `next` method. When it observes a value, it will store that\n * value for a time determined by the configuration of the `ReplaySubject`, as passed to its constructor.\n *\n * When a new subscriber subscribes to the `ReplaySubject` instance, it will synchronously emit all values in its buffer in\n * a First-In-First-Out (FIFO) manner. The `ReplaySubject` will also complete, if it has observed completion; and it will\n * error if it has observed an error.\n *\n * There are two main configuration items to be concerned with:\n *\n * 1. `bufferSize` - This will determine how many items are stored in the buffer, defaults to infinite.\n * 2. `windowTime` - The amount of time to hold a value in the buffer before removing it from the buffer.\n *\n * Both configurations may exist simultaneously. So if you would like to buffer a maximum of 3 values, as long as the values\n * are less than 2 seconds old, you could do so with a `new ReplaySubject(3, 2000)`.\n *\n * ### Differences with BehaviorSubject\n *\n * `BehaviorSubject` is similar to `new ReplaySubject(1)`, with a couple of exceptions:\n *\n * 1. `BehaviorSubject` comes \"primed\" with a single value upon construction.\n * 2. `ReplaySubject` will replay values, even after observing an error, where `BehaviorSubject` will not.\n *\n * @see {@link Subject}\n * @see {@link BehaviorSubject}\n * @see {@link shareReplay}\n */\nexport class ReplaySubject<T> extends Subject<T> {\n  private _buffer: (T | number)[] = [];\n  private _infiniteTimeWindow = true;\n\n  /**\n   * @param bufferSize The size of the buffer to replay on subscription\n   * @param windowTime The amount of time the buffered items will stay buffered\n   * @param timestampProvider An object with a `now()` method that provides the current timestamp. This is used to\n   * calculate the amount of time something has been buffered.\n   */\n  constructor(\n    private _bufferSize = Infinity,\n    private _windowTime = Infinity,\n    private _timestampProvider: TimestampProvider = dateTimestampProvider\n  ) {\n    super();\n    this._infiniteTimeWindow = _windowTime === Infinity;\n    this._bufferSize = Math.max(1, _bufferSize);\n    this._windowTime = Math.max(1, _windowTime);\n  }\n\n  next(value: T): void {\n    const { isStopped, _buffer, _infiniteTimeWindow, _timestampProvider, _windowTime } = this;\n    if (!isStopped) {\n      _buffer.push(value);\n      !_infiniteTimeWindow && _buffer.push(_timestampProvider.now() + _windowTime);\n    }\n    this._trimBuffer();\n    super.next(value);\n  }\n\n  /** @internal */\n  protected _subscribe(subscriber: Subscriber<T>): Subscription {\n    this._throwIfClosed();\n    this._trimBuffer();\n\n    const subscription = this._innerSubscribe(subscriber);\n\n    const { _infiniteTimeWindow, _buffer } = this;\n    // We use a copy here, so reentrant code does not mutate our array while we're\n    // emitting it to a new subscriber.\n    const copy = _buffer.slice();\n    for (let i = 0; i < copy.length && !subscriber.closed; i += _infiniteTimeWindow ? 1 : 2) {\n      subscriber.next(copy[i] as T);\n    }\n\n    this._checkFinalizedStatuses(subscriber);\n\n    return subscription;\n  }\n\n  private _trimBuffer() {\n    const { _bufferSize, _timestampProvider, _buffer, _infiniteTimeWindow } = this;\n    // If we don't have an infinite buffer size, and we're over the length,\n    // use splice to truncate the old buffer values off. Note that we have to\n    // double the size for instances where we're not using an infinite time window\n    // because we're storing the values and the timestamps in the same array.\n    const adjustedBufferSize = (_infiniteTimeWindow ? 1 : 2) * _bufferSize;\n    _bufferSize < Infinity && adjustedBufferSize < _buffer.length && _buffer.splice(0, _buffer.length - adjustedBufferSize);\n\n    // Now, if we're not in an infinite time window, remove all values where the time is\n    // older than what is allowed.\n    if (!_infiniteTimeWindow) {\n      const now = _timestampProvider.now();\n      let last = 0;\n      // Search the array for the first timestamp that isn't expired and\n      // truncate the buffer up to that point.\n      for (let i = 1; i < _buffer.length && (_buffer[i] as number) <= now; i += 2) {\n        last = i;\n      }\n      last && _buffer.splice(0, last + 1);\n    }\n  }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Subscription } from '../Subscription';\nimport { SchedulerAction } from '../types';\n\n/**\n * A unit of work to be executed in a `scheduler`. An action is typically\n * created from within a {@link SchedulerLike} and an RxJS user does not need to concern\n * themselves about creating and manipulating an Action.\n *\n * ```ts\n * class Action<T> extends Subscription {\n *   new (scheduler: Scheduler, work: (state?: T) => void);\n *   schedule(state?: T, delay: number = 0): Subscription;\n * }\n * ```\n *\n * @class Action<T>\n */\nexport class Action<T> extends Subscription {\n  constructor(scheduler: Scheduler, work: (this: SchedulerAction<T>, state?: T) => void) {\n    super();\n  }\n  /**\n   * Schedules this action on its parent {@link SchedulerLike} for execution. May be passed\n   * some context object, `state`. May happen at some point in the future,\n   * according to the `delay` parameter, if specified.\n   * @param {T} [state] Some contextual data that the `work` function uses when\n   * called by the Scheduler.\n   * @param {number} [delay] Time to wait before executing the work, where the\n   * time unit is implicit and defined by the Scheduler.\n   * @return {void}\n   */\n  public schedule(state?: T, delay: number = 0): Subscription {\n    return this;\n  }\n}\n", "import type { TimerHandle } from './timerHandle';\ntype SetIntervalFunction = (handler: () => void, timeout?: number, ...args: any[]) => TimerHandle;\ntype ClearIntervalFunction = (handle: TimerHandle) => void;\n\ninterface IntervalProvider {\n  setInterval: SetIntervalFunction;\n  clearInterval: ClearIntervalFunction;\n  delegate:\n    | {\n        setInterval: SetIntervalFunction;\n        clearInterval: ClearIntervalFunction;\n      }\n    | undefined;\n}\n\nexport const intervalProvider: IntervalProvider = {\n  // When accessing the delegate, use the variable rather than `this` so that\n  // the functions can be called without being bound to the provider.\n  setInterval(handler: () => void, timeout?: number, ...args) {\n    const { delegate } = intervalProvider;\n    if (delegate?.setInterval) {\n      return delegate.setInterval(handler, timeout, ...args);\n    }\n    return setInterval(handler, timeout, ...args);\n  },\n  clearInterval(handle) {\n    const { delegate } = intervalProvider;\n    return (delegate?.clearInterval || clearInterval)(handle as any);\n  },\n  delegate: undefined,\n};\n", "import { Action } from './Action';\nimport { SchedulerAction } from '../types';\nimport { Subscription } from '../Subscription';\nimport { AsyncScheduler } from './AsyncScheduler';\nimport { intervalProvider } from './intervalProvider';\nimport { arrRemove } from '../util/arrRemove';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncAction<T> extends Action<T> {\n  public id: TimerHandle | undefined;\n  public state?: T;\n  // @ts-ignore: Property has no initializer and is not definitely assigned\n  public delay: number;\n  protected pending: boolean = false;\n\n  constructor(protected scheduler: AsyncScheduler, protected work: (this: SchedulerAction<T>, state?: T) => void) {\n    super(scheduler, work);\n  }\n\n  public schedule(state?: T, delay: number = 0): Subscription {\n    if (this.closed) {\n      return this;\n    }\n\n    // Always replace the current state with the new state.\n    this.state = state;\n\n    const id = this.id;\n    const scheduler = this.scheduler;\n\n    //\n    // Important implementation note:\n    //\n    // Actions only execute once by default, unless rescheduled from within the\n    // scheduled callback. This allows us to implement single and repeat\n    // actions via the same code path, without adding API surface area, as well\n    // as mimic traditional recursion but across asynchronous boundaries.\n    //\n    // However, JS runtimes and timers distinguish between intervals achieved by\n    // serial `setTimeout` calls vs. a single `setInterval` call. An interval of\n    // serial `setTimeout` calls can be individually delayed, which delays\n    // scheduling the next `setTimeout`, and so on. `setInterval` attempts to\n    // guarantee the interval callback will be invoked more precisely to the\n    // interval period, regardless of load.\n    //\n    // Therefore, we use `setInterval` to schedule single and repeat actions.\n    // If the action reschedules itself with the same delay, the interval is not\n    // canceled. If the action doesn't reschedule, or reschedules with a\n    // different delay, the interval will be canceled after scheduled callback\n    // execution.\n    //\n    if (id != null) {\n      this.id = this.recycleAsyncId(scheduler, id, delay);\n    }\n\n    // Set the pending flag indicating that this action has been scheduled, or\n    // has recursively rescheduled itself.\n    this.pending = true;\n\n    this.delay = delay;\n    // If this action has already an async Id, don't request a new one.\n    this.id = this.id ?? this.requestAsyncId(scheduler, this.id, delay);\n\n    return this;\n  }\n\n  protected requestAsyncId(scheduler: AsyncScheduler, _id?: TimerHandle, delay: number = 0): TimerHandle {\n    return intervalProvider.setInterval(scheduler.flush.bind(scheduler, this), delay);\n  }\n\n  protected recycleAsyncId(_scheduler: AsyncScheduler, id?: TimerHandle, delay: number | null = 0): TimerHandle | undefined {\n    // If this action is rescheduled with the same delay time, don't clear the interval id.\n    if (delay != null && this.delay === delay && this.pending === false) {\n      return id;\n    }\n    // Otherwise, if the action's delay time is different from the current delay,\n    // or the action has been rescheduled before it's executed, clear the interval id\n    if (id != null) {\n      intervalProvider.clearInterval(id);\n    }\n\n    return undefined;\n  }\n\n  /**\n   * Immediately executes this action and the `work` it contains.\n   * @return {any}\n   */\n  public execute(state: T, delay: number): any {\n    if (this.closed) {\n      return new Error('executing a cancelled action');\n    }\n\n    this.pending = false;\n    const error = this._execute(state, delay);\n    if (error) {\n      return error;\n    } else if (this.pending === false && this.id != null) {\n      // Dequeue if the action didn't reschedule itself. Don't call\n      // unsubscribe(), because the action could reschedule later.\n      // For example:\n      // ```\n      // scheduler.schedule(function doWork(counter) {\n      //   /* ... I'm a busy worker bee ... */\n      //   var originalAction = this;\n      //   /* wait 100ms before rescheduling the action */\n      //   setTimeout(function () {\n      //     originalAction.schedule(counter + 1);\n      //   }, 100);\n      // }, 1000);\n      // ```\n      this.id = this.recycleAsyncId(this.scheduler, this.id, null);\n    }\n  }\n\n  protected _execute(state: T, _delay: number): any {\n    let errored: boolean = false;\n    let errorValue: any;\n    try {\n      this.work(state);\n    } catch (e) {\n      errored = true;\n      // HACK: Since code elsewhere is relying on the \"truthiness\" of the\n      // return here, we can't have it return \"\" or 0 or false.\n      // TODO: Clean this up when we refactor schedulers mid-version-8 or so.\n      errorValue = e ? e : new Error('Scheduled action threw falsy error');\n    }\n    if (errored) {\n      this.unsubscribe();\n      return errorValue;\n    }\n  }\n\n  unsubscribe() {\n    if (!this.closed) {\n      const { id, scheduler } = this;\n      const { actions } = scheduler;\n\n      this.work = this.state = this.scheduler = null!;\n      this.pending = false;\n\n      arrRemove(actions, this);\n      if (id != null) {\n        this.id = this.recycleAsyncId(scheduler, id, null);\n      }\n\n      this.delay = null!;\n      super.unsubscribe();\n    }\n  }\n}\n", "import { Action } from './scheduler/Action';\nimport { Subscription } from './Subscription';\nimport { SchedulerLike, SchedulerAction } from './types';\nimport { dateTimestampProvider } from './scheduler/dateTimestampProvider';\n\n/**\n * An execution context and a data structure to order tasks and schedule their\n * execution. Provides a notion of (potentially virtual) time, through the\n * `now()` getter method.\n *\n * Each unit of work in a Scheduler is called an `Action`.\n *\n * ```ts\n * class Scheduler {\n *   now(): number;\n *   schedule(work, delay?, state?): Subscription;\n * }\n * ```\n *\n * @class Scheduler\n * @deprecated Scheduler is an internal implementation detail of RxJS, and\n * should not be used directly. Rather, create your own class and implement\n * {@link SchedulerLike}. Will be made internal in v8.\n */\nexport class Scheduler implements SchedulerLike {\n  public static now: () => number = dateTimestampProvider.now;\n\n  constructor(private schedulerActionCtor: typeof Action, now: () => number = Scheduler.now) {\n    this.now = now;\n  }\n\n  /**\n   * A getter method that returns a number representing the current time\n   * (at the time this function was called) according to the scheduler's own\n   * internal clock.\n   * @return {number} A number that represents the current time. May or may not\n   * have a relation to wall-clock time. May or may not refer to a time unit\n   * (e.g. milliseconds).\n   */\n  public now: () => number;\n\n  /**\n   * Schedules a function, `work`, for execution. May happen at some point in\n   * the future, according to the `delay` parameter, if specified. May be passed\n   * some context object, `state`, which will be passed to the `work` function.\n   *\n   * The given arguments will be processed an stored as an Action object in a\n   * queue of actions.\n   *\n   * @param {function(state: ?T): ?Subscription} work A function representing a\n   * task, or some unit of work to be executed by the Scheduler.\n   * @param {number} [delay] Time to wait before executing the work, where the\n   * time unit is implicit and defined by the Scheduler itself.\n   * @param {T} [state] Some contextual data that the `work` function uses when\n   * called by the Scheduler.\n   * @return {Subscription} A subscription in order to be able to unsubscribe\n   * the scheduled work.\n   */\n  public schedule<T>(work: (this: SchedulerAction<T>, state?: T) => void, delay: number = 0, state?: T): Subscription {\n    return new this.schedulerActionCtor<T>(this, work).schedule(state, delay);\n  }\n}\n", "import { Scheduler } from '../Scheduler';\nimport { Action } from './Action';\nimport { AsyncAction } from './AsyncAction';\nimport { TimerHandle } from './timerHandle';\n\nexport class AsyncScheduler extends Scheduler {\n  public actions: Array<AsyncAction<any>> = [];\n  /**\n   * A flag to indicate whether the Scheduler is currently executing a batch of\n   * queued actions.\n   * @type {boolean}\n   * @internal\n   */\n  public _active: boolean = false;\n  /**\n   * An internal ID used to track the latest asynchronous task such as those\n   * coming from `setTimeout`, `setInterval`, `requestAnimationFrame`, and\n   * others.\n   * @type {any}\n   * @internal\n   */\n  public _scheduled: TimerHandle | undefined;\n\n  constructor(SchedulerAction: typeof Action, now: () => number = Scheduler.now) {\n    super(SchedulerAction, now);\n  }\n\n  public flush(action: AsyncAction<any>): void {\n    const { actions } = this;\n\n    if (this._active) {\n      actions.push(action);\n      return;\n    }\n\n    let error: any;\n    this._active = true;\n\n    do {\n      if ((error = action.execute(action.state, action.delay))) {\n        break;\n      }\n    } while ((action = actions.shift()!)); // exhaust the scheduler queue\n\n    this._active = false;\n\n    if (error) {\n      while ((action = actions.shift()!)) {\n        action.unsubscribe();\n      }\n      throw error;\n    }\n  }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\n/**\n *\n * Async Scheduler\n *\n * <span class=\"informal\">Schedule task as if you used setTimeout(task, duration)</span>\n *\n * `async` scheduler schedules tasks asynchronously, by putting them on the JavaScript\n * event loop queue. It is best used to delay tasks in time or to schedule tasks repeating\n * in intervals.\n *\n * If you just want to \"defer\" task, that is to perform it right after currently\n * executing synchronous code ends (commonly achieved by `setTimeout(deferredTask, 0)`),\n * better choice will be the {@link asapScheduler} scheduler.\n *\n * ## Examples\n * Use async scheduler to delay task\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * const task = () => console.log('it works!');\n *\n * asyncScheduler.schedule(task, 2000);\n *\n * // After 2 seconds logs:\n * // \"it works!\"\n * ```\n *\n * Use async scheduler to repeat task in intervals\n * ```ts\n * import { asyncScheduler } from 'rxjs';\n *\n * function task(state) {\n *   console.log(state);\n *   this.schedule(state + 1, 1000); // `this` references currently executing Action,\n *                                   // which we reschedule with new state and delay\n * }\n *\n * asyncScheduler.schedule(task, 3000, 0);\n *\n * // Logs:\n * // 0 after 3s\n * // 1 after 4s\n * // 2 after 5s\n * // 3 after 6s\n * ```\n */\n\nexport const asyncScheduler = new AsyncScheduler(AsyncAction);\n\n/**\n * @deprecated Renamed to {@link asyncScheduler}. Will be removed in v8.\n */\nexport const async = asyncScheduler;\n", "import { AsyncAction } from './AsyncAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\nimport { SchedulerAction } from '../types';\nimport { animationFrameProvider } from './animationFrameProvider';\nimport { TimerHandle } from './timerHandle';\n\nexport class AnimationFrameAction<T> extends AsyncAction<T> {\n  constructor(protected scheduler: AnimationFrameScheduler, protected work: (this: SchedulerAction<T>, state?: T) => void) {\n    super(scheduler, work);\n  }\n\n  protected requestAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle {\n    // If delay is greater than 0, request as an async action.\n    if (delay !== null && delay > 0) {\n      return super.requestAsyncId(scheduler, id, delay);\n    }\n    // Push the action to the end of the scheduler queue.\n    scheduler.actions.push(this);\n    // If an animation frame has already been requested, don't request another\n    // one. If an animation frame hasn't been requested yet, request one. Return\n    // the current animation frame request id.\n    return scheduler._scheduled || (scheduler._scheduled = animationFrameProvider.requestAnimationFrame(() => scheduler.flush(undefined)));\n  }\n\n  protected recycleAsyncId(scheduler: AnimationFrameScheduler, id?: TimerHandle, delay: number = 0): TimerHandle | undefined {\n    // If delay exists and is greater than 0, or if the delay is null (the\n    // action wasn't rescheduled) but was originally scheduled as an async\n    // action, then recycle as an async action.\n    if (delay != null ? delay > 0 : this.delay > 0) {\n      return super.recycleAsyncId(scheduler, id, delay);\n    }\n    // If the scheduler queue has no remaining actions with the same async id,\n    // cancel the requested animation frame and set the scheduled flag to\n    // undefined so the next AnimationFrameAction will request its own.\n    const { actions } = scheduler;\n    if (id != null && actions[actions.length - 1]?.id !== id) {\n      animationFrameProvider.cancelAnimationFrame(id as number);\n      scheduler._scheduled = undefined;\n    }\n    // Return undefined so the action knows to request a new async id if it's rescheduled.\n    return undefined;\n  }\n}\n", "import { AsyncAction } from './AsyncAction';\nimport { AsyncScheduler } from './AsyncScheduler';\n\nexport class AnimationFrameScheduler extends AsyncScheduler {\n  public flush(action?: AsyncAction<any>): void {\n    this._active = true;\n    // The async id that effects a call to flush is stored in _scheduled.\n    // Before executing an action, it's necessary to check the action's async\n    // id to determine whether it's supposed to be executed in the current\n    // flush.\n    // Previous implementations of this method used a count to determine this,\n    // but that was unsound, as actions that are unsubscribed - i.e. cancelled -\n    // are removed from the actions array and that can shift actions that are\n    // scheduled to be executed in a subsequent flush into positions at which\n    // they are executed within the current flush.\n    const flushId = this._scheduled;\n    this._scheduled = undefined;\n\n    const { actions } = this;\n    let error: any;\n    action = action || actions.shift()!;\n\n    do {\n      if ((error = action.execute(action.state, action.delay))) {\n        break;\n      }\n    } while ((action = actions[0]) && action.id === flushId && actions.shift());\n\n    this._active = false;\n\n    if (error) {\n      while ((action = actions[0]) && action.id === flushId && actions.shift()) {\n        action.unsubscribe();\n      }\n      throw error;\n    }\n  }\n}\n", "import { AnimationFrameAction } from './AnimationFrameAction';\nimport { AnimationFrameScheduler } from './AnimationFrameScheduler';\n\n/**\n *\n * Animation Frame Scheduler\n *\n * <span class=\"informal\">Perform task when `window.requestAnimationFrame` would fire</span>\n *\n * When `animationFrame` scheduler is used with delay, it will fall back to {@link asyncScheduler} scheduler\n * behaviour.\n *\n * Without delay, `animationFrame` scheduler can be used to create smooth browser animations.\n * It makes sure scheduled task will happen just before next browser content repaint,\n * thus performing animations as efficiently as possible.\n *\n * ## Example\n * Schedule div height animation\n * ```ts\n * // html: <div style=\"background: #0ff;\"></div>\n * import { animationFrameScheduler } from 'rxjs';\n *\n * const div = document.querySelector('div');\n *\n * animationFrameScheduler.schedule(function(height) {\n *   div.style.height = height + \"px\";\n *\n *   this.schedule(height + 1);  // `this` references currently executing Action,\n *                               // which we reschedule with new state\n * }, 0, 0);\n *\n * // You will see a div element growing in height\n * ```\n */\n\nexport const animationFrameScheduler = new AnimationFrameScheduler(AnimationFrameAction);\n\n/**\n * @deprecated Renamed to {@link animationFrameScheduler}. Will be removed in v8.\n */\nexport const animationFrame = animationFrameScheduler;\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\n\n/**\n * A simple Observable that emits no items to the Observer and immediately\n * emits a complete notification.\n *\n * <span class=\"informal\">Just emits 'complete', and nothing else.</span>\n *\n * ![](empty.png)\n *\n * A simple Observable that only emits the complete notification. It can be used\n * for composing with other Observables, such as in a {@link mergeMap}.\n *\n * ## Examples\n *\n * Log complete notification\n *\n * ```ts\n * import { EMPTY } from 'rxjs';\n *\n * EMPTY.subscribe({\n *   next: () => console.log('Next'),\n *   complete: () => console.log('Complete!')\n * });\n *\n * // Outputs\n * // Complete!\n * ```\n *\n * Emit the number 7, then complete\n *\n * ```ts\n * import { EMPTY, startWith } from 'rxjs';\n *\n * const result = EMPTY.pipe(startWith(7));\n * result.subscribe(x => console.log(x));\n *\n * // Outputs\n * // 7\n * ```\n *\n * Map and flatten only odd numbers to the sequence `'a'`, `'b'`, `'c'`\n *\n * ```ts\n * import { interval, mergeMap, of, EMPTY } from 'rxjs';\n *\n * const interval$ = interval(1000);\n * const result = interval$.pipe(\n *   mergeMap(x => x % 2 === 1 ? of('a', 'b', 'c') : EMPTY),\n * );\n * result.subscribe(x => console.log(x));\n *\n * // Results in the following to the console:\n * // x is equal to the count on the interval, e.g. (0, 1, 2, 3, ...)\n * // x will occur every 1000ms\n * // if x % 2 is equal to 1, print a, b, c (each on its own)\n * // if x % 2 is not equal to 1, nothing will be output\n * ```\n *\n * @see {@link Observable}\n * @see {@link NEVER}\n * @see {@link of}\n * @see {@link throwError}\n */\nexport const EMPTY = new Observable<never>((subscriber) => subscriber.complete());\n\n/**\n * @param scheduler A {@link SchedulerLike} to use for scheduling\n * the emission of the complete notification.\n * @deprecated Replaced with the {@link EMPTY} constant or {@link scheduled} (e.g. `scheduled([], scheduler)`). Will be removed in v8.\n */\nexport function empty(scheduler?: SchedulerLike) {\n  return scheduler ? emptyScheduled(scheduler) : EMPTY;\n}\n\nfunction emptyScheduled(scheduler: SchedulerLike) {\n  return new Observable<never>((subscriber) => scheduler.schedule(() => subscriber.complete()));\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport function isScheduler(value: any): value is SchedulerLike {\n  return value && isFunction(value.schedule);\n}\n", "import { SchedulerLike } from '../types';\nimport { isFunction } from './isFunction';\nimport { isScheduler } from './isScheduler';\n\nfunction last<T>(arr: T[]): T | undefined {\n  return arr[arr.length - 1];\n}\n\nexport function popResultSelector(args: any[]): ((...args: unknown[]) => unknown) | undefined {\n  return isFunction(last(args)) ? args.pop() : undefined;\n}\n\nexport function popScheduler(args: any[]): SchedulerLike | undefined {\n  return isScheduler(last(args)) ? args.pop() : undefined;\n}\n\nexport function popNumber(args: any[], defaultValue: number): number {\n  return typeof last(args) === 'number' ? args.pop()! : defaultValue;\n}\n", "export const isArrayLike = (<T>(x: any): x is ArrayLike<T> => x && typeof x.length === 'number' && typeof x !== 'function');", "import { isFunction } from \"./isFunction\";\n\n/**\n * Tests to see if the object is \"thennable\".\n * @param value the object to test\n */\nexport function isPromise(value: any): value is PromiseLike<any> {\n  return isFunction(value?.then);\n}\n", "import { InteropObservable } from '../types';\nimport { observable as Symbol_observable } from '../symbol/observable';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being Observable (but not necessary an Rx Observable) */\nexport function isInteropObservable(input: any): input is InteropObservable<any> {\n  return isFunction(input[Symbol_observable]);\n}\n", "import { isFunction } from './isFunction';\n\nexport function isAsyncIterable<T>(obj: any): obj is AsyncIterable<T> {\n  return Symbol.asyncIterator && isFunction(obj?.[Symbol.asyncIterator]);\n}\n", "/**\n * Creates the TypeError to throw if an invalid object is passed to `from` or `scheduled`.\n * @param input The object that was passed.\n */\nexport function createInvalidObservableTypeError(input: any) {\n  // TODO: We should create error codes that can be looked up, so this can be less verbose.\n  return new TypeError(\n    `You provided ${\n      input !== null && typeof input === 'object' ? 'an invalid object' : `'${input}'`\n    } where a stream was expected. You can provide an Observable, Promise, ReadableStream, Array, AsyncIterable, or Iterable.`\n  );\n}\n", "export function getSymbolIterator(): symbol {\n  if (typeof Symbol !== 'function' || !Symbol.iterator) {\n    return '@@iterator' as any;\n  }\n\n  return Symbol.iterator;\n}\n\nexport const iterator = getSymbolIterator();\n", "import { iterator as Symbol_iterator } from '../symbol/iterator';\nimport { isFunction } from './isFunction';\n\n/** Identifies an input as being an Iterable */\nexport function isIterable(input: any): input is Iterable<any> {\n  return isFunction(input?.[Symbol_iterator]);\n}\n", "import { ReadableStreamLike } from '../types';\nimport { isFunction } from './isFunction';\n\nexport async function* readableStreamLikeToAsyncGenerator<T>(readableStream: ReadableStreamLike<T>): AsyncGenerator<T> {\n  const reader = readableStream.getReader();\n  try {\n    while (true) {\n      const { value, done } = await reader.read();\n      if (done) {\n        return;\n      }\n      yield value!;\n    }\n  } finally {\n    reader.releaseLock();\n  }\n}\n\nexport function isReadableStreamLike<T>(obj: any): obj is ReadableStreamLike<T> {\n  // We don't want to use instanceof checks because they would return\n  // false for instances from another Realm, like an <iframe>.\n  return isFunction(obj?.getReader);\n}\n", "import { isArrayLike } from '../util/isArrayLike';\nimport { isPromise } from '../util/isPromise';\nimport { Observable } from '../Observable';\nimport { ObservableInput, ObservedValueOf, ReadableStreamLike } from '../types';\nimport { isInteropObservable } from '../util/isInteropObservable';\nimport { isAsyncIterable } from '../util/isAsyncIterable';\nimport { createInvalidObservableTypeError } from '../util/throwUnobservableError';\nimport { isIterable } from '../util/isIterable';\nimport { isReadableStreamLike, readableStreamLikeToAsyncGenerator } from '../util/isReadableStreamLike';\nimport { Subscriber } from '../Subscriber';\nimport { isFunction } from '../util/isFunction';\nimport { reportUnhandledError } from '../util/reportUnhandledError';\nimport { observable as Symbol_observable } from '../symbol/observable';\n\nexport function innerFrom<O extends ObservableInput<any>>(input: O): Observable<ObservedValueOf<O>>;\nexport function innerFrom<T>(input: ObservableInput<T>): Observable<T> {\n  if (input instanceof Observable) {\n    return input;\n  }\n  if (input != null) {\n    if (isInteropObservable(input)) {\n      return fromInteropObservable(input);\n    }\n    if (isArrayLike(input)) {\n      return fromArrayLike(input);\n    }\n    if (isPromise(input)) {\n      return fromPromise(input);\n    }\n    if (isAsyncIterable(input)) {\n      return fromAsyncIterable(input);\n    }\n    if (isIterable(input)) {\n      return fromIterable(input);\n    }\n    if (isReadableStreamLike(input)) {\n      return fromReadableStreamLike(input);\n    }\n  }\n\n  throw createInvalidObservableTypeError(input);\n}\n\n/**\n * Creates an RxJS Observable from an object that implements `Symbol.observable`.\n * @param obj An object that properly implements `Symbol.observable`.\n */\nexport function fromInteropObservable<T>(obj: any) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    const obs = obj[Symbol_observable]();\n    if (isFunction(obs.subscribe)) {\n      return obs.subscribe(subscriber);\n    }\n    // Should be caught by observable subscribe function error handling.\n    throw new TypeError('Provided object does not correctly implement Symbol.observable');\n  });\n}\n\n/**\n * Synchronously emits the values of an array like and completes.\n * This is exported because there are creation functions and operators that need to\n * make direct use of the same logic, and there's no reason to make them run through\n * `from` conditionals because we *know* they're dealing with an array.\n * @param array The array to emit values from\n */\nexport function fromArrayLike<T>(array: ArrayLike<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    // Loop over the array and emit each value. Note two things here:\n    // 1. We're making sure that the subscriber is not closed on each loop.\n    //    This is so we don't continue looping over a very large array after\n    //    something like a `take`, `takeWhile`, or other synchronous unsubscription\n    //    has already unsubscribed.\n    // 2. In this form, reentrant code can alter that array we're looping over.\n    //    This is a known issue, but considered an edge case. The alternative would\n    //    be to copy the array before executing the loop, but this has\n    //    performance implications.\n    for (let i = 0; i < array.length && !subscriber.closed; i++) {\n      subscriber.next(array[i]);\n    }\n    subscriber.complete();\n  });\n}\n\nexport function fromPromise<T>(promise: PromiseLike<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    promise\n      .then(\n        (value) => {\n          if (!subscriber.closed) {\n            subscriber.next(value);\n            subscriber.complete();\n          }\n        },\n        (err: any) => subscriber.error(err)\n      )\n      .then(null, reportUnhandledError);\n  });\n}\n\nexport function fromIterable<T>(iterable: Iterable<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    for (const value of iterable) {\n      subscriber.next(value);\n      if (subscriber.closed) {\n        return;\n      }\n    }\n    subscriber.complete();\n  });\n}\n\nexport function fromAsyncIterable<T>(asyncIterable: AsyncIterable<T>) {\n  return new Observable((subscriber: Subscriber<T>) => {\n    process(asyncIterable, subscriber).catch((err) => subscriber.error(err));\n  });\n}\n\nexport function fromReadableStreamLike<T>(readableStream: ReadableStreamLike<T>) {\n  return fromAsyncIterable(readableStreamLikeToAsyncGenerator(readableStream));\n}\n\nasync function process<T>(asyncIterable: AsyncIterable<T>, subscriber: Subscriber<T>) {\n  for await (const value of asyncIterable) {\n    subscriber.next(value);\n    // A side-effect may have closed our subscriber,\n    // check before the next iteration.\n    if (subscriber.closed) {\n      return;\n    }\n  }\n  subscriber.complete();\n}\n", "import { Subscription } from '../Subscription';\nimport { SchedulerAction, SchedulerLike } from '../types';\n\nexport function executeSchedule(\n  parentSubscription: Subscription,\n  scheduler: SchedulerLike,\n  work: () => void,\n  delay: number,\n  repeat: true\n): void;\nexport function executeSchedule(\n  parentSubscription: Subscription,\n  scheduler: SchedulerLike,\n  work: () => void,\n  delay?: number,\n  repeat?: false\n): Subscription;\n\nexport function executeSchedule(\n  parentSubscription: Subscription,\n  scheduler: SchedulerLike,\n  work: () => void,\n  delay = 0,\n  repeat = false\n): Subscription | void {\n  const scheduleSubscription = scheduler.schedule(function (this: SchedulerAction<any>) {\n    work();\n    if (repeat) {\n      parentSubscription.add(this.schedule(null, delay));\n    } else {\n      this.unsubscribe();\n    }\n  }, delay);\n\n  parentSubscription.add(scheduleSubscription);\n\n  if (!repeat) {\n    // Because user-land scheduler implementations are unlikely to properly reuse\n    // Actions for repeat scheduling, we can't trust that the returned subscription\n    // will control repeat subscription scenarios. So we're trying to avoid using them\n    // incorrectly within this library.\n    return scheduleSubscription;\n  }\n}\n", "/** @prettier */\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { executeSchedule } from '../util/executeSchedule';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Re-emits all notifications from source Observable with specified scheduler.\n *\n * <span class=\"informal\">Ensure a specific scheduler is used, from outside of an Observable.</span>\n *\n * `observeOn` is an operator that accepts a scheduler as a first parameter, which will be used to reschedule\n * notifications emitted by the source Observable. It might be useful, if you do not have control over\n * internal scheduler of a given Observable, but want to control when its values are emitted nevertheless.\n *\n * Returned Observable emits the same notifications (nexted values, complete and error events) as the source Observable,\n * but rescheduled with provided scheduler. Note that this doesn't mean that source Observables internal\n * scheduler will be replaced in any way. Original scheduler still will be used, but when the source Observable emits\n * notification, it will be immediately scheduled again - this time with scheduler passed to `observeOn`.\n * An anti-pattern would be calling `observeOn` on Observable that emits lots of values synchronously, to split\n * that emissions into asynchronous chunks. For this to happen, scheduler would have to be passed into the source\n * Observable directly (usually into the operator that creates it). `observeOn` simply delays notifications a\n * little bit more, to ensure that they are emitted at expected moments.\n *\n * As a matter of fact, `observeOn` accepts second parameter, which specifies in milliseconds with what delay notifications\n * will be emitted. The main difference between {@link delay} operator and `observeOn` is that `observeOn`\n * will delay all notifications - including error notifications - while `delay` will pass through error\n * from source Observable immediately when it is emitted. In general it is highly recommended to use `delay` operator\n * for any kind of delaying of values in the stream, while using `observeOn` to specify which scheduler should be used\n * for notification emissions in general.\n *\n * ## Example\n *\n * Ensure values in subscribe are called just before browser repaint\n *\n * ```ts\n * import { interval, observeOn, animationFrameScheduler } from 'rxjs';\n *\n * const someDiv = document.createElement('div');\n * someDiv.style.cssText = 'width: 200px;background: #09c';\n * document.body.appendChild(someDiv);\n * const intervals = interval(10);      // Intervals are scheduled\n *                                      // with async scheduler by default...\n * intervals.pipe(\n *   observeOn(animationFrameScheduler) // ...but we will observe on animationFrame\n * )                                    // scheduler to ensure smooth animation.\n * .subscribe(val => {\n *   someDiv.style.height = val + 'px';\n * });\n * ```\n *\n * @see {@link delay}\n *\n * @param scheduler Scheduler that will be used to reschedule notifications from source Observable.\n * @param delay Number of milliseconds that states with what delay every notification should be rescheduled.\n * @return A function that returns an Observable that emits the same\n * notifications as the source Observable, but with provided scheduler.\n */\nexport function observeOn<T>(scheduler: SchedulerLike, delay = 0): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => executeSchedule(subscriber, scheduler, () => subscriber.next(value), delay),\n        () => executeSchedule(subscriber, scheduler, () => subscriber.complete(), delay),\n        (err) => executeSchedule(subscriber, scheduler, () => subscriber.error(err), delay)\n      )\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { operate } from '../util/lift';\n\n/**\n * Asynchronously subscribes Observers to this Observable on the specified {@link SchedulerLike}.\n *\n * With `subscribeOn` you can decide what type of scheduler a specific Observable will be using when it is subscribed to.\n *\n * Schedulers control the speed and order of emissions to observers from an Observable stream.\n *\n * ![](subscribeOn.png)\n *\n * ## Example\n *\n * Given the following code:\n *\n * ```ts\n * import { of, merge } from 'rxjs';\n *\n * const a = of(1, 2, 3);\n * const b = of(4, 5, 6);\n *\n * merge(a, b).subscribe(console.log);\n *\n * // Outputs\n * // 1\n * // 2\n * // 3\n * // 4\n * // 5\n * // 6\n * ```\n *\n * Both Observable `a` and `b` will emit their values directly and synchronously once they are subscribed to.\n *\n * If we instead use the `subscribeOn` operator declaring that we want to use the {@link asyncScheduler} for values emitted by Observable `a`:\n *\n * ```ts\n * import { of, subscribeOn, asyncScheduler, merge } from 'rxjs';\n *\n * const a = of(1, 2, 3).pipe(subscribeOn(asyncScheduler));\n * const b = of(4, 5, 6);\n *\n * merge(a, b).subscribe(console.log);\n *\n * // Outputs\n * // 4\n * // 5\n * // 6\n * // 1\n * // 2\n * // 3\n * ```\n *\n * The reason for this is that Observable `b` emits its values directly and synchronously like before\n * but the emissions from `a` are scheduled on the event loop because we are now using the {@link asyncScheduler} for that specific Observable.\n *\n * @param scheduler The {@link SchedulerLike} to perform subscription actions on.\n * @param delay A delay to pass to the scheduler to delay subscriptions\n * @return A function that returns an Observable modified so that its\n * subscriptions happen on the specified {@link SchedulerLike}.\n */\nexport function subscribeOn<T>(scheduler: SchedulerLike, delay: number = 0): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    subscriber.add(scheduler.schedule(() => source.subscribe(subscriber), delay));\n  });\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { observeOn } from '../operators/observeOn';\nimport { subscribeOn } from '../operators/subscribeOn';\nimport { InteropObservable, SchedulerLike } from '../types';\n\nexport function scheduleObservable<T>(input: InteropObservable<T>, scheduler: SchedulerLike) {\n  return innerFrom(input).pipe(subscribeOn(scheduler), observeOn(scheduler));\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { observeOn } from '../operators/observeOn';\nimport { subscribeOn } from '../operators/subscribeOn';\nimport { SchedulerLike } from '../types';\n\nexport function schedulePromise<T>(input: PromiseLike<T>, scheduler: SchedulerLike) {\n  return innerFrom(input).pipe(subscribeOn(scheduler), observeOn(scheduler));\n}\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\n\nexport function scheduleArray<T>(input: ArrayLike<T>, scheduler: SchedulerLike) {\n  return new Observable<T>((subscriber) => {\n    // The current array index.\n    let i = 0;\n    // Start iterating over the array like on a schedule.\n    return scheduler.schedule(function () {\n      if (i === input.length) {\n        // If we have hit the end of the array like in the\n        // previous job, we can complete.\n        subscriber.complete();\n      } else {\n        // Otherwise let's next the value at the current index,\n        // then increment our index.\n        subscriber.next(input[i++]);\n        // If the last emission didn't cause us to close the subscriber\n        // (via take or some side effect), reschedule the job and we'll\n        // make another pass.\n        if (!subscriber.closed) {\n          this.schedule();\n        }\n      }\n    });\n  });\n}\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\nimport { iterator as Symbol_iterator } from '../symbol/iterator';\nimport { isFunction } from '../util/isFunction';\nimport { executeSchedule } from '../util/executeSchedule';\n\n/**\n * Used in {@link scheduled} to create an observable from an Iterable.\n * @param input The iterable to create an observable from\n * @param scheduler The scheduler to use\n */\nexport function scheduleIterable<T>(input: Iterable<T>, scheduler: SchedulerLike) {\n  return new Observable<T>((subscriber) => {\n    let iterator: Iterator<T, T>;\n\n    // Schedule the initial creation of the iterator from\n    // the iterable. This is so the code in the iterable is\n    // not called until the scheduled job fires.\n    executeSchedule(subscriber, scheduler, () => {\n      // Create the iterator.\n      iterator = (input as any)[Symbol_iterator]();\n\n      executeSchedule(\n        subscriber,\n        scheduler,\n        () => {\n          let value: T;\n          let done: boolean | undefined;\n          try {\n            // Pull the value out of the iterator\n            ({ value, done } = iterator.next());\n          } catch (err) {\n            // We got an error while pulling from the iterator\n            subscriber.error(err);\n            return;\n          }\n\n          if (done) {\n            // If it is \"done\" we just complete. This mimics the\n            // behavior of JavaScript's `for..of` consumption of\n            // iterables, which will not emit the value from an iterator\n            // result of `{ done: true: value: 'here' }`.\n            subscriber.complete();\n          } else {\n            // The iterable is not done, emit the value.\n            subscriber.next(value);\n          }\n        },\n        0,\n        true\n      );\n    });\n\n    // During finalization, if we see this iterator has a `return` method,\n    // then we know it is a Generator, and not just an Iterator. So we call\n    // the `return()` function. This will ensure that any `finally { }` blocks\n    // inside of the generator we can hit will be hit properly.\n    return () => isFunction(iterator?.return) && iterator.return();\n  });\n}\n", "import { SchedulerLike } from '../types';\nimport { Observable } from '../Observable';\nimport { executeSchedule } from '../util/executeSchedule';\n\nexport function scheduleAsyncIterable<T>(input: AsyncIterable<T>, scheduler: SchedulerLike) {\n  if (!input) {\n    throw new Error('Iterable cannot be null');\n  }\n  return new Observable<T>((subscriber) => {\n    executeSchedule(subscriber, scheduler, () => {\n      const iterator = input[Symbol.asyncIterator]();\n      executeSchedule(\n        subscriber,\n        scheduler,\n        () => {\n          iterator.next().then((result) => {\n            if (result.done) {\n              // This will remove the subscriptions from\n              // the parent subscription.\n              subscriber.complete();\n            } else {\n              subscriber.next(result.value);\n            }\n          });\n        },\n        0,\n        true\n      );\n    });\n  });\n}\n", "import { SchedulerLike, ReadableStreamLike } from '../types';\nimport { Observable } from '../Observable';\nimport { scheduleAsyncIterable } from './scheduleAsyncIterable';\nimport { readableStreamLikeToAsyncGenerator } from '../util/isReadableStreamLike';\n\nexport function scheduleReadableStreamLike<T>(input: ReadableStreamLike<T>, scheduler: SchedulerLike): Observable<T> {\n  return scheduleAsyncIterable(readableStreamLikeToAsyncGenerator(input), scheduler);\n}\n", "import { scheduleObservable } from './scheduleObservable';\nimport { schedulePromise } from './schedulePromise';\nimport { scheduleArray } from './scheduleArray';\nimport { scheduleIterable } from './scheduleIterable';\nimport { scheduleAsyncIterable } from './scheduleAsyncIterable';\nimport { isInteropObservable } from '../util/isInteropObservable';\nimport { isPromise } from '../util/isPromise';\nimport { isArrayLike } from '../util/isArrayLike';\nimport { isIterable } from '../util/isIterable';\nimport { ObservableInput, SchedulerLike } from '../types';\nimport { Observable } from '../Observable';\nimport { isAsyncIterable } from '../util/isAsyncIterable';\nimport { createInvalidObservableTypeError } from '../util/throwUnobservableError';\nimport { isReadableStreamLike } from '../util/isReadableStreamLike';\nimport { scheduleReadableStreamLike } from './scheduleReadableStreamLike';\n\n/**\n * Converts from a common {@link ObservableInput} type to an observable where subscription and emissions\n * are scheduled on the provided scheduler.\n *\n * @see {@link from}\n * @see {@link of}\n *\n * @param input The observable, array, promise, iterable, etc you would like to schedule\n * @param scheduler The scheduler to use to schedule the subscription and emissions from\n * the returned observable.\n */\nexport function scheduled<T>(input: ObservableInput<T>, scheduler: SchedulerLike): Observable<T> {\n  if (input != null) {\n    if (isInteropObservable(input)) {\n      return scheduleObservable(input, scheduler);\n    }\n    if (isArrayLike(input)) {\n      return scheduleArray(input, scheduler);\n    }\n    if (isPromise(input)) {\n      return schedulePromise(input, scheduler);\n    }\n    if (isAsyncIterable(input)) {\n      return scheduleAsyncIterable(input, scheduler);\n    }\n    if (isIterable(input)) {\n      return scheduleIterable(input, scheduler);\n    }\n    if (isReadableStreamLike(input)) {\n      return scheduleReadableStreamLike(input, scheduler);\n    }\n  }\n  throw createInvalidObservableTypeError(input);\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInput, SchedulerLike, ObservedValueOf } from '../types';\nimport { scheduled } from '../scheduled/scheduled';\nimport { innerFrom } from './innerFrom';\n\nexport function from<O extends ObservableInput<any>>(input: O): Observable<ObservedValueOf<O>>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function from<O extends ObservableInput<any>>(input: O, scheduler: SchedulerLike | undefined): Observable<ObservedValueOf<O>>;\n\n/**\n * Creates an Observable from an Array, an array-like object, a Promise, an iterable object, or an Observable-like object.\n *\n * <span class=\"informal\">Converts almost anything to an Observable.</span>\n *\n * ![](from.png)\n *\n * `from` converts various other objects and data types into Observables. It also converts a Promise, an array-like, or an\n * <a href=\"https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Iteration_protocols#iterable\" target=\"_blank\">iterable</a>\n * object into an Observable that emits the items in that promise, array, or iterable. A String, in this context, is treated\n * as an array of characters. Observable-like objects (contains a function named with the ES2015 Symbol for Observable) can also be\n * converted through this operator.\n *\n * ## Examples\n *\n * Converts an array to an Observable\n *\n * ```ts\n * import { from } from 'rxjs';\n *\n * const array = [10, 20, 30];\n * const result = from(array);\n *\n * result.subscribe(x => console.log(x));\n *\n * // Logs:\n * // 10\n * // 20\n * // 30\n * ```\n *\n * Convert an infinite iterable (from a generator) to an Observable\n *\n * ```ts\n * import { from, take } from 'rxjs';\n *\n * function* generateDoubles(seed) {\n *    let i = seed;\n *    while (true) {\n *      yield i;\n *      i = 2 * i; // double it\n *    }\n * }\n *\n * const iterator = generateDoubles(3);\n * const result = from(iterator).pipe(take(10));\n *\n * result.subscribe(x => console.log(x));\n *\n * // Logs:\n * // 3\n * // 6\n * // 12\n * // 24\n * // 48\n * // 96\n * // 192\n * // 384\n * // 768\n * // 1536\n * ```\n *\n * With `asyncScheduler`\n *\n * ```ts\n * import { from, asyncScheduler } from 'rxjs';\n *\n * console.log('start');\n *\n * const array = [10, 20, 30];\n * const result = from(array, asyncScheduler);\n *\n * result.subscribe(x => console.log(x));\n *\n * console.log('end');\n *\n * // Logs:\n * // 'start'\n * // 'end'\n * // 10\n * // 20\n * // 30\n * ```\n *\n * @see {@link fromEvent}\n * @see {@link fromEventPattern}\n *\n * @param {ObservableInput<T>} A subscription object, a Promise, an Observable-like,\n * an Array, an iterable, or an array-like object to be converted.\n * @param {SchedulerLike} An optional {@link SchedulerLike} on which to schedule the emission of values.\n * @return {Observable<T>}\n */\nexport function from<T>(input: ObservableInput<T>, scheduler?: SchedulerLike): Observable<T> {\n  return scheduler ? scheduled(input, scheduler) : innerFrom(input);\n}\n", "import { SchedulerLike, ValueFromArray } from '../types';\nimport { Observable } from '../Observable';\nimport { popScheduler } from '../util/args';\nimport { from } from './from';\n\n// Devs are more likely to pass null or undefined than they are a scheduler\n// without accompanying values. To make things easier for (naughty) devs who\n// use the `strictNullChecks: false` TypeScript compiler option, these\n// overloads with explicit null and undefined values are included.\n\nexport function of(value: null): Observable<null>;\nexport function of(value: undefined): Observable<undefined>;\n\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function of(scheduler: SchedulerLike): Observable<never>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function of<A extends readonly unknown[]>(...valuesAndScheduler: [...A, SchedulerLike]): Observable<ValueFromArray<A>>;\n\nexport function of(): Observable<never>;\n/** @deprecated Do not specify explicit type parameters. Signatures with type parameters that cannot be inferred will be removed in v8. */\nexport function of<T>(): Observable<T>;\nexport function of<T>(value: T): Observable<T>;\nexport function of<A extends readonly unknown[]>(...values: A): Observable<ValueFromArray<A>>;\n\n/**\n * Converts the arguments to an observable sequence.\n *\n * <span class=\"informal\">Each argument becomes a `next` notification.</span>\n *\n * ![](of.png)\n *\n * Unlike {@link from}, it does not do any flattening and emits each argument in whole\n * as a separate `next` notification.\n *\n * ## Examples\n *\n * Emit the values `10, 20, 30`\n *\n * ```ts\n * import { of } from 'rxjs';\n *\n * of(10, 20, 30)\n *   .subscribe({\n *     next: value => console.log('next:', value),\n *     error: err => console.log('error:', err),\n *     complete: () => console.log('the end'),\n *   });\n *\n * // Outputs\n * // next: 10\n * // next: 20\n * // next: 30\n * // the end\n * ```\n *\n * Emit the array `[1, 2, 3]`\n *\n * ```ts\n * import { of } from 'rxjs';\n *\n * of([1, 2, 3])\n *   .subscribe({\n *     next: value => console.log('next:', value),\n *     error: err => console.log('error:', err),\n *     complete: () => console.log('the end'),\n *   });\n *\n * // Outputs\n * // next: [1, 2, 3]\n * // the end\n * ```\n *\n * @see {@link from}\n * @see {@link range}\n *\n * @param {...T} values A comma separated list of arguments you want to be emitted\n * @return {Observable} An Observable that emits the arguments\n * described above and then completes.\n */\nexport function of<T>(...args: Array<T | SchedulerLike>): Observable<T> {\n  const scheduler = popScheduler(args);\n  return from(args as T[], scheduler);\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { SchedulerLike } from '../types';\nimport { isFunction } from '../util/isFunction';\n\n/**\n * Creates an observable that will create an error instance and push it to the consumer as an error\n * immediately upon subscription.\n *\n * <span class=\"informal\">Just errors and does nothing else</span>\n *\n * ![](throw.png)\n *\n * This creation function is useful for creating an observable that will create an error and error every\n * time it is subscribed to. Generally, inside of most operators when you might want to return an errored\n * observable, this is unnecessary. In most cases, such as in the inner return of {@link concatMap},\n * {@link mergeMap}, {@link defer}, and many others, you can simply throw the error, and RxJS will pick\n * that up and notify the consumer of the error.\n *\n * ## Example\n *\n * Create a simple observable that will create a new error with a timestamp and log it\n * and the message every time you subscribe to it\n *\n * ```ts\n * import { throwError } from 'rxjs';\n *\n * let errorCount = 0;\n *\n * const errorWithTimestamp$ = throwError(() => {\n *   const error: any = new Error(`This is error number ${ ++errorCount }`);\n *   error.timestamp = Date.now();\n *   return error;\n * });\n *\n * errorWithTimestamp$.subscribe({\n *   error: err => console.log(err.timestamp, err.message)\n * });\n *\n * errorWithTimestamp$.subscribe({\n *   error: err => console.log(err.timestamp, err.message)\n * });\n *\n * // Logs the timestamp and a new error message for each subscription\n * ```\n *\n * ### Unnecessary usage\n *\n * Using `throwError` inside of an operator or creation function\n * with a callback, is usually not necessary\n *\n * ```ts\n * import { of, concatMap, timer, throwError } from 'rxjs';\n *\n * const delays$ = of(1000, 2000, Infinity, 3000);\n *\n * delays$.pipe(\n *   concatMap(ms => {\n *     if (ms < 10000) {\n *       return timer(ms);\n *     } else {\n *       // This is probably overkill.\n *       return throwError(() => new Error(`Invalid time ${ ms }`));\n *     }\n *   })\n * )\n * .subscribe({\n *   next: console.log,\n *   error: console.error\n * });\n * ```\n *\n * You can just throw the error instead\n *\n * ```ts\n * import { of, concatMap, timer } from 'rxjs';\n *\n * const delays$ = of(1000, 2000, Infinity, 3000);\n *\n * delays$.pipe(\n *   concatMap(ms => {\n *     if (ms < 10000) {\n *       return timer(ms);\n *     } else {\n *       // Cleaner and easier to read for most folks.\n *       throw new Error(`Invalid time ${ ms }`);\n *     }\n *   })\n * )\n * .subscribe({\n *   next: console.log,\n *   error: console.error\n * });\n * ```\n *\n * @param errorFactory A factory function that will create the error instance that is pushed.\n */\nexport function throwError(errorFactory: () => any): Observable<never>;\n\n/**\n * Returns an observable that will error with the specified error immediately upon subscription.\n *\n * @param error The error instance to emit\n * @deprecated Support for passing an error value will be removed in v8. Instead, pass a factory function to `throwError(() => new Error('test'))`. This is\n * because it will create the error at the moment it should be created and capture a more appropriate stack trace. If\n * for some reason you need to create the error ahead of time, you can still do that: `const err = new Error('test'); throwError(() => err);`.\n */\nexport function throwError(error: any): Observable<never>;\n\n/**\n * Notifies the consumer of an error using a given scheduler by scheduling it at delay `0` upon subscription.\n *\n * @param errorOrErrorFactory An error instance or error factory\n * @param scheduler A scheduler to use to schedule the error notification\n * @deprecated The `scheduler` parameter will be removed in v8.\n * Use `throwError` in combination with {@link observeOn}: `throwError(() => new Error('test')).pipe(observeOn(scheduler));`.\n * Details: https://rxjs.dev/deprecations/scheduler-argument\n */\nexport function throwError(errorOrErrorFactory: any, scheduler: SchedulerLike): Observable<never>;\n\nexport function throwError(errorOrErrorFactory: any, scheduler?: SchedulerLike): Observable<never> {\n  const errorFactory = isFunction(errorOrErrorFactory) ? errorOrErrorFactory : () => errorOrErrorFactory;\n  const init = (subscriber: Subscriber<never>) => subscriber.error(errorFactory());\n  return new Observable(scheduler ? (subscriber) => scheduler.schedule(init as any, 0, subscriber) : init);\n}\n", "import { createErrorClass } from './createErrorClass';\n\nexport interface EmptyError extends Error {}\n\nexport interface EmptyErrorCtor {\n  /**\n   * @deprecated Internal implementation detail. Do not construct error instances.\n   * Cannot be tagged as internal: https://github.com/ReactiveX/rxjs/issues/6269\n   */\n  new (): EmptyError;\n}\n\n/**\n * An error thrown when an Observable or a sequence was queried but has no\n * elements.\n *\n * @see {@link first}\n * @see {@link last}\n * @see {@link single}\n * @see {@link firstValueFrom}\n * @see {@link lastValueFrom}\n *\n * @class EmptyError\n */\nexport const EmptyError: EmptyErrorCtor = createErrorClass((_super) => function EmptyErrorImpl(this: any) {\n  _super(this);\n  this.name = 'EmptyError';\n  this.message = 'no elements in sequence';\n});\n", "/**\n * Checks to see if a value is not only a `Date` object,\n * but a *valid* `Date` object that can be converted to a\n * number. For example, `new Date('blah')` is indeed an\n * `instanceof Date`, however it cannot be converted to a\n * number.\n */\nexport function isValidDate(value: any): value is Date {\n  return value instanceof Date && !isNaN(value as any);\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\nexport function map<T, R>(project: (value: T, index: number) => R): OperatorFunction<T, R>;\n/** @deprecated Use a closure instead of a `thisArg`. Signatures accepting a `thisArg` will be removed in v8. */\nexport function map<T, R, A>(project: (this: A, value: T, index: number) => R, thisArg: A): OperatorFunction<T, R>;\n\n/**\n * Applies a given `project` function to each value emitted by the source\n * Observable, and emits the resulting values as an Observable.\n *\n * <span class=\"informal\">Like [Array.prototype.map()](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/map),\n * it passes each source value through a transformation function to get\n * corresponding output values.</span>\n *\n * ![](map.png)\n *\n * Similar to the well known `Array.prototype.map` function, this operator\n * applies a projection to each value and emits that projection in the output\n * Observable.\n *\n * ## Example\n *\n * Map every click to the `clientX` position of that click\n *\n * ```ts\n * import { fromEvent, map } from 'rxjs';\n *\n * const clicks = fromEvent<PointerEvent>(document, 'click');\n * const positions = clicks.pipe(map(ev => ev.clientX));\n *\n * positions.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link mapTo}\n * @see {@link pluck}\n *\n * @param {function(value: T, index: number): R} project The function to apply\n * to each `value` emitted by the source Observable. The `index` parameter is\n * the number `i` for the i-th emission that has happened since the\n * subscription, starting from the number `0`.\n * @param {any} [thisArg] An optional argument to define what `this` is in the\n * `project` function.\n * @return A function that returns an Observable that emits the values from the\n * source Observable transformed by the given `project` function.\n */\nexport function map<T, R>(project: (value: T, index: number) => R, thisArg?: any): OperatorFunction<T, R> {\n  return operate((source, subscriber) => {\n    // The index of the value from the source. Used with projection.\n    let index = 0;\n    // Subscribe to the source, all errors and completions are sent along\n    // to the consumer.\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value: T) => {\n        // Call the projection function with the appropriate this context,\n        // and send the resulting value to the consumer.\n        subscriber.next(project.call(thisArg, value, index++));\n      })\n    );\n  });\n}\n", "import { OperatorFunction } from \"../types\";\nimport { map } from \"../operators/map\";\n\nconst { isArray } = Array;\n\nfunction callOrApply<T, R>(fn: ((...values: T[]) => R), args: T|T[]): R {\n    return isArray(args) ? fn(...args) : fn(args);\n}\n\n/**\n * Used in several -- mostly deprecated -- situations where we need to \n * apply a list of arguments or a single argument to a result selector.\n */\nexport function mapOneOrManyArgs<T, R>(fn: ((...values: T[]) => R)): OperatorFunction<T|T[], R> {\n    return map(args => callOrApply(fn, args))\n}", "const { isArray } = Array;\nconst { getPrototypeOf, prototype: objectProto, keys: getKeys } = Object;\n\n/**\n * Used in functions where either a list of arguments, a single array of arguments, or a\n * dictionary of arguments can be returned. Returns an object with an `args` property with\n * the arguments in an array, if it is a dictionary, it will also return the `keys` in another\n * property.\n */\nexport function argsArgArrayOrObject<T, O extends Record<string, T>>(args: T[] | [O] | [T[]]): { args: T[]; keys: string[] | null } {\n  if (args.length === 1) {\n    const first = args[0];\n    if (isArray(first)) {\n      return { args: first, keys: null };\n    }\n    if (isPOJO(first)) {\n      const keys = getKeys(first);\n      return {\n        args: keys.map((key) => first[key]),\n        keys,\n      };\n    }\n  }\n\n  return { args: args as T[], keys: null };\n}\n\nfunction isPOJO(obj: any): obj is object {\n  return obj && typeof obj === 'object' && getPrototypeOf(obj) === objectProto;\n}\n", "export function createObject(keys: string[], values: any[]) {\n  return keys.reduce((result, key, i) => ((result[key] = values[i]), result), {} as any);\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInput, SchedulerLike, ObservedValueOf, ObservableInputTuple } from '../types';\nimport { argsArgArrayOrObject } from '../util/argsArgArrayOrObject';\nimport { Subscriber } from '../Subscriber';\nimport { from } from './from';\nimport { identity } from '../util/identity';\nimport { Subscription } from '../Subscription';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\nimport { popResultSelector, popScheduler } from '../util/args';\nimport { createObject } from '../util/createObject';\nimport { createOperatorSubscriber } from '../operators/OperatorSubscriber';\nimport { AnyCatcher } from '../AnyCatcher';\nimport { executeSchedule } from '../util/executeSchedule';\n\n// combineLatest(any)\n// We put this first because we need to catch cases where the user has supplied\n// _exactly `any`_ as the argument. Since `any` literally matches _anything_,\n// we don't want it to randomly hit one of the other type signatures below,\n// as we have no idea at build-time what type we should be returning when given an any.\n\n/**\n * You have passed `any` here, we can't figure out if it is\n * an array or an object, so you're getting `unknown`. Use better types.\n * @param arg Something typed as `any`\n */\nexport function combineLatest<T extends AnyCatcher>(arg: T): Observable<unknown>;\n\n// combineLatest([a, b, c])\nexport function combineLatest(sources: []): Observable<never>;\nexport function combineLatest<A extends readonly unknown[]>(sources: readonly [...ObservableInputTuple<A>]): Observable<A>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[], R>(\n  sources: readonly [...ObservableInputTuple<A>],\n  resultSelector: (...values: A) => R,\n  scheduler: SchedulerLike\n): Observable<R>;\nexport function combineLatest<A extends readonly unknown[], R>(\n  sources: readonly [...ObservableInputTuple<A>],\n  resultSelector: (...values: A) => R\n): Observable<R>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[]>(\n  sources: readonly [...ObservableInputTuple<A>],\n  scheduler: SchedulerLike\n): Observable<A>;\n\n// combineLatest(a, b, c)\n/** @deprecated Pass an array of sources instead. The rest-parameters signature will be removed in v8. Details: https://rxjs.dev/deprecations/array-argument */\nexport function combineLatest<A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): Observable<A>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[], R>(\n  ...sourcesAndResultSelectorAndScheduler: [...ObservableInputTuple<A>, (...values: A) => R, SchedulerLike]\n): Observable<R>;\n/** @deprecated Pass an array of sources instead. The rest-parameters signature will be removed in v8. Details: https://rxjs.dev/deprecations/array-argument */\nexport function combineLatest<A extends readonly unknown[], R>(\n  ...sourcesAndResultSelector: [...ObservableInputTuple<A>, (...values: A) => R]\n): Observable<R>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `combineLatestAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function combineLatest<A extends readonly unknown[]>(\n  ...sourcesAndScheduler: [...ObservableInputTuple<A>, SchedulerLike]\n): Observable<A>;\n\n// combineLatest({a, b, c})\nexport function combineLatest(sourcesObject: { [K in any]: never }): Observable<never>;\nexport function combineLatest<T extends Record<string, ObservableInput<any>>>(\n  sourcesObject: T\n): Observable<{ [K in keyof T]: ObservedValueOf<T[K]> }>;\n\n/**\n * Combines multiple Observables to create an Observable whose values are\n * calculated from the latest values of each of its input Observables.\n *\n * <span class=\"informal\">Whenever any input Observable emits a value, it\n * computes a formula using the latest values from all the inputs, then emits\n * the output of that formula.</span>\n *\n * ![](combineLatest.png)\n *\n * `combineLatest` combines the values from all the Observables passed in the\n * observables array. This is done by subscribing to each Observable in order and,\n * whenever any Observable emits, collecting an array of the most recent\n * values from each Observable. So if you pass `n` Observables to this operator,\n * the returned Observable will always emit an array of `n` values, in an order\n * corresponding to the order of the passed Observables (the value from the first Observable\n * will be at index 0 of the array and so on).\n *\n * Static version of `combineLatest` accepts an array of Observables. Note that an array of\n * Observables is a good choice, if you don't know beforehand how many Observables\n * you will combine. Passing an empty array will result in an Observable that\n * completes immediately.\n *\n * To ensure the output array always has the same length, `combineLatest` will\n * actually wait for all input Observables to emit at least once,\n * before it starts emitting results. This means if some Observable emits\n * values before other Observables started emitting, all these values but the last\n * will be lost. On the other hand, if some Observable does not emit a value but\n * completes, resulting Observable will complete at the same moment without\n * emitting anything, since it will now be impossible to include a value from the\n * completed Observable in the resulting array. Also, if some input Observable does\n * not emit any value and never completes, `combineLatest` will also never emit\n * and never complete, since, again, it will wait for all streams to emit some\n * value.\n *\n * If at least one Observable was passed to `combineLatest` and all passed Observables\n * emitted something, the resulting Observable will complete when all combined\n * streams complete. So even if some Observable completes, the result of\n * `combineLatest` will still emit values when other Observables do. In case\n * of a completed Observable, its value from now on will always be the last\n * emitted value. On the other hand, if any Observable errors, `combineLatest`\n * will error immediately as well, and all other Observables will be unsubscribed.\n *\n * ## Examples\n *\n * Combine two timer Observables\n *\n * ```ts\n * import { timer, combineLatest } from 'rxjs';\n *\n * const firstTimer = timer(0, 1000); // emit 0, 1, 2... after every second, starting from now\n * const secondTimer = timer(500, 1000); // emit 0, 1, 2... after every second, starting 0,5s from now\n * const combinedTimers = combineLatest([firstTimer, secondTimer]);\n * combinedTimers.subscribe(value => console.log(value));\n * // Logs\n * // [0, 0] after 0.5s\n * // [1, 0] after 1s\n * // [1, 1] after 1.5s\n * // [2, 1] after 2s\n * ```\n *\n * Combine a dictionary of Observables\n *\n * ```ts\n * import { of, delay, startWith, combineLatest } from 'rxjs';\n *\n * const observables = {\n *   a: of(1).pipe(delay(1000), startWith(0)),\n *   b: of(5).pipe(delay(5000), startWith(0)),\n *   c: of(10).pipe(delay(10000), startWith(0))\n * };\n * const combined = combineLatest(observables);\n * combined.subscribe(value => console.log(value));\n * // Logs\n * // { a: 0, b: 0, c: 0 } immediately\n * // { a: 1, b: 0, c: 0 } after 1s\n * // { a: 1, b: 5, c: 0 } after 5s\n * // { a: 1, b: 5, c: 10 } after 10s\n * ```\n *\n * Combine an array of Observables\n *\n * ```ts\n * import { of, delay, startWith, combineLatest } from 'rxjs';\n *\n * const observables = [1, 5, 10].map(\n *   n => of(n).pipe(\n *     delay(n * 1000), // emit 0 and then emit n after n seconds\n *     startWith(0)\n *   )\n * );\n * const combined = combineLatest(observables);\n * combined.subscribe(value => console.log(value));\n * // Logs\n * // [0, 0, 0] immediately\n * // [1, 0, 0] after 1s\n * // [1, 5, 0] after 5s\n * // [1, 5, 10] after 10s\n * ```\n *\n * Use map operator to dynamically calculate the Body-Mass Index\n *\n * ```ts\n * import { of, combineLatest, map } from 'rxjs';\n *\n * const weight = of(70, 72, 76, 79, 75);\n * const height = of(1.76, 1.77, 1.78);\n * const bmi = combineLatest([weight, height]).pipe(\n *   map(([w, h]) => w / (h * h)),\n * );\n * bmi.subscribe(x => console.log('BMI is ' + x));\n *\n * // With output to console:\n * // BMI is 24.212293388429753\n * // BMI is 23.93948099205209\n * // BMI is 23.671253629592222\n * ```\n *\n * @see {@link combineLatestAll}\n * @see {@link merge}\n * @see {@link withLatestFrom}\n *\n * @param {ObservableInput} [observables] An array of input Observables to combine with each other.\n * An array of Observables must be given as the first argument.\n * @param {function} [project] An optional function to project the values from\n * the combined latest values into a new value on the output Observable.\n * @param {SchedulerLike} [scheduler=null] The {@link SchedulerLike} to use for subscribing to\n * each input Observable.\n * @return {Observable} An Observable of projected values from the most recent\n * values from each input Observable, or an array of the most recent values from\n * each input Observable.\n */\nexport function combineLatest<O extends ObservableInput<any>, R>(...args: any[]): Observable<R> | Observable<ObservedValueOf<O>[]> {\n  const scheduler = popScheduler(args);\n  const resultSelector = popResultSelector(args);\n\n  const { args: observables, keys } = argsArgArrayOrObject(args);\n\n  if (observables.length === 0) {\n    // If no observables are passed, or someone has passed an empty array\n    // of observables, or even an empty object POJO, we need to just\n    // complete (EMPTY), but we have to honor the scheduler provided if any.\n    return from([], scheduler as any);\n  }\n\n  const result = new Observable<ObservedValueOf<O>[]>(\n    combineLatestInit(\n      observables as ObservableInput<ObservedValueOf<O>>[],\n      scheduler,\n      keys\n        ? // A handler for scrubbing the array of args into a dictionary.\n          (values) => createObject(keys, values)\n        : // A passthrough to just return the array\n          identity\n    )\n  );\n\n  return resultSelector ? (result.pipe(mapOneOrManyArgs(resultSelector)) as Observable<R>) : result;\n}\n\nexport function combineLatestInit(\n  observables: ObservableInput<any>[],\n  scheduler?: SchedulerLike,\n  valueTransform: (values: any[]) => any = identity\n) {\n  return (subscriber: Subscriber<any>) => {\n    // The outer subscription. We're capturing this in a function\n    // because we may have to schedule it.\n    maybeSchedule(\n      scheduler,\n      () => {\n        const { length } = observables;\n        // A store for the values each observable has emitted so far. We match observable to value on index.\n        const values = new Array(length);\n        // The number of currently active subscriptions, as they complete, we decrement this number to see if\n        // we are all done combining values, so we can complete the result.\n        let active = length;\n        // The number of inner sources that still haven't emitted the first value\n        // We need to track this because all sources need to emit one value in order\n        // to start emitting values.\n        let remainingFirstValues = length;\n        // The loop to kick off subscription. We're keying everything on index `i` to relate the observables passed\n        // in to the slot in the output array or the key in the array of keys in the output dictionary.\n        for (let i = 0; i < length; i++) {\n          maybeSchedule(\n            scheduler,\n            () => {\n              const source = from(observables[i], scheduler as any);\n              let hasFirstValue = false;\n              source.subscribe(\n                createOperatorSubscriber(\n                  subscriber,\n                  (value) => {\n                    // When we get a value, record it in our set of values.\n                    values[i] = value;\n                    if (!hasFirstValue) {\n                      // If this is our first value, record that.\n                      hasFirstValue = true;\n                      remainingFirstValues--;\n                    }\n                    if (!remainingFirstValues) {\n                      // We're not waiting for any more\n                      // first values, so we can emit!\n                      subscriber.next(valueTransform(values.slice()));\n                    }\n                  },\n                  () => {\n                    if (!--active) {\n                      // We only complete the result if we have no more active\n                      // inner observables.\n                      subscriber.complete();\n                    }\n                  }\n                )\n              );\n            },\n            subscriber\n          );\n        }\n      },\n      subscriber\n    );\n  };\n}\n\n/**\n * A small utility to handle the couple of locations where we want to schedule if a scheduler was provided,\n * but we don't if there was no scheduler.\n */\nfunction maybeSchedule(scheduler: SchedulerLike | undefined, execute: () => void, subscription: Subscription) {\n  if (scheduler) {\n    executeSchedule(subscription, scheduler, execute);\n  } else {\n    execute();\n  }\n}\n", "import { Observable } from '../Observable';\nimport { innerFrom } from '../observable/innerFrom';\nimport { Subscriber } from '../Subscriber';\nimport { ObservableInput, SchedulerLike } from '../types';\nimport { executeSchedule } from '../util/executeSchedule';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * A process embodying the general \"merge\" strategy. This is used in\n * `mergeMap` and `mergeScan` because the logic is otherwise nearly identical.\n * @param source The original source observable\n * @param subscriber The consumer subscriber\n * @param project The projection function to get our inner sources\n * @param concurrent The number of concurrent inner subscriptions\n * @param onBeforeNext Additional logic to apply before nexting to our consumer\n * @param expand If `true` this will perform an \"expand\" strategy, which differs only\n * in that it recurses, and the inner subscription must be schedule-able.\n * @param innerSubScheduler A scheduler to use to schedule inner subscriptions,\n * this is to support the expand strategy, mostly, and should be deprecated\n */\nexport function mergeInternals<T, R>(\n  source: Observable<T>,\n  subscriber: Subscriber<R>,\n  project: (value: T, index: number) => ObservableInput<R>,\n  concurrent: number,\n  onBeforeNext?: (innerValue: R) => void,\n  expand?: boolean,\n  innerSubScheduler?: SchedulerLike,\n  additionalFinalizer?: () => void\n) {\n  // Buffered values, in the event of going over our concurrency limit\n  const buffer: T[] = [];\n  // The number of active inner subscriptions.\n  let active = 0;\n  // An index to pass to our accumulator function\n  let index = 0;\n  // Whether or not the outer source has completed.\n  let isComplete = false;\n\n  /**\n   * Checks to see if we can complete our result or not.\n   */\n  const checkComplete = () => {\n    // If the outer has completed, and nothing is left in the buffer,\n    // and we don't have any active inner subscriptions, then we can\n    // Emit the state and complete.\n    if (isComplete && !buffer.length && !active) {\n      subscriber.complete();\n    }\n  };\n\n  // If we're under our concurrency limit, just start the inner subscription, otherwise buffer and wait.\n  const outerNext = (value: T) => (active < concurrent ? doInnerSub(value) : buffer.push(value));\n\n  const doInnerSub = (value: T) => {\n    // If we're expanding, we need to emit the outer values and the inner values\n    // as the inners will \"become outers\" in a way as they are recursively fed\n    // back to the projection mechanism.\n    expand && subscriber.next(value as any);\n\n    // Increment the number of active subscriptions so we can track it\n    // against our concurrency limit later.\n    active++;\n\n    // A flag used to show that the inner observable completed.\n    // This is checked during finalization to see if we should\n    // move to the next item in the buffer, if there is on.\n    let innerComplete = false;\n\n    // Start our inner subscription.\n    innerFrom(project(value, index++)).subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (innerValue) => {\n          // `mergeScan` has additional handling here. For example\n          // taking the inner value and updating state.\n          onBeforeNext?.(innerValue);\n\n          if (expand) {\n            // If we're expanding, then just recurse back to our outer\n            // handler. It will emit the value first thing.\n            outerNext(innerValue as any);\n          } else {\n            // Otherwise, emit the inner value.\n            subscriber.next(innerValue);\n          }\n        },\n        () => {\n          // Flag that we have completed, so we know to check the buffer\n          // during finalization.\n          innerComplete = true;\n        },\n        // Errors are passed to the destination.\n        undefined,\n        () => {\n          // During finalization, if the inner completed (it wasn't errored or\n          // cancelled), then we want to try the next item in the buffer if\n          // there is one.\n          if (innerComplete) {\n            // We have to wrap this in a try/catch because it happens during\n            // finalization, possibly asynchronously, and we want to pass\n            // any errors that happen (like in a projection function) to\n            // the outer Subscriber.\n            try {\n              // INNER SOURCE COMPLETE\n              // Decrement the active count to ensure that the next time\n              // we try to call `doInnerSub`, the number is accurate.\n              active--;\n              // If we have more values in the buffer, try to process those\n              // Note that this call will increment `active` ahead of the\n              // next conditional, if there were any more inner subscriptions\n              // to start.\n              while (buffer.length && active < concurrent) {\n                const bufferedValue = buffer.shift()!;\n                // Particularly for `expand`, we need to check to see if a scheduler was provided\n                // for when we want to start our inner subscription. Otherwise, we just start\n                // are next inner subscription.\n                if (innerSubScheduler) {\n                  executeSchedule(subscriber, innerSubScheduler, () => doInnerSub(bufferedValue));\n                } else {\n                  doInnerSub(bufferedValue);\n                }\n              }\n              // Check to see if we can complete, and complete if so.\n              checkComplete();\n            } catch (err) {\n              subscriber.error(err);\n            }\n          }\n        }\n      )\n    );\n  };\n\n  // Subscribe to our source observable.\n  source.subscribe(\n    createOperatorSubscriber(subscriber, outerNext, () => {\n      // Outer completed, make a note of it, and check to see if we can complete everything.\n      isComplete = true;\n      checkComplete();\n    })\n  );\n\n  // Additional finalization (for when the destination is torn down).\n  // Other finalization is added implicitly via subscription above.\n  return () => {\n    additionalFinalizer?.();\n  };\n}\n", "import { ObservableInput, OperatorFunction, ObservedValueOf } from '../types';\nimport { map } from './map';\nimport { innerFrom } from '../observable/innerFrom';\nimport { operate } from '../util/lift';\nimport { mergeInternals } from './mergeInternals';\nimport { isFunction } from '../util/isFunction';\n\n/* tslint:disable:max-line-length */\nexport function mergeMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  concurrent?: number\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function mergeMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: undefined,\n  concurrent?: number\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function mergeMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: (outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R,\n  concurrent?: number\n): OperatorFunction<T, R>;\n/* tslint:enable:max-line-length */\n\n/**\n * Projects each source value to an Observable which is merged in the output\n * Observable.\n *\n * <span class=\"informal\">Maps each value to an Observable, then flattens all of\n * these inner Observables using {@link mergeAll}.</span>\n *\n * ![](mergeMap.png)\n *\n * Returns an Observable that emits items based on applying a function that you\n * supply to each item emitted by the source Observable, where that function\n * returns an Observable, and then merging those resulting Observables and\n * emitting the results of this merger.\n *\n * ## Example\n *\n * Map and flatten each letter to an Observable ticking every 1 second\n *\n * ```ts\n * import { of, mergeMap, interval, map } from 'rxjs';\n *\n * const letters = of('a', 'b', 'c');\n * const result = letters.pipe(\n *   mergeMap(x => interval(1000).pipe(map(i => x + i)))\n * );\n *\n * result.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // a0\n * // b0\n * // c0\n * // a1\n * // b1\n * // c1\n * // continues to list a, b, c every second with respective ascending integers\n * ```\n *\n * @see {@link concatMap}\n * @see {@link exhaustMap}\n * @see {@link merge}\n * @see {@link mergeAll}\n * @see {@link mergeMapTo}\n * @see {@link mergeScan}\n * @see {@link switchMap}\n *\n * @param {function(value: T, ?index: number): ObservableInput} project A function\n * that, when applied to an item emitted by the source Observable, returns an\n * Observable.\n * @param {number} [concurrent=Infinity] Maximum number of input\n * Observables being subscribed to concurrently.\n * @return A function that returns an Observable that emits the result of\n * applying the projection function (and the optional deprecated\n * `resultSelector`) to each item emitted by the source Observable and merging\n * the results of the Observables obtained from this transformation.\n */\nexport function mergeMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector?: ((outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R) | number,\n  concurrent: number = Infinity\n): OperatorFunction<T, ObservedValueOf<O> | R> {\n  if (isFunction(resultSelector)) {\n    // DEPRECATED PATH\n    return mergeMap((a, i) => map((b: any, ii: number) => resultSelector(a, b, i, ii))(innerFrom(project(a, i))), concurrent);\n  } else if (typeof resultSelector === 'number') {\n    concurrent = resultSelector;\n  }\n\n  return operate((source, subscriber) => mergeInternals(source, subscriber, project, concurrent));\n}\n", "import { mergeMap } from './mergeMap';\nimport { identity } from '../util/identity';\nimport { OperatorFunction, ObservableInput, ObservedValueOf } from '../types';\n\n/**\n * Converts a higher-order Observable into a first-order Observable which\n * concurrently delivers all values that are emitted on the inner Observables.\n *\n * <span class=\"informal\">Flattens an Observable-of-Observables.</span>\n *\n * ![](mergeAll.png)\n *\n * `mergeAll` subscribes to an Observable that emits Observables, also known as\n * a higher-order Observable. Each time it observes one of these emitted inner\n * Observables, it subscribes to that and delivers all the values from the\n * inner Observable on the output Observable. The output Observable only\n * completes once all inner Observables have completed. Any error delivered by\n * a inner Observable will be immediately emitted on the output Observable.\n *\n * ## Examples\n *\n * Spawn a new interval Observable for each click event, and blend their outputs as one Observable\n *\n * ```ts\n * import { fromEvent, map, interval, mergeAll } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const higherOrder = clicks.pipe(map(() => interval(1000)));\n * const firstOrder = higherOrder.pipe(mergeAll());\n *\n * firstOrder.subscribe(x => console.log(x));\n * ```\n *\n * Count from 0 to 9 every second for each click, but only allow 2 concurrent timers\n *\n * ```ts\n * import { fromEvent, map, interval, take, mergeAll } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const higherOrder = clicks.pipe(\n *   map(() => interval(1000).pipe(take(10)))\n * );\n * const firstOrder = higherOrder.pipe(mergeAll(2));\n *\n * firstOrder.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link combineLatestAll}\n * @see {@link concatAll}\n * @see {@link exhaustAll}\n * @see {@link merge}\n * @see {@link mergeMap}\n * @see {@link mergeMapTo}\n * @see {@link mergeScan}\n * @see {@link switchAll}\n * @see {@link switchMap}\n * @see {@link zipAll}\n *\n * @param {number} [concurrent=Infinity] Maximum number of inner\n * Observables being subscribed to concurrently.\n * @return A function that returns an Observable that emits values coming from\n * all the inner Observables emitted by the source Observable.\n */\nexport function mergeAll<O extends ObservableInput<any>>(concurrent: number = Infinity): OperatorFunction<O, ObservedValueOf<O>> {\n  return mergeMap(identity, concurrent);\n}\n", "import { mergeAll } from './mergeAll';\nimport { OperatorFunction, ObservableInput, ObservedValueOf } from '../types';\n\n/**\n * Converts a higher-order Observable into a first-order Observable by\n * concatenating the inner Observables in order.\n *\n * <span class=\"informal\">Flattens an Observable-of-Observables by putting one\n * inner Observable after the other.</span>\n *\n * ![](concatAll.svg)\n *\n * Joins every Observable emitted by the source (a higher-order Observable), in\n * a serial fashion. It subscribes to each inner Observable only after the\n * previous inner Observable has completed, and merges all of their values into\n * the returned observable.\n *\n * __Warning:__ If the source Observable emits Observables quickly and\n * endlessly, and the inner Observables it emits generally complete slower than\n * the source emits, you can run into memory issues as the incoming Observables\n * collect in an unbounded buffer.\n *\n * Note: `concatAll` is equivalent to `mergeAll` with concurrency parameter set\n * to `1`.\n *\n * ## Example\n *\n * For each click event, tick every second from 0 to 3, with no concurrency\n *\n * ```ts\n * import { fromEvent, map, interval, take, concatAll } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const higherOrder = clicks.pipe(\n *   map(() => interval(1000).pipe(take(4)))\n * );\n * const firstOrder = higherOrder.pipe(concatAll());\n * firstOrder.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // (results are not concurrent)\n * // For every click on the \"document\" it will emit values 0 to 3 spaced\n * // on a 1000ms interval\n * // one click = 1000ms-> 0 -1000ms-> 1 -1000ms-> 2 -1000ms-> 3\n * ```\n *\n * @see {@link combineLatestAll}\n * @see {@link concat}\n * @see {@link concatMap}\n * @see {@link concatMapTo}\n * @see {@link exhaustAll}\n * @see {@link mergeAll}\n * @see {@link switchAll}\n * @see {@link switchMap}\n * @see {@link zipAll}\n *\n * @return A function that returns an Observable emitting values from all the\n * inner Observables concatenated.\n */\nexport function concatAll<O extends ObservableInput<any>>(): OperatorFunction<O, ObservedValueOf<O>> {\n  return mergeAll(1);\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInputTuple, SchedulerLike } from '../types';\nimport { concatAll } from '../operators/concatAll';\nimport { popScheduler } from '../util/args';\nimport { from } from './from';\n\nexport function concat<T extends readonly unknown[]>(...inputs: [...ObservableInputTuple<T>]): Observable<T[number]>;\nexport function concat<T extends readonly unknown[]>(\n  ...inputsAndScheduler: [...ObservableInputTuple<T>, SchedulerLike]\n): Observable<T[number]>;\n\n/**\n * Creates an output Observable which sequentially emits all values from the first given\n * Observable and then moves on to the next.\n *\n * <span class=\"informal\">Concatenates multiple Observables together by\n * sequentially emitting their values, one Observable after the other.</span>\n *\n * ![](concat.png)\n *\n * `concat` joins multiple Observables together, by subscribing to them one at a time and\n * merging their results into the output Observable. You can pass either an array of\n * Observables, or put them directly as arguments. Passing an empty array will result\n * in Observable that completes immediately.\n *\n * `concat` will subscribe to first input Observable and emit all its values, without\n * changing or affecting them in any way. When that Observable completes, it will\n * subscribe to then next Observable passed and, again, emit its values. This will be\n * repeated, until the operator runs out of Observables. When last input Observable completes,\n * `concat` will complete as well. At any given moment only one Observable passed to operator\n * emits values. If you would like to emit values from passed Observables concurrently, check out\n * {@link merge} instead, especially with optional `concurrent` parameter. As a matter of fact,\n * `concat` is an equivalent of `merge` operator with `concurrent` parameter set to `1`.\n *\n * Note that if some input Observable never completes, `concat` will also never complete\n * and Observables following the one that did not complete will never be subscribed. On the other\n * hand, if some Observable simply completes immediately after it is subscribed, it will be\n * invisible for `concat`, which will just move on to the next Observable.\n *\n * If any Observable in chain errors, instead of passing control to the next Observable,\n * `concat` will error immediately as well. Observables that would be subscribed after\n * the one that emitted error, never will.\n *\n * If you pass to `concat` the same Observable many times, its stream of values\n * will be \"replayed\" on every subscription, which means you can repeat given Observable\n * as many times as you like. If passing the same Observable to `concat` 1000 times becomes tedious,\n * you can always use {@link repeat}.\n *\n * ## Examples\n *\n * Concatenate a timer counting from 0 to 3 with a synchronous sequence from 1 to 10\n *\n * ```ts\n * import { interval, take, range, concat } from 'rxjs';\n *\n * const timer = interval(1000).pipe(take(4));\n * const sequence = range(1, 10);\n * const result = concat(timer, sequence);\n * result.subscribe(x => console.log(x));\n *\n * // results in:\n * // 0 -1000ms-> 1 -1000ms-> 2 -1000ms-> 3 -immediate-> 1 ... 10\n * ```\n *\n * Concatenate 3 Observables\n *\n * ```ts\n * import { interval, take, concat } from 'rxjs';\n *\n * const timer1 = interval(1000).pipe(take(10));\n * const timer2 = interval(2000).pipe(take(6));\n * const timer3 = interval(500).pipe(take(10));\n *\n * const result = concat(timer1, timer2, timer3);\n * result.subscribe(x => console.log(x));\n *\n * // results in the following:\n * // (Prints to console sequentially)\n * // -1000ms-> 0 -1000ms-> 1 -1000ms-> ... 9\n * // -2000ms-> 0 -2000ms-> 1 -2000ms-> ... 5\n * // -500ms-> 0 -500ms-> 1 -500ms-> ... 9\n * ```\n *\n * Concatenate the same Observable to repeat it\n *\n * ```ts\n * import { interval, take, concat } from 'rxjs';\n *\n * const timer = interval(1000).pipe(take(2));\n *\n * concat(timer, timer) // concatenating the same Observable!\n *   .subscribe({\n *     next: value => console.log(value),\n *     complete: () => console.log('...and it is done!')\n *   });\n *\n * // Logs:\n * // 0 after 1s\n * // 1 after 2s\n * // 0 after 3s\n * // 1 after 4s\n * // '...and it is done!' also after 4s\n * ```\n *\n * @see {@link concatAll}\n * @see {@link concatMap}\n * @see {@link concatMapTo}\n * @see {@link startWith}\n * @see {@link endWith}\n *\n * @param args Input Observables to concatenate.\n */\nexport function concat(...args: any[]): Observable<unknown> {\n  return concatAll()(from(args, popScheduler(args)));\n}\n", "import { Observable } from '../Observable';\nimport { ObservedValueOf, ObservableInput } from '../types';\nimport { innerFrom } from './innerFrom';\n\n/**\n * Creates an Observable that, on subscribe, calls an Observable factory to\n * make an Observable for each new Observer.\n *\n * <span class=\"informal\">Creates the Observable lazily, that is, only when it\n * is subscribed.\n * </span>\n *\n * ![](defer.png)\n *\n * `defer` allows you to create an Observable only when the Observer\n * subscribes. It waits until an Observer subscribes to it, calls the given\n * factory function to get an Observable -- where a factory function typically\n * generates a new Observable -- and subscribes the Observer to this Observable.\n * In case the factory function returns a falsy value, then EMPTY is used as\n * Observable instead. Last but not least, an exception during the factory\n * function call is transferred to the Observer by calling `error`.\n *\n * ## Example\n *\n * Subscribe to either an Observable of clicks or an Observable of interval, at random\n *\n * ```ts\n * import { defer, fromEvent, interval } from 'rxjs';\n *\n * const clicksOrInterval = defer(() => {\n *   return Math.random() > 0.5\n *     ? fromEvent(document, 'click')\n *     : interval(1000);\n * });\n * clicksOrInterval.subscribe(x => console.log(x));\n *\n * // Results in the following behavior:\n * // If the result of Math.random() is greater than 0.5 it will listen\n * // for clicks anywhere on the \"document\"; when document is clicked it\n * // will log a MouseEvent object to the console. If the result is less\n * // than 0.5 it will emit ascending numbers, one every second(1000ms).\n * ```\n *\n * @see {@link Observable}\n *\n * @param {function(): ObservableInput} observableFactory The Observable\n * factory function to invoke for each Observer that subscribes to the output\n * Observable. May also return a Promise, which will be converted on the fly\n * to an Observable.\n * @return {Observable} An Observable whose Observers' subscriptions trigger\n * an invocation of the given Observable factory function.\n */\nexport function defer<R extends ObservableInput<any>>(observableFactory: () => R): Observable<ObservedValueOf<R>> {\n  return new Observable<ObservedValueOf<R>>((subscriber) => {\n    innerFrom(observableFactory()).subscribe(subscriber);\n  });\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { Observable } from '../Observable';\nimport { mergeMap } from '../operators/mergeMap';\nimport { isArrayLike } from '../util/isArrayLike';\nimport { isFunction } from '../util/isFunction';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\n\n// These constants are used to create handler registry functions using array mapping below.\nconst nodeEventEmitterMethods = ['addListener', 'removeListener'] as const;\nconst eventTargetMethods = ['addEventListener', 'removeEventListener'] as const;\nconst jqueryMethods = ['on', 'off'] as const;\n\nexport interface NodeStyleEventEmitter {\n  addListener(eventName: string | symbol, handler: NodeEventHandler): this;\n  removeListener(eventName: string | symbol, handler: NodeEventHandler): this;\n}\n\nexport type NodeEventHandler = (...args: any[]) => void;\n\n// For APIs that implement `addListener` and `removeListener` methods that may\n// not use the same arguments or return EventEmitter values\n// such as React Native\nexport interface NodeCompatibleEventEmitter {\n  addListener(eventName: string, handler: NodeEventHandler): void | {};\n  removeListener(eventName: string, handler: NodeEventHandler): void | {};\n}\n\n// Use handler types like those in @types/jquery. See:\n// https://github.com/DefinitelyTyped/DefinitelyTyped/blob/847731ba1d7fa6db6b911c0e43aa0afe596e7723/types/jquery/misc.d.ts#L6395\nexport interface JQueryStyleEventEmitter<TContext, T> {\n  on(eventName: string, handler: (this: TContext, t: T, ...args: any[]) => any): void;\n  off(eventName: string, handler: (this: TContext, t: T, ...args: any[]) => any): void;\n}\n\nexport interface EventListenerObject<E> {\n  handleEvent(evt: E): void;\n}\n\nexport interface HasEventTargetAddRemove<E> {\n  addEventListener(\n    type: string,\n    listener: ((evt: E) => void) | EventListenerObject<E> | null,\n    options?: boolean | AddEventListenerOptions\n  ): void;\n  removeEventListener(\n    type: string,\n    listener: ((evt: E) => void) | EventListenerObject<E> | null,\n    options?: EventListenerOptions | boolean\n  ): void;\n}\n\nexport interface EventListenerOptions {\n  capture?: boolean;\n  passive?: boolean;\n  once?: boolean;\n}\n\nexport interface AddEventListenerOptions extends EventListenerOptions {\n  once?: boolean;\n  passive?: boolean;\n}\n\nexport function fromEvent<T>(target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>, eventName: string): Observable<T>;\nexport function fromEvent<T, R>(\n  target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>,\n  eventName: string,\n  resultSelector: (event: T) => R\n): Observable<R>;\nexport function fromEvent<T>(\n  target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>,\n  eventName: string,\n  options: EventListenerOptions\n): Observable<T>;\nexport function fromEvent<T, R>(\n  target: HasEventTargetAddRemove<T> | ArrayLike<HasEventTargetAddRemove<T>>,\n  eventName: string,\n  options: EventListenerOptions,\n  resultSelector: (event: T) => R\n): Observable<R>;\n\nexport function fromEvent(target: NodeStyleEventEmitter | ArrayLike<NodeStyleEventEmitter>, eventName: string): Observable<unknown>;\n/** @deprecated Do not specify explicit type parameters. Signatures with type parameters that cannot be inferred will be removed in v8. */\nexport function fromEvent<T>(target: NodeStyleEventEmitter | ArrayLike<NodeStyleEventEmitter>, eventName: string): Observable<T>;\nexport function fromEvent<R>(\n  target: NodeStyleEventEmitter | ArrayLike<NodeStyleEventEmitter>,\n  eventName: string,\n  resultSelector: (...args: any[]) => R\n): Observable<R>;\n\nexport function fromEvent(\n  target: NodeCompatibleEventEmitter | ArrayLike<NodeCompatibleEventEmitter>,\n  eventName: string\n): Observable<unknown>;\n/** @deprecated Do not specify explicit type parameters. Signatures with type parameters that cannot be inferred will be removed in v8. */\nexport function fromEvent<T>(target: NodeCompatibleEventEmitter | ArrayLike<NodeCompatibleEventEmitter>, eventName: string): Observable<T>;\nexport function fromEvent<R>(\n  target: NodeCompatibleEventEmitter | ArrayLike<NodeCompatibleEventEmitter>,\n  eventName: string,\n  resultSelector: (...args: any[]) => R\n): Observable<R>;\n\nexport function fromEvent<T>(\n  target: JQueryStyleEventEmitter<any, T> | ArrayLike<JQueryStyleEventEmitter<any, T>>,\n  eventName: string\n): Observable<T>;\nexport function fromEvent<T, R>(\n  target: JQueryStyleEventEmitter<any, T> | ArrayLike<JQueryStyleEventEmitter<any, T>>,\n  eventName: string,\n  resultSelector: (value: T, ...args: any[]) => R\n): Observable<R>;\n\n/**\n * Creates an Observable that emits events of a specific type coming from the\n * given event target.\n *\n * <span class=\"informal\">Creates an Observable from DOM events, or Node.js\n * EventEmitter events or others.</span>\n *\n * ![](fromEvent.png)\n *\n * `fromEvent` accepts as a first argument event target, which is an object with methods\n * for registering event handler functions. As a second argument it takes string that indicates\n * type of event we want to listen for. `fromEvent` supports selected types of event targets,\n * which are described in detail below. If your event target does not match any of the ones listed,\n * you should use {@link fromEventPattern}, which can be used on arbitrary APIs.\n * When it comes to APIs supported by `fromEvent`, their methods for adding and removing event\n * handler functions have different names, but they all accept a string describing event type\n * and function itself, which will be called whenever said event happens.\n *\n * Every time resulting Observable is subscribed, event handler function will be registered\n * to event target on given event type. When that event fires, value\n * passed as a first argument to registered function will be emitted by output Observable.\n * When Observable is unsubscribed, function will be unregistered from event target.\n *\n * Note that if event target calls registered function with more than one argument, second\n * and following arguments will not appear in resulting stream. In order to get access to them,\n * you can pass to `fromEvent` optional project function, which will be called with all arguments\n * passed to event handler. Output Observable will then emit value returned by project function,\n * instead of the usual value.\n *\n * Remember that event targets listed below are checked via duck typing. It means that\n * no matter what kind of object you have and no matter what environment you work in,\n * you can safely use `fromEvent` on that object if it exposes described methods (provided\n * of course they behave as was described above). So for example if Node.js library exposes\n * event target which has the same method names as DOM EventTarget, `fromEvent` is still\n * a good choice.\n *\n * If the API you use is more callback then event handler oriented (subscribed\n * callback function fires only once and thus there is no need to manually\n * unregister it), you should use {@link bindCallback} or {@link bindNodeCallback}\n * instead.\n *\n * `fromEvent` supports following types of event targets:\n *\n * **DOM EventTarget**\n *\n * This is an object with `addEventListener` and `removeEventListener` methods.\n *\n * In the browser, `addEventListener` accepts - apart from event type string and event\n * handler function arguments - optional third parameter, which is either an object or boolean,\n * both used for additional configuration how and when passed function will be called. When\n * `fromEvent` is used with event target of that type, you can provide this values\n * as third parameter as well.\n *\n * **Node.js EventEmitter**\n *\n * An object with `addListener` and `removeListener` methods.\n *\n * **JQuery-style event target**\n *\n * An object with `on` and `off` methods\n *\n * **DOM NodeList**\n *\n * List of DOM Nodes, returned for example by `document.querySelectorAll` or `Node.childNodes`.\n *\n * Although this collection is not event target in itself, `fromEvent` will iterate over all Nodes\n * it contains and install event handler function in every of them. When returned Observable\n * is unsubscribed, function will be removed from all Nodes.\n *\n * **DOM HtmlCollection**\n *\n * Just as in case of NodeList it is a collection of DOM nodes. Here as well event handler function is\n * installed and removed in each of elements.\n *\n *\n * ## Examples\n *\n * Emit clicks happening on the DOM document\n *\n * ```ts\n * import { fromEvent } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * clicks.subscribe(x => console.log(x));\n *\n * // Results in:\n * // MouseEvent object logged to console every time a click\n * // occurs on the document.\n * ```\n *\n * Use `addEventListener` with capture option\n *\n * ```ts\n * import { fromEvent } from 'rxjs';\n *\n * const div = document.createElement('div');\n * div.style.cssText = 'width: 200px; height: 200px; background: #09c;';\n * document.body.appendChild(div);\n *\n * // note optional configuration parameter which will be passed to addEventListener\n * const clicksInDocument = fromEvent(document, 'click', { capture: true });\n * const clicksInDiv = fromEvent(div, 'click');\n *\n * clicksInDocument.subscribe(() => console.log('document'));\n * clicksInDiv.subscribe(() => console.log('div'));\n *\n * // By default events bubble UP in DOM tree, so normally\n * // when we would click on div in document\n * // \"div\" would be logged first and then \"document\".\n * // Since we specified optional `capture` option, document\n * // will catch event when it goes DOWN DOM tree, so console\n * // will log \"document\" and then \"div\".\n * ```\n *\n * @see {@link bindCallback}\n * @see {@link bindNodeCallback}\n * @see {@link fromEventPattern}\n *\n * @param {FromEventTarget<T>} target The DOM EventTarget, Node.js\n * EventEmitter, JQuery-like event target, NodeList or HTMLCollection to attach the event handler to.\n * @param {string} eventName The event name of interest, being emitted by the\n * `target`.\n * @param {EventListenerOptions} [options] Options to pass through to addEventListener\n * @return {Observable<T>}\n */\nexport function fromEvent<T>(\n  target: any,\n  eventName: string,\n  options?: EventListenerOptions | ((...args: any[]) => T),\n  resultSelector?: (...args: any[]) => T\n): Observable<T> {\n  if (isFunction(options)) {\n    resultSelector = options;\n    options = undefined;\n  }\n  if (resultSelector) {\n    return fromEvent<T>(target, eventName, options as EventListenerOptions).pipe(mapOneOrManyArgs(resultSelector));\n  }\n\n  // Figure out our add and remove methods. In order to do this,\n  // we are going to analyze the target in a preferred order, if\n  // the target matches a given signature, we take the two \"add\" and \"remove\"\n  // method names and apply them to a map to create opposite versions of the\n  // same function. This is because they all operate in duplicate pairs,\n  // `addListener(name, handler)`, `removeListener(name, handler)`, for example.\n  // The call only differs by method name, as to whether or not you're adding or removing.\n  const [add, remove] =\n    // If it is an EventTarget, we need to use a slightly different method than the other two patterns.\n    isEventTarget(target)\n      ? eventTargetMethods.map((methodName) => (handler: any) => target[methodName](eventName, handler, options as EventListenerOptions))\n      : // In all other cases, the call pattern is identical with the exception of the method names.\n      isNodeStyleEventEmitter(target)\n      ? nodeEventEmitterMethods.map(toCommonHandlerRegistry(target, eventName))\n      : isJQueryStyleEventEmitter(target)\n      ? jqueryMethods.map(toCommonHandlerRegistry(target, eventName))\n      : [];\n\n  // If add is falsy, it's because we didn't match a pattern above.\n  // Check to see if it is an ArrayLike, because if it is, we want to\n  // try to apply fromEvent to all of it's items. We do this check last,\n  // because there are may be some types that are both ArrayLike *and* implement\n  // event registry points, and we'd rather delegate to that when possible.\n  if (!add) {\n    if (isArrayLike(target)) {\n      return mergeMap((subTarget: any) => fromEvent(subTarget, eventName, options as EventListenerOptions))(\n        innerFrom(target)\n      ) as Observable<T>;\n    }\n  }\n\n  // If add is falsy and we made it here, it's because we didn't\n  // match any valid target objects above.\n  if (!add) {\n    throw new TypeError('Invalid event target');\n  }\n\n  return new Observable<T>((subscriber) => {\n    // The handler we are going to register. Forwards the event object, by itself, or\n    // an array of arguments to the event handler, if there is more than one argument,\n    // to the consumer.\n    const handler = (...args: any[]) => subscriber.next(1 < args.length ? args : args[0]);\n    // Do the work of adding the handler to the target.\n    add(handler);\n    // When we finalize, we want to remove the handler and free up memory.\n    return () => remove!(handler);\n  });\n}\n\n/**\n * Used to create `add` and `remove` functions to register and unregister event handlers\n * from a target in the most common handler pattern, where there are only two arguments.\n * (e.g.  `on(name, fn)`, `off(name, fn)`, `addListener(name, fn)`, or `removeListener(name, fn)`)\n * @param target The target we're calling methods on\n * @param eventName The event name for the event we're creating register or unregister functions for\n */\nfunction toCommonHandlerRegistry(target: any, eventName: string) {\n  return (methodName: string) => (handler: any) => target[methodName](eventName, handler);\n}\n\n/**\n * Checks to see if the target implements the required node-style EventEmitter methods\n * for adding and removing event handlers.\n * @param target the object to check\n */\nfunction isNodeStyleEventEmitter(target: any): target is NodeStyleEventEmitter {\n  return isFunction(target.addListener) && isFunction(target.removeListener);\n}\n\n/**\n * Checks to see if the target implements the required jQuery-style EventEmitter methods\n * for adding and removing event handlers.\n * @param target the object to check\n */\nfunction isJQueryStyleEventEmitter(target: any): target is JQueryStyleEventEmitter<any, any> {\n  return isFunction(target.on) && isFunction(target.off);\n}\n\n/**\n * Checks to see if the target implements the required EventTarget methods\n * for adding and removing event handlers.\n * @param target the object to check\n */\nfunction isEventTarget(target: any): target is HasEventTargetAddRemove<any> {\n  return isFunction(target.addEventListener) && isFunction(target.removeEventListener);\n}\n", "import { Observable } from '../Observable';\nimport { isFunction } from '../util/isFunction';\nimport { NodeEventHandler } from './fromEvent';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\n\n/* tslint:disable:max-line-length */\nexport function fromEventPattern<T>(\n  addHandler: (handler: NodeEventHandler) => any,\n  removeHandler?: (handler: NodeEventHandler, signal?: any) => void\n): Observable<T>;\nexport function fromEventPattern<T>(\n  addHandler: (handler: NodeEventHandler) => any,\n  removeHandler?: (handler: NodeEventHandler, signal?: any) => void,\n  resultSelector?: (...args: any[]) => T\n): Observable<T>;\n/* tslint:enable:max-line-length */\n\n/**\n * Creates an Observable from an arbitrary API for registering event handlers.\n *\n * <span class=\"informal\">When that method for adding event handler was something {@link fromEvent}\n * was not prepared for.</span>\n *\n * ![](fromEventPattern.png)\n *\n * `fromEventPattern` allows you to convert into an Observable any API that supports registering handler functions\n * for events. It is similar to {@link fromEvent}, but far\n * more flexible. In fact, all use cases of {@link fromEvent} could be easily handled by\n * `fromEventPattern` (although in slightly more verbose way).\n *\n * This operator accepts as a first argument an `addHandler` function, which will be injected with\n * handler parameter. That handler is actually an event handler function that you now can pass\n * to API expecting it. `addHandler` will be called whenever Observable\n * returned by the operator is subscribed, so registering handler in API will not\n * necessarily happen when `fromEventPattern` is called.\n *\n * After registration, every time an event that we listen to happens,\n * Observable returned by `fromEventPattern` will emit value that event handler\n * function was called with. Note that if event handler was called with more\n * than one argument, second and following arguments will not appear in the Observable.\n *\n * If API you are using allows to unregister event handlers as well, you can pass to `fromEventPattern`\n * another function - `removeHandler` - as a second parameter. It will be injected\n * with the same handler function as before, which now you can use to unregister\n * it from the API. `removeHandler` will be called when consumer of resulting Observable\n * unsubscribes from it.\n *\n * In some APIs unregistering is actually handled differently. Method registering an event handler\n * returns some kind of token, which is later used to identify which function should\n * be unregistered or it itself has method that unregisters event handler.\n * If that is the case with your API, make sure token returned\n * by registering method is returned by `addHandler`. Then it will be passed\n * as a second argument to `removeHandler`, where you will be able to use it.\n *\n * If you need access to all event handler parameters (not only the first one),\n * or you need to transform them in any way, you can call `fromEventPattern` with optional\n * third parameter - project function which will accept all arguments passed to\n * event handler when it is called. Whatever is returned from project function will appear on\n * resulting stream instead of usual event handlers first argument. This means\n * that default project can be thought of as function that takes its first parameter\n * and ignores the rest.\n *\n * ## Examples\n *\n * Emits clicks happening on the DOM document\n *\n * ```ts\n * import { fromEventPattern } from 'rxjs';\n *\n * function addClickHandler(handler) {\n *   document.addEventListener('click', handler);\n * }\n *\n * function removeClickHandler(handler) {\n *   document.removeEventListener('click', handler);\n * }\n *\n * const clicks = fromEventPattern(\n *   addClickHandler,\n *   removeClickHandler\n * );\n * clicks.subscribe(x => console.log(x));\n *\n * // Whenever you click anywhere in the browser, DOM MouseEvent\n * // object will be logged.\n * ```\n *\n * Use with API that returns cancellation token\n *\n * ```ts\n * import { fromEventPattern } from 'rxjs';\n *\n * const token = someAPI.registerEventHandler(function() {});\n * someAPI.unregisterEventHandler(token); // this APIs cancellation method accepts\n *                                        // not handler itself, but special token.\n *\n * const someAPIObservable = fromEventPattern(\n *   function(handler) { return someAPI.registerEventHandler(handler); }, // Note that we return the token here...\n *   function(handler, token) { someAPI.unregisterEventHandler(token); }  // ...to then use it here.\n * );\n * ```\n *\n * Use with project function\n *\n * ```ts\n * import { fromEventPattern } from 'rxjs';\n *\n * someAPI.registerEventHandler((eventType, eventMessage) => {\n *   console.log(eventType, eventMessage); // Logs 'EVENT_TYPE' 'EVENT_MESSAGE' to console.\n * });\n *\n * const someAPIObservable = fromEventPattern(\n *   handler => someAPI.registerEventHandler(handler),\n *   handler => someAPI.unregisterEventHandler(handler)\n *   (eventType, eventMessage) => eventType + ' --- ' + eventMessage // without that function only 'EVENT_TYPE'\n * );                                                                // would be emitted by the Observable\n *\n * someAPIObservable.subscribe(value => console.log(value));\n *\n * // Logs:\n * // 'EVENT_TYPE --- EVENT_MESSAGE'\n * ```\n *\n * @see {@link fromEvent}\n * @see {@link bindCallback}\n * @see {@link bindNodeCallback}\n *\n * @param {function(handler: Function): any} addHandler A function that takes\n * a `handler` function as argument and attaches it somehow to the actual\n * source of events.\n * @param {function(handler: Function, token?: any): void} [removeHandler] A function that\n * takes a `handler` function as an argument and removes it from the event source. If `addHandler`\n * returns some kind of token, `removeHandler` function will have it as a second parameter.\n * @param {function(...args: any): T} [project] A function to\n * transform results. It takes the arguments from the event handler and\n * should return a single value.\n * @return {Observable<T>} Observable which, when an event happens, emits first parameter\n * passed to registered event handler. Alternatively it emits whatever project function returns\n * at that moment.\n */\nexport function fromEventPattern<T>(\n  addHandler: (handler: NodeEventHandler) => any,\n  removeHandler?: (handler: NodeEventHandler, signal?: any) => void,\n  resultSelector?: (...args: any[]) => T\n): Observable<T | T[]> {\n  if (resultSelector) {\n    return fromEventPattern<T>(addHandler, removeHandler).pipe(mapOneOrManyArgs(resultSelector));\n  }\n\n  return new Observable<T | T[]>((subscriber) => {\n    const handler = (...e: T[]) => subscriber.next(e.length === 1 ? e[0] : e);\n    const retValue = addHandler(handler);\n    return isFunction(removeHandler) ? () => removeHandler(handler, retValue) : undefined;\n  });\n}\n", "import { Observable } from '../Observable';\nimport { SchedulerLike } from '../types';\nimport { async as asyncScheduler } from '../scheduler/async';\nimport { isScheduler } from '../util/isScheduler';\nimport { isValidDate } from '../util/isDate';\n\n/**\n * Creates an observable that will wait for a specified time period, or exact date, before\n * emitting the number 0.\n *\n * <span class=\"informal\">Used to emit a notification after a delay.</span>\n *\n * This observable is useful for creating delays in code, or racing against other values\n * for ad-hoc timeouts.\n *\n * The `delay` is specified by default in milliseconds, however providing a custom scheduler could\n * create a different behavior.\n *\n * ## Examples\n *\n * Wait 3 seconds and start another observable\n *\n * You might want to use `timer` to delay subscription to an\n * observable by a set amount of time. Here we use a timer with\n * {@link concatMapTo} or {@link concatMap} in order to wait\n * a few seconds and start a subscription to a source.\n *\n * ```ts\n * import { of, timer, concatMap } from 'rxjs';\n *\n * // This could be any observable\n * const source = of(1, 2, 3);\n *\n * timer(3000)\n *   .pipe(concatMap(() => source))\n *   .subscribe(console.log);\n * ```\n *\n * Take all values until the start of the next minute\n *\n * Using a `Date` as the trigger for the first emission, you can\n * do things like wait until midnight to fire an event, or in this case,\n * wait until a new minute starts (chosen so the example wouldn't take\n * too long to run) in order to stop watching a stream. Leveraging\n * {@link takeUntil}.\n *\n * ```ts\n * import { interval, takeUntil, timer } from 'rxjs';\n *\n * // Build a Date object that marks the\n * // next minute.\n * const currentDate = new Date();\n * const startOfNextMinute = new Date(\n *   currentDate.getFullYear(),\n *   currentDate.getMonth(),\n *   currentDate.getDate(),\n *   currentDate.getHours(),\n *   currentDate.getMinutes() + 1\n * );\n *\n * // This could be any observable stream\n * const source = interval(1000);\n *\n * const result = source.pipe(\n *   takeUntil(timer(startOfNextMinute))\n * );\n *\n * result.subscribe(console.log);\n * ```\n *\n * ### Known Limitations\n *\n * - The {@link asyncScheduler} uses `setTimeout` which has limitations for how far in the future it can be scheduled.\n *\n * - If a `scheduler` is provided that returns a timestamp other than an epoch from `now()`, and\n * a `Date` object is passed to the `dueTime` argument, the calculation for when the first emission\n * should occur will be incorrect. In this case, it would be best to do your own calculations\n * ahead of time, and pass a `number` in as the `dueTime`.\n *\n * @param due If a `number`, the amount of time in milliseconds to wait before emitting.\n * If a `Date`, the exact time at which to emit.\n * @param scheduler The scheduler to use to schedule the delay. Defaults to {@link asyncScheduler}.\n */\nexport function timer(due: number | Date, scheduler?: SchedulerLike): Observable<0>;\n\n/**\n * Creates an observable that starts an interval after a specified delay, emitting incrementing numbers -- starting at `0` --\n * on each interval after words.\n *\n * The `delay` and `intervalDuration` are specified by default in milliseconds, however providing a custom scheduler could\n * create a different behavior.\n *\n * ## Example\n *\n * ### Start an interval that starts right away\n *\n * Since {@link interval} waits for the passed delay before starting,\n * sometimes that's not ideal. You may want to start an interval immediately.\n * `timer` works well for this. Here we have both side-by-side so you can\n * see them in comparison.\n *\n * Note that this observable will never complete.\n *\n * ```ts\n * import { timer, interval } from 'rxjs';\n *\n * timer(0, 1000).subscribe(n => console.log('timer', n));\n * interval(1000).subscribe(n => console.log('interval', n));\n * ```\n *\n * ### Known Limitations\n *\n * - The {@link asyncScheduler} uses `setTimeout` which has limitations for how far in the future it can be scheduled.\n *\n * - If a `scheduler` is provided that returns a timestamp other than an epoch from `now()`, and\n * a `Date` object is passed to the `dueTime` argument, the calculation for when the first emission\n * should occur will be incorrect. In this case, it would be best to do your own calculations\n * ahead of time, and pass a `number` in as the `startDue`.\n * @param startDue If a `number`, is the time to wait before starting the interval.\n * If a `Date`, is the exact time at which to start the interval.\n * @param intervalDuration The delay between each value emitted in the interval. Passing a\n * negative number here will result in immediate completion after the first value is emitted, as though\n * no `intervalDuration` was passed at all.\n * @param scheduler The scheduler to use to schedule the delay. Defaults to {@link asyncScheduler}.\n */\nexport function timer(startDue: number | Date, intervalDuration: number, scheduler?: SchedulerLike): Observable<number>;\n\n/**\n * @deprecated The signature allowing `undefined` to be passed for `intervalDuration` will be removed in v8. Use the `timer(dueTime, scheduler?)` signature instead.\n */\nexport function timer(dueTime: number | Date, unused: undefined, scheduler?: SchedulerLike): Observable<0>;\n\nexport function timer(\n  dueTime: number | Date = 0,\n  intervalOrScheduler?: number | SchedulerLike,\n  scheduler: SchedulerLike = asyncScheduler\n): Observable<number> {\n  // Since negative intervalDuration is treated as though no\n  // interval was specified at all, we start with a negative number.\n  let intervalDuration = -1;\n\n  if (intervalOrScheduler != null) {\n    // If we have a second argument, and it's a scheduler,\n    // override the scheduler we had defaulted. Otherwise,\n    // it must be an interval.\n    if (isScheduler(intervalOrScheduler)) {\n      scheduler = intervalOrScheduler;\n    } else {\n      // Note that this *could* be negative, in which case\n      // it's like not passing an intervalDuration at all.\n      intervalDuration = intervalOrScheduler;\n    }\n  }\n\n  return new Observable((subscriber) => {\n    // If a valid date is passed, calculate how long to wait before\n    // executing the first value... otherwise, if it's a number just schedule\n    // that many milliseconds (or scheduler-specified unit size) in the future.\n    let due = isValidDate(dueTime) ? +dueTime - scheduler!.now() : dueTime;\n\n    if (due < 0) {\n      // Ensure we don't schedule in the future.\n      due = 0;\n    }\n\n    // The incrementing value we emit.\n    let n = 0;\n\n    // Start the timer.\n    return scheduler.schedule(function () {\n      if (!subscriber.closed) {\n        // Emit the next value and increment.\n        subscriber.next(n++);\n\n        if (0 <= intervalDuration) {\n          // If we have a interval after the initial timer,\n          // reschedule with the period.\n          this.schedule(undefined, intervalDuration);\n        } else {\n          // We didn't have an interval. So just complete.\n          subscriber.complete();\n        }\n      }\n    }, due);\n  });\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInput, ObservableInputTuple, SchedulerLike } from '../types';\nimport { mergeAll } from '../operators/mergeAll';\nimport { innerFrom } from './innerFrom';\nimport { EMPTY } from './empty';\nimport { popNumber, popScheduler } from '../util/args';\nimport { from } from './from';\n\nexport function merge<A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): Observable<A[number]>;\nexport function merge<A extends readonly unknown[]>(...sourcesAndConcurrency: [...ObservableInputTuple<A>, number?]): Observable<A[number]>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `mergeAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function merge<A extends readonly unknown[]>(\n  ...sourcesAndScheduler: [...ObservableInputTuple<A>, SchedulerLike?]\n): Observable<A[number]>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `mergeAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function merge<A extends readonly unknown[]>(\n  ...sourcesAndConcurrencyAndScheduler: [...ObservableInputTuple<A>, number?, SchedulerLike?]\n): Observable<A[number]>;\n\n/**\n * Creates an output Observable which concurrently emits all values from every\n * given input Observable.\n *\n * <span class=\"informal\">Flattens multiple Observables together by blending\n * their values into one Observable.</span>\n *\n * ![](merge.png)\n *\n * `merge` subscribes to each given input Observable (as arguments), and simply\n * forwards (without doing any transformation) all the values from all the input\n * Observables to the output Observable. The output Observable only completes\n * once all input Observables have completed. Any error delivered by an input\n * Observable will be immediately emitted on the output Observable.\n *\n * ## Examples\n *\n * Merge together two Observables: 1s interval and clicks\n *\n * ```ts\n * import { merge, fromEvent, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const timer = interval(1000);\n * const clicksOrTimer = merge(clicks, timer);\n * clicksOrTimer.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // timer will emit ascending values, one every second(1000ms) to console\n * // clicks logs MouseEvents to console every time the \"document\" is clicked\n * // Since the two streams are merged you see these happening\n * // as they occur.\n * ```\n *\n * Merge together 3 Observables, but run only 2 concurrently\n *\n * ```ts\n * import { interval, take, merge } from 'rxjs';\n *\n * const timer1 = interval(1000).pipe(take(10));\n * const timer2 = interval(2000).pipe(take(6));\n * const timer3 = interval(500).pipe(take(10));\n *\n * const concurrent = 2; // the argument\n * const merged = merge(timer1, timer2, timer3, concurrent);\n * merged.subscribe(x => console.log(x));\n *\n * // Results in the following:\n * // - First timer1 and timer2 will run concurrently\n * // - timer1 will emit a value every 1000ms for 10 iterations\n * // - timer2 will emit a value every 2000ms for 6 iterations\n * // - after timer1 hits its max iteration, timer2 will\n * //   continue, and timer3 will start to run concurrently with timer2\n * // - when timer2 hits its max iteration it terminates, and\n * //   timer3 will continue to emit a value every 500ms until it is complete\n * ```\n *\n * @see {@link mergeAll}\n * @see {@link mergeMap}\n * @see {@link mergeMapTo}\n * @see {@link mergeScan}\n *\n * @param {...ObservableInput} observables Input Observables to merge together.\n * @param {number} [concurrent=Infinity] Maximum number of input\n * Observables being subscribed to concurrently.\n * @param {SchedulerLike} [scheduler=null] The {@link SchedulerLike} to use for managing\n * concurrency of input Observables.\n * @return {Observable} an Observable that emits items that are the result of\n * every input Observable.\n */\nexport function merge(...args: (ObservableInput<unknown> | number | SchedulerLike)[]): Observable<unknown> {\n  const scheduler = popScheduler(args);\n  const concurrent = popNumber(args, Infinity);\n  const sources = args as ObservableInput<unknown>[];\n  return !sources.length\n    ? // No source provided\n      EMPTY\n    : sources.length === 1\n    ? // One source? Just return it.\n      innerFrom(sources[0])\n    : // Merge all sources\n      mergeAll(concurrent)(from(sources, scheduler));\n}\n", "import { Observable } from '../Observable';\nimport { noop } from '../util/noop';\n\n/**\n * An Observable that emits no items to the Observer and never completes.\n *\n * ![](never.png)\n *\n * A simple Observable that emits neither values nor errors nor the completion\n * notification. It can be used for testing purposes or for composing with other\n * Observables. Please note that by never emitting a complete notification, this\n * Observable keeps the subscription from being disposed automatically.\n * Subscriptions need to be manually disposed.\n *\n * ##  Example\n *\n * Emit the number 7, then never emit anything else (not even complete)\n *\n * ```ts\n * import { NEVER, startWith } from 'rxjs';\n *\n * const info = () => console.log('Will not be called');\n *\n * const result = NEVER.pipe(startWith(7));\n * result.subscribe({\n *   next: x => console.log(x),\n *   error: info,\n *   complete: info\n * });\n * ```\n *\n * @see {@link Observable}\n * @see {@link EMPTY}\n * @see {@link of}\n * @see {@link throwError}\n */\nexport const NEVER = new Observable<never>(noop);\n\n/**\n * @deprecated Replaced with the {@link NEVER} constant. Will be removed in v8.\n */\nexport function never() {\n  return NEVER;\n}\n", "const { isArray } = Array;\n\n/**\n * Used in operators and functions that accept either a list of arguments, or an array of arguments\n * as a single argument.\n */\nexport function argsOrArgArray<T>(args: (T | T[])[]): T[] {\n  return args.length === 1 && isArray(args[0]) ? args[0] : (args as T[]);\n}\n", "import { OperatorFunction, MonoTypeOperatorFunction, TruthyTypesOf } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/** @deprecated Use a closure instead of a `thisArg`. Signatures accepting a `thisArg` will be removed in v8. */\nexport function filter<T, S extends T, A>(predicate: (this: A, value: T, index: number) => value is S, thisArg: A): OperatorFunction<T, S>;\nexport function filter<T, S extends T>(predicate: (value: T, index: number) => value is S): OperatorFunction<T, S>;\nexport function filter<T>(predicate: BooleanConstructor): OperatorFunction<T, TruthyTypesOf<T>>;\n/** @deprecated Use a closure instead of a `thisArg`. Signatures accepting a `thisArg` will be removed in v8. */\nexport function filter<T, A>(predicate: (this: A, value: T, index: number) => boolean, thisArg: A): MonoTypeOperatorFunction<T>;\nexport function filter<T>(predicate: (value: T, index: number) => boolean): MonoTypeOperatorFunction<T>;\n\n/**\n * Filter items emitted by the source Observable by only emitting those that\n * satisfy a specified predicate.\n *\n * <span class=\"informal\">Like\n * [Array.prototype.filter()](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/filter),\n * it only emits a value from the source if it passes a criterion function.</span>\n *\n * ![](filter.png)\n *\n * Similar to the well-known `Array.prototype.filter` method, this operator\n * takes values from the source Observable, passes them through a `predicate`\n * function and only emits those values that yielded `true`.\n *\n * ## Example\n *\n * Emit only click events whose target was a DIV element\n *\n * ```ts\n * import { fromEvent, filter } from 'rxjs';\n *\n * const div = document.createElement('div');\n * div.style.cssText = 'width: 200px; height: 200px; background: #09c;';\n * document.body.appendChild(div);\n *\n * const clicks = fromEvent(document, 'click');\n * const clicksOnDivs = clicks.pipe(filter(ev => (<HTMLElement>ev.target).tagName === 'DIV'));\n * clicksOnDivs.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link distinct}\n * @see {@link distinctUntilChanged}\n * @see {@link distinctUntilKeyChanged}\n * @see {@link ignoreElements}\n * @see {@link partition}\n * @see {@link skip}\n *\n * @param predicate A function that\n * evaluates each value emitted by the source Observable. If it returns `true`,\n * the value is emitted, if `false` the value is not passed to the output\n * Observable. The `index` parameter is the number `i` for the i-th source\n * emission that has happened since the subscription, starting from the number\n * `0`.\n * @param thisArg An optional argument to determine the value of `this`\n * in the `predicate` function.\n * @return A function that returns an Observable that emits items from the\n * source Observable that satisfy the specified `predicate`.\n */\nexport function filter<T>(predicate: (value: T, index: number) => boolean, thisArg?: any): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    // An index passed to our predicate function on each call.\n    let index = 0;\n\n    // Subscribe to the source, all errors and completions are\n    // forwarded to the consumer.\n    source.subscribe(\n      // Call the predicate with the appropriate `this` context,\n      // if the predicate returns `true`, then send the value\n      // to the consumer.\n      createOperatorSubscriber(subscriber, (value) => predicate.call(thisArg, value, index++) && subscriber.next(value))\n    );\n  });\n}\n", "import { Observable } from '../Observable';\nimport { ObservableInputTuple } from '../types';\nimport { innerFrom } from './innerFrom';\nimport { argsOrArgArray } from '../util/argsOrArgArray';\nimport { EMPTY } from './empty';\nimport { createOperatorSubscriber } from '../operators/OperatorSubscriber';\nimport { popResultSelector } from '../util/args';\n\nexport function zip<A extends readonly unknown[]>(sources: [...ObservableInputTuple<A>]): Observable<A>;\nexport function zip<A extends readonly unknown[], R>(\n  sources: [...ObservableInputTuple<A>],\n  resultSelector: (...values: A) => R\n): Observable<R>;\nexport function zip<A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): Observable<A>;\nexport function zip<A extends readonly unknown[], R>(\n  ...sourcesAndResultSelector: [...ObservableInputTuple<A>, (...values: A) => R]\n): Observable<R>;\n\n/**\n * Combines multiple Observables to create an Observable whose values are calculated from the values, in order, of each\n * of its input Observables.\n *\n * If the last parameter is a function, this function is used to compute the created value from the input values.\n * Otherwise, an array of the input values is returned.\n *\n * ## Example\n *\n * Combine age and name from different sources\n *\n * ```ts\n * import { of, zip, map } from 'rxjs';\n *\n * const age$ = of(27, 25, 29);\n * const name$ = of('Foo', 'Bar', 'Beer');\n * const isDev$ = of(true, true, false);\n *\n * zip(age$, name$, isDev$).pipe(\n *   map(([age, name, isDev]) => ({ age, name, isDev }))\n * )\n * .subscribe(x => console.log(x));\n *\n * // Outputs\n * // { age: 27, name: 'Foo', isDev: true }\n * // { age: 25, name: 'Bar', isDev: true }\n * // { age: 29, name: 'Beer', isDev: false }\n * ```\n *\n * @param sources\n * @return {Observable<R>}\n */\nexport function zip(...args: unknown[]): Observable<unknown> {\n  const resultSelector = popResultSelector(args);\n\n  const sources = argsOrArgArray(args) as Observable<unknown>[];\n\n  return sources.length\n    ? new Observable<unknown[]>((subscriber) => {\n        // A collection of buffers of values from each source.\n        // Keyed by the same index with which the sources were passed in.\n        let buffers: unknown[][] = sources.map(() => []);\n\n        // An array of flags of whether or not the sources have completed.\n        // This is used to check to see if we should complete the result.\n        // Keyed by the same index with which the sources were passed in.\n        let completed = sources.map(() => false);\n\n        // When everything is done, release the arrays above.\n        subscriber.add(() => {\n          buffers = completed = null!;\n        });\n\n        // Loop over our sources and subscribe to each one. The index `i` is\n        // especially important here, because we use it in closures below to\n        // access the related buffers and completion properties\n        for (let sourceIndex = 0; !subscriber.closed && sourceIndex < sources.length; sourceIndex++) {\n          innerFrom(sources[sourceIndex]).subscribe(\n            createOperatorSubscriber(\n              subscriber,\n              (value) => {\n                buffers[sourceIndex].push(value);\n                // if every buffer has at least one value in it, then we\n                // can shift out the oldest value from each buffer and emit\n                // them as an array.\n                if (buffers.every((buffer) => buffer.length)) {\n                  const result: any = buffers.map((buffer) => buffer.shift()!);\n                  // Emit the array. If theres' a result selector, use that.\n                  subscriber.next(resultSelector ? resultSelector(...result) : result);\n                  // If any one of the sources is both complete and has an empty buffer\n                  // then we complete the result. This is because we cannot possibly have\n                  // any more values to zip together.\n                  if (buffers.some((buffer, i) => !buffer.length && completed[i])) {\n                    subscriber.complete();\n                  }\n                }\n              },\n              () => {\n                // This source completed. Mark it as complete so we can check it later\n                // if we have to.\n                completed[sourceIndex] = true;\n                // But, if this complete source has nothing in its buffer, then we\n                // can complete the result, because we can't possibly have any more\n                // values from this to zip together with the other values.\n                !buffers[sourceIndex].length && subscriber.complete();\n              }\n            )\n          );\n        }\n\n        // When everything is done, release the arrays above.\n        return () => {\n          buffers = completed = null!;\n        };\n      })\n    : EMPTY;\n}\n", "import { Subscriber } from '../Subscriber';\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\n\nimport { operate } from '../util/lift';\nimport { innerFrom } from '../observable/innerFrom';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Ignores source values for a duration determined by another Observable, then\n * emits the most recent value from the source Observable, then repeats this\n * process.\n *\n * <span class=\"informal\">It's like {@link auditTime}, but the silencing\n * duration is determined by a second Observable.</span>\n *\n * ![](audit.svg)\n *\n * `audit` is similar to `throttle`, but emits the last value from the silenced\n * time window, instead of the first value. `audit` emits the most recent value\n * from the source Observable on the output Observable as soon as its internal\n * timer becomes disabled, and ignores source values while the timer is enabled.\n * Initially, the timer is disabled. As soon as the first source value arrives,\n * the timer is enabled by calling the `durationSelector` function with the\n * source value, which returns the \"duration\" Observable. When the duration\n * Observable emits a value, the timer is disabled, then the most\n * recent source value is emitted on the output Observable, and this process\n * repeats for the next source value.\n *\n * ## Example\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, audit, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(audit(ev => interval(1000)));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link auditTime}\n * @see {@link debounce}\n * @see {@link delayWhen}\n * @see {@link sample}\n * @see {@link throttle}\n *\n * @param durationSelector A function\n * that receives a value from the source Observable, for computing the silencing\n * duration, returned as an Observable or a Promise.\n * @return A function that returns an Observable that performs rate-limiting of\n * emissions from the source Observable.\n */\nexport function audit<T>(durationSelector: (value: T) => ObservableInput<any>): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let hasValue = false;\n    let lastValue: T | null = null;\n    let durationSubscriber: Subscriber<any> | null = null;\n    let isComplete = false;\n\n    const endDuration = () => {\n      durationSubscriber?.unsubscribe();\n      durationSubscriber = null;\n      if (hasValue) {\n        hasValue = false;\n        const value = lastValue!;\n        lastValue = null;\n        subscriber.next(value);\n      }\n      isComplete && subscriber.complete();\n    };\n\n    const cleanupDuration = () => {\n      durationSubscriber = null;\n      isComplete && subscriber.complete();\n    };\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          hasValue = true;\n          lastValue = value;\n          if (!durationSubscriber) {\n            innerFrom(durationSelector(value)).subscribe(\n              (durationSubscriber = createOperatorSubscriber(subscriber, endDuration, cleanupDuration))\n            );\n          }\n        },\n        () => {\n          isComplete = true;\n          (!hasValue || !durationSubscriber || durationSubscriber.closed) && subscriber.complete();\n        }\n      )\n    );\n  });\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { audit } from './audit';\nimport { timer } from '../observable/timer';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\n\n/**\n * Ignores source values for `duration` milliseconds, then emits the most recent\n * value from the source Observable, then repeats this process.\n *\n * <span class=\"informal\">When it sees a source value, it ignores that plus\n * the next ones for `duration` milliseconds, and then it emits the most recent\n * value from the source.</span>\n *\n * ![](auditTime.png)\n *\n * `auditTime` is similar to `throttleTime`, but emits the last value from the\n * silenced time window, instead of the first value. `auditTime` emits the most\n * recent value from the source Observable on the output Observable as soon as\n * its internal timer becomes disabled, and ignores source values while the\n * timer is enabled. Initially, the timer is disabled. As soon as the first\n * source value arrives, the timer is enabled. After `duration` milliseconds (or\n * the time unit determined internally by the optional `scheduler`) has passed,\n * the timer is disabled, then the most recent source value is emitted on the\n * output Observable, and this process repeats for the next source value.\n * Optionally takes a {@link SchedulerLike} for managing timers.\n *\n * ## Example\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, auditTime } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(auditTime(1000));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link audit}\n * @see {@link debounceTime}\n * @see {@link delay}\n * @see {@link sampleTime}\n * @see {@link throttleTime}\n *\n * @param {number} duration Time to wait before emitting the most recent source\n * value, measured in milliseconds or the time unit determined internally\n * by the optional `scheduler`.\n * @param {SchedulerLike} [scheduler=async] The {@link SchedulerLike} to use for\n * managing the timers that handle the rate-limiting behavior.\n * @return A function that returns an Observable that performs rate-limiting of\n * emissions from the source Observable.\n */\nexport function auditTime<T>(duration: number, scheduler: SchedulerLike = asyncScheduler): MonoTypeOperatorFunction<T> {\n  return audit(() => timer(duration, scheduler));\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { arrRemove } from '../util/arrRemove';\n\n/**\n * Buffers the source Observable values until the size hits the maximum\n * `bufferSize` given.\n *\n * <span class=\"informal\">Collects values from the past as an array, and emits\n * that array only when its size reaches `bufferSize`.</span>\n *\n * ![](bufferCount.png)\n *\n * Buffers a number of values from the source Observable by `bufferSize` then\n * emits the buffer and clears it, and starts a new buffer each\n * `startBufferEvery` values. If `startBufferEvery` is not provided or is\n * `null`, then new buffers are started immediately at the start of the source\n * and when each buffer closes and is emitted.\n *\n * ## Examples\n *\n * Emit the last two click events as an array\n *\n * ```ts\n * import { fromEvent, bufferCount } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const buffered = clicks.pipe(bufferCount(2));\n * buffered.subscribe(x => console.log(x));\n * ```\n *\n * On every click, emit the last two click events as an array\n *\n * ```ts\n * import { fromEvent, bufferCount } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const buffered = clicks.pipe(bufferCount(2, 1));\n * buffered.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link buffer}\n * @see {@link bufferTime}\n * @see {@link bufferToggle}\n * @see {@link bufferWhen}\n * @see {@link pairwise}\n * @see {@link windowCount}\n *\n * @param {number} bufferSize The maximum size of the buffer emitted.\n * @param {number} [startBufferEvery] Interval at which to start a new buffer.\n * For example if `startBufferEvery` is `2`, then a new buffer will be started\n * on every other value from the source. A new buffer is started at the\n * beginning of the source by default.\n * @return A function that returns an Observable of arrays of buffered values.\n */\nexport function bufferCount<T>(bufferSize: number, startBufferEvery: number | null = null): OperatorFunction<T, T[]> {\n  // If no `startBufferEvery` value was supplied, then we're\n  // opening and closing on the bufferSize itself.\n  startBufferEvery = startBufferEvery ?? bufferSize;\n\n  return operate((source, subscriber) => {\n    let buffers: T[][] = [];\n    let count = 0;\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          let toEmit: T[][] | null = null;\n\n          // Check to see if we need to start a buffer.\n          // This will start one at the first value, and then\n          // a new one every N after that.\n          if (count++ % startBufferEvery! === 0) {\n            buffers.push([]);\n          }\n\n          // Push our value into our active buffers.\n          for (const buffer of buffers) {\n            buffer.push(value);\n            // Check to see if we're over the bufferSize\n            // if we are, record it so we can emit it later.\n            // If we emitted it now and removed it, it would\n            // mutate the `buffers` array while we're looping\n            // over it.\n            if (bufferSize <= buffer.length) {\n              toEmit = toEmit ?? [];\n              toEmit.push(buffer);\n            }\n          }\n\n          if (toEmit) {\n            // We have found some buffers that are over the\n            // `bufferSize`. Emit them, and remove them from our\n            // buffers list.\n            for (const buffer of toEmit) {\n              arrRemove(buffers, buffer);\n              subscriber.next(buffer);\n            }\n          }\n        },\n        () => {\n          // When the source completes, emit all of our\n          // active buffers.\n          for (const buffer of buffers) {\n            subscriber.next(buffer);\n          }\n          subscriber.complete();\n        },\n        // Pass all errors through to consumer.\n        undefined,\n        () => {\n          // Clean up our memory when we finalize\n          buffers = null!;\n        }\n      )\n    );\n  });\n}\n", "import { Observable } from '../Observable';\n\nimport { ObservableInput, OperatorFunction, ObservedValueOf } from '../types';\nimport { Subscription } from '../Subscription';\nimport { innerFrom } from '../observable/innerFrom';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { operate } from '../util/lift';\n\n/* tslint:disable:max-line-length */\nexport function catchError<T, O extends ObservableInput<any>>(\n  selector: (err: any, caught: Observable<T>) => O\n): OperatorFunction<T, T | ObservedValueOf<O>>;\n/* tslint:enable:max-line-length */\n\n/**\n * Catches errors on the observable to be handled by returning a new observable or throwing an error.\n *\n * <span class=\"informal\">\n * It only listens to the error channel and ignores notifications.\n * Handles errors from the source observable, and maps them to a new observable.\n * The error may also be rethrown, or a new error can be thrown to emit an error from the result.\n * </span>\n *\n * ![](catch.png)\n *\n * This operator handles errors, but forwards along all other events to the resulting observable.\n * If the source observable terminates with an error, it will map that error to a new observable,\n * subscribe to it, and forward all of its events to the resulting observable.\n *\n * ## Examples\n *\n * Continue with a different Observable when there's an error\n *\n * ```ts\n * import { of, map, catchError } from 'rxjs';\n *\n * of(1, 2, 3, 4, 5)\n *   .pipe(\n *     map(n => {\n *       if (n === 4) {\n *         throw 'four!';\n *       }\n *       return n;\n *     }),\n *     catchError(err => of('I', 'II', 'III', 'IV', 'V'))\n *   )\n *   .subscribe(x => console.log(x));\n *   // 1, 2, 3, I, II, III, IV, V\n * ```\n *\n * Retry the caught source Observable again in case of error, similar to `retry()` operator\n *\n * ```ts\n * import { of, map, catchError, take } from 'rxjs';\n *\n * of(1, 2, 3, 4, 5)\n *   .pipe(\n *     map(n => {\n *       if (n === 4) {\n *         throw 'four!';\n *       }\n *       return n;\n *     }),\n *     catchError((err, caught) => caught),\n *     take(30)\n *   )\n *   .subscribe(x => console.log(x));\n *   // 1, 2, 3, 1, 2, 3, ...\n * ```\n *\n * Throw a new error when the source Observable throws an error\n *\n * ```ts\n * import { of, map, catchError } from 'rxjs';\n *\n * of(1, 2, 3, 4, 5)\n *   .pipe(\n *     map(n => {\n *       if (n === 4) {\n *         throw 'four!';\n *       }\n *       return n;\n *     }),\n *     catchError(err => {\n *       throw 'error in source. Details: ' + err;\n *     })\n *   )\n *   .subscribe({\n *     next: x => console.log(x),\n *     error: err => console.log(err)\n *   });\n *   // 1, 2, 3, error in source. Details: four!\n * ```\n *\n * @see {@link onErrorResumeNext}\n * @see {@link repeat}\n * @see {@link repeatWhen}\n * @see {@link retry }\n * @see {@link retryWhen}\n *\n * @param {function} selector a function that takes as arguments `err`, which is the error, and `caught`, which\n * is the source observable, in case you'd like to \"retry\" that observable by returning it again. Whatever observable\n * is returned by the `selector` will be used to continue the observable chain.\n * @return A function that returns an Observable that originates from either\n * the source or the Observable returned by the `selector` function.\n */\nexport function catchError<T, O extends ObservableInput<any>>(\n  selector: (err: any, caught: Observable<T>) => O\n): OperatorFunction<T, T | ObservedValueOf<O>> {\n  return operate((source, subscriber) => {\n    let innerSub: Subscription | null = null;\n    let syncUnsub = false;\n    let handledResult: Observable<ObservedValueOf<O>>;\n\n    innerSub = source.subscribe(\n      createOperatorSubscriber(subscriber, undefined, undefined, (err) => {\n        handledResult = innerFrom(selector(err, catchError(selector)(source)));\n        if (innerSub) {\n          innerSub.unsubscribe();\n          innerSub = null;\n          handledResult.subscribe(subscriber);\n        } else {\n          // We don't have an innerSub yet, that means the error was synchronous\n          // because the subscribe call hasn't returned yet.\n          syncUnsub = true;\n        }\n      })\n    );\n\n    if (syncUnsub) {\n      // We have a synchronous error, we need to make sure to\n      // finalize right away. This ensures that callbacks in the `finalize` operator are called\n      // at the right time, and that finalization occurs at the expected\n      // time between the source error and the subscription to the\n      // next observable.\n      innerSub.unsubscribe();\n      innerSub = null;\n      handledResult!.subscribe(subscriber);\n    }\n  });\n}\n", "import { Observable } from '../Observable';\nimport { Subscriber } from '../Subscriber';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * A basic scan operation. This is used for `scan` and `reduce`.\n * @param accumulator The accumulator to use\n * @param seed The seed value for the state to accumulate\n * @param hasSeed Whether or not a seed was provided\n * @param emitOnNext Whether or not to emit the state on next\n * @param emitBeforeComplete Whether or not to emit the before completion\n */\n\nexport function scanInternals<V, A, S>(\n  accumulator: (acc: V | A | S, value: V, index: number) => A,\n  seed: S,\n  hasSeed: boolean,\n  emitOnNext: boolean,\n  emitBeforeComplete?: undefined | true\n) {\n  return (source: Observable<V>, subscriber: Subscriber<any>) => {\n    // Whether or not we have state yet. This will only be\n    // false before the first value arrives if we didn't get\n    // a seed value.\n    let hasState = hasSeed;\n    // The state that we're tracking, starting with the seed,\n    // if there is one, and then updated by the return value\n    // from the accumulator on each emission.\n    let state: any = seed;\n    // An index to pass to the accumulator function.\n    let index = 0;\n\n    // Subscribe to our source. All errors and completions are passed through.\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          // Always increment the index.\n          const i = index++;\n          // Set the state\n          state = hasState\n            ? // We already have state, so we can get the new state from the accumulator\n              accumulator(state, value, i)\n            : // We didn't have state yet, a seed value was not provided, so\n\n              // we set the state to the first value, and mark that we have state now\n              ((hasState = true), value);\n\n          // Maybe send it to the consumer.\n          emitOnNext && subscriber.next(state);\n        },\n        // If an onComplete was given, call it, otherwise\n        // just pass through the complete notification to the consumer.\n        emitBeforeComplete &&\n          (() => {\n            hasState && subscriber.next(state);\n            subscriber.complete();\n          })\n      )\n    );\n  };\n}\n", "import { combineLatestInit } from '../observable/combineLatest';\nimport { ObservableInput, ObservableInputTuple, OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { argsOrArgArray } from '../util/argsOrArgArray';\nimport { mapOneOrManyArgs } from '../util/mapOneOrManyArgs';\nimport { pipe } from '../util/pipe';\nimport { popResultSelector } from '../util/args';\n\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(\n  sources: [...ObservableInputTuple<A>],\n  project: (...values: [T, ...A]) => R\n): OperatorFunction<T, R>;\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(sources: [...ObservableInputTuple<A>]): OperatorFunction<T, [T, ...A]>;\n\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(\n  ...sourcesAndProject: [...ObservableInputTuple<A>, (...values: [T, ...A]) => R]\n): OperatorFunction<T, R>;\n/** @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8. */\nexport function combineLatest<T, A extends readonly unknown[], R>(...sources: [...ObservableInputTuple<A>]): OperatorFunction<T, [T, ...A]>;\n\n/**\n * @deprecated Replaced with {@link combineLatestWith}. Will be removed in v8.\n */\nexport function combineLatest<T, R>(...args: (ObservableInput<any> | ((...values: any[]) => R))[]): OperatorFunction<T, unknown> {\n  const resultSelector = popResultSelector(args);\n  return resultSelector\n    ? pipe(combineLatest(...(args as Array<ObservableInput<any>>)), mapOneOrManyArgs(resultSelector))\n    : operate((source, subscriber) => {\n        combineLatestInit([source, ...argsOrArgArray(args)])(subscriber);\n      });\n}\n", "import { ObservableInputTuple, OperatorFunction, Cons } from '../types';\nimport { combineLatest } from './combineLatest';\n\n/**\n * Create an observable that combines the latest values from all passed observables and the source\n * into arrays and emits them.\n *\n * Returns an observable, that when subscribed to, will subscribe to the source observable and all\n * sources provided as arguments. Once all sources emit at least one value, all of the latest values\n * will be emitted as an array. After that, every time any source emits a value, all of the latest values\n * will be emitted as an array.\n *\n * This is a useful operator for eagerly calculating values based off of changed inputs.\n *\n * ## Example\n *\n * Simple concatenation of values from two inputs\n *\n * ```ts\n * import { fromEvent, combineLatestWith, map } from 'rxjs';\n *\n * // Setup: Add two inputs to the page\n * const input1 = document.createElement('input');\n * document.body.appendChild(input1);\n * const input2 = document.createElement('input');\n * document.body.appendChild(input2);\n *\n * // Get streams of changes\n * const input1Changes$ = fromEvent(input1, 'change');\n * const input2Changes$ = fromEvent(input2, 'change');\n *\n * // Combine the changes by adding them together\n * input1Changes$.pipe(\n *   combineLatestWith(input2Changes$),\n *   map(([e1, e2]) => (<HTMLInputElement>e1.target).value + ' - ' + (<HTMLInputElement>e2.target).value)\n * )\n * .subscribe(x => console.log(x));\n * ```\n *\n * @param otherSources the other sources to subscribe to.\n * @return A function that returns an Observable that emits the latest\n * emissions from both source and provided Observables.\n */\nexport function combineLatestWith<T, A extends readonly unknown[]>(\n  ...otherSources: [...ObservableInputTuple<A>]\n): OperatorFunction<T, Cons<T, A>> {\n  return combineLatest(...otherSources);\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { Subscription } from '../Subscription';\nimport { MonoTypeOperatorFunction, SchedulerAction, SchedulerLike } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Emits a notification from the source Observable only after a particular time span\n * has passed without another source emission.\n *\n * <span class=\"informal\">It's like {@link delay}, but passes only the most\n * recent notification from each burst of emissions.</span>\n *\n * ![](debounceTime.png)\n *\n * `debounceTime` delays notifications emitted by the source Observable, but drops\n * previous pending delayed emissions if a new notification arrives on the source\n * Observable. This operator keeps track of the most recent notification from the\n * source Observable, and emits that only when `dueTime` has passed\n * without any other notification appearing on the source Observable. If a new value\n * appears before `dueTime` silence occurs, the previous notification will be dropped\n * and will not be emitted and a new `dueTime` is scheduled.\n * If the completing event happens during `dueTime` the last cached notification\n * is emitted before the completion event is forwarded to the output observable.\n * If the error event happens during `dueTime` or after it only the error event is\n * forwarded to the output observable. The cache notification is not emitted in this case.\n *\n * This is a rate-limiting operator, because it is impossible for more than one\n * notification to be emitted in any time window of duration `dueTime`, but it is also\n * a delay-like operator since output emissions do not occur at the same time as\n * they did on the source Observable. Optionally takes a {@link SchedulerLike} for\n * managing timers.\n *\n * ## Example\n *\n * Emit the most recent click after a burst of clicks\n *\n * ```ts\n * import { fromEvent, debounceTime } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(debounceTime(1000));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link audit}\n * @see {@link auditTime}\n * @see {@link debounce}\n * @see {@link sample}\n * @see {@link sampleTime}\n * @see {@link throttle}\n * @see {@link throttleTime}\n *\n * @param {number} dueTime The timeout duration in milliseconds (or the time\n * unit determined internally by the optional `scheduler`) for the window of\n * time required to wait for emission silence before emitting the most recent\n * source value.\n * @param {SchedulerLike} [scheduler=async] The {@link SchedulerLike} to use for\n * managing the timers that handle the timeout for each value.\n * @return A function that returns an Observable that delays the emissions of\n * the source Observable by the specified `dueTime`, and may drop some values\n * if they occur too frequently.\n */\nexport function debounceTime<T>(dueTime: number, scheduler: SchedulerLike = asyncScheduler): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let activeTask: Subscription | null = null;\n    let lastValue: T | null = null;\n    let lastTime: number | null = null;\n\n    const emit = () => {\n      if (activeTask) {\n        // We have a value! Free up memory first, then emit the value.\n        activeTask.unsubscribe();\n        activeTask = null;\n        const value = lastValue!;\n        lastValue = null;\n        subscriber.next(value);\n      }\n    };\n    function emitWhenIdle(this: SchedulerAction<unknown>) {\n      // This is called `dueTime` after the first value\n      // but we might have received new values during this window!\n\n      const targetTime = lastTime! + dueTime;\n      const now = scheduler.now();\n      if (now < targetTime) {\n        // On that case, re-schedule to the new target\n        activeTask = this.schedule(undefined, targetTime - now);\n        subscriber.add(activeTask);\n        return;\n      }\n\n      emit();\n    }\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value: T) => {\n          lastValue = value;\n          lastTime = scheduler.now();\n\n          // Only set up a task if it's not already up\n          if (!activeTask) {\n            activeTask = scheduler.schedule(emitWhenIdle, dueTime);\n            subscriber.add(activeTask);\n          }\n        },\n        () => {\n          // Source completed.\n          // Emit any pending debounced values then complete\n          emit();\n          subscriber.complete();\n        },\n        // Pass all errors through to consumer.\n        undefined,\n        () => {\n          // Finalization.\n          lastValue = activeTask = null;\n        }\n      )\n    );\n  });\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Emits a given value if the source Observable completes without emitting any\n * `next` value, otherwise mirrors the source Observable.\n *\n * <span class=\"informal\">If the source Observable turns out to be empty, then\n * this operator will emit a default value.</span>\n *\n * ![](defaultIfEmpty.png)\n *\n * `defaultIfEmpty` emits the values emitted by the source Observable or a\n * specified default value if the source Observable is empty (completes without\n * having emitted any `next` value).\n *\n * ## Example\n *\n * If no clicks happen in 5 seconds, then emit 'no clicks'\n *\n * ```ts\n * import { fromEvent, takeUntil, interval, defaultIfEmpty } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const clicksBeforeFive = clicks.pipe(takeUntil(interval(5000)));\n * const result = clicksBeforeFive.pipe(defaultIfEmpty('no clicks'));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link empty}\n * @see {@link last}\n *\n * @param defaultValue The default value used if the source\n * Observable is empty.\n * @return A function that returns an Observable that emits either the\n * specified `defaultValue` if the source Observable emits no items, or the\n * values emitted by the source Observable.\n */\nexport function defaultIfEmpty<T, R>(defaultValue: R): OperatorFunction<T, T | R> {\n  return operate((source, subscriber) => {\n    let hasValue = false;\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          hasValue = true;\n          subscriber.next(value);\n        },\n        () => {\n          if (!hasValue) {\n            subscriber.next(defaultValue!);\n          }\n          subscriber.complete();\n        }\n      )\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { EMPTY } from '../observable/empty';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Emits only the first `count` values emitted by the source Observable.\n *\n * <span class=\"informal\">Takes the first `count` values from the source, then\n * completes.</span>\n *\n * ![](take.png)\n *\n * `take` returns an Observable that emits only the first `count` values emitted\n * by the source Observable. If the source emits fewer than `count` values then\n * all of its values are emitted. After that, it completes, regardless if the\n * source completes.\n *\n * ## Example\n *\n * Take the first 5 seconds of an infinite 1-second interval Observable\n *\n * ```ts\n * import { interval, take } from 'rxjs';\n *\n * const intervalCount = interval(1000);\n * const takeFive = intervalCount.pipe(take(5));\n * takeFive.subscribe(x => console.log(x));\n *\n * // Logs:\n * // 0\n * // 1\n * // 2\n * // 3\n * // 4\n * ```\n *\n * @see {@link takeLast}\n * @see {@link takeUntil}\n * @see {@link takeWhile}\n * @see {@link skip}\n *\n * @param count The maximum number of `next` values to emit.\n * @return A function that returns an Observable that emits only the first\n * `count` values emitted by the source Observable, or all of the values from\n * the source if the source emits fewer than `count` values.\n */\nexport function take<T>(count: number): MonoTypeOperatorFunction<T> {\n  return count <= 0\n    ? // If we are taking no values, that's empty.\n      () => EMPTY\n    : operate((source, subscriber) => {\n        let seen = 0;\n        source.subscribe(\n          createOperatorSubscriber(subscriber, (value) => {\n            // Increment the number of values we have seen,\n            // then check it against the allowed count to see\n            // if we are still letting values through.\n            if (++seen <= count) {\n              subscriber.next(value);\n              // If we have met or passed our allowed count,\n              // we need to complete. We have to do <= here,\n              // because re-entrant code will increment `seen` twice.\n              if (count <= seen) {\n                subscriber.complete();\n              }\n            }\n          })\n        );\n      });\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { noop } from '../util/noop';\n\n/**\n * Ignores all items emitted by the source Observable and only passes calls of `complete` or `error`.\n *\n * ![](ignoreElements.png)\n *\n * The `ignoreElements` operator suppresses all items emitted by the source Observable,\n * but allows its termination notification (either `error` or `complete`) to pass through unchanged.\n *\n * If you do not care about the items being emitted by an Observable, but you do want to be notified\n * when it completes or when it terminates with an error, you can apply the `ignoreElements` operator\n * to the Observable, which will ensure that it will never call its observers\u2019 `next` handlers.\n *\n * ## Example\n *\n * Ignore all `next` emissions from the source\n *\n * ```ts\n * import { of, ignoreElements } from 'rxjs';\n *\n * of('you', 'talking', 'to', 'me')\n *   .pipe(ignoreElements())\n *   .subscribe({\n *     next: word => console.log(word),\n *     error: err => console.log('error:', err),\n *     complete: () => console.log('the end'),\n *   });\n *\n * // result:\n * // 'the end'\n * ```\n *\n * @return A function that returns an empty Observable that only calls\n * `complete` or `error`, based on which one is called by the source\n * Observable.\n */\nexport function ignoreElements(): OperatorFunction<unknown, never> {\n  return operate((source, subscriber) => {\n    source.subscribe(createOperatorSubscriber(subscriber, noop));\n  });\n}\n", "import { OperatorFunction } from '../types';\nimport { map } from './map';\n\n/** @deprecated To be removed in v9. Use {@link map} instead: `map(() => value)`. */\nexport function mapTo<R>(value: R): OperatorFunction<unknown, R>;\n/**\n * @deprecated Do not specify explicit type parameters. Signatures with type parameters\n * that cannot be inferred will be removed in v8. `mapTo` itself will be removed in v9,\n * use {@link map} instead: `map(() => value)`.\n * */\nexport function mapTo<T, R>(value: R): OperatorFunction<T, R>;\n\n/**\n * Emits the given constant value on the output Observable every time the source\n * Observable emits a value.\n *\n * <span class=\"informal\">Like {@link map}, but it maps every source value to\n * the same output value every time.</span>\n *\n * ![](mapTo.png)\n *\n * Takes a constant `value` as argument, and emits that whenever the source\n * Observable emits a value. In other words, ignores the actual source value,\n * and simply uses the emission moment to know when to emit the given `value`.\n *\n * ## Example\n *\n * Map every click to the string `'Hi'`\n *\n * ```ts\n * import { fromEvent, mapTo } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const greetings = clicks.pipe(mapTo('Hi'));\n *\n * greetings.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link map}\n *\n * @param value The value to map each source value to.\n * @return A function that returns an Observable that emits the given `value`\n * every time the source Observable emits.\n * @deprecated To be removed in v9. Use {@link map} instead: `map(() => value)`.\n */\nexport function mapTo<R>(value: R): OperatorFunction<unknown, R> {\n  return map(() => value);\n}\n", "import { Observable } from '../Observable';\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { concat } from '../observable/concat';\nimport { take } from './take';\nimport { ignoreElements } from './ignoreElements';\nimport { mapTo } from './mapTo';\nimport { mergeMap } from './mergeMap';\nimport { innerFrom } from '../observable/innerFrom';\n\n/** @deprecated The `subscriptionDelay` parameter will be removed in v8. */\nexport function delayWhen<T>(\n  delayDurationSelector: (value: T, index: number) => ObservableInput<any>,\n  subscriptionDelay: Observable<any>\n): MonoTypeOperatorFunction<T>;\nexport function delayWhen<T>(delayDurationSelector: (value: T, index: number) => ObservableInput<any>): MonoTypeOperatorFunction<T>;\n\n/**\n * Delays the emission of items from the source Observable by a given time span\n * determined by the emissions of another Observable.\n *\n * <span class=\"informal\">It's like {@link delay}, but the time span of the\n * delay duration is determined by a second Observable.</span>\n *\n * ![](delayWhen.png)\n *\n * `delayWhen` operator shifts each emitted value from the source Observable by\n * a time span determined by another Observable. When the source emits a value,\n * the `delayDurationSelector` function is called with the value emitted from\n * the source Observable as the first argument to the `delayDurationSelector`.\n * The `delayDurationSelector` function should return an {@link ObservableInput},\n * that is internally converted to an Observable that is called the \"duration\"\n * Observable.\n *\n * The source value is emitted on the output Observable only when the \"duration\"\n * Observable emits ({@link guide/glossary-and-semantics#next next}s) any value.\n * Upon that, the \"duration\" Observable gets unsubscribed.\n *\n * Before RxJS V7, the {@link guide/glossary-and-semantics#complete completion}\n * of the \"duration\" Observable would have been triggering the emission of the\n * source value to the output Observable, but with RxJS V7, this is not the case\n * anymore.\n *\n * Only next notifications (from the \"duration\" Observable) trigger values from\n * the source Observable to be passed to the output Observable. If the \"duration\"\n * Observable only emits the complete notification (without next), the value\n * emitted by the source Observable will never get to the output Observable - it\n * will be swallowed. If the \"duration\" Observable errors, the error will be\n * propagated to the output Observable.\n *\n * Optionally, `delayWhen` takes a second argument, `subscriptionDelay`, which\n * is an Observable. When `subscriptionDelay` emits its first value or\n * completes, the source Observable is subscribed to and starts behaving like\n * described in the previous paragraph. If `subscriptionDelay` is not provided,\n * `delayWhen` will subscribe to the source Observable as soon as the output\n * Observable is subscribed.\n *\n * ## Example\n *\n * Delay each click by a random amount of time, between 0 and 5 seconds\n *\n * ```ts\n * import { fromEvent, delayWhen, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const delayedClicks = clicks.pipe(\n *   delayWhen(() => interval(Math.random() * 5000))\n * );\n * delayedClicks.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link delay}\n * @see {@link throttle}\n * @see {@link throttleTime}\n * @see {@link debounce}\n * @see {@link debounceTime}\n * @see {@link sample}\n * @see {@link sampleTime}\n * @see {@link audit}\n * @see {@link auditTime}\n *\n * @param delayDurationSelector A function that returns an `ObservableInput` for\n * each `value` emitted by the source Observable, which is then used to delay the\n * emission of that `value` on the output Observable until the `ObservableInput`\n * returned from this function emits a next value. When called, beside `value`,\n * this function receives a zero-based `index` of the emission order.\n * @param subscriptionDelay An Observable that triggers the subscription to the\n * source Observable once it emits any value.\n * @return A function that returns an Observable that delays the emissions of\n * the source Observable by an amount of time specified by the Observable\n * returned by `delayDurationSelector`.\n */\nexport function delayWhen<T>(\n  delayDurationSelector: (value: T, index: number) => ObservableInput<any>,\n  subscriptionDelay?: Observable<any>\n): MonoTypeOperatorFunction<T> {\n  if (subscriptionDelay) {\n    // DEPRECATED PATH\n    return (source: Observable<T>) =>\n      concat(subscriptionDelay.pipe(take(1), ignoreElements()), source.pipe(delayWhen(delayDurationSelector)));\n  }\n\n  return mergeMap((value, index) => innerFrom(delayDurationSelector(value, index)).pipe(take(1), mapTo(value)));\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { delayWhen } from './delayWhen';\nimport { timer } from '../observable/timer';\n\n/**\n * Delays the emission of items from the source Observable by a given timeout or\n * until a given Date.\n *\n * <span class=\"informal\">Time shifts each item by some specified amount of\n * milliseconds.</span>\n *\n * ![](delay.svg)\n *\n * If the delay argument is a Number, this operator time shifts the source\n * Observable by that amount of time expressed in milliseconds. The relative\n * time intervals between the values are preserved.\n *\n * If the delay argument is a Date, this operator time shifts the start of the\n * Observable execution until the given date occurs.\n *\n * ## Examples\n *\n * Delay each click by one second\n *\n * ```ts\n * import { fromEvent, delay } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const delayedClicks = clicks.pipe(delay(1000)); // each click emitted after 1 second\n * delayedClicks.subscribe(x => console.log(x));\n * ```\n *\n * Delay all clicks until a future date happens\n *\n * ```ts\n * import { fromEvent, delay } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const date = new Date('March 15, 2050 12:00:00'); // in the future\n * const delayedClicks = clicks.pipe(delay(date)); // click emitted only after that date\n * delayedClicks.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link delayWhen}\n * @see {@link throttle}\n * @see {@link throttleTime}\n * @see {@link debounce}\n * @see {@link debounceTime}\n * @see {@link sample}\n * @see {@link sampleTime}\n * @see {@link audit}\n * @see {@link auditTime}\n *\n * @param {number|Date} due The delay duration in milliseconds (a `number`) or\n * a `Date` until which the emission of the source items is delayed.\n * @param {SchedulerLike} [scheduler=async] The {@link SchedulerLike} to use for\n * managing the timers that handle the time-shift for each item.\n * @return A function that returns an Observable that delays the emissions of\n * the source Observable by the specified timeout or Date.\n */\nexport function delay<T>(due: number | Date, scheduler: SchedulerLike = asyncScheduler): MonoTypeOperatorFunction<T> {\n  const duration = timer(due, scheduler);\n  return delayWhen(() => duration);\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { identity } from '../util/identity';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\nexport function distinctUntilChanged<T>(comparator?: (previous: T, current: T) => boolean): MonoTypeOperatorFunction<T>;\nexport function distinctUntilChanged<T, K>(\n  comparator: (previous: K, current: K) => boolean,\n  keySelector: (value: T) => K\n): MonoTypeOperatorFunction<T>;\n\n/**\n * Returns a result {@link Observable} that emits all values pushed by the source observable if they\n * are distinct in comparison to the last value the result observable emitted.\n *\n * When provided without parameters or with the first parameter (`{@link distinctUntilChanged#comparator comparator}`),\n * it behaves like this:\n *\n * 1. It will always emit the first value from the source.\n * 2. For all subsequent values pushed by the source, they will be compared to the previously emitted values\n *    using the provided `comparator` or an `===` equality check.\n * 3. If the value pushed by the source is determined to be unequal by this check, that value is emitted and\n *    becomes the new \"previously emitted value\" internally.\n *\n * When the second parameter (`{@link distinctUntilChanged#keySelector keySelector}`) is provided, the behavior\n * changes:\n *\n * 1. It will always emit the first value from the source.\n * 2. The `keySelector` will be run against all values, including the first value.\n * 3. For all values after the first, the selected key will be compared against the key selected from\n *    the previously emitted value using the `comparator`.\n * 4. If the keys are determined to be unequal by this check, the value (not the key), is emitted\n *    and the selected key from that value is saved for future comparisons against other keys.\n *\n * ## Examples\n *\n * A very basic example with no `{@link distinctUntilChanged#comparator comparator}`. Note that `1` is emitted more than once,\n * because it's distinct in comparison to the _previously emitted_ value,\n * not in comparison to _all other emitted values_.\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * of(1, 1, 1, 2, 2, 2, 1, 1, 3, 3)\n *   .pipe(distinctUntilChanged())\n *   .subscribe(console.log);\n * // Logs: 1, 2, 1, 3\n * ```\n *\n * With a `{@link distinctUntilChanged#comparator comparator}`, you can do custom comparisons. Let's say\n * you only want to emit a value when all of its components have\n * changed:\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * const totallyDifferentBuilds$ = of(\n *   { engineVersion: '1.1.0', transmissionVersion: '1.2.0' },\n *   { engineVersion: '1.1.0', transmissionVersion: '1.4.0' },\n *   { engineVersion: '1.3.0', transmissionVersion: '1.4.0' },\n *   { engineVersion: '1.3.0', transmissionVersion: '1.5.0' },\n *   { engineVersion: '2.0.0', transmissionVersion: '1.5.0' }\n * ).pipe(\n *   distinctUntilChanged((prev, curr) => {\n *     return (\n *       prev.engineVersion === curr.engineVersion ||\n *       prev.transmissionVersion === curr.transmissionVersion\n *     );\n *   })\n * );\n *\n * totallyDifferentBuilds$.subscribe(console.log);\n *\n * // Logs:\n * // { engineVersion: '1.1.0', transmissionVersion: '1.2.0' }\n * // { engineVersion: '1.3.0', transmissionVersion: '1.4.0' }\n * // { engineVersion: '2.0.0', transmissionVersion: '1.5.0' }\n * ```\n *\n * You can also provide a custom `{@link distinctUntilChanged#comparator comparator}` to check that emitted\n * changes are only in one direction. Let's say you only want to get\n * the next record temperature:\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * const temps$ = of(30, 31, 20, 34, 33, 29, 35, 20);\n *\n * const recordHighs$ = temps$.pipe(\n *   distinctUntilChanged((prevHigh, temp) => {\n *     // If the current temp is less than\n *     // or the same as the previous record,\n *     // the record hasn't changed.\n *     return temp <= prevHigh;\n *   })\n * );\n *\n * recordHighs$.subscribe(console.log);\n * // Logs: 30, 31, 34, 35\n * ```\n *\n * Selecting update events only when the `updatedBy` field shows\n * the account changed hands.\n *\n * ```ts\n * import { of, distinctUntilChanged } from 'rxjs';\n *\n * // A stream of updates to a given account\n * const accountUpdates$ = of(\n *   { updatedBy: 'blesh', data: [] },\n *   { updatedBy: 'blesh', data: [] },\n *   { updatedBy: 'ncjamieson', data: [] },\n *   { updatedBy: 'ncjamieson', data: [] },\n *   { updatedBy: 'blesh', data: [] }\n * );\n *\n * // We only want the events where it changed hands\n * const changedHands$ = accountUpdates$.pipe(\n *   distinctUntilChanged(undefined, update => update.updatedBy)\n * );\n *\n * changedHands$.subscribe(console.log);\n * // Logs:\n * // { updatedBy: 'blesh', data: Array[0] }\n * // { updatedBy: 'ncjamieson', data: Array[0] }\n * // { updatedBy: 'blesh', data: Array[0] }\n * ```\n *\n * @see {@link distinct}\n * @see {@link distinctUntilKeyChanged}\n *\n * @param comparator A function used to compare the previous and current keys for\n * equality. Defaults to a `===` check.\n * @param keySelector Used to select a key value to be passed to the `comparator`.\n *\n * @return A function that returns an Observable that emits items from the\n * source Observable with distinct values.\n */\nexport function distinctUntilChanged<T, K>(\n  comparator?: (previous: K, current: K) => boolean,\n  keySelector: (value: T) => K = identity as (value: T) => K\n): MonoTypeOperatorFunction<T> {\n  // We've been allowing `null` do be passed as the `compare`, so we can't do\n  // a default value for the parameter, because that will only work\n  // for `undefined`.\n  comparator = comparator ?? defaultCompare;\n\n  return operate((source, subscriber) => {\n    // The previous key, used to compare against keys selected\n    // from new arrivals to determine \"distinctiveness\".\n    let previousKey: K;\n    // Whether or not this is the first value we've gotten.\n    let first = true;\n\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value) => {\n        // We always call the key selector.\n        const currentKey = keySelector(value);\n\n        // If it's the first value, we always emit it.\n        // Otherwise, we compare this key to the previous key, and\n        // if the comparer returns false, we emit.\n        if (first || !comparator!(previousKey, currentKey)) {\n          // Update our state *before* we emit the value\n          // as emission can be the source of re-entrant code\n          // in functional libraries like this. We only really\n          // need to do this if it's the first value, or if the\n          // key we're tracking in previous needs to change.\n          first = false;\n          previousKey = currentKey;\n\n          // Emit the value!\n          subscriber.next(value);\n        }\n      })\n    );\n  });\n}\n\nfunction defaultCompare(a: any, b: any) {\n  return a === b;\n}\n", "import { distinctUntilChanged } from './distinctUntilChanged';\nimport { MonoTypeOperatorFunction } from '../types';\n\n/* tslint:disable:max-line-length */\nexport function distinctUntilKeyChanged<T>(key: keyof T): MonoTypeOperatorFunction<T>;\nexport function distinctUntilKeyChanged<T, K extends keyof T>(key: K, compare: (x: T[K], y: T[K]) => boolean): MonoTypeOperatorFunction<T>;\n/* tslint:enable:max-line-length */\n\n/**\n * Returns an Observable that emits all items emitted by the source Observable that are distinct by comparison from the previous item,\n * using a property accessed by using the key provided to check if the two items are distinct.\n *\n * If a comparator function is provided, then it will be called for each item to test for whether or not that value should be emitted.\n *\n * If a comparator function is not provided, an equality check is used by default.\n *\n * ## Examples\n *\n * An example comparing the name of persons\n *\n * ```ts\n * import { of, distinctUntilKeyChanged } from 'rxjs';\n *\n * of(\n *   { age: 4, name: 'Foo' },\n *   { age: 7, name: 'Bar' },\n *   { age: 5, name: 'Foo' },\n *   { age: 6, name: 'Foo' }\n * ).pipe(\n *   distinctUntilKeyChanged('name')\n * )\n * .subscribe(x => console.log(x));\n *\n * // displays:\n * // { age: 4, name: 'Foo' }\n * // { age: 7, name: 'Bar' }\n * // { age: 5, name: 'Foo' }\n * ```\n *\n * An example comparing the first letters of the name\n *\n * ```ts\n * import { of, distinctUntilKeyChanged } from 'rxjs';\n *\n * of(\n *   { age: 4, name: 'Foo1' },\n *   { age: 7, name: 'Bar' },\n *   { age: 5, name: 'Foo2' },\n *   { age: 6, name: 'Foo3' }\n * ).pipe(\n *   distinctUntilKeyChanged('name', (x, y) => x.substring(0, 3) === y.substring(0, 3))\n * )\n * .subscribe(x => console.log(x));\n *\n * // displays:\n * // { age: 4, name: 'Foo1' }\n * // { age: 7, name: 'Bar' }\n * // { age: 5, name: 'Foo2' }\n * ```\n *\n * @see {@link distinct}\n * @see {@link distinctUntilChanged}\n *\n * @param {string} key String key for object property lookup on each item.\n * @param {function} [compare] Optional comparison function called to test if an item is distinct from the previous item in the source.\n * @return A function that returns an Observable that emits items from the\n * source Observable with distinct values based on the key specified.\n */\nexport function distinctUntilKeyChanged<T, K extends keyof T>(key: K, compare?: (x: T[K], y: T[K]) => boolean): MonoTypeOperatorFunction<T> {\n  return distinctUntilChanged((x: T, y: T) => compare ? compare(x[key], y[key]) : x[key] === y[key]);\n}\n", "import { EmptyError } from '../util/EmptyError';\nimport { MonoTypeOperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * If the source observable completes without emitting a value, it will emit\n * an error. The error will be created at that time by the optional\n * `errorFactory` argument, otherwise, the error will be {@link EmptyError}.\n *\n * ![](throwIfEmpty.png)\n *\n * ## Example\n *\n * Throw an error if the document wasn't clicked within 1 second\n *\n * ```ts\n * import { fromEvent, takeUntil, timer, throwIfEmpty } from 'rxjs';\n *\n * const click$ = fromEvent(document, 'click');\n *\n * click$.pipe(\n *   takeUntil(timer(1000)),\n *   throwIfEmpty(() => new Error('The document was not clicked within 1 second'))\n * )\n * .subscribe({\n *   next() {\n *    console.log('The document was clicked');\n *   },\n *   error(err) {\n *     console.error(err.message);\n *   }\n * });\n * ```\n *\n * @param errorFactory A factory function called to produce the\n * error to be thrown when the source observable completes without emitting a\n * value.\n * @return A function that returns an Observable that throws an error if the\n * source Observable completed without emitting.\n */\nexport function throwIfEmpty<T>(errorFactory: () => any = defaultErrorFactory): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let hasValue = false;\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          hasValue = true;\n          subscriber.next(value);\n        },\n        () => (hasValue ? subscriber.complete() : subscriber.error(errorFactory()))\n      )\n    );\n  });\n}\n\nfunction defaultErrorFactory() {\n  return new EmptyError();\n}\n", "/** prettier */\nimport { Observable } from '../Observable';\nimport { concat } from '../observable/concat';\nimport { of } from '../observable/of';\nimport { MonoTypeOperatorFunction, SchedulerLike, OperatorFunction, ValueFromArray } from '../types';\n\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `concatAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function endWith<T>(scheduler: SchedulerLike): MonoTypeOperatorFunction<T>;\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `concatAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function endWith<T, A extends unknown[] = T[]>(\n  ...valuesAndScheduler: [...A, SchedulerLike]\n): OperatorFunction<T, T | ValueFromArray<A>>;\n\nexport function endWith<T, A extends unknown[] = T[]>(...values: A): OperatorFunction<T, T | ValueFromArray<A>>;\n\n/**\n * Returns an observable that will emit all values from the source, then synchronously emit\n * the provided value(s) immediately after the source completes.\n *\n * NOTE: Passing a last argument of a Scheduler is _deprecated_, and may result in incorrect\n * types in TypeScript.\n *\n * This is useful for knowing when an observable ends. Particularly when paired with an\n * operator like {@link takeUntil}\n *\n * ![](endWith.png)\n *\n * ## Example\n *\n * Emit values to know when an interval starts and stops. The interval will\n * stop when a user clicks anywhere on the document.\n *\n * ```ts\n * import { interval, map, fromEvent, startWith, takeUntil, endWith } from 'rxjs';\n *\n * const ticker$ = interval(5000).pipe(\n *   map(() => 'tick')\n * );\n *\n * const documentClicks$ = fromEvent(document, 'click');\n *\n * ticker$.pipe(\n *   startWith('interval started'),\n *   takeUntil(documentClicks$),\n *   endWith('interval ended by click')\n * )\n * .subscribe(x => console.log(x));\n *\n * // Result (assuming a user clicks after 15 seconds)\n * // 'interval started'\n * // 'tick'\n * // 'tick'\n * // 'tick'\n * // 'interval ended by click'\n * ```\n *\n * @see {@link startWith}\n * @see {@link concat}\n * @see {@link takeUntil}\n *\n * @param values Items you want the modified Observable to emit last.\n * @return A function that returns an Observable that emits all values from the\n * source, then synchronously emits the provided value(s) immediately after the\n * source completes.\n */\nexport function endWith<T>(...values: Array<T | SchedulerLike>): MonoTypeOperatorFunction<T> {\n  return (source: Observable<T>) => concat(source, of(...values)) as Observable<T>;\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { operate } from '../util/lift';\n\n/**\n * Returns an Observable that mirrors the source Observable, but will call a specified function when\n * the source terminates on complete or error.\n * The specified function will also be called when the subscriber explicitly unsubscribes.\n *\n * ## Examples\n *\n * Execute callback function when the observable completes\n *\n * ```ts\n * import { interval, take, finalize } from 'rxjs';\n *\n * // emit value in sequence every 1 second\n * const source = interval(1000);\n * const example = source.pipe(\n *   take(5), //take only the first 5 values\n *   finalize(() => console.log('Sequence complete')) // Execute when the observable completes\n * );\n * const subscribe = example.subscribe(val => console.log(val));\n *\n * // results:\n * // 0\n * // 1\n * // 2\n * // 3\n * // 4\n * // 'Sequence complete'\n * ```\n *\n * Execute callback function when the subscriber explicitly unsubscribes\n *\n * ```ts\n * import { interval, finalize, tap, noop, timer } from 'rxjs';\n *\n * const source = interval(100).pipe(\n *   finalize(() => console.log('[finalize] Called')),\n *   tap({\n *     next: () => console.log('[next] Called'),\n *     error: () => console.log('[error] Not called'),\n *     complete: () => console.log('[tap complete] Not called')\n *   })\n * );\n *\n * const sub = source.subscribe({\n *   next: x => console.log(x),\n *   error: noop,\n *   complete: () => console.log('[complete] Not called')\n * });\n *\n * timer(150).subscribe(() => sub.unsubscribe());\n *\n * // results:\n * // '[next] Called'\n * // 0\n * // '[finalize] Called'\n * ```\n *\n * @param {function} callback Function to be called when source terminates.\n * @return A function that returns an Observable that mirrors the source, but\n * will call the specified function on termination.\n */\nexport function finalize<T>(callback: () => void): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    // TODO: This try/finally was only added for `useDeprecatedSynchronousErrorHandling`.\n    // REMOVE THIS WHEN THAT HOT GARBAGE IS REMOVED IN V8.\n    try {\n      source.subscribe(subscriber);\n    } finally {\n      subscriber.add(callback);\n    }\n  });\n}\n", "import { Observable } from '../Observable';\nimport { EmptyError } from '../util/EmptyError';\nimport { OperatorFunction, TruthyTypesOf } from '../types';\nimport { filter } from './filter';\nimport { take } from './take';\nimport { defaultIfEmpty } from './defaultIfEmpty';\nimport { throwIfEmpty } from './throwIfEmpty';\nimport { identity } from '../util/identity';\n\nexport function first<T, D = T>(predicate?: null, defaultValue?: D): OperatorFunction<T, T | D>;\nexport function first<T>(predicate: BooleanConstructor): OperatorFunction<T, TruthyTypesOf<T>>;\nexport function first<T, D>(predicate: BooleanConstructor, defaultValue: D): OperatorFunction<T, TruthyTypesOf<T> | D>;\nexport function first<T, S extends T>(\n  predicate: (value: T, index: number, source: Observable<T>) => value is S,\n  defaultValue?: S\n): OperatorFunction<T, S>;\nexport function first<T, S extends T, D>(\n  predicate: (value: T, index: number, source: Observable<T>) => value is S,\n  defaultValue: D\n): OperatorFunction<T, S | D>;\nexport function first<T, D = T>(\n  predicate: (value: T, index: number, source: Observable<T>) => boolean,\n  defaultValue?: D\n): OperatorFunction<T, T | D>;\n\n/**\n * Emits only the first value (or the first value that meets some condition)\n * emitted by the source Observable.\n *\n * <span class=\"informal\">Emits only the first value. Or emits only the first\n * value that passes some test.</span>\n *\n * ![](first.png)\n *\n * If called with no arguments, `first` emits the first value of the source\n * Observable, then completes. If called with a `predicate` function, `first`\n * emits the first value of the source that matches the specified condition. Throws an error if\n * `defaultValue` was not provided and a matching element is not found.\n *\n * ## Examples\n *\n * Emit only the first click that happens on the DOM\n *\n * ```ts\n * import { fromEvent, first } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(first());\n * result.subscribe(x => console.log(x));\n * ```\n *\n * Emits the first click that happens on a DIV\n *\n * ```ts\n * import { fromEvent, first } from 'rxjs';\n *\n * const div = document.createElement('div');\n * div.style.cssText = 'width: 200px; height: 200px; background: #09c;';\n * document.body.appendChild(div);\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(first(ev => (<HTMLElement>ev.target).tagName === 'DIV'));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link filter}\n * @see {@link find}\n * @see {@link take}\n *\n * @throws {EmptyError} Delivers an EmptyError to the Observer's `error`\n * callback if the Observable completes before any `next` notification was sent.\n * This is how `first()` is different from {@link take}(1) which completes instead.\n *\n * @param {function(value: T, index: number, source: Observable<T>): boolean} [predicate]\n * An optional function called with each item to test for condition matching.\n * @param {D} [defaultValue] The default value emitted in case no valid value\n * was found on the source.\n * @return A function that returns an Observable that emits the first item that\n * matches the condition.\n */\nexport function first<T, D>(\n  predicate?: ((value: T, index: number, source: Observable<T>) => boolean) | null,\n  defaultValue?: D\n): OperatorFunction<T, T | D> {\n  const hasDefaultValue = arguments.length >= 2;\n  return (source: Observable<T>) =>\n    source.pipe(\n      predicate ? filter((v, i) => predicate(v, i, source)) : identity,\n      take(1),\n      hasDefaultValue ? defaultIfEmpty(defaultValue!) : throwIfEmpty(() => new EmptyError())\n    );\n}\n", "import { EMPTY } from '../observable/empty';\nimport { MonoTypeOperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Waits for the source to complete, then emits the last N values from the source,\n * as specified by the `count` argument.\n *\n * ![](takeLast.png)\n *\n * `takeLast` results in an observable that will hold values up to `count` values in memory,\n * until the source completes. It then pushes all values in memory to the consumer, in the\n * order they were received from the source, then notifies the consumer that it is\n * complete.\n *\n * If for some reason the source completes before the `count` supplied to `takeLast` is reached,\n * all values received until that point are emitted, and then completion is notified.\n *\n * **Warning**: Using `takeLast` with an observable that never completes will result\n * in an observable that never emits a value.\n *\n * ## Example\n *\n * Take the last 3 values of an Observable with many values\n *\n * ```ts\n * import { range, takeLast } from 'rxjs';\n *\n * const many = range(1, 100);\n * const lastThree = many.pipe(takeLast(3));\n * lastThree.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link take}\n * @see {@link takeUntil}\n * @see {@link takeWhile}\n * @see {@link skip}\n *\n * @param count The maximum number of values to emit from the end of\n * the sequence of values emitted by the source Observable.\n * @return A function that returns an Observable that emits at most the last\n * `count` values emitted by the source Observable.\n */\nexport function takeLast<T>(count: number): MonoTypeOperatorFunction<T> {\n  return count <= 0\n    ? () => EMPTY\n    : operate((source, subscriber) => {\n        // This buffer will hold the values we are going to emit\n        // when the source completes. Since we only want to take the\n        // last N values, we can't emit until we're sure we're not getting\n        // any more values.\n        let buffer: T[] = [];\n        source.subscribe(\n          createOperatorSubscriber(\n            subscriber,\n            (value) => {\n              // Add the most recent value onto the end of our buffer.\n              buffer.push(value);\n              // If our buffer is now larger than the number of values we\n              // want to take, we remove the oldest value from the buffer.\n              count < buffer.length && buffer.shift();\n            },\n            () => {\n              // The source completed, we now know what are last values\n              // are, emit them in the order they were received.\n              for (const value of buffer) {\n                subscriber.next(value);\n              }\n              subscriber.complete();\n            },\n            // Errors are passed through to the consumer\n            undefined,\n            () => {\n              // During finalization release the values in our buffer.\n              buffer = null!;\n            }\n          )\n        );\n      });\n}\n", "import { ObservableInput, ObservableInputTuple, OperatorFunction, SchedulerLike } from '../types';\nimport { operate } from '../util/lift';\nimport { argsOrArgArray } from '../util/argsOrArgArray';\nimport { mergeAll } from './mergeAll';\nimport { popNumber, popScheduler } from '../util/args';\nimport { from } from '../observable/from';\n\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(...sources: [...ObservableInputTuple<A>]): OperatorFunction<T, T | A[number]>;\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(\n  ...sourcesAndConcurrency: [...ObservableInputTuple<A>, number]\n): OperatorFunction<T, T | A[number]>;\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(\n  ...sourcesAndScheduler: [...ObservableInputTuple<A>, SchedulerLike]\n): OperatorFunction<T, T | A[number]>;\n/** @deprecated Replaced with {@link mergeWith}. Will be removed in v8. */\nexport function merge<T, A extends readonly unknown[]>(\n  ...sourcesAndConcurrencyAndScheduler: [...ObservableInputTuple<A>, number, SchedulerLike]\n): OperatorFunction<T, T | A[number]>;\n\nexport function merge<T>(...args: unknown[]): OperatorFunction<T, unknown> {\n  const scheduler = popScheduler(args);\n  const concurrent = popNumber(args, Infinity);\n  args = argsOrArgArray(args);\n\n  return operate((source, subscriber) => {\n    mergeAll(concurrent)(from([source, ...(args as ObservableInput<T>[])], scheduler)).subscribe(subscriber);\n  });\n}\n", "import { ObservableInputTuple, OperatorFunction } from '../types';\nimport { merge } from './merge';\n\n/**\n * Merge the values from all observables to a single observable result.\n *\n * Creates an observable, that when subscribed to, subscribes to the source\n * observable, and all other sources provided as arguments. All values from\n * every source are emitted from the resulting subscription.\n *\n * When all sources complete, the resulting observable will complete.\n *\n * When any source errors, the resulting observable will error.\n *\n * ## Example\n *\n * Joining all outputs from multiple user input event streams\n *\n * ```ts\n * import { fromEvent, map, mergeWith } from 'rxjs';\n *\n * const clicks$ = fromEvent(document, 'click').pipe(map(() => 'click'));\n * const mousemoves$ = fromEvent(document, 'mousemove').pipe(map(() => 'mousemove'));\n * const dblclicks$ = fromEvent(document, 'dblclick').pipe(map(() => 'dblclick'));\n *\n * mousemoves$\n *   .pipe(mergeWith(clicks$, dblclicks$))\n *   .subscribe(x => console.log(x));\n *\n * // result (assuming user interactions)\n * // 'mousemove'\n * // 'mousemove'\n * // 'mousemove'\n * // 'click'\n * // 'click'\n * // 'dblclick'\n * ```\n *\n * @see {@link merge}\n *\n * @param otherSources the sources to combine the current source with.\n * @return A function that returns an Observable that merges the values from\n * all given Observables.\n */\nexport function mergeWith<T, A extends readonly unknown[]>(\n  ...otherSources: [...ObservableInputTuple<A>]\n): OperatorFunction<T, T | A[number]> {\n  return merge(...otherSources);\n}\n", "import { Subscription } from '../Subscription';\nimport { EMPTY } from '../observable/empty';\nimport { operate } from '../util/lift';\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { timer } from '../observable/timer';\n\nexport interface RepeatConfig {\n  /**\n   * The number of times to repeat the source. Defaults to `Infinity`.\n   */\n  count?: number;\n\n  /**\n   * If a `number`, will delay the repeat of the source by that number of milliseconds.\n   * If a function, it will provide the number of times the source has been subscribed to,\n   * and the return value should be a valid observable input that will notify when the source\n   * should be repeated. If the notifier observable is empty, the result will complete.\n   */\n  delay?: number | ((count: number) => ObservableInput<any>);\n}\n\n/**\n * Returns an Observable that will resubscribe to the source stream when the source stream completes.\n *\n * <span class=\"informal\">Repeats all values emitted on the source. It's like {@link retry}, but for non error cases.</span>\n *\n * ![](repeat.png)\n *\n * Repeat will output values from a source until the source completes, then it will resubscribe to the\n * source a specified number of times, with a specified delay. Repeat can be particularly useful in\n * combination with closing operators like {@link take}, {@link takeUntil}, {@link first}, or {@link takeWhile},\n * as it can be used to restart a source again from scratch.\n *\n * Repeat is very similar to {@link retry}, where {@link retry} will resubscribe to the source in the error case, but\n * `repeat` will resubscribe if the source completes.\n *\n * Note that `repeat` will _not_ catch errors. Use {@link retry} for that.\n *\n * - `repeat(0)` returns an empty observable\n * - `repeat()` will repeat forever\n * - `repeat({ delay: 200 })` will repeat forever, with a delay of 200ms between repetitions.\n * - `repeat({ count: 2, delay: 400 })` will repeat twice, with a delay of 400ms between repetitions.\n * - `repeat({ delay: (count) => timer(count * 1000) })` will repeat forever, but will have a delay that grows by one second for each repetition.\n *\n * ## Example\n *\n * Repeat a message stream\n *\n * ```ts\n * import { of, repeat } from 'rxjs';\n *\n * const source = of('Repeat message');\n * const result = source.pipe(repeat(3));\n *\n * result.subscribe(x => console.log(x));\n *\n * // Results\n * // 'Repeat message'\n * // 'Repeat message'\n * // 'Repeat message'\n * ```\n *\n * Repeat 3 values, 2 times\n *\n * ```ts\n * import { interval, take, repeat } from 'rxjs';\n *\n * const source = interval(1000);\n * const result = source.pipe(take(3), repeat(2));\n *\n * result.subscribe(x => console.log(x));\n *\n * // Results every second\n * // 0\n * // 1\n * // 2\n * // 0\n * // 1\n * // 2\n * ```\n *\n * Defining two complex repeats with delays on the same source.\n * Note that the second repeat cannot be called until the first\n * repeat as exhausted it's count.\n *\n * ```ts\n * import { defer, of, repeat } from 'rxjs';\n *\n * const source = defer(() => {\n *    return of(`Hello, it is ${new Date()}`)\n * });\n *\n * source.pipe(\n *    // Repeat 3 times with a delay of 1 second between repetitions\n *    repeat({\n *      count: 3,\n *      delay: 1000,\n *    }),\n *\n *    // *Then* repeat forever, but with an exponential step-back\n *    // maxing out at 1 minute.\n *    repeat({\n *      delay: (count) => timer(Math.min(60000, 2 ^ count * 1000))\n *    })\n * )\n * ```\n *\n * @see {@link repeatWhen}\n * @see {@link retry}\n *\n * @param count The number of times the source Observable items are repeated, a count of 0 will yield\n * an empty Observable.\n */\nexport function repeat<T>(countOrConfig?: number | RepeatConfig): MonoTypeOperatorFunction<T> {\n  let count = Infinity;\n  let delay: RepeatConfig['delay'];\n\n  if (countOrConfig != null) {\n    if (typeof countOrConfig === 'object') {\n      ({ count = Infinity, delay } = countOrConfig);\n    } else {\n      count = countOrConfig;\n    }\n  }\n\n  return count <= 0\n    ? () => EMPTY\n    : operate((source, subscriber) => {\n        let soFar = 0;\n        let sourceSub: Subscription | null;\n\n        const resubscribe = () => {\n          sourceSub?.unsubscribe();\n          sourceSub = null;\n          if (delay != null) {\n            const notifier = typeof delay === 'number' ? timer(delay) : innerFrom(delay(soFar));\n            const notifierSubscriber = createOperatorSubscriber(subscriber, () => {\n              notifierSubscriber.unsubscribe();\n              subscribeToSource();\n            });\n            notifier.subscribe(notifierSubscriber);\n          } else {\n            subscribeToSource();\n          }\n        };\n\n        const subscribeToSource = () => {\n          let syncUnsub = false;\n          sourceSub = source.subscribe(\n            createOperatorSubscriber(subscriber, undefined, () => {\n              if (++soFar < count) {\n                if (sourceSub) {\n                  resubscribe();\n                } else {\n                  syncUnsub = true;\n                }\n              } else {\n                subscriber.complete();\n              }\n            })\n          );\n\n          if (syncUnsub) {\n            resubscribe();\n          }\n        };\n\n        subscribeToSource();\n      });\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\nimport { noop } from '../util/noop';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/**\n * Emits the most recently emitted value from the source Observable whenever\n * another Observable, the `notifier`, emits.\n *\n * <span class=\"informal\">It's like {@link sampleTime}, but samples whenever\n * the `notifier` `ObservableInput` emits something.</span>\n *\n * ![](sample.png)\n *\n * Whenever the `notifier` `ObservableInput` emits a value, `sample`\n * looks at the source Observable and emits whichever value it has most recently\n * emitted since the previous sampling, unless the source has not emitted\n * anything since the previous sampling. The `notifier` is subscribed to as soon\n * as the output Observable is subscribed.\n *\n * ## Example\n *\n * On every click, sample the most recent `seconds` timer\n *\n * ```ts\n * import { fromEvent, interval, sample } from 'rxjs';\n *\n * const seconds = interval(1000);\n * const clicks = fromEvent(document, 'click');\n * const result = seconds.pipe(sample(clicks));\n *\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link audit}\n * @see {@link debounce}\n * @see {@link sampleTime}\n * @see {@link throttle}\n *\n * @param notifier The `ObservableInput` to use for sampling the\n * source Observable.\n * @return A function that returns an Observable that emits the results of\n * sampling the values emitted by the source Observable whenever the notifier\n * Observable emits value or completes.\n */\nexport function sample<T>(notifier: ObservableInput<any>): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let hasValue = false;\n    let lastValue: T | null = null;\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value) => {\n        hasValue = true;\n        lastValue = value;\n      })\n    );\n    innerFrom(notifier).subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        () => {\n          if (hasValue) {\n            hasValue = false;\n            const value = lastValue!;\n            lastValue = null;\n            subscriber.next(value);\n          }\n        },\n        noop\n      )\n    );\n  });\n}\n", "import { OperatorFunction } from '../types';\nimport { operate } from '../util/lift';\nimport { scanInternals } from './scanInternals';\n\nexport function scan<V, A = V>(accumulator: (acc: A | V, value: V, index: number) => A): OperatorFunction<V, V | A>;\nexport function scan<V, A>(accumulator: (acc: A, value: V, index: number) => A, seed: A): OperatorFunction<V, A>;\nexport function scan<V, A, S>(accumulator: (acc: A | S, value: V, index: number) => A, seed: S): OperatorFunction<V, A>;\n\n// TODO: link to a \"redux pattern\" section in the guide (location TBD)\n\n/**\n * Useful for encapsulating and managing state. Applies an accumulator (or \"reducer function\")\n * to each value from the source after an initial state is established -- either via\n * a `seed` value (second argument), or from the first value from the source.\n *\n * <span class=\"informal\">It's like {@link reduce}, but emits the current\n * accumulation state after each update</span>\n *\n * ![](scan.png)\n *\n * This operator maintains an internal state and emits it after processing each value as follows:\n *\n * 1. First value arrives\n *   - If a `seed` value was supplied (as the second argument to `scan`), let `state = seed` and `value = firstValue`.\n *   - If NO `seed` value was supplied (no second argument), let `state = firstValue` and go to 3.\n * 2. Let `state = accumulator(state, value)`.\n *   - If an error is thrown by `accumulator`, notify the consumer of an error. The process ends.\n * 3. Emit `state`.\n * 4. Next value arrives, let `value = nextValue`, go to 2.\n *\n * ## Examples\n *\n * An average of previous numbers. This example shows how\n * not providing a `seed` can prime the stream with the\n * first value from the source.\n *\n * ```ts\n * import { of, scan, map } from 'rxjs';\n *\n * const numbers$ = of(1, 2, 3);\n *\n * numbers$\n *   .pipe(\n *     // Get the sum of the numbers coming in.\n *     scan((total, n) => total + n),\n *     // Get the average by dividing the sum by the total number\n *     // received so far (which is 1 more than the zero-based index).\n *     map((sum, index) => sum / (index + 1))\n *   )\n *   .subscribe(console.log);\n * ```\n *\n * The Fibonacci sequence. This example shows how you can use\n * a seed to prime accumulation process. Also... you know... Fibonacci.\n * So important to like, computers and stuff that its whiteboarded\n * in job interviews. Now you can show them the Rx version! (Please don't, haha)\n *\n * ```ts\n * import { interval, scan, map, startWith } from 'rxjs';\n *\n * const firstTwoFibs = [0, 1];\n * // An endless stream of Fibonacci numbers.\n * const fibonacci$ = interval(1000).pipe(\n *   // Scan to get the fibonacci numbers (after 0, 1)\n *   scan(([a, b]) => [b, a + b], firstTwoFibs),\n *   // Get the second number in the tuple, it's the one you calculated\n *   map(([, n]) => n),\n *   // Start with our first two digits :)\n *   startWith(...firstTwoFibs)\n * );\n *\n * fibonacci$.subscribe(console.log);\n * ```\n *\n * @see {@link expand}\n * @see {@link mergeScan}\n * @see {@link reduce}\n * @see {@link switchScan}\n *\n * @param accumulator A \"reducer function\". This will be called for each value after an initial state is\n * acquired.\n * @param seed The initial state. If this is not provided, the first value from the source will\n * be used as the initial state, and emitted without going through the accumulator. All subsequent values\n * will be processed by the accumulator function. If this is provided, all values will go through\n * the accumulator function.\n * @return A function that returns an Observable of the accumulated values.\n */\nexport function scan<V, A, S>(accumulator: (acc: V | A | S, value: V, index: number) => A, seed?: S): OperatorFunction<V, V | A> {\n  // providing a seed of `undefined` *should* be valid and trigger\n  // hasSeed! so don't use `seed !== undefined` checks!\n  // For this reason, we have to check it here at the original call site\n  // otherwise inside Operator/Subscriber we won't know if `undefined`\n  // means they didn't provide anything or if they literally provided `undefined`\n  return operate(scanInternals(accumulator, seed as S, arguments.length >= 2, true));\n}\n", "import { innerFrom } from '../observable/innerFrom';\nimport { Subject } from '../Subject';\nimport { SafeSubscriber } from '../Subscriber';\nimport { Subscription } from '../Subscription';\nimport { MonoTypeOperatorFunction, SubjectLike, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\n\nexport interface ShareConfig<T> {\n  /**\n   * The factory used to create the subject that will connect the source observable to\n   * multicast consumers.\n   */\n  connector?: () => SubjectLike<T>;\n  /**\n   * If `true`, the resulting observable will reset internal state on error from source and return to a \"cold\" state. This\n   * allows the resulting observable to be \"retried\" in the event of an error.\n   * If `false`, when an error comes from the source it will push the error into the connecting subject, and the subject\n   * will remain the connecting subject, meaning the resulting observable will not go \"cold\" again, and subsequent retries\n   * or resubscriptions will resubscribe to that same subject. In all cases, RxJS subjects will emit the same error again, however\n   * {@link ReplaySubject} will also push its buffered values before pushing the error.\n   * It is also possible to pass a notifier factory returning an `ObservableInput` instead which grants more fine-grained\n   * control over how and when the reset should happen. This allows behaviors like conditional or delayed resets.\n   */\n  resetOnError?: boolean | ((error: any) => ObservableInput<any>);\n  /**\n   * If `true`, the resulting observable will reset internal state on completion from source and return to a \"cold\" state. This\n   * allows the resulting observable to be \"repeated\" after it is done.\n   * If `false`, when the source completes, it will push the completion through the connecting subject, and the subject\n   * will remain the connecting subject, meaning the resulting observable will not go \"cold\" again, and subsequent repeats\n   * or resubscriptions will resubscribe to that same subject.\n   * It is also possible to pass a notifier factory returning an `ObservableInput` instead which grants more fine-grained\n   * control over how and when the reset should happen. This allows behaviors like conditional or delayed resets.\n   */\n  resetOnComplete?: boolean | (() => ObservableInput<any>);\n  /**\n   * If `true`, when the number of subscribers to the resulting observable reaches zero due to those subscribers unsubscribing, the\n   * internal state will be reset and the resulting observable will return to a \"cold\" state. This means that the next\n   * time the resulting observable is subscribed to, a new subject will be created and the source will be subscribed to\n   * again.\n   * If `false`, when the number of subscribers to the resulting observable reaches zero due to unsubscription, the subject\n   * will remain connected to the source, and new subscriptions to the result will be connected through that same subject.\n   * It is also possible to pass a notifier factory returning an `ObservableInput` instead which grants more fine-grained\n   * control over how and when the reset should happen. This allows behaviors like conditional or delayed resets.\n   */\n  resetOnRefCountZero?: boolean | (() => ObservableInput<any>);\n}\n\nexport function share<T>(): MonoTypeOperatorFunction<T>;\n\nexport function share<T>(options: ShareConfig<T>): MonoTypeOperatorFunction<T>;\n\n/**\n * Returns a new Observable that multicasts (shares) the original Observable. As long as there is at least one\n * Subscriber this Observable will be subscribed and emitting data. When all subscribers have unsubscribed it will\n * unsubscribe from the source Observable. Because the Observable is multicasting it makes the stream `hot`.\n * This is an alias for `multicast(() => new Subject()), refCount()`.\n *\n * The subscription to the underlying source Observable can be reset (unsubscribe and resubscribe for new subscribers),\n * if the subscriber count to the shared observable drops to 0, or if the source Observable errors or completes. It is\n * possible to use notifier factories for the resets to allow for behaviors like conditional or delayed resets. Please\n * note that resetting on error or complete of the source Observable does not behave like a transparent retry or restart\n * of the source because the error or complete will be forwarded to all subscribers and their subscription will be\n * closed. Only new subscribers after a reset on error or complete happened will cause a fresh subscription to the\n * source. To achieve transparent retries or restarts pipe the source through appropriate operators before sharing.\n *\n * ![](share.png)\n *\n * ## Example\n *\n * Generate new multicast Observable from the `source` Observable value\n *\n * ```ts\n * import { interval, tap, map, take, share } from 'rxjs';\n *\n * const source = interval(1000).pipe(\n *   tap(x => console.log('Processing: ', x)),\n *   map(x => x * x),\n *   take(6),\n *   share()\n * );\n *\n * source.subscribe(x => console.log('subscription 1: ', x));\n * source.subscribe(x => console.log('subscription 2: ', x));\n *\n * // Logs:\n * // Processing: 0\n * // subscription 1: 0\n * // subscription 2: 0\n * // Processing: 1\n * // subscription 1: 1\n * // subscription 2: 1\n * // Processing: 2\n * // subscription 1: 4\n * // subscription 2: 4\n * // Processing: 3\n * // subscription 1: 9\n * // subscription 2: 9\n * // Processing: 4\n * // subscription 1: 16\n * // subscription 2: 16\n * // Processing: 5\n * // subscription 1: 25\n * // subscription 2: 25\n * ```\n *\n * ## Example with notifier factory: Delayed reset\n *\n * ```ts\n * import { interval, take, share, timer } from 'rxjs';\n *\n * const source = interval(1000).pipe(\n *   take(3),\n *   share({\n *     resetOnRefCountZero: () => timer(1000)\n *   })\n * );\n *\n * const subscriptionOne = source.subscribe(x => console.log('subscription 1: ', x));\n * setTimeout(() => subscriptionOne.unsubscribe(), 1300);\n *\n * setTimeout(() => source.subscribe(x => console.log('subscription 2: ', x)), 1700);\n *\n * setTimeout(() => source.subscribe(x => console.log('subscription 3: ', x)), 5000);\n *\n * // Logs:\n * // subscription 1:  0\n * // (subscription 1 unsubscribes here)\n * // (subscription 2 subscribes here ~400ms later, source was not reset)\n * // subscription 2:  1\n * // subscription 2:  2\n * // (subscription 2 unsubscribes here)\n * // (subscription 3 subscribes here ~2000ms later, source did reset before)\n * // subscription 3:  0\n * // subscription 3:  1\n * // subscription 3:  2\n * ```\n *\n * @see {@link shareReplay}\n *\n * @return A function that returns an Observable that mirrors the source.\n */\nexport function share<T>(options: ShareConfig<T> = {}): MonoTypeOperatorFunction<T> {\n  const { connector = () => new Subject<T>(), resetOnError = true, resetOnComplete = true, resetOnRefCountZero = true } = options;\n  // It's necessary to use a wrapper here, as the _operator_ must be\n  // referentially transparent. Otherwise, it cannot be used in calls to the\n  // static `pipe` function - to create a partial pipeline.\n  //\n  // The _operator function_ - the function returned by the _operator_ - will\n  // not be referentially transparent - as it shares its source - but the\n  // _operator function_ is called when the complete pipeline is composed via a\n  // call to a source observable's `pipe` method - not when the static `pipe`\n  // function is called.\n  return (wrapperSource) => {\n    let connection: SafeSubscriber<T> | undefined;\n    let resetConnection: Subscription | undefined;\n    let subject: SubjectLike<T> | undefined;\n    let refCount = 0;\n    let hasCompleted = false;\n    let hasErrored = false;\n\n    const cancelReset = () => {\n      resetConnection?.unsubscribe();\n      resetConnection = undefined;\n    };\n    // Used to reset the internal state to a \"cold\"\n    // state, as though it had never been subscribed to.\n    const reset = () => {\n      cancelReset();\n      connection = subject = undefined;\n      hasCompleted = hasErrored = false;\n    };\n    const resetAndUnsubscribe = () => {\n      // We need to capture the connection before\n      // we reset (if we need to reset).\n      const conn = connection;\n      reset();\n      conn?.unsubscribe();\n    };\n\n    return operate<T, T>((source, subscriber) => {\n      refCount++;\n      if (!hasErrored && !hasCompleted) {\n        cancelReset();\n      }\n\n      // Create the subject if we don't have one yet. Grab a local reference to\n      // it as well, which avoids non-null assertions when using it and, if we\n      // connect to it now, then error/complete need a reference after it was\n      // reset.\n      const dest = (subject = subject ?? connector());\n\n      // Add the finalization directly to the subscriber - instead of returning it -\n      // so that the handling of the subscriber's unsubscription will be wired\n      // up _before_ the subscription to the source occurs. This is done so that\n      // the assignment to the source connection's `closed` property will be seen\n      // by synchronous firehose sources.\n      subscriber.add(() => {\n        refCount--;\n\n        // If we're resetting on refCount === 0, and it's 0, we only want to do\n        // that on \"unsubscribe\", really. Resetting on error or completion is a different\n        // configuration.\n        if (refCount === 0 && !hasErrored && !hasCompleted) {\n          resetConnection = handleReset(resetAndUnsubscribe, resetOnRefCountZero);\n        }\n      });\n\n      // The following line adds the subscription to the subscriber passed.\n      // Basically, `subscriber === dest.subscribe(subscriber)` is `true`.\n      dest.subscribe(subscriber);\n\n      if (\n        !connection &&\n        // Check this shareReplay is still activate - it can be reset to 0\n        // and be \"unsubscribed\" _before_ it actually subscribes.\n        // If we were to subscribe then, it'd leak and get stuck.\n        refCount > 0\n      ) {\n        // We need to create a subscriber here - rather than pass an observer and\n        // assign the returned subscription to connection - because it's possible\n        // for reentrant subscriptions to the shared observable to occur and in\n        // those situations we want connection to be already-assigned so that we\n        // don't create another connection to the source.\n        connection = new SafeSubscriber({\n          next: (value) => dest.next(value),\n          error: (err) => {\n            hasErrored = true;\n            cancelReset();\n            resetConnection = handleReset(reset, resetOnError, err);\n            dest.error(err);\n          },\n          complete: () => {\n            hasCompleted = true;\n            cancelReset();\n            resetConnection = handleReset(reset, resetOnComplete);\n            dest.complete();\n          },\n        });\n        innerFrom(source).subscribe(connection);\n      }\n    })(wrapperSource);\n  };\n}\n\nfunction handleReset<T extends unknown[] = never[]>(\n  reset: () => void,\n  on: boolean | ((...args: T) => ObservableInput<any>),\n  ...args: T\n): Subscription | undefined {\n  if (on === true) {\n    reset();\n    return;\n  }\n\n  if (on === false) {\n    return;\n  }\n\n  const onSubscriber = new SafeSubscriber({\n    next: () => {\n      onSubscriber.unsubscribe();\n      reset();\n    },\n  });\n\n  return innerFrom(on(...args)).subscribe(onSubscriber);\n}\n", "import { ReplaySubject } from '../ReplaySubject';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { share } from './share';\n\nexport interface ShareReplayConfig {\n  bufferSize?: number;\n  windowTime?: number;\n  refCount: boolean;\n  scheduler?: SchedulerLike;\n}\n\nexport function shareReplay<T>(config: ShareReplayConfig): MonoTypeOperatorFunction<T>;\nexport function shareReplay<T>(bufferSize?: number, windowTime?: number, scheduler?: SchedulerLike): MonoTypeOperatorFunction<T>;\n\n/**\n * Share source and replay specified number of emissions on subscription.\n *\n * This operator is a specialization of `replay` that connects to a source observable\n * and multicasts through a `ReplaySubject` constructed with the specified arguments.\n * A successfully completed source will stay cached in the `shareReplay`ed observable forever,\n * but an errored source can be retried.\n *\n * ## Why use `shareReplay`?\n *\n * You generally want to use `shareReplay` when you have side-effects or taxing computations\n * that you do not wish to be executed amongst multiple subscribers.\n * It may also be valuable in situations where you know you will have late subscribers to\n * a stream that need access to previously emitted values.\n * This ability to replay values on subscription is what differentiates {@link share} and `shareReplay`.\n *\n * ## Reference counting\n *\n * By default `shareReplay` will use `refCount` of false, meaning that it will _not_ unsubscribe the\n * source when the reference counter drops to zero, i.e. the inner `ReplaySubject` will _not_ be unsubscribed\n * (and potentially run for ever).\n * This is the default as it is expected that `shareReplay` is often used to keep around expensive to setup\n * observables which we want to keep running instead of having to do the expensive setup again.\n *\n * As of RXJS version 6.4.0 a new overload signature was added to allow for manual control over what\n * happens when the operators internal reference counter drops to zero.\n * If `refCount` is true, the source will be unsubscribed from once the reference count drops to zero, i.e.\n * the inner `ReplaySubject` will be unsubscribed. All new subscribers will receive value emissions from a\n * new `ReplaySubject` which in turn will cause a new subscription to the source observable.\n *\n * ## Examples\n *\n * Example with a third subscriber coming late to the party\n *\n * ```ts\n * import { interval, take, shareReplay } from 'rxjs';\n *\n * const shared$ = interval(2000).pipe(\n *   take(6),\n *   shareReplay(3)\n * );\n *\n * shared$.subscribe(x => console.log('sub A: ', x));\n * shared$.subscribe(y => console.log('sub B: ', y));\n *\n * setTimeout(() => {\n *   shared$.subscribe(y => console.log('sub C: ', y));\n * }, 11000);\n *\n * // Logs:\n * // (after ~2000 ms)\n * // sub A: 0\n * // sub B: 0\n * // (after ~4000 ms)\n * // sub A: 1\n * // sub B: 1\n * // (after ~6000 ms)\n * // sub A: 2\n * // sub B: 2\n * // (after ~8000 ms)\n * // sub A: 3\n * // sub B: 3\n * // (after ~10000 ms)\n * // sub A: 4\n * // sub B: 4\n * // (after ~11000 ms, sub C gets the last 3 values)\n * // sub C: 2\n * // sub C: 3\n * // sub C: 4\n * // (after ~12000 ms)\n * // sub A: 5\n * // sub B: 5\n * // sub C: 5\n * ```\n *\n * Example for `refCount` usage\n *\n * ```ts\n * import { Observable, tap, interval, shareReplay, take } from 'rxjs';\n *\n * const log = <T>(name: string, source: Observable<T>) => source.pipe(\n *   tap({\n *     subscribe: () => console.log(`${ name }: subscribed`),\n *     next: value => console.log(`${ name }: ${ value }`),\n *     complete: () => console.log(`${ name }: completed`),\n *     finalize: () => console.log(`${ name }: unsubscribed`)\n *   })\n * );\n *\n * const obs$ = log('source', interval(1000));\n *\n * const shared$ = log('shared', obs$.pipe(\n *   shareReplay({ bufferSize: 1, refCount: true }),\n *   take(2)\n * ));\n *\n * shared$.subscribe(x => console.log('sub A: ', x));\n * shared$.subscribe(y => console.log('sub B: ', y));\n *\n * // PRINTS:\n * // shared: subscribed <-- reference count = 1\n * // source: subscribed\n * // shared: subscribed <-- reference count = 2\n * // source: 0\n * // shared: 0\n * // sub A: 0\n * // shared: 0\n * // sub B: 0\n * // source: 1\n * // shared: 1\n * // sub A: 1\n * // shared: completed <-- take(2) completes the subscription for sub A\n * // shared: unsubscribed <-- reference count = 1\n * // shared: 1\n * // sub B: 1\n * // shared: completed <-- take(2) completes the subscription for sub B\n * // shared: unsubscribed <-- reference count = 0\n * // source: unsubscribed <-- replaySubject unsubscribes from source observable because the reference count dropped to 0 and refCount is true\n *\n * // In case of refCount being false, the unsubscribe is never called on the source and the source would keep on emitting, even if no subscribers\n * // are listening.\n * // source: 2\n * // source: 3\n * // source: 4\n * // ...\n * ```\n *\n * @see {@link publish}\n * @see {@link share}\n * @see {@link publishReplay}\n *\n * @param configOrBufferSize Maximum element count of the replay buffer or {@link ShareReplayConfig configuration}\n * object.\n * @param windowTime Maximum time length of the replay buffer in milliseconds.\n * @param scheduler Scheduler where connected observers within the selector function\n * will be invoked on.\n * @return A function that returns an Observable sequence that contains the\n * elements of a sequence produced by multicasting the source sequence within a\n * selector function.\n */\nexport function shareReplay<T>(\n  configOrBufferSize?: ShareReplayConfig | number,\n  windowTime?: number,\n  scheduler?: SchedulerLike\n): MonoTypeOperatorFunction<T> {\n  let bufferSize: number;\n  let refCount = false;\n  if (configOrBufferSize && typeof configOrBufferSize === 'object') {\n    ({ bufferSize = Infinity, windowTime = Infinity, refCount = false, scheduler } = configOrBufferSize);\n  } else {\n    bufferSize = (configOrBufferSize ?? Infinity) as number;\n  }\n  return share<T>({\n    connector: () => new ReplaySubject(bufferSize, windowTime, scheduler),\n    resetOnError: true,\n    resetOnComplete: false,\n    resetOnRefCountZero: refCount,\n  });\n}\n", "import { MonoTypeOperatorFunction } from '../types';\nimport { filter } from './filter';\n\n/**\n * Returns an Observable that skips the first `count` items emitted by the source Observable.\n *\n * ![](skip.png)\n *\n * Skips the values until the sent notifications are equal or less than provided skip count. It raises\n * an error if skip count is equal or more than the actual number of emits and source raises an error.\n *\n * ## Example\n *\n * Skip the values before the emission\n *\n * ```ts\n * import { interval, skip } from 'rxjs';\n *\n * // emit every half second\n * const source = interval(500);\n * // skip the first 10 emitted values\n * const result = source.pipe(skip(10));\n *\n * result.subscribe(value => console.log(value));\n * // output: 10...11...12...13...\n * ```\n *\n * @see {@link last}\n * @see {@link skipWhile}\n * @see {@link skipUntil}\n * @see {@link skipLast}\n *\n * @param {Number} count - The number of times, items emitted by source Observable should be skipped.\n * @return A function that returns an Observable that skips the first `count`\n * values emitted by the source Observable.\n */\nexport function skip<T>(count: number): MonoTypeOperatorFunction<T> {\n  return filter((_, index) => count <= index);\n}\n", "import { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { noop } from '../util/noop';\n\n/**\n * Returns an Observable that skips items emitted by the source Observable until a second Observable emits an item.\n *\n * The `skipUntil` operator causes the observable stream to skip the emission of values until the passed in observable\n * emits the first value. This can be particularly useful in combination with user interactions, responses of HTTP\n * requests or waiting for specific times to pass by.\n *\n * ![](skipUntil.png)\n *\n * Internally, the `skipUntil` operator subscribes to the passed in `notifier` `ObservableInput` (which gets converted\n * to an Observable) in order to recognize the emission of its first value. When `notifier` emits next, the operator\n * unsubscribes from it and starts emitting the values of the *source* observable until it completes or errors. It\n * will never let the *source* observable emit any values if the `notifier` completes or throws an error without\n * emitting a value before.\n *\n * ## Example\n *\n * In the following example, all emitted values of the interval observable are skipped until the user clicks anywhere\n * within the page\n *\n * ```ts\n * import { interval, fromEvent, skipUntil } from 'rxjs';\n *\n * const intervalObservable = interval(1000);\n * const click = fromEvent(document, 'click');\n *\n * const emitAfterClick = intervalObservable.pipe(\n *   skipUntil(click)\n * );\n * // clicked at 4.6s. output: 5...6...7...8........ or\n * // clicked at 7.3s. output: 8...9...10..11.......\n * emitAfterClick.subscribe(value => console.log(value));\n * ```\n *\n * @see {@link last}\n * @see {@link skip}\n * @see {@link skipWhile}\n * @see {@link skipLast}\n *\n * @param notifier An `ObservableInput` that has to emit an item before the source Observable elements begin to\n * be mirrored by the resulting Observable.\n * @return A function that returns an Observable that skips items from the\n * source Observable until the `notifier` Observable emits an item, then emits the\n * remaining items.\n */\nexport function skipUntil<T>(notifier: ObservableInput<any>): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let taking = false;\n\n    const skipSubscriber = createOperatorSubscriber(\n      subscriber,\n      () => {\n        skipSubscriber?.unsubscribe();\n        taking = true;\n      },\n      noop\n    );\n\n    innerFrom(notifier).subscribe(skipSubscriber);\n\n    source.subscribe(createOperatorSubscriber(subscriber, (value) => taking && subscriber.next(value)));\n  });\n}\n", "import { concat } from '../observable/concat';\nimport { OperatorFunction, SchedulerLike, ValueFromArray } from '../types';\nimport { popScheduler } from '../util/args';\nimport { operate } from '../util/lift';\n\n// Devs are more likely to pass null or undefined than they are a scheduler\n// without accompanying values. To make things easier for (naughty) devs who\n// use the `strictNullChecks: false` TypeScript compiler option, these\n// overloads with explicit null and undefined values are included.\n\nexport function startWith<T>(value: null): OperatorFunction<T, T | null>;\nexport function startWith<T>(value: undefined): OperatorFunction<T, T | undefined>;\n\n/** @deprecated The `scheduler` parameter will be removed in v8. Use `scheduled` and `concatAll`. Details: https://rxjs.dev/deprecations/scheduler-argument */\nexport function startWith<T, A extends readonly unknown[] = T[]>(\n  ...valuesAndScheduler: [...A, SchedulerLike]\n): OperatorFunction<T, T | ValueFromArray<A>>;\nexport function startWith<T, A extends readonly unknown[] = T[]>(...values: A): OperatorFunction<T, T | ValueFromArray<A>>;\n\n/**\n * Returns an observable that, at the moment of subscription, will synchronously emit all\n * values provided to this operator, then subscribe to the source and mirror all of its emissions\n * to subscribers.\n *\n * This is a useful way to know when subscription has occurred on an existing observable.\n *\n * <span class=\"informal\">First emits its arguments in order, and then any\n * emissions from the source.</span>\n *\n * ![](startWith.png)\n *\n * ## Examples\n *\n * Emit a value when a timer starts.\n *\n * ```ts\n * import { timer, map, startWith } from 'rxjs';\n *\n * timer(1000)\n *   .pipe(\n *     map(() => 'timer emit'),\n *     startWith('timer start')\n *   )\n *   .subscribe(x => console.log(x));\n *\n * // results:\n * // 'timer start'\n * // 'timer emit'\n * ```\n *\n * @param values Items you want the modified Observable to emit first.\n * @return A function that returns an Observable that synchronously emits\n * provided values before subscribing to the source Observable.\n *\n * @see {@link endWith}\n * @see {@link finalize}\n * @see {@link concat}\n */\nexport function startWith<T, D>(...values: D[]): OperatorFunction<T, T | D> {\n  const scheduler = popScheduler(values);\n  return operate((source, subscriber) => {\n    // Here we can't pass `undefined` as a scheduler, because if we did, the\n    // code inside of `concat` would be confused by the `undefined`, and treat it\n    // like an invalid observable. So we have to split it two different ways.\n    (scheduler ? concat(values, source, scheduler) : concat(values, source)).subscribe(subscriber);\n  });\n}\n", "import { Subscriber } from '../Subscriber';\nimport { ObservableInput, OperatorFunction, ObservedValueOf } from '../types';\nimport { innerFrom } from '../observable/innerFrom';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\n/* tslint:disable:max-line-length */\nexport function switchMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function switchMap<T, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: undefined\n): OperatorFunction<T, ObservedValueOf<O>>;\n/** @deprecated The `resultSelector` parameter will be removed in v8. Use an inner `map` instead. Details: https://rxjs.dev/deprecations/resultSelector */\nexport function switchMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector: (outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R\n): OperatorFunction<T, R>;\n/* tslint:enable:max-line-length */\n\n/**\n * Projects each source value to an Observable which is merged in the output\n * Observable, emitting values only from the most recently projected Observable.\n *\n * <span class=\"informal\">Maps each value to an Observable, then flattens all of\n * these inner Observables using {@link switchAll}.</span>\n *\n * ![](switchMap.png)\n *\n * Returns an Observable that emits items based on applying a function that you\n * supply to each item emitted by the source Observable, where that function\n * returns an (so-called \"inner\") Observable. Each time it observes one of these\n * inner Observables, the output Observable begins emitting the items emitted by\n * that inner Observable. When a new inner Observable is emitted, `switchMap`\n * stops emitting items from the earlier-emitted inner Observable and begins\n * emitting items from the new one. It continues to behave like this for\n * subsequent inner Observables.\n *\n * ## Example\n *\n * Generate new Observable according to source Observable values\n *\n * ```ts\n * import { of, switchMap } from 'rxjs';\n *\n * const switched = of(1, 2, 3).pipe(switchMap(x => of(x, x ** 2, x ** 3)));\n * switched.subscribe(x => console.log(x));\n * // outputs\n * // 1\n * // 1\n * // 1\n * // 2\n * // 4\n * // 8\n * // 3\n * // 9\n * // 27\n * ```\n *\n * Restart an interval Observable on every click event\n *\n * ```ts\n * import { fromEvent, switchMap, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(switchMap(() => interval(1000)));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link concatMap}\n * @see {@link exhaustMap}\n * @see {@link mergeMap}\n * @see {@link switchAll}\n * @see {@link switchMapTo}\n *\n * @param {function(value: T, index: number): ObservableInput} project A function\n * that, when applied to an item emitted by the source Observable, returns an\n * Observable.\n * @return A function that returns an Observable that emits the result of\n * applying the projection function (and the optional deprecated\n * `resultSelector`) to each item emitted by the source Observable and taking\n * only the values from the most recently projected inner Observable.\n */\nexport function switchMap<T, R, O extends ObservableInput<any>>(\n  project: (value: T, index: number) => O,\n  resultSelector?: (outerValue: T, innerValue: ObservedValueOf<O>, outerIndex: number, innerIndex: number) => R\n): OperatorFunction<T, ObservedValueOf<O> | R> {\n  return operate((source, subscriber) => {\n    let innerSubscriber: Subscriber<ObservedValueOf<O>> | null = null;\n    let index = 0;\n    // Whether or not the source subscription has completed\n    let isComplete = false;\n\n    // We only complete the result if the source is complete AND we don't have an active inner subscription.\n    // This is called both when the source completes and when the inners complete.\n    const checkComplete = () => isComplete && !innerSubscriber && subscriber.complete();\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        (value) => {\n          // Cancel the previous inner subscription if there was one\n          innerSubscriber?.unsubscribe();\n          let innerIndex = 0;\n          const outerIndex = index++;\n          // Start the next inner subscription\n          innerFrom(project(value, outerIndex)).subscribe(\n            (innerSubscriber = createOperatorSubscriber(\n              subscriber,\n              // When we get a new inner value, next it through. Note that this is\n              // handling the deprecate result selector here. This is because with this architecture\n              // it ends up being smaller than using the map operator.\n              (innerValue) => subscriber.next(resultSelector ? resultSelector(value, innerValue, outerIndex, innerIndex++) : innerValue),\n              () => {\n                // The inner has completed. Null out the inner subscriber to\n                // free up memory and to signal that we have no inner subscription\n                // currently.\n                innerSubscriber = null!;\n                checkComplete();\n              }\n            ))\n          );\n        },\n        () => {\n          isComplete = true;\n          checkComplete();\n        }\n      )\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { noop } from '../util/noop';\n\n/**\n * Emits the values emitted by the source Observable until a `notifier`\n * Observable emits a value.\n *\n * <span class=\"informal\">Lets values pass until a second Observable,\n * `notifier`, emits a value. Then, it completes.</span>\n *\n * ![](takeUntil.png)\n *\n * `takeUntil` subscribes and begins mirroring the source Observable. It also\n * monitors a second Observable, `notifier` that you provide. If the `notifier`\n * emits a value, the output Observable stops mirroring the source Observable\n * and completes. If the `notifier` doesn't emit any value and completes\n * then `takeUntil` will pass all values.\n *\n * ## Example\n *\n * Tick every second until the first click happens\n *\n * ```ts\n * import { interval, fromEvent, takeUntil } from 'rxjs';\n *\n * const source = interval(1000);\n * const clicks = fromEvent(document, 'click');\n * const result = source.pipe(takeUntil(clicks));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link take}\n * @see {@link takeLast}\n * @see {@link takeWhile}\n * @see {@link skip}\n *\n * @param {Observable} notifier The Observable whose first emitted value will\n * cause the output Observable of `takeUntil` to stop emitting values from the\n * source Observable.\n * @return A function that returns an Observable that emits the values from the\n * source Observable until `notifier` emits its first value.\n */\nexport function takeUntil<T>(notifier: ObservableInput<any>): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    innerFrom(notifier).subscribe(createOperatorSubscriber(subscriber, () => subscriber.complete(), noop));\n    !subscriber.closed && source.subscribe(subscriber);\n  });\n}\n", "import { OperatorFunction, MonoTypeOperatorFunction, TruthyTypesOf } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\n\nexport function takeWhile<T>(predicate: BooleanConstructor, inclusive: true): MonoTypeOperatorFunction<T>;\nexport function takeWhile<T>(predicate: BooleanConstructor, inclusive: false): OperatorFunction<T, TruthyTypesOf<T>>;\nexport function takeWhile<T>(predicate: BooleanConstructor): OperatorFunction<T, TruthyTypesOf<T>>;\nexport function takeWhile<T, S extends T>(predicate: (value: T, index: number) => value is S): OperatorFunction<T, S>;\nexport function takeWhile<T, S extends T>(predicate: (value: T, index: number) => value is S, inclusive: false): OperatorFunction<T, S>;\nexport function takeWhile<T>(predicate: (value: T, index: number) => boolean, inclusive?: boolean): MonoTypeOperatorFunction<T>;\n\n/**\n * Emits values emitted by the source Observable so long as each value satisfies\n * the given `predicate`, and then completes as soon as this `predicate` is not\n * satisfied.\n *\n * <span class=\"informal\">Takes values from the source only while they pass the\n * condition given. When the first value does not satisfy, it completes.</span>\n *\n * ![](takeWhile.png)\n *\n * `takeWhile` subscribes and begins mirroring the source Observable. Each value\n * emitted on the source is given to the `predicate` function which returns a\n * boolean, representing a condition to be satisfied by the source values. The\n * output Observable emits the source values until such time as the `predicate`\n * returns false, at which point `takeWhile` stops mirroring the source\n * Observable and completes the output Observable.\n *\n * ## Example\n *\n * Emit click events only while the clientX property is greater than 200\n *\n * ```ts\n * import { fromEvent, takeWhile } from 'rxjs';\n *\n * const clicks = fromEvent<PointerEvent>(document, 'click');\n * const result = clicks.pipe(takeWhile(ev => ev.clientX > 200));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link take}\n * @see {@link takeLast}\n * @see {@link takeUntil}\n * @see {@link skip}\n *\n * @param {function(value: T, index: number): boolean} predicate A function that\n * evaluates a value emitted by the source Observable and returns a boolean.\n * Also takes the (zero-based) index as the second argument.\n * @param {boolean} inclusive When set to `true` the value that caused\n * `predicate` to return `false` will also be emitted.\n * @return A function that returns an Observable that emits values from the\n * source Observable so long as each value satisfies the condition defined by\n * the `predicate`, then completes.\n */\nexport function takeWhile<T>(predicate: (value: T, index: number) => boolean, inclusive = false): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    let index = 0;\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value) => {\n        const result = predicate(value, index++);\n        (result || inclusive) && subscriber.next(value);\n        !result && subscriber.complete();\n      })\n    );\n  });\n}\n", "import { MonoTypeOperatorFunction, Observer } from '../types';\nimport { isFunction } from '../util/isFunction';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { identity } from '../util/identity';\n\n/**\n * An extension to the {@link Observer} interface used only by the {@link tap} operator.\n *\n * It provides a useful set of callbacks a user can register to do side-effects in\n * cases other than what the usual {@link Observer} callbacks are\n * ({@link guide/glossary-and-semantics#next next},\n * {@link guide/glossary-and-semantics#error error} and/or\n * {@link guide/glossary-and-semantics#complete complete}).\n *\n * ## Example\n *\n * ```ts\n * import { fromEvent, switchMap, tap, interval, take } from 'rxjs';\n *\n * const source$ = fromEvent(document, 'click');\n * const result$ = source$.pipe(\n *   switchMap((_, i) => i % 2 === 0\n *     ? fromEvent(document, 'mousemove').pipe(\n *         tap({\n *           subscribe: () => console.log('Subscribed to the mouse move events after click #' + i),\n *           unsubscribe: () => console.log('Mouse move events #' + i + ' unsubscribed'),\n *           finalize: () => console.log('Mouse move events #' + i + ' finalized')\n *         })\n *       )\n *     : interval(1_000).pipe(\n *         take(5),\n *         tap({\n *           subscribe: () => console.log('Subscribed to the 1-second interval events after click #' + i),\n *           unsubscribe: () => console.log('1-second interval events #' + i + ' unsubscribed'),\n *           finalize: () => console.log('1-second interval events #' + i + ' finalized')\n *         })\n *       )\n *   )\n * );\n *\n * const subscription = result$.subscribe({\n *   next: console.log\n * });\n *\n * setTimeout(() => {\n *   console.log('Unsubscribe after 60 seconds');\n *   subscription.unsubscribe();\n * }, 60_000);\n * ```\n */\nexport interface TapObserver<T> extends Observer<T> {\n  /**\n   * The callback that `tap` operator invokes at the moment when the source Observable\n   * gets subscribed to.\n   */\n  subscribe: () => void;\n  /**\n   * The callback that `tap` operator invokes when an explicit\n   * {@link guide/glossary-and-semantics#unsubscription unsubscribe} happens. It won't get invoked on\n   * `error` or `complete` events.\n   */\n  unsubscribe: () => void;\n  /**\n   * The callback that `tap` operator invokes when any kind of\n   * {@link guide/glossary-and-semantics#finalization finalization} happens - either when\n   * the source Observable `error`s or `complete`s or when it gets explicitly unsubscribed\n   * by the user. There is no difference in using this callback or the {@link finalize}\n   * operator, but if you're already using `tap` operator, you can use this callback\n   * instead. You'd get the same result in either case.\n   */\n  finalize: () => void;\n}\nexport function tap<T>(observerOrNext?: Partial<TapObserver<T>> | ((value: T) => void)): MonoTypeOperatorFunction<T>;\n/** @deprecated Instead of passing separate callback arguments, use an observer argument. Signatures taking separate callback arguments will be removed in v8. Details: https://rxjs.dev/deprecations/subscribe-arguments */\nexport function tap<T>(\n  next?: ((value: T) => void) | null,\n  error?: ((error: any) => void) | null,\n  complete?: (() => void) | null\n): MonoTypeOperatorFunction<T>;\n\n/**\n * Used to perform side-effects for notifications from the source observable\n *\n * <span class=\"informal\">Used when you want to affect outside state with a notification without altering the notification</span>\n *\n * ![](tap.png)\n *\n * Tap is designed to allow the developer a designated place to perform side effects. While you _could_ perform side-effects\n * inside of a `map` or a `mergeMap`, that would make their mapping functions impure, which isn't always a big deal, but will\n * make it so you can't do things like memoize those functions. The `tap` operator is designed solely for such side-effects to\n * help you remove side-effects from other operations.\n *\n * For any notification, next, error, or complete, `tap` will call the appropriate callback you have provided to it, via a function\n * reference, or a partial observer, then pass that notification down the stream.\n *\n * The observable returned by `tap` is an exact mirror of the source, with one exception: Any error that occurs -- synchronously -- in a handler\n * provided to `tap` will be emitted as an error from the returned observable.\n *\n * > Be careful! You can mutate objects as they pass through the `tap` operator's handlers.\n *\n * The most common use of `tap` is actually for debugging. You can place a `tap(console.log)` anywhere\n * in your observable `pipe`, log out the notifications as they are emitted by the source returned by the previous\n * operation.\n *\n * ## Examples\n *\n * Check a random number before it is handled. Below is an observable that will use a random number between 0 and 1,\n * and emit `'big'` or `'small'` depending on the size of that number. But we wanted to log what the original number\n * was, so we have added a `tap(console.log)`.\n *\n * ```ts\n * import { of, tap, map } from 'rxjs';\n *\n * of(Math.random()).pipe(\n *   tap(console.log),\n *   map(n => n > 0.5 ? 'big' : 'small')\n * ).subscribe(console.log);\n * ```\n *\n * Using `tap` to analyze a value and force an error. Below is an observable where in our system we only\n * want to emit numbers 3 or less we get from another source. We can force our observable to error\n * using `tap`.\n *\n * ```ts\n * import { of, tap } from 'rxjs';\n *\n * const source = of(1, 2, 3, 4, 5);\n *\n * source.pipe(\n *   tap(n => {\n *     if (n > 3) {\n *       throw new TypeError(`Value ${ n } is greater than 3`);\n *     }\n *   })\n * )\n * .subscribe({ next: console.log, error: err => console.log(err.message) });\n * ```\n *\n * We want to know when an observable completes before moving on to the next observable. The system\n * below will emit a random series of `'X'` characters from 3 different observables in sequence. The\n * only way we know when one observable completes and moves to the next one, in this case, is because\n * we have added a `tap` with the side effect of logging to console.\n *\n * ```ts\n * import { of, concatMap, interval, take, map, tap } from 'rxjs';\n *\n * of(1, 2, 3).pipe(\n *   concatMap(n => interval(1000).pipe(\n *     take(Math.round(Math.random() * 10)),\n *     map(() => 'X'),\n *     tap({ complete: () => console.log(`Done with ${ n }`) })\n *   ))\n * )\n * .subscribe(console.log);\n * ```\n *\n * @see {@link finalize}\n * @see {@link TapObserver}\n *\n * @param observerOrNext A next handler or partial observer\n * @param error An error handler\n * @param complete A completion handler\n * @return A function that returns an Observable identical to the source, but\n * runs the specified Observer or callback(s) for each item.\n */\nexport function tap<T>(\n  observerOrNext?: Partial<TapObserver<T>> | ((value: T) => void) | null,\n  error?: ((e: any) => void) | null,\n  complete?: (() => void) | null\n): MonoTypeOperatorFunction<T> {\n  // We have to check to see not only if next is a function,\n  // but if error or complete were passed. This is because someone\n  // could technically call tap like `tap(null, fn)` or `tap(null, null, fn)`.\n  const tapObserver =\n    isFunction(observerOrNext) || error || complete\n      ? // tslint:disable-next-line: no-object-literal-type-assertion\n        ({ next: observerOrNext as Exclude<typeof observerOrNext, Partial<TapObserver<T>>>, error, complete } as Partial<TapObserver<T>>)\n      : observerOrNext;\n\n  return tapObserver\n    ? operate((source, subscriber) => {\n        tapObserver.subscribe?.();\n        let isUnsub = true;\n        source.subscribe(\n          createOperatorSubscriber(\n            subscriber,\n            (value) => {\n              tapObserver.next?.(value);\n              subscriber.next(value);\n            },\n            () => {\n              isUnsub = false;\n              tapObserver.complete?.();\n              subscriber.complete();\n            },\n            (err) => {\n              isUnsub = false;\n              tapObserver.error?.(err);\n              subscriber.error(err);\n            },\n            () => {\n              if (isUnsub) {\n                tapObserver.unsubscribe?.();\n              }\n              tapObserver.finalize?.();\n            }\n          )\n        );\n      })\n    : // Tap was called with no valid tap observer or handler\n      // (e.g. `tap(null, null, null)` or `tap(null)` or `tap()`)\n      // so we're going to just mirror the source.\n      identity;\n}\n", "import { Subscription } from '../Subscription';\n\nimport { MonoTypeOperatorFunction, ObservableInput } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\n\n/**\n * An object interface used by {@link throttle} or {@link throttleTime} that ensure\n * configuration options of these operators.\n *\n * @see {@link throttle}\n * @see {@link throttleTime}\n */\nexport interface ThrottleConfig {\n  /**\n   * If `true`, the resulting Observable will emit the first value from the source\n   * Observable at the **start** of the \"throttling\" process (when starting an\n   * internal timer that prevents other emissions from the source to pass through).\n   * If `false`, it will not emit the first value from the source Observable at the\n   * start of the \"throttling\" process.\n   *\n   * If not provided, defaults to: `true`.\n   */\n  leading?: boolean;\n  /**\n   * If `true`, the resulting Observable will emit the last value from the source\n   * Observable at the **end** of the \"throttling\" process (when ending an internal\n   * timer that prevents other emissions from the source to pass through).\n   * If `false`, it will not emit the last value from the source Observable at the\n   * end of the \"throttling\" process.\n   *\n   * If not provided, defaults to: `false`.\n   */\n  trailing?: boolean;\n}\n\n/**\n * Emits a value from the source Observable, then ignores subsequent source\n * values for a duration determined by another Observable, then repeats this\n * process.\n *\n * <span class=\"informal\">It's like {@link throttleTime}, but the silencing\n * duration is determined by a second Observable.</span>\n *\n * ![](throttle.svg)\n *\n * `throttle` emits the source Observable values on the output Observable\n * when its internal timer is disabled, and ignores source values when the timer\n * is enabled. Initially, the timer is disabled. As soon as the first source\n * value arrives, it is forwarded to the output Observable, and then the timer\n * is enabled by calling the `durationSelector` function with the source value,\n * which returns the \"duration\" Observable. When the duration Observable emits a\n * value, the timer is disabled, and this process repeats for the\n * next source value.\n *\n * ## Example\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, throttle, interval } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(throttle(() => interval(1000)));\n *\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link audit}\n * @see {@link debounce}\n * @see {@link delayWhen}\n * @see {@link sample}\n * @see {@link throttleTime}\n *\n * @param durationSelector A function that receives a value from the source\n * Observable, for computing the silencing duration for each source value,\n * returned as an `ObservableInput`.\n * @param config A configuration object to define `leading` and `trailing`\n * behavior. Defaults to `{ leading: true, trailing: false }`.\n * @return A function that returns an Observable that performs the throttle\n * operation to limit the rate of emissions from the source.\n */\nexport function throttle<T>(durationSelector: (value: T) => ObservableInput<any>, config?: ThrottleConfig): MonoTypeOperatorFunction<T> {\n  return operate((source, subscriber) => {\n    const { leading = true, trailing = false } = config ?? {};\n    let hasValue = false;\n    let sendValue: T | null = null;\n    let throttled: Subscription | null = null;\n    let isComplete = false;\n\n    const endThrottling = () => {\n      throttled?.unsubscribe();\n      throttled = null;\n      if (trailing) {\n        send();\n        isComplete && subscriber.complete();\n      }\n    };\n\n    const cleanupThrottling = () => {\n      throttled = null;\n      isComplete && subscriber.complete();\n    };\n\n    const startThrottle = (value: T) =>\n      (throttled = innerFrom(durationSelector(value)).subscribe(createOperatorSubscriber(subscriber, endThrottling, cleanupThrottling)));\n\n    const send = () => {\n      if (hasValue) {\n        // Ensure we clear out our value and hasValue flag\n        // before we emit, otherwise reentrant code can cause\n        // issues here.\n        hasValue = false;\n        const value = sendValue!;\n        sendValue = null;\n        // Emit the value.\n        subscriber.next(value);\n        !isComplete && startThrottle(value);\n      }\n    };\n\n    source.subscribe(\n      createOperatorSubscriber(\n        subscriber,\n        // Regarding the presence of throttled.closed in the following\n        // conditions, if a synchronous duration selector is specified - weird,\n        // but legal - an already-closed subscription will be assigned to\n        // throttled, so the subscription's closed property needs to be checked,\n        // too.\n        (value) => {\n          hasValue = true;\n          sendValue = value;\n          !(throttled && !throttled.closed) && (leading ? send() : startThrottle(value));\n        },\n        () => {\n          isComplete = true;\n          !(trailing && hasValue && throttled && !throttled.closed) && subscriber.complete();\n        }\n      )\n    );\n  });\n}\n", "import { asyncScheduler } from '../scheduler/async';\nimport { throttle, ThrottleConfig } from './throttle';\nimport { MonoTypeOperatorFunction, SchedulerLike } from '../types';\nimport { timer } from '../observable/timer';\n\n/**\n * Emits a value from the source Observable, then ignores subsequent source\n * values for `duration` milliseconds, then repeats this process.\n *\n * <span class=\"informal\">Lets a value pass, then ignores source values for the\n * next `duration` milliseconds.</span>\n *\n * ![](throttleTime.png)\n *\n * `throttleTime` emits the source Observable values on the output Observable\n * when its internal timer is disabled, and ignores source values when the timer\n * is enabled. Initially, the timer is disabled. As soon as the first source\n * value arrives, it is forwarded to the output Observable, and then the timer\n * is enabled. After `duration` milliseconds (or the time unit determined\n * internally by the optional `scheduler`) has passed, the timer is disabled,\n * and this process repeats for the next source value. Optionally takes a\n * {@link SchedulerLike} for managing timers.\n *\n * ## Examples\n *\n * ### Limit click rate\n *\n * Emit clicks at a rate of at most one click per second\n *\n * ```ts\n * import { fromEvent, throttleTime } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const result = clicks.pipe(throttleTime(1000));\n *\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link auditTime}\n * @see {@link debounceTime}\n * @see {@link delay}\n * @see {@link sampleTime}\n * @see {@link throttle}\n *\n * @param duration Time to wait before emitting another value after\n * emitting the last value, measured in milliseconds or the time unit determined\n * internally by the optional `scheduler`.\n * @param scheduler The {@link SchedulerLike} to use for\n * managing the timers that handle the throttling. Defaults to {@link asyncScheduler}.\n * @param config A configuration object to define `leading` and\n * `trailing` behavior. Defaults to `{ leading: true, trailing: false }`.\n * @return A function that returns an Observable that performs the throttle\n * operation to limit the rate of emissions from the source.\n */\nexport function throttleTime<T>(\n  duration: number,\n  scheduler: SchedulerLike = asyncScheduler,\n  config?: ThrottleConfig\n): MonoTypeOperatorFunction<T> {\n  const duration$ = timer(duration, scheduler);\n  return throttle(() => duration$, config);\n}\n", "import { OperatorFunction, ObservableInputTuple } from '../types';\nimport { operate } from '../util/lift';\nimport { createOperatorSubscriber } from './OperatorSubscriber';\nimport { innerFrom } from '../observable/innerFrom';\nimport { identity } from '../util/identity';\nimport { noop } from '../util/noop';\nimport { popResultSelector } from '../util/args';\n\nexport function withLatestFrom<T, O extends unknown[]>(...inputs: [...ObservableInputTuple<O>]): OperatorFunction<T, [T, ...O]>;\n\nexport function withLatestFrom<T, O extends unknown[], R>(\n  ...inputs: [...ObservableInputTuple<O>, (...value: [T, ...O]) => R]\n): OperatorFunction<T, R>;\n\n/**\n * Combines the source Observable with other Observables to create an Observable\n * whose values are calculated from the latest values of each, only when the\n * source emits.\n *\n * <span class=\"informal\">Whenever the source Observable emits a value, it\n * computes a formula using that value plus the latest values from other input\n * Observables, then emits the output of that formula.</span>\n *\n * ![](withLatestFrom.png)\n *\n * `withLatestFrom` combines each value from the source Observable (the\n * instance) with the latest values from the other input Observables only when\n * the source emits a value, optionally using a `project` function to determine\n * the value to be emitted on the output Observable. All input Observables must\n * emit at least one value before the output Observable will emit a value.\n *\n * ## Example\n *\n * On every click event, emit an array with the latest timer event plus the click event\n *\n * ```ts\n * import { fromEvent, interval, withLatestFrom } from 'rxjs';\n *\n * const clicks = fromEvent(document, 'click');\n * const timer = interval(1000);\n * const result = clicks.pipe(withLatestFrom(timer));\n * result.subscribe(x => console.log(x));\n * ```\n *\n * @see {@link combineLatest}\n *\n * @param {ObservableInput} other An input Observable to combine with the source\n * Observable. More than one input Observables may be given as argument.\n * @param {Function} [project] Projection function for combining values\n * together. Receives all values in order of the Observables passed, where the\n * first parameter is a value from the source Observable. (e.g.\n * `a.pipe(withLatestFrom(b, c), map(([a1, b1, c1]) => a1 + b1 + c1))`). If this is not\n * passed, arrays will be emitted on the output Observable.\n * @return A function that returns an Observable of projected values from the\n * most recent values from each input Observable, or an array of the most\n * recent values from each input Observable.\n */\nexport function withLatestFrom<T, R>(...inputs: any[]): OperatorFunction<T, R | any[]> {\n  const project = popResultSelector(inputs) as ((...args: any[]) => R) | undefined;\n\n  return operate((source, subscriber) => {\n    const len = inputs.length;\n    const otherValues = new Array(len);\n    // An array of whether or not the other sources have emitted. Matched with them by index.\n    // TODO: At somepoint, we should investigate the performance implications here, and look\n    // into using a `Set()` and checking the `size` to see if we're ready.\n    let hasValue = inputs.map(() => false);\n    // Flipped true when we have at least one value from all other sources and\n    // we are ready to start emitting values.\n    let ready = false;\n\n    // Other sources. Note that here we are not checking `subscriber.closed`,\n    // this causes all inputs to be subscribed to, even if nothing can be emitted\n    // from them. This is an important distinction because subscription constitutes\n    // a side-effect.\n    for (let i = 0; i < len; i++) {\n      innerFrom(inputs[i]).subscribe(\n        createOperatorSubscriber(\n          subscriber,\n          (value) => {\n            otherValues[i] = value;\n            if (!ready && !hasValue[i]) {\n              // If we're not ready yet, flag to show this observable has emitted.\n              hasValue[i] = true;\n              // Intentionally terse code.\n              // If all of our other observables have emitted, set `ready` to `true`,\n              // so we know we can start emitting values, then clean up the `hasValue` array,\n              // because we don't need it anymore.\n              (ready = hasValue.every(identity)) && (hasValue = null!);\n            }\n          },\n          // Completing one of the other sources has\n          // no bearing on the completion of our result.\n          noop\n        )\n      );\n    }\n\n    // Source subscription\n    source.subscribe(\n      createOperatorSubscriber(subscriber, (value) => {\n        if (ready) {\n          // We have at least one value from the other sources. Go ahead and emit.\n          const values = [value, ...otherValues];\n          subscriber.next(project ? project(...values) : values);\n        }\n      })\n    );\n  });\n}\n", "import { zip as zipStatic } from '../observable/zip';\nimport { ObservableInput, ObservableInputTuple, OperatorFunction, Cons } from '../types';\nimport { operate } from '../util/lift';\n\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[]>(otherInputs: [...ObservableInputTuple<A>]): OperatorFunction<T, Cons<T, A>>;\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[], R>(\n  otherInputsAndProject: [...ObservableInputTuple<A>],\n  project: (...values: Cons<T, A>) => R\n): OperatorFunction<T, R>;\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[]>(...otherInputs: [...ObservableInputTuple<A>]): OperatorFunction<T, Cons<T, A>>;\n/** @deprecated Replaced with {@link zipWith}. Will be removed in v8. */\nexport function zip<T, A extends readonly unknown[], R>(\n  ...otherInputsAndProject: [...ObservableInputTuple<A>, (...values: Cons<T, A>) => R]\n): OperatorFunction<T, R>;\n\n/**\n * @deprecated Replaced with {@link zipWith}. Will be removed in v8.\n */\nexport function zip<T, R>(...sources: Array<ObservableInput<any> | ((...values: Array<any>) => R)>): OperatorFunction<T, any> {\n  return operate((source, subscriber) => {\n    zipStatic(source as ObservableInput<any>, ...(sources as Array<ObservableInput<any>>)).subscribe(subscriber);\n  });\n}\n", "import { ObservableInputTuple, OperatorFunction, Cons } from '../types';\nimport { zip } from './zip';\n\n/**\n * Subscribes to the source, and the observable inputs provided as arguments, and combines their values, by index, into arrays.\n *\n * What is meant by \"combine by index\": The first value from each will be made into a single array, then emitted,\n * then the second value from each will be combined into a single array and emitted, then the third value\n * from each will be combined into a single array and emitted, and so on.\n *\n * This will continue until it is no longer able to combine values of the same index into an array.\n *\n * After the last value from any one completed source is emitted in an array, the resulting observable will complete,\n * as there is no way to continue \"zipping\" values together by index.\n *\n * Use-cases for this operator are limited. There are memory concerns if one of the streams is emitting\n * values at a much faster rate than the others. Usage should likely be limited to streams that emit\n * at a similar pace, or finite streams of known length.\n *\n * In many cases, authors want `combineLatestWith` and not `zipWith`.\n *\n * @param otherInputs other observable inputs to collate values from.\n * @return A function that returns an Observable that emits items by index\n * combined from the source Observable and provided Observables, in form of an\n * array.\n */\nexport function zipWith<T, A extends readonly unknown[]>(...otherInputs: [...ObservableInputTuple<A>]): OperatorFunction<T, Cons<T, A>> {\n  return zip(...otherInputs);\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  ReplaySubject,\n  Subject,\n  fromEvent\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch document\n *\n * Documents are implemented as subjects, so all downstream observables are\n * automatically updated when a new document is emitted.\n *\n * @returns Document subject\n */\nexport function watchDocument(): Subject<Document> {\n  const document$ = new ReplaySubject<Document>(1)\n  fromEvent(document, \"DOMContentLoaded\", { once: true })\n    .subscribe(() => document$.next(document))\n\n  /* Return document */\n  return document$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve all elements matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getElements<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T][]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T[]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T[] {\n  return Array.from(node.querySelectorAll<T>(selector))\n}\n\n/**\n * Retrieve an element matching a query selector or throw a reference error\n *\n * Note that this function assumes that the element is present. If unsure if an\n * element is existent, use the `getOptionalElement` function instead.\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T]\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T {\n  const el = getOptionalElement<T>(selector, node)\n  if (typeof el === \"undefined\")\n    throw new ReferenceError(\n      `Missing element: expected \"${selector}\" to be present`\n    )\n\n  /* Return element */\n  return el\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Retrieve an optional element matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element or nothing\n */\nexport function getOptionalElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T] | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T | undefined {\n  return node.querySelector<T>(selector) || undefined\n}\n\n/**\n * Retrieve the currently active element\n *\n * @returns Element or nothing\n */\nexport function getActiveElement(): HTMLElement | undefined {\n  return document.activeElement instanceof HTMLElement\n    ? document.activeElement || undefined\n    : undefined\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  debounceTime,\n  distinctUntilChanged,\n  fromEvent,\n  map,\n  merge,\n  shareReplay,\n  startWith\n} from \"rxjs\"\n\nimport { getActiveElement } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Focus observable\n *\n * Previously, this observer used `focus` and `blur` events to determine whether\n * an element is focused, but this doesn't work if there are focusable elements\n * within the elements itself. A better solutions are `focusin` and `focusout`\n * events, which bubble up the tree and allow for more fine-grained control.\n *\n * `debounceTime` is necessary, because when a focus change happens inside an\n * element, the observable would first emit `false` and then `true` again.\n */\nconst observer$ = merge(\n  fromEvent(document.body, \"focusin\"),\n  fromEvent(document.body, \"focusout\")\n)\n  .pipe(\n    debounceTime(1),\n    startWith(undefined),\n    map(() => getActiveElement() || document.body),\n    shareReplay(1)\n  )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch element focus\n *\n * @param el - Element\n *\n * @returns Element focus observable\n */\nexport function watchElementFocus(\n  el: HTMLElement\n): Observable<boolean> {\n  return observer$\n    .pipe(\n      map(active => el.contains(active)),\n      distinctUntilChanged()\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  debounceTime,\n  fromEvent,\n  identity,\n  map,\n  merge,\n  startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch element hover\n *\n * @param el - Element\n * @param duration - Debounce duration\n *\n * @returns Element hover observable\n */\nexport function watchElementHover(\n  el: HTMLElement, duration?: number\n): Observable<boolean> {\n  return merge(\n    fromEvent(el, \"mouseenter\").pipe(map(() => true)),\n    fromEvent(el, \"mouseleave\").pipe(map(() => false))\n  )\n    .pipe(\n      duration ? debounceTime(duration) : identity,\n      startWith(false)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  animationFrameScheduler,\n  auditTime,\n  fromEvent,\n  map,\n  merge,\n  startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Element offset\n */\nexport interface ElementOffset {\n  x: number                            /* Horizontal offset */\n  y: number                            /* Vertical offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element offset\n *\n * @param el - Element\n *\n * @returns Element offset\n */\nexport function getElementOffset(\n  el: HTMLElement\n): ElementOffset {\n  return {\n    x: el.offsetLeft,\n    y: el.offsetTop\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element offset\n *\n * @param el - Element\n *\n * @returns Element offset observable\n */\nexport function watchElementOffset(\n  el: HTMLElement\n): Observable<ElementOffset> {\n  return merge(\n    fromEvent(window, \"load\"),\n    fromEvent(window, \"resize\")\n  )\n    .pipe(\n      auditTime(0, animationFrameScheduler),\n      map(() => getElementOffset(el)),\n      startWith(getElementOffset(el))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  animationFrameScheduler,\n  auditTime,\n  fromEvent,\n  map,\n  merge,\n  startWith\n} from \"rxjs\"\n\nimport { ElementOffset } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element content offset (= scroll offset)\n *\n * @param el - Element\n *\n * @returns Element content offset\n */\nexport function getElementContentOffset(\n  el: HTMLElement\n): ElementOffset {\n  return {\n    x: el.scrollLeft,\n    y: el.scrollTop\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element content offset\n *\n * @param el - Element\n *\n * @returns Element content offset observable\n */\nexport function watchElementContentOffset(\n  el: HTMLElement\n): Observable<ElementOffset> {\n  return merge(\n    fromEvent(el, \"scroll\"),\n    fromEvent(window, \"resize\")\n  )\n    .pipe(\n      auditTime(0, animationFrameScheduler),\n      map(() => getElementContentOffset(el)),\n      startWith(getElementContentOffset(el))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { JSX as JSXInternal } from \"preact\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * HTML attributes\n */\ntype Attributes =\n  & JSXInternal.HTMLAttributes\n  & JSXInternal.SVGAttributes\n  & Record<string, any>\n\n/**\n * Child element\n */\ntype Child =\n  | ChildNode\n  | HTMLElement\n  | Text\n  | string\n  | number\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Append a child node to an element\n *\n * @param el - Element\n * @param child - Child node(s)\n */\nfunction appendChild(el: HTMLElement, child: Child | Child[]): void {\n\n  /* Handle primitive types (including raw HTML) */\n  if (typeof child === \"string\" || typeof child === \"number\") {\n    el.innerHTML += child.toString()\n\n  /* Handle nodes */\n  } else if (child instanceof Node) {\n    el.appendChild(child)\n\n  /* Handle nested children */\n  } else if (Array.isArray(child)) {\n    for (const node of child)\n      appendChild(el, node)\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * JSX factory\n *\n * @template T - Element type\n *\n * @param tag - HTML tag\n * @param attributes - HTML attributes\n * @param children - Child elements\n *\n * @returns Element\n */\nexport function h<T extends keyof HTMLElementTagNameMap>(\n  tag: T, attributes?: Attributes | null, ...children: Child[]\n): HTMLElementTagNameMap[T]\n\nexport function h<T extends h.JSX.Element>(\n  tag: string, attributes?: Attributes | null, ...children: Child[]\n): T\n\nexport function h<T extends h.JSX.Element>(\n  tag: string, attributes?: Attributes | null, ...children: Child[]\n): T {\n  const el = document.createElement(tag)\n\n  /* Set attributes, if any */\n  if (attributes)\n    for (const attr of Object.keys(attributes)) {\n      if (typeof attributes[attr] === \"undefined\")\n        continue\n\n      /* Set default attribute or boolean */\n      if (typeof attributes[attr] !== \"boolean\")\n        el.setAttribute(attr, attributes[attr])\n      else\n        el.setAttribute(attr, \"\")\n    }\n\n  /* Append child nodes */\n  for (const child of children)\n    appendChild(el, child)\n\n  /* Return element */\n  return el as T\n}\n\n/* ----------------------------------------------------------------------------\n * Namespace\n * ------------------------------------------------------------------------- */\n\nexport declare namespace h {\n  namespace JSX {\n    type Element = HTMLElement\n    type IntrinsicElements = JSXInternal.IntrinsicElements\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Round a number for display with repository facts\n *\n * This is a reverse-engineered version of GitHub's weird rounding algorithm\n * for stars, forks and all other numbers. While all numbers below `1,000` are\n * returned as-is, bigger numbers are converted to fixed numbers:\n *\n * - `1,049` => `1k`\n * - `1,050` => `1.1k`\n * - `1,949` => `1.9k`\n * - `1,950` => `2k`\n *\n * @param value - Original value\n *\n * @returns Rounded value\n */\nexport function round(value: number): string {\n  if (value > 999) {\n    const digits = +((value - 950) % 1000 > 99)\n    return `${((value + 0.000001) / 1000).toFixed(digits)}k`\n  } else {\n    return value.toString()\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  defer,\n  finalize,\n  fromEvent,\n  map,\n  merge,\n  switchMap,\n  take,\n  throwError\n} from \"rxjs\"\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create and load a `script` element\n *\n * This function returns an observable that will emit when the script was\n * successfully loaded, or throw an error if it wasn't.\n *\n * @param src - Script URL\n *\n * @returns Script observable\n */\nexport function watchScript(src: string): Observable<void> {\n  const script = h(\"script\", { src })\n  return defer(() => {\n    document.head.appendChild(script)\n    return merge(\n      fromEvent(script, \"load\"),\n      fromEvent(script, \"error\")\n        .pipe(\n          switchMap(() => (\n            throwError(() => new ReferenceError(`Invalid script: ${src}`))\n          ))\n        )\n    )\n      .pipe(\n        map(() => undefined),\n        finalize(() => document.head.removeChild(script)),\n        take(1)\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  NEVER,\n  Observable,\n  Subject,\n  defer,\n  filter,\n  finalize,\n  map,\n  merge,\n  of,\n  shareReplay,\n  startWith,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { watchScript } from \"../../../script\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Element offset\n */\nexport interface ElementSize {\n  width: number                        /* Element width */\n  height: number                       /* Element height */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Resize observer entry subject\n */\nconst entry$ = new Subject<ResizeObserverEntry>()\n\n/**\n * Resize observer observable\n *\n * This observable will create a `ResizeObserver` on the first subscription\n * and will automatically terminate it when there are no more subscribers.\n * It's quite important to centralize observation in a single `ResizeObserver`,\n * as the performance difference can be quite dramatic, as the link shows.\n *\n * If the browser doesn't have a `ResizeObserver` implementation available, a\n * polyfill is automatically downloaded from unpkg.com. This is also compatible\n * with the built-in privacy plugin, which will download the polyfill and put\n * it alongside the built site for self-hosting.\n *\n * @see https://bit.ly/3iIYfEm - Google Groups on performance\n */\nconst observer$ = defer(() => (\n  typeof ResizeObserver === \"undefined\"\n    ? watchScript(\"https://unpkg.com/resize-observer-polyfill\")\n    : of(undefined)\n))\n  .pipe(\n    map(() => new ResizeObserver(entries => {\n      for (const entry of entries)\n        entry$.next(entry)\n    })),\n    switchMap(observer => merge(NEVER, of(observer))\n      .pipe(\n        finalize(() => observer.disconnect())\n      )\n    ),\n    shareReplay(1)\n  )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element size\n *\n * @param el - Element\n *\n * @returns Element size\n */\nexport function getElementSize(\n  el: HTMLElement\n): ElementSize {\n  return {\n    width:  el.offsetWidth,\n    height: el.offsetHeight\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch element size\n *\n * This function returns an observable that subscribes to a single internal\n * instance of `ResizeObserver` upon subscription, and emit resize events until\n * termination. Note that this function should not be called with the same\n * element twice, as the first unsubscription will terminate observation.\n *\n * Sadly, we can't use the `DOMRect` objects returned by the observer, because\n * we need the emitted values to be consistent with `getElementSize`, which will\n * return the used values (rounded) and not actual values (unrounded). Thus, we\n * use the `offset*` properties. See the linked GitHub issue.\n *\n * @see https://bit.ly/3m0k3he - GitHub issue\n *\n * @param el - Element\n *\n * @returns Element size observable\n */\nexport function watchElementSize(\n  el: HTMLElement\n): Observable<ElementSize> {\n  return observer$\n    .pipe(\n      tap(observer => observer.observe(el)),\n      switchMap(observer => entry$\n        .pipe(\n          filter(({ target }) => target === el),\n          finalize(() => observer.unobserve(el)),\n          map(() => getElementSize(el))\n        )\n      ),\n      startWith(getElementSize(el))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ElementSize } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve element content size (= scroll width and height)\n *\n * @param el - Element\n *\n * @returns Element content size\n */\nexport function getElementContentSize(\n  el: HTMLElement\n): ElementSize {\n  return {\n    width:  el.scrollWidth,\n    height: el.scrollHeight\n  }\n}\n\n/**\n * Retrieve the overflowing container of an element, if any\n *\n * @param el - Element\n *\n * @returns Overflowing container or nothing\n */\nexport function getElementContainer(\n  el: HTMLElement\n): HTMLElement | undefined {\n  let parent = el.parentElement\n  while (parent)\n    if (\n      el.scrollWidth  <= parent.scrollWidth &&\n      el.scrollHeight <= parent.scrollHeight\n    )\n      parent = (el = parent).parentElement\n    else\n      break\n\n  /* Return overflowing container */\n  return parent ? el : undefined\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  NEVER,\n  Observable,\n  Subject,\n  defer,\n  distinctUntilChanged,\n  filter,\n  finalize,\n  map,\n  merge,\n  of,\n  shareReplay,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport {\n  getElementContentSize,\n  getElementSize,\n  watchElementContentOffset\n} from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Intersection observer entry subject\n */\nconst entry$ = new Subject<IntersectionObserverEntry>()\n\n/**\n * Intersection observer observable\n *\n * This observable will create an `IntersectionObserver` on first subscription\n * and will automatically terminate it when there are no more subscribers.\n *\n * @see https://bit.ly/3iIYfEm - Google Groups on performance\n */\nconst observer$ = defer(() => of(\n  new IntersectionObserver(entries => {\n    for (const entry of entries)\n      entry$.next(entry)\n  }, {\n    threshold: 0\n  })\n))\n  .pipe(\n    switchMap(observer => merge(NEVER, of(observer))\n      .pipe(\n        finalize(() => observer.disconnect())\n      )\n    ),\n    shareReplay(1)\n  )\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch element visibility\n *\n * @param el - Element\n *\n * @returns Element visibility observable\n */\nexport function watchElementVisibility(\n  el: HTMLElement\n): Observable<boolean> {\n  return observer$\n    .pipe(\n      tap(observer => observer.observe(el)),\n      switchMap(observer => entry$\n        .pipe(\n          filter(({ target }) => target === el),\n          finalize(() => observer.unobserve(el)),\n          map(({ isIntersecting }) => isIntersecting)\n        )\n      )\n    )\n}\n\n/**\n * Watch element boundary\n *\n * This function returns an observable which emits whether the bottom content\n * boundary (= scroll offset) of an element is within a certain threshold.\n *\n * @param el - Element\n * @param threshold - Threshold\n *\n * @returns Element boundary observable\n */\nexport function watchElementBoundary(\n  el: HTMLElement, threshold = 16\n): Observable<boolean> {\n  return watchElementContentOffset(el)\n    .pipe(\n      map(({ y }) => {\n        const visible = getElementSize(el)\n        const content = getElementContentSize(el)\n        return y >= (\n          content.height - visible.height - threshold\n        )\n      }),\n      distinctUntilChanged()\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  startWith\n} from \"rxjs\"\n\nimport { getElement } from \"../element\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Toggle\n */\nexport type Toggle =\n  | \"drawer\"                           /* Toggle for drawer */\n  | \"search\"                           /* Toggle for search */\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Toggle map\n */\nconst toggles: Record<Toggle, HTMLInputElement> = {\n  drawer: getElement(\"[data-md-toggle=drawer]\"),\n  search: getElement(\"[data-md-toggle=search]\")\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve the value of a toggle\n *\n * @param name - Toggle\n *\n * @returns Toggle value\n */\nexport function getToggle(name: Toggle): boolean {\n  return toggles[name].checked\n}\n\n/**\n * Set toggle\n *\n * Simulating a click event seems to be the most cross-browser compatible way\n * of changing the value while also emitting a `change` event. Before, Material\n * used `CustomEvent` to programmatically change the value of a toggle, but this\n * is a much simpler and cleaner solution which doesn't require a polyfill.\n *\n * @param name - Toggle\n * @param value - Toggle value\n */\nexport function setToggle(name: Toggle, value: boolean): void {\n  if (toggles[name].checked !== value)\n    toggles[name].click()\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch toggle\n *\n * @param name - Toggle\n *\n * @returns Toggle value observable\n */\nexport function watchToggle(name: Toggle): Observable<boolean> {\n  const el = toggles[name]\n  return fromEvent(el, \"change\")\n    .pipe(\n      map(() => el.checked),\n      startWith(el.checked)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  filter,\n  fromEvent,\n  map,\n  merge,\n  share,\n  startWith,\n  switchMap\n} from \"rxjs\"\n\nimport { getActiveElement } from \"../element\"\nimport { getToggle } from \"../toggle\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Keyboard mode\n */\nexport type KeyboardMode =\n  | \"global\"                           /* Global */\n  | \"search\"                           /* Search is open */\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Keyboard\n */\nexport interface Keyboard {\n  mode: KeyboardMode                   /* Keyboard mode */\n  type: string                         /* Key type */\n  claim(): void                        /* Key claim */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Check whether an element may receive keyboard input\n *\n * @param el - Element\n * @param type - Key type\n *\n * @returns Test result\n */\nfunction isSusceptibleToKeyboard(\n  el: HTMLElement, type: string\n): boolean {\n  switch (el.constructor) {\n\n    /* Input elements */\n    case HTMLInputElement:\n      /* @ts-expect-error - omit unnecessary type cast */\n      if (el.type === \"radio\")\n        return /^Arrow/.test(type)\n      else\n        return true\n\n    /* Select element and textarea */\n    case HTMLSelectElement:\n    case HTMLTextAreaElement:\n      return true\n\n    /* Everything else */\n    default:\n      return el.isContentEditable\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch composition events\n *\n * @returns Composition observable\n */\nexport function watchComposition(): Observable<boolean> {\n  return merge(\n    fromEvent(window, \"compositionstart\").pipe(map(() => true)),\n    fromEvent(window, \"compositionend\").pipe(map(() => false))\n  )\n    .pipe(\n      startWith(false)\n    )\n}\n\n/**\n * Watch keyboard\n *\n * @returns Keyboard observable\n */\nexport function watchKeyboard(): Observable<Keyboard> {\n  const keyboard$ = fromEvent<KeyboardEvent>(window, \"keydown\")\n    .pipe(\n      filter(ev => !(ev.metaKey || ev.ctrlKey)),\n      map(ev => ({\n        mode: getToggle(\"search\") ? \"search\" : \"global\",\n        type: ev.key,\n        claim() {\n          ev.preventDefault()\n          ev.stopPropagation()\n        }\n      } as Keyboard)),\n      filter(({ mode, type }) => {\n        if (mode === \"global\") {\n          const active = getActiveElement()\n          if (typeof active !== \"undefined\")\n            return !isSusceptibleToKeyboard(active, type)\n        }\n        return true\n      }),\n      share()\n    )\n\n  /* Don't emit during composition events - see https://bit.ly/3te3Wl8 */\n  return watchComposition()\n    .pipe(\n      switchMap(active => !active ? keyboard$ : EMPTY)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Subject } from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve location\n *\n * This function returns a `URL` object (and not `Location`) to normalize the\n * typings across the application. Furthermore, locations need to be tracked\n * without setting them and `Location` is a singleton which represents the\n * current location.\n *\n * @returns URL\n */\nexport function getLocation(): URL {\n  return new URL(location.href)\n}\n\n/**\n * Set location\n *\n * If instant navigation is enabled, this function creates a temporary anchor\n * element, sets the `href` attribute, appends it to the body, clicks it, and\n * then removes it again. The event will bubble up the DOM and trigger be\n * intercepted by the instant loading business logic.\n *\n * Note that we must append and remove the anchor element, or the event will\n * not bubble up the DOM, making it impossible to intercept it.\n *\n * @param url - URL to navigate to\n * @param navigate - Force navigation\n */\nexport function setLocation(\n  url: URL | HTMLLinkElement, navigate = false\n): void {\n  if (feature(\"navigation.instant\") && !navigate) {\n    const el = h(\"a\", { href: url.href })\n    document.body.appendChild(el)\n    el.click()\n    el.remove()\n\n  // If we're not using instant navigation, and the page should not be reloaded\n  // just instruct the browser to navigate to the given URL\n  } else {\n    location.href = url.href\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch location\n *\n * @returns Location subject\n */\nexport function watchLocation(): Subject<URL> {\n  return new Subject<URL>()\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  filter,\n  fromEvent,\n  map,\n  merge,\n  shareReplay,\n  startWith\n} from \"rxjs\"\n\nimport { getOptionalElement } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve location hash\n *\n * @returns Location hash\n */\nexport function getLocationHash(): string {\n  return location.hash.slice(1)\n}\n\n/**\n * Set location hash\n *\n * Setting a new fragment identifier via `location.hash` will have no effect\n * if the value doesn't change. When a new fragment identifier is set, we want\n * the browser to target the respective element at all times, which is why we\n * use this dirty little trick.\n *\n * @param hash - Location hash\n */\nexport function setLocationHash(hash: string): void {\n  const el = h(\"a\", { href: hash })\n  el.addEventListener(\"click\", ev => ev.stopPropagation())\n  el.click()\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch location hash\n *\n * @param location$ - Location observable\n *\n * @returns Location hash observable\n */\nexport function watchLocationHash(\n  location$: Observable<URL>\n): Observable<string> {\n  return merge(\n    fromEvent<HashChangeEvent>(window, \"hashchange\"),\n    location$\n  )\n    .pipe(\n      map(getLocationHash),\n      startWith(getLocationHash()),\n      filter(hash => hash.length > 0),\n      shareReplay(1)\n    )\n}\n\n/**\n * Watch location target\n *\n * @param location$ - Location observable\n *\n * @returns Location target observable\n */\nexport function watchLocationTarget(\n  location$: Observable<URL>\n): Observable<HTMLElement> {\n  return watchLocationHash(location$)\n    .pipe(\n      map(id => getOptionalElement(`[id=\"${id}\"]`)!),\n      filter(el => typeof el !== \"undefined\")\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  fromEvent,\n  fromEventPattern,\n  map,\n  merge,\n  startWith,\n  switchMap\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch media query\n *\n * Note that although `MediaQueryList.addListener` is deprecated we have to\n * use it, because it's the only way to ensure proper downward compatibility.\n *\n * @see https://bit.ly/3dUBH2m - GitHub issue\n *\n * @param query - Media query\n *\n * @returns Media observable\n */\nexport function watchMedia(query: string): Observable<boolean> {\n  const media = matchMedia(query)\n  return fromEventPattern<boolean>(next => (\n    media.addListener(() => next(media.matches))\n  ))\n    .pipe(\n      startWith(media.matches)\n    )\n}\n\n/**\n * Watch print mode\n *\n * @returns Print observable\n */\nexport function watchPrint(): Observable<boolean> {\n  const media = matchMedia(\"print\")\n  return merge(\n    fromEvent(window, \"beforeprint\").pipe(map(() => true)),\n    fromEvent(window, \"afterprint\").pipe(map(() => false))\n  )\n    .pipe(\n      startWith(media.matches)\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Toggle an observable with a media observable\n *\n * @template T - Data type\n *\n * @param query$ - Media observable\n * @param factory - Observable factory\n *\n * @returns Toggled observable\n */\nexport function at<T>(\n  query$: Observable<boolean>, factory: () => Observable<T>\n): Observable<T> {\n  return query$\n    .pipe(\n      switchMap(active => active ? factory() : EMPTY)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  map,\n  shareReplay,\n  switchMap\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Options\n */\ninterface Options {\n  progress$?: Subject<number>          // Progress subject\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch the given URL\n *\n * If the request fails (e.g. when dispatched from `file://` locations), the\n * observable will complete without emitting a value.\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Response observable\n */\nexport function request(\n  url: URL | string, options?: Options\n): Observable<Blob> {\n  return new Observable<Blob>(observer => {\n    const req = new XMLHttpRequest()\n    req.open(\"GET\",  `${url}`)\n    req.responseType = \"blob\"\n\n    // Handle response\n    req.addEventListener(\"load\", () => {\n      if (req.status >= 200 && req.status < 300) {\n        observer.next(req.response)\n        observer.complete()\n      } else {\n        observer.error(new Error(req.statusText))\n      }\n    })\n\n    // Handle network errors\n    req.addEventListener(\"error\", () => {\n      observer.error(new Error(\"Network Error\"))\n    })\n\n    // Handle aborted requests\n    req.addEventListener(\"abort\", () => {\n      observer.error(new Error(\"Request aborted\"))\n    })\n\n    // Handle download progress\n    if (typeof options?.progress$ !== \"undefined\") {\n      req.addEventListener(\"progress\", event => {\n        if (event.lengthComputable) {\n          options.progress$!.next((event.loaded / event.total) * 100)\n        } else { // https://bugs.chromium.org/p/chromium/issues/detail?id=463622\n          const totalFromHeader = Number(req.getResponseHeader(\"Content-Length\")) || 0\n          options.progress$!.next((event.loaded / totalFromHeader) * 100)\n        }\n      })\n\n      // Immediately set progress to 5% to indicate that we're loading\n      options.progress$.next(5)\n    }\n\n    // Send request\n    req.send()\n  })\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Fetch JSON from the given URL\n *\n * @template T - Data type\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Data observable\n */\nexport function requestJSON<T>(\n  url: URL | string, options?: Options\n): Observable<T> {\n  return request(url, options)\n    .pipe(\n      switchMap(res => res.text()),\n      map(body => JSON.parse(body) as T),\n      shareReplay(1)\n    )\n}\n\n/**\n * Fetch XML from the given URL\n *\n * @param url - Request URL\n * @param options - Options\n *\n * @returns Data observable\n */\nexport function requestXML(\n  url: URL | string, options?: Options\n): Observable<Document> {\n  const dom = new DOMParser()\n  return request(url, options)\n    .pipe(\n      switchMap(res => res.text()),\n      map(res => dom.parseFromString(res, \"text/xml\")),\n      shareReplay(1)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  merge,\n  startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport offset\n */\nexport interface ViewportOffset {\n  x: number                            /* Horizontal offset */\n  y: number                            /* Vertical offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve viewport offset\n *\n * On iOS Safari, viewport offset can be negative due to overflow scrolling.\n * As this may induce strange behaviors downstream, we'll just limit it to 0.\n *\n * @returns Viewport offset\n */\nexport function getViewportOffset(): ViewportOffset {\n  return {\n    x: Math.max(0, scrollX),\n    y: Math.max(0, scrollY)\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport offset\n *\n * @returns Viewport offset observable\n */\nexport function watchViewportOffset(): Observable<ViewportOffset> {\n  return merge(\n    fromEvent(window, \"scroll\", { passive: true }),\n    fromEvent(window, \"resize\", { passive: true })\n  )\n    .pipe(\n      map(getViewportOffset),\n      startWith(getViewportOffset())\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  startWith\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport size\n */\nexport interface ViewportSize {\n  width: number                        /* Viewport width */\n  height: number                       /* Viewport height */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve viewport size\n *\n * @returns Viewport size\n */\nexport function getViewportSize(): ViewportSize {\n  return {\n    width:  innerWidth,\n    height: innerHeight\n  }\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport size\n *\n * @returns Viewport size observable\n */\nexport function watchViewportSize(): Observable<ViewportSize> {\n  return fromEvent(window, \"resize\", { passive: true })\n    .pipe(\n      map(getViewportSize),\n      startWith(getViewportSize())\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  map,\n  shareReplay\n} from \"rxjs\"\n\nimport {\n  ViewportOffset,\n  watchViewportOffset\n} from \"../offset\"\nimport {\n  ViewportSize,\n  watchViewportSize\n} from \"../size\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Viewport\n */\nexport interface Viewport {\n  offset: ViewportOffset               /* Viewport offset */\n  size: ViewportSize                   /* Viewport size */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport\n *\n * @returns Viewport observable\n */\nexport function watchViewport(): Observable<Viewport> {\n  return combineLatest([\n    watchViewportOffset(),\n    watchViewportSize()\n  ])\n    .pipe(\n      map(([offset, size]) => ({ offset, size })),\n      shareReplay(1)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  distinctUntilKeyChanged,\n  map\n} from \"rxjs\"\n\nimport { Header } from \"~/components\"\n\nimport { getElementOffset } from \"../../element\"\nimport { Viewport } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch viewport relative to element\n *\n * @param el - Element\n * @param options - Options\n *\n * @returns Viewport observable\n */\nexport function watchViewportAt(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<Viewport> {\n  const size$ = viewport$\n    .pipe(\n      distinctUntilKeyChanged(\"size\")\n    )\n\n  /* Compute element offset */\n  const offset$ = combineLatest([size$, header$])\n    .pipe(\n      map(() => getElementOffset(el))\n    )\n\n  /* Compute relative viewport, return hot observable */\n  return combineLatest([header$, viewport$, offset$])\n    .pipe(\n      map(([{ height }, { offset, size }, { x, y }]) => ({\n        offset: {\n          x: offset.x - x,\n          y: offset.y - y + height\n        },\n        size\n      }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  endWith,\n  fromEvent,\n  ignoreElements,\n  mergeWith,\n  share,\n  takeUntil\n} from \"rxjs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Worker message\n */\nexport interface WorkerMessage {\n  type: unknown                        /* Message type */\n  data?: unknown                       /* Message data */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create an observable for receiving from a web worker\n *\n * @template T - Data type\n *\n * @param worker - Web worker\n *\n * @returns Message observable\n */\nfunction recv<T>(worker: Worker): Observable<T> {\n  return fromEvent<MessageEvent<T>, T>(worker, \"message\", ev => ev.data)\n}\n\n/**\n * Create a subject for sending to a web worker\n *\n * @template T - Data type\n *\n * @param worker - Web worker\n *\n * @returns Message subject\n */\nfunction send<T>(worker: Worker): Subject<T> {\n  const send$ = new Subject<T>()\n  send$.subscribe(data => worker.postMessage(data))\n\n  /* Return message subject */\n  return send$\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a bidirectional communication channel to a web worker\n *\n * @template T - Data type\n *\n * @param url - Worker URL\n * @param worker - Worker\n *\n * @returns Worker subject\n */\nexport function watchWorker<T extends WorkerMessage>(\n  url: string, worker = new Worker(url)\n): Subject<T> {\n  const recv$ = recv<T>(worker)\n  const send$ = send<T>(worker)\n\n  /* Create worker subject and forward messages */\n  const worker$ = new Subject<T>()\n  worker$.subscribe(send$)\n\n  /* Return worker subject */\n  const done$ = send$.pipe(ignoreElements(), endWith(true))\n  return worker$\n    .pipe(\n      ignoreElements(),\n      mergeWith(recv$.pipe(takeUntil(done$))),\n      share()\n    ) as Subject<T>\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { getElement, getLocation } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Feature flag\n */\nexport type Flag =\n  | \"announce.dismiss\"                 /* Dismissable announcement bar */\n  | \"content.code.annotate\"            /* Code annotations */\n  | \"content.code.copy\"                /* Code copy button */\n  | \"content.lazy\"                     /* Lazy content elements */\n  | \"content.tabs.link\"                /* Link content tabs */\n  | \"content.tooltips\"                 /* Tooltips */\n  | \"header.autohide\"                  /* Hide header */\n  | \"navigation.expand\"                /* Automatic expansion */\n  | \"navigation.indexes\"               /* Section pages */\n  | \"navigation.instant\"               /* Instant navigation */\n  | \"navigation.instant.progress\"      /* Instant navigation progress */\n  | \"navigation.sections\"              /* Section navigation */\n  | \"navigation.tabs\"                  /* Tabs navigation */\n  | \"navigation.tabs.sticky\"           /* Tabs navigation (sticky) */\n  | \"navigation.top\"                   /* Back-to-top button */\n  | \"navigation.tracking\"              /* Anchor tracking */\n  | \"search.highlight\"                 /* Search highlighting */\n  | \"search.share\"                     /* Search sharing */\n  | \"search.suggest\"                   /* Search suggestions */\n  | \"toc.follow\"                       /* Following table of contents */\n  | \"toc.integrate\"                    /* Integrated table of contents */\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Translation\n */\nexport type Translation =\n  | \"clipboard.copy\"                   /* Copy to clipboard */\n  | \"clipboard.copied\"                 /* Copied to clipboard */\n  | \"search.result.placeholder\"        /* Type to start searching */\n  | \"search.result.none\"               /* No matching documents */\n  | \"search.result.one\"                /* 1 matching document */\n  | \"search.result.other\"              /* # matching documents */\n  | \"search.result.more.one\"           /* 1 more on this page */\n  | \"search.result.more.other\"         /* # more on this page */\n  | \"search.result.term.missing\"       /* Missing */\n  | \"select.version\"                   /* Version selector */\n\n/**\n * Translations\n */\nexport type Translations =\n  Record<Translation, string>\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Versioning\n */\nexport interface Versioning {\n  provider: \"mike\"                     /* Version provider */\n  default?: string | string[]          /* Default version */\n}\n\n/**\n * Configuration\n */\nexport interface Config {\n  base: string                         /* Base URL */\n  features: Flag[]                     /* Feature flags */\n  translations: Translations           /* Translations */\n  search: string                       /* Search worker URL */\n  tags?: Record<string, string>        /* Tags mapping */\n  version?: Versioning                 /* Versioning */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve global configuration and make base URL absolute\n */\nconst script = getElement(\"#__config\")\nconst config: Config = JSON.parse(script.textContent!)\nconfig.base = `${new URL(config.base, getLocation())}`\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve global configuration\n *\n * @returns Global configuration\n */\nexport function configuration(): Config {\n  return config\n}\n\n/**\n * Check whether a feature flag is enabled\n *\n * @param flag - Feature flag\n *\n * @returns Test result\n */\nexport function feature(flag: Flag): boolean {\n  return config.features.includes(flag)\n}\n\n/**\n * Retrieve the translation for the given key\n *\n * @param key - Key to be translated\n * @param value - Positional value, if any\n *\n * @returns Translation\n */\nexport function translation(\n  key: Translation, value?: string | number\n): string {\n  return typeof value !== \"undefined\"\n    ? config.translations[key].replace(\"#\", value.toString())\n    : config.translations[key]\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { getElement, getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Component type\n */\nexport type ComponentType =\n  | \"announce\"                         /* Announcement bar */\n  | \"container\"                        /* Container */\n  | \"consent\"                          /* Consent */\n  | \"content\"                          /* Content */\n  | \"dialog\"                           /* Dialog */\n  | \"header\"                           /* Header */\n  | \"header-title\"                     /* Header title */\n  | \"header-topic\"                     /* Header topic */\n  | \"main\"                             /* Main area */\n  | \"outdated\"                         /* Version warning */\n  | \"palette\"                          /* Color palette */\n  | \"progress\"                         /* Progress indicator */\n  | \"search\"                           /* Search */\n  | \"search-query\"                     /* Search input */\n  | \"search-result\"                    /* Search results */\n  | \"search-share\"                     /* Search sharing */\n  | \"search-suggest\"                   /* Search suggestions */\n  | \"sidebar\"                          /* Sidebar */\n  | \"skip\"                             /* Skip link */\n  | \"source\"                           /* Repository information */\n  | \"tabs\"                             /* Navigation tabs */\n  | \"toc\"                              /* Table of contents */\n  | \"top\"                              /* Back-to-top button */\n\n/**\n * Component\n *\n * @template T - Component type\n * @template U - Reference type\n */\nexport type Component<\n  T extends {} = {},\n  U extends HTMLElement = HTMLElement\n> =\n  T & {\n    ref: U                             /* Component reference */\n  }\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Component type map\n */\ninterface ComponentTypeMap {\n  \"announce\": HTMLElement              /* Announcement bar */\n  \"container\": HTMLElement             /* Container */\n  \"consent\": HTMLElement               /* Consent */\n  \"content\": HTMLElement               /* Content */\n  \"dialog\": HTMLElement                /* Dialog */\n  \"header\": HTMLElement                /* Header */\n  \"header-title\": HTMLElement          /* Header title */\n  \"header-topic\": HTMLElement          /* Header topic */\n  \"main\": HTMLElement                  /* Main area */\n  \"outdated\": HTMLElement              /* Version warning */\n  \"palette\": HTMLElement               /* Color palette */\n  \"progress\": HTMLElement              /* Progress indicator */\n  \"search\": HTMLElement                /* Search */\n  \"search-query\": HTMLInputElement     /* Search input */\n  \"search-result\": HTMLElement         /* Search results */\n  \"search-share\": HTMLAnchorElement    /* Search sharing */\n  \"search-suggest\": HTMLElement        /* Search suggestions */\n  \"sidebar\": HTMLElement               /* Sidebar */\n  \"skip\": HTMLAnchorElement            /* Skip link */\n  \"source\": HTMLAnchorElement          /* Repository information */\n  \"tabs\": HTMLElement                  /* Navigation tabs */\n  \"toc\": HTMLElement                   /* Table of contents */\n  \"top\": HTMLAnchorElement             /* Back-to-top button */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve the element for a given component or throw a reference error\n *\n * @template T - Component type\n *\n * @param type - Component type\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getComponentElement<T extends ComponentType>(\n  type: T, node: ParentNode = document\n): ComponentTypeMap[T] {\n  return getElement(`[data-md-component=${type}]`, node)\n}\n\n/**\n * Retrieve all elements for a given component\n *\n * @template T - Component type\n *\n * @param type - Component type\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getComponentElements<T extends ComponentType>(\n  type: T, node: ParentNode = document\n): ComponentTypeMap[T][] {\n  return getElements(`[data-md-component=${type}]`, node)\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  finalize,\n  fromEvent,\n  map,\n  tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport { getElement } from \"~/browser\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Announcement bar\n */\nexport interface Announce {\n  hash: number                        /* Content hash */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch announcement bar\n *\n * @param el - Announcement bar element\n *\n * @returns Announcement bar observable\n */\nexport function watchAnnounce(\n  el: HTMLElement\n): Observable<Announce> {\n  const button = getElement(\".md-typeset > :first-child\", el)\n  return fromEvent(button, \"click\", { once: true })\n    .pipe(\n      map(() => getElement(\".md-typeset\", el)),\n      map(content => ({ hash: __md_hash(content.innerHTML) }))\n    )\n}\n\n/**\n * Mount announcement bar\n *\n * @param el - Announcement bar element\n *\n * @returns Announcement bar component observable\n */\nexport function mountAnnounce(\n  el: HTMLElement\n): Observable<Component<Announce>> {\n  if (!feature(\"announce.dismiss\") || !el.childElementCount)\n    return EMPTY\n\n  /* Support instant navigation - see https://t.ly/3FTme */\n  if (!el.hidden) {\n    const content = getElement(\".md-typeset\", el)\n    if (__md_hash(content.innerHTML) === __md_get(\"__announce\"))\n      el.hidden = true\n  }\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject<Announce>()\n    push$.subscribe(({ hash }) => {\n      el.hidden = true\n\n      /* Persist preference in local storage */\n      __md_set<number>(\"__announce\", hash)\n    })\n\n    /* Create and return component */\n    return watchAnnounce(el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  finalize,\n  map,\n  tap\n} from \"rxjs\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Consent\n */\nexport interface Consent {\n  hidden: boolean                      /* Consent is hidden */\n}\n\n/**\n * Consent defaults\n */\nexport interface ConsentDefaults {\n  analytics?: boolean                  /* Consent for Analytics */\n  github?: boolean                     /* Consent for GitHub */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  target$: Observable<HTMLElement>     /* Target observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch consent\n *\n * @param el - Consent element\n * @param options - Options\n *\n * @returns Consent observable\n */\nexport function watchConsent(\n  el: HTMLElement, { target$ }: WatchOptions\n): Observable<Consent> {\n  return target$\n    .pipe(\n      map(target => ({ hidden: target !== el }))\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount consent\n *\n * @param el - Consent element\n * @param options - Options\n *\n * @returns Consent component observable\n */\nexport function mountConsent(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Consent>> {\n  const internal$ = new Subject<Consent>()\n  internal$.subscribe(({ hidden }) => {\n    el.hidden = hidden\n  })\n\n  /* Create and return component */\n  return watchConsent(el, options)\n    .pipe(\n      tap(state => internal$.next(state)),\n      finalize(() => internal$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Tooltip style\n */\nexport type TooltipStyle =\n  | \"inline\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a tooltip\n *\n * @param id - Tooltip identifier\n * @param style - Tooltip style\n *\n * @returns Element\n */\nexport function renderTooltip(\n  id?: string, style?: TooltipStyle\n): HTMLElement {\n  if (style === \"inline\") { // @todo refactor control flow\n    return (\n      <div class=\"md-tooltip md-tooltip--inline\" id={id} role=\"tooltip\">\n        <div class=\"md-tooltip__inner md-typeset\"></div>\n      </div>\n    )\n  } else {\n    return (\n      <div class=\"md-tooltip\" id={id} role=\"tooltip\">\n        <div class=\"md-tooltip__inner md-typeset\"></div>\n      </div>\n    )\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\nimport { renderTooltip } from \"../tooltip\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render an annotation\n *\n * @param id - Annotation identifier\n * @param prefix - Tooltip identifier prefix\n *\n * @returns Element\n */\nexport function renderAnnotation(\n  id: string | number, prefix?: string\n): HTMLElement {\n  prefix = prefix ? `${prefix}_annotation_${id}` : undefined\n\n  /* Render tooltip with anchor, if given */\n  if (prefix) {\n    const anchor = prefix ? `#${prefix}` : undefined\n    return (\n      <aside class=\"md-annotation\" tabIndex={0}>\n        {renderTooltip(prefix)}\n        <a href={anchor} class=\"md-annotation__index\" tabIndex={-1}>\n          <span data-md-annotation-id={id}></span>\n        </a>\n      </aside>\n    )\n  } else {\n    return (\n      <aside class=\"md-annotation\" tabIndex={0}>\n        {renderTooltip(prefix)}\n        <span class=\"md-annotation__index\" tabIndex={-1}>\n          <span data-md-annotation-id={id}></span>\n        </span>\n      </aside>\n    )\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { translation } from \"~/_\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a 'copy-to-clipboard' button\n *\n * @param id - Unique identifier\n *\n * @returns Element\n */\nexport function renderClipboardButton(id: string): HTMLElement {\n  return (\n    <button\n      class=\"md-clipboard md-icon\"\n      title={translation(\"clipboard.copy\")}\n      data-clipboard-target={`#${id} > code`}\n    ></button>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ComponentChild } from \"preact\"\n\nimport { configuration, feature, translation } from \"~/_\"\nimport { SearchItem } from \"~/integrations/search\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Render flag\n */\nconst enum Flag {\n  TEASER = 1,                          /* Render teaser */\n  PARENT = 2                           /* Render as parent */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper function\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a search document\n *\n * @param document - Search document\n * @param flag - Render flags\n *\n * @returns Element\n */\nfunction renderSearchDocument(\n  document: SearchItem, flag: Flag\n): HTMLElement {\n  const parent = flag & Flag.PARENT\n  const teaser = flag & Flag.TEASER\n\n  /* Render missing query terms */\n  const missing = Object.keys(document.terms)\n    .filter(key => !document.terms[key])\n    .reduce<ComponentChild[]>((list, key) => [\n      ...list, <del>{key}</del>, \" \"\n    ], [])\n    .slice(0, -1)\n\n  /* Assemble query string for highlighting */\n  const config = configuration()\n  const url = new URL(document.location, config.base)\n  if (feature(\"search.highlight\"))\n    url.searchParams.set(\"h\", Object.entries(document.terms)\n      .filter(([, match]) => match)\n      .reduce((highlight, [value]) => `${highlight} ${value}`.trim(), \"\")\n    )\n\n  /* Render article or section, depending on flags */\n  const { tags } = configuration()\n  return (\n    <a href={`${url}`} class=\"md-search-result__link\" tabIndex={-1}>\n      <article\n        class=\"md-search-result__article md-typeset\"\n        data-md-score={document.score.toFixed(2)}\n      >\n        {parent > 0 && <div class=\"md-search-result__icon md-icon\"></div>}\n        {parent > 0 && <h1>{document.title}</h1>}\n        {parent <= 0 && <h2>{document.title}</h2>}\n        {teaser > 0 && document.text.length > 0 &&\n          document.text\n        }\n        {document.tags && document.tags.map(tag => {\n          const type = tags\n            ? tag in tags\n              ? `md-tag-icon md-tag--${tags[tag]}`\n              : \"md-tag-icon\"\n            : \"\"\n          return (\n            <span class={`md-tag ${type}`}>{tag}</span>\n          )\n        })}\n        {teaser > 0 && missing.length > 0 &&\n          <p class=\"md-search-result__terms\">\n            {translation(\"search.result.term.missing\")}: {...missing}\n          </p>\n        }\n      </article>\n    </a>\n  )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a search result\n *\n * @param result - Search result\n *\n * @returns Element\n */\nexport function renderSearchResultItem(\n  result: SearchItem[]\n): HTMLElement {\n  const threshold = result[0].score\n  const docs = [...result]\n\n  const config = configuration()\n\n  /* Find and extract parent article */\n  const parent = docs.findIndex(doc => {\n    const l = `${new URL(doc.location, config.base)}` // @todo hacky\n    return !l.includes(\"#\")\n  })\n  const [article] = docs.splice(parent, 1)\n\n  /* Determine last index above threshold */\n  let index = docs.findIndex(doc => doc.score < threshold)\n  if (index === -1)\n    index = docs.length\n\n  /* Partition sections */\n  const best = docs.slice(0, index)\n  const more = docs.slice(index)\n\n  /* Render children */\n  const children = [\n    renderSearchDocument(article, Flag.PARENT | +(!parent && index === 0)),\n    ...best.map(section => renderSearchDocument(section, Flag.TEASER)),\n    ...more.length ? [\n      <details class=\"md-search-result__more\">\n        <summary tabIndex={-1}>\n          <div>\n            {more.length > 0 && more.length === 1\n              ? translation(\"search.result.more.one\")\n              : translation(\"search.result.more.other\", more.length)\n            }\n          </div>\n        </summary>\n        {...more.map(section => renderSearchDocument(section, Flag.TEASER))}\n      </details>\n    ] : []\n  ]\n\n  /* Render search result */\n  return (\n    <li class=\"md-search-result__item\">\n      {children}\n    </li>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { SourceFacts } from \"~/components\"\nimport { h, round } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render repository facts\n *\n * @param facts - Repository facts\n *\n * @returns Element\n */\nexport function renderSourceFacts(facts: SourceFacts): HTMLElement {\n  return (\n    <ul class=\"md-source__facts\">\n      {Object.entries(facts).map(([key, value]) => (\n        <li class={`md-source__fact md-source__fact--${key}`}>\n          {typeof value === \"number\" ? round(value) : value}\n        </li>\n      ))}\n    </ul>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Tabbed control type\n */\ntype TabbedControlType =\n  | \"prev\"\n  | \"next\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render control for content tabs\n *\n * @param type - Control type\n *\n * @returns Element\n */\nexport function renderTabbedControl(\n  type: TabbedControlType\n): HTMLElement {\n  const classes = `tabbed-control tabbed-control--${type}`\n  return (\n    <div class={classes} hidden>\n      <button class=\"tabbed-button\" tabIndex={-1} aria-hidden=\"true\"></button>\n    </div>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a table inside a wrapper to improve scrolling on mobile\n *\n * @param table - Table element\n *\n * @returns Element\n */\nexport function renderTable(table: HTMLElement): HTMLElement {\n  return (\n    <div class=\"md-typeset__scrollwrap\">\n      <div class=\"md-typeset__table\">\n        {table}\n      </div>\n    </div>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { configuration, translation } from \"~/_\"\nimport { h } from \"~/utilities\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Version\n */\nexport interface Version {\n  version: string                      /* Version identifier */\n  title: string                        /* Version title */\n  aliases: string[]                    /* Version aliases */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a version\n *\n * @param version - Version\n *\n * @returns Element\n */\nfunction renderVersion(version: Version): HTMLElement {\n  const config = configuration()\n\n  /* Ensure trailing slash - see https://bit.ly/3rL5u3f */\n  const url = new URL(`../${version.version}/`, config.base)\n  return (\n    <li class=\"md-version__item\">\n      <a href={`${url}`} class=\"md-version__link\">\n        {version.title}\n      </a>\n    </li>\n  )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Render a version selector\n *\n * @param versions - Versions\n * @param active - Active version\n *\n * @returns Element\n */\nexport function renderVersionSelector(\n  versions: Version[], active: Version\n): HTMLElement {\n  return (\n    <div class=\"md-version\">\n      <button\n        class=\"md-version__current\"\n        aria-label={translation(\"select.version\")}\n      >\n        {active.title}\n      </button>\n      <ul class=\"md-version__list\">\n        {versions.map(renderVersion)}\n      </ul>\n    </div>\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  animationFrameScheduler,\n  asyncScheduler,\n  auditTime,\n  combineLatest,\n  debounceTime,\n  defer,\n  distinctUntilChanged,\n  filter,\n  finalize,\n  map,\n  merge,\n  of,\n  subscribeOn,\n  tap,\n  throttleTime\n} from \"rxjs\"\n\nimport {\n  ElementOffset,\n  getElement,\n  getElementContainer,\n  getElementOffset,\n  getElementSize,\n  watchElementContentOffset,\n  watchElementFocus,\n  watchElementHover\n} from \"~/browser\"\nimport { renderTooltip } from \"~/templates\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Tooltip\n */\nexport interface Tooltip {\n  active: boolean                      /* Tooltip is active */\n  offset: ElementOffset                /* Tooltip offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Global sequence number for tooltips\n */\nlet sequence = 0\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch tooltip\n *\n * This function will append the tooltip temporarily to compute its width,\n * which is necessary for correct centering, and then removing it again.\n *\n * @param el - Tooltip element\n * @param host - Host element\n *\n * @returns Tooltip observable\n */\nexport function watchTooltip(\n  el: HTMLElement, host: HTMLElement\n): Observable<Tooltip> {\n  document.body.append(el)\n\n  /* Compute width and remove tooltip immediately */\n  const { width } = getElementSize(el)\n  el.style.setProperty(\"--md-tooltip-width\", `${width}px`)\n  el.remove()\n\n  /* Retrieve and watch containing element */\n  const container = getElementContainer(host)\n  const scroll$ =\n    typeof container !== \"undefined\"\n      ? watchElementContentOffset(container)\n      : of({ x: 0, y: 0 })\n\n  /* Compute tooltip visibility */\n  const active$ = merge(\n    watchElementFocus(host),\n    watchElementHover(host)\n  )\n    .pipe(\n      distinctUntilChanged()\n    )\n\n  /* Compute tooltip offset */\n  return combineLatest([active$, scroll$])\n    .pipe(\n      map(([active, scroll]) => {\n        let { x, y } = getElementOffset(host)\n        const size = getElementSize(host)\n\n        /**\n         * Experimental: fix handling of tables - see https://bit.ly/3TQEj5O\n         *\n         * If this proves to be a viable fix, we should refactor tooltip\n         * positioning and somehow streamline the current process. This might\n         * also fix positioning for annotations inside tables, which is another\n         * limitation.\n         */\n        const table = host.closest(\"table\")\n        if (table && host.parentElement) {\n          x += table.offsetLeft + host.parentElement.offsetLeft\n          y += table.offsetTop  + host.parentElement.offsetTop\n        }\n        return {\n          active,\n          offset: {\n            x: x - scroll.x + size.width  / 2 - width / 2,\n            y: y - scroll.y + size.height + 8\n          }\n        }\n      })\n    )\n}\n\n/**\n * Mount tooltip\n *\n * @param el - Host element\n *\n * @returns Tooltip component observable\n */\nexport function mountTooltip(\n  el: HTMLElement\n): Observable<Component<Tooltip>> {\n  const title = el.title\n  if (!title.length)\n    return EMPTY\n\n  /* Render tooltip and set title from host element */\n  const id = `__tooltip_${sequence++}`\n  const tooltip = renderTooltip(id, \"inline\")\n  const typeset = getElement(\".md-typeset\", tooltip)\n  typeset.innerHTML = title\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject<Tooltip>()\n    push$.subscribe({\n\n      /* Handle emission */\n      next({ offset }) {\n        tooltip.style.setProperty(\"--md-tooltip-x\", `${offset.x}px`)\n        tooltip.style.setProperty(\"--md-tooltip-y\", `${offset.y}px`)\n      },\n\n      /* Handle complete */\n      complete() {\n        tooltip.style.removeProperty(\"--md-tooltip-x\")\n        tooltip.style.removeProperty(\"--md-tooltip-y\")\n      }\n    })\n\n    /* Toggle tooltip presence to mitigate empty lines when copying */\n    merge(\n      push$.pipe(filter(({ active }) => active)),\n      push$.pipe(debounceTime(250), filter(({ active }) => !active))\n    )\n      .subscribe({\n\n        /* Handle emission */\n        next({ active }) {\n          if (active) {\n            el.insertAdjacentElement(\"afterend\", tooltip)\n            el.setAttribute(\"aria-describedby\", id)\n            el.removeAttribute(\"title\")\n          } else {\n            tooltip.remove()\n            el.removeAttribute(\"aria-describedby\")\n            el.setAttribute(\"title\", title)\n          }\n        },\n\n        /* Handle complete */\n        complete() {\n          tooltip.remove()\n          el.removeAttribute(\"aria-describedby\")\n          el.setAttribute(\"title\", title)\n        }\n      })\n\n    /* Toggle tooltip visibility */\n    push$\n      .pipe(\n        auditTime(16, animationFrameScheduler)\n      )\n        .subscribe(({ active }) => {\n          tooltip.classList.toggle(\"md-tooltip--active\", active)\n        })\n\n    // @todo - refactor positioning together with annotations \u2013 there are\n    // several things that overlap and are identical in handling\n\n    /* Track relative origin of tooltip */\n    push$\n      .pipe(\n        throttleTime(125, animationFrameScheduler),\n        filter(() => !!el.offsetParent),\n        map(() => el.offsetParent!.getBoundingClientRect()),\n        map(({ x }) => x)\n      )\n      .subscribe({\n\n        /* Handle emission */\n        next(origin) {\n          if (origin)\n            tooltip.style.setProperty(\"--md-tooltip-0\", `${-origin}px`)\n          else\n            tooltip.style.removeProperty(\"--md-tooltip-0\")\n        },\n\n        /* Handle complete */\n        complete() {\n          tooltip.style.removeProperty(\"--md-tooltip-0\")\n        }\n      })\n\n    /* Create and return component */\n    return watchTooltip(tooltip, el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n    .pipe(\n      subscribeOn(asyncScheduler)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  animationFrameScheduler,\n  auditTime,\n  combineLatest,\n  debounceTime,\n  defer,\n  delay,\n  endWith,\n  filter,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  merge,\n  switchMap,\n  take,\n  takeUntil,\n  tap,\n  throttleTime,\n  withLatestFrom\n} from \"rxjs\"\n\nimport {\n  ElementOffset,\n  getActiveElement,\n  getElementSize,\n  watchElementContentOffset,\n  watchElementFocus,\n  watchElementOffset,\n  watchElementVisibility\n} from \"~/browser\"\n\nimport { Component } from \"../../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Annotation\n */\nexport interface Annotation {\n  active: boolean                      /* Annotation is active */\n  offset: ElementOffset                /* Annotation offset */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch annotation\n *\n * @param el - Annotation element\n * @param container - Containing element\n *\n * @returns Annotation observable\n */\nexport function watchAnnotation(\n  el: HTMLElement, container: HTMLElement\n): Observable<Annotation> {\n  const offset$ = defer(() => combineLatest([\n    watchElementOffset(el),\n    watchElementContentOffset(container)\n  ]))\n    .pipe(\n      map(([{ x, y }, scroll]): ElementOffset => {\n        const { width, height } = getElementSize(el)\n        return ({\n          x: x - scroll.x + width  / 2,\n          y: y - scroll.y + height / 2\n        })\n      })\n    )\n\n  /* Actively watch annotation on focus */\n  return watchElementFocus(el)\n    .pipe(\n      switchMap(active => offset$\n        .pipe(\n          map(offset => ({ active, offset })),\n          take(+!active || Infinity)\n        )\n      )\n    )\n}\n\n/**\n * Mount annotation\n *\n * @param el - Annotation element\n * @param container - Containing element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotation(\n  el: HTMLElement, container: HTMLElement, { target$ }: MountOptions\n): Observable<Component<Annotation>> {\n  const [tooltip, index] = Array.from(el.children)\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject<Annotation>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    push$.subscribe({\n\n      /* Handle emission */\n      next({ offset }) {\n        el.style.setProperty(\"--md-tooltip-x\", `${offset.x}px`)\n        el.style.setProperty(\"--md-tooltip-y\", `${offset.y}px`)\n      },\n\n      /* Handle complete */\n      complete() {\n        el.style.removeProperty(\"--md-tooltip-x\")\n        el.style.removeProperty(\"--md-tooltip-y\")\n      }\n    })\n\n    /* Start animation only when annotation is visible */\n    watchElementVisibility(el)\n      .pipe(\n        takeUntil(done$)\n      )\n        .subscribe(visible => {\n          el.toggleAttribute(\"data-md-visible\", visible)\n        })\n\n    /* Toggle tooltip presence to mitigate empty lines when copying */\n    merge(\n      push$.pipe(filter(({ active }) => active)),\n      push$.pipe(debounceTime(250), filter(({ active }) => !active))\n    )\n      .subscribe({\n\n        /* Handle emission */\n        next({ active }) {\n          if (active)\n            el.prepend(tooltip)\n          else\n            tooltip.remove()\n        },\n\n        /* Handle complete */\n        complete() {\n          el.prepend(tooltip)\n        }\n      })\n\n    /* Toggle tooltip visibility */\n    push$\n      .pipe(\n        auditTime(16, animationFrameScheduler)\n      )\n        .subscribe(({ active }) => {\n          tooltip.classList.toggle(\"md-tooltip--active\", active)\n        })\n\n    /* Track relative origin of tooltip */\n    push$\n      .pipe(\n        throttleTime(125, animationFrameScheduler),\n        filter(() => !!el.offsetParent),\n        map(() => el.offsetParent!.getBoundingClientRect()),\n        map(({ x }) => x)\n      )\n        .subscribe({\n\n          /* Handle emission */\n          next(origin) {\n            if (origin)\n              el.style.setProperty(\"--md-tooltip-0\", `${-origin}px`)\n            else\n              el.style.removeProperty(\"--md-tooltip-0\")\n          },\n\n          /* Handle complete */\n          complete() {\n            el.style.removeProperty(\"--md-tooltip-0\")\n          }\n        })\n\n    /* Allow to copy link without scrolling to anchor */\n    fromEvent<MouseEvent>(index, \"click\")\n      .pipe(\n        takeUntil(done$),\n        filter(ev => !(ev.metaKey || ev.ctrlKey))\n      )\n        .subscribe(ev => {\n          ev.stopPropagation()\n          ev.preventDefault()\n        })\n\n    /* Allow to open link in new tab or blur on close */\n    fromEvent<MouseEvent>(index, \"mousedown\")\n      .pipe(\n        takeUntil(done$),\n        withLatestFrom(push$)\n      )\n        .subscribe(([ev, { active }]) => {\n\n          /* Open in new tab */\n          if (ev.button !== 0 || ev.metaKey || ev.ctrlKey) {\n            ev.preventDefault()\n\n          /* Close annotation */\n          } else if (active) {\n            ev.preventDefault()\n\n            /* Focus parent annotation, if any */\n            const parent = el.parentElement!.closest(\".md-annotation\")\n            if (parent instanceof HTMLElement)\n              parent.focus()\n            else\n              getActiveElement()?.blur()\n          }\n        })\n\n    /* Open and focus annotation on location target */\n    target$\n      .pipe(\n        takeUntil(done$),\n        filter(target => target === tooltip),\n        delay(125)\n      )\n        .subscribe(() => el.focus())\n\n    /* Create and return component */\n    return watchAnnotation(el, container)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  endWith,\n  finalize,\n  ignoreElements,\n  merge,\n  share,\n  takeUntil\n} from \"rxjs\"\n\nimport {\n  getElement,\n  getElements,\n  getOptionalElement\n} from \"~/browser\"\nimport { renderAnnotation } from \"~/templates\"\n\nimport { Component } from \"../../../_\"\nimport {\n  Annotation,\n  mountAnnotation\n} from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find all annotation hosts in the containing element\n *\n * @param container - Containing element\n *\n * @returns Annotation hosts\n */\nfunction findHosts(container: HTMLElement): HTMLElement[] {\n  return container.tagName === \"CODE\"\n    ? getElements(\".c, .c1, .cm\", container)\n    : [container]\n}\n\n/**\n * Find all annotation markers in the containing element\n *\n * @param container - Containing element\n *\n * @returns Annotation markers\n */\nfunction findMarkers(container: HTMLElement): Text[] {\n  const markers: Text[] = []\n  for (const el of findHosts(container)) {\n    const nodes: Text[] = []\n\n    /* Find all text nodes in current element */\n    const it = document.createNodeIterator(el, NodeFilter.SHOW_TEXT)\n    for (let node = it.nextNode(); node; node = it.nextNode())\n      nodes.push(node as Text)\n\n    /* Find all markers in each text node */\n    for (let text of nodes) {\n      let match: RegExpExecArray | null\n\n      /* Split text at marker and add to list */\n      while ((match = /(\\(\\d+\\))(!)?/.exec(text.textContent!))) {\n        const [, id, force] = match\n        if (typeof force === \"undefined\") {\n          const marker = text.splitText(match.index)\n          text = marker.splitText(id.length)\n          markers.push(marker)\n\n        /* Replace entire text with marker */\n        } else {\n          text.textContent = id\n          markers.push(text)\n          break\n        }\n      }\n    }\n  }\n  return markers\n}\n\n/**\n * Swap the child nodes of two elements\n *\n * @param source - Source element\n * @param target - Target element\n */\nfunction swap(source: HTMLElement, target: HTMLElement): void {\n  target.append(...Array.from(source.childNodes))\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount annotation list\n *\n * This function analyzes the containing code block and checks for markers\n * referring to elements in the given annotation list. If no markers are found,\n * the list is left untouched. Otherwise, list elements are rendered as\n * annotations inside the code block.\n *\n * @param el - Annotation list element\n * @param container - Containing element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotationList(\n  el: HTMLElement, container: HTMLElement, { target$, print$ }: MountOptions\n): Observable<Component<Annotation>> {\n\n  /* Compute prefix for tooltip anchors */\n  const parent = container.closest(\"[id]\")\n  const prefix = parent?.id\n\n  /* Find and replace all markers with empty annotations */\n  const annotations = new Map<string, HTMLElement>()\n  for (const marker of findMarkers(container)) {\n    const [, id] = marker.textContent!.match(/\\((\\d+)\\)/)!\n    if (getOptionalElement(`:scope > li:nth-child(${id})`, el)) {\n      annotations.set(id, renderAnnotation(id, prefix))\n      marker.replaceWith(annotations.get(id)!)\n    }\n  }\n\n  /* Keep list if there are no annotations to render */\n  if (annotations.size === 0)\n    return EMPTY\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n\n    /* Retrieve container pairs for swapping */\n    const pairs: [HTMLElement, HTMLElement][] = []\n    for (const [id, annotation] of annotations)\n      pairs.push([\n        getElement(\".md-typeset\", annotation),\n        getElement(`:scope > li:nth-child(${id})`, el)\n      ])\n\n    /* Handle print mode - see https://bit.ly/3rgPdpt */\n    print$.pipe(takeUntil(done$))\n      .subscribe(active => {\n        el.hidden = !active\n\n        /* Add class to discern list element */\n        el.classList.toggle(\"md-annotation-list\", active)\n\n        /* Show annotations in code block or list (print) */\n        for (const [inner, child] of pairs)\n          if (!active)\n            swap(child, inner)\n          else\n            swap(inner, child)\n      })\n\n    /* Create and return component */\n    return merge(...[...annotations]\n      .map(([, annotation]) => (\n        mountAnnotation(annotation, container, { target$ })\n      ))\n    )\n      .pipe(\n        finalize(() => push$.complete()),\n        share()\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { EMPTY, Observable, defer } from \"rxjs\"\n\nimport { Component } from \"../../../_\"\nimport { Annotation } from \"../_\"\nimport { mountAnnotationList } from \"../list\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find list element directly following a block\n *\n * @param el - Annotation block element\n *\n * @returns List element or nothing\n */\nfunction findList(el: HTMLElement): HTMLElement | undefined {\n  if (el.nextElementSibling) {\n    const sibling = el.nextElementSibling as HTMLElement\n    if (sibling.tagName === \"OL\")\n      return sibling\n\n    /* Skip empty paragraphs - see https://bit.ly/3r4ZJ2O */\n    else if (sibling.tagName === \"P\" && !sibling.children.length)\n      return findList(sibling)\n  }\n\n  /* Everything else */\n  return undefined\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount annotation block\n *\n * @param el - Annotation block element\n * @param options - Options\n *\n * @returns Annotation component observable\n */\nexport function mountAnnotationBlock(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Annotation>> {\n  return defer(() => {\n    const list = findList(el)\n    return typeof list !== \"undefined\"\n      ? mountAnnotationList(list, el, options)\n      : EMPTY\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport ClipboardJS from \"clipboard\"\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  filter,\n  finalize,\n  map,\n  mergeWith,\n  switchMap,\n  take,\n  takeLast,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  getElementContentSize,\n  watchElementSize,\n  watchElementVisibility\n} from \"~/browser\"\nimport { renderClipboardButton } from \"~/templates\"\n\nimport { Component } from \"../../../_\"\nimport {\n  Tooltip,\n  mountTooltip\n} from \"../../../tooltip\"\nimport {\n  Annotation,\n  mountAnnotationList\n} from \"../../annotation\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Code block overflow\n */\nexport interface Overflow {\n  scrollable: boolean                  /* Code block overflows */\n}\n\n/**\n * Code block\n */\nexport type CodeBlock =\n  | Overflow\n  | Annotation\n  | Tooltip\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Global sequence number for code blocks\n */\nlet sequence = 0\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Find candidate list element directly following a code block\n *\n * @param el - Code block element\n *\n * @returns List element or nothing\n */\nfunction findCandidateList(el: HTMLElement): HTMLElement | undefined {\n  if (el.nextElementSibling) {\n    const sibling = el.nextElementSibling as HTMLElement\n    if (sibling.tagName === \"OL\")\n      return sibling\n\n    /* Skip empty paragraphs - see https://bit.ly/3r4ZJ2O */\n    else if (sibling.tagName === \"P\" && !sibling.children.length)\n      return findCandidateList(sibling)\n  }\n\n  /* Everything else */\n  return undefined\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch code block\n *\n * This function monitors size changes of the viewport, as well as switches of\n * content tabs with embedded code blocks, as both may trigger overflow.\n *\n * @param el - Code block element\n *\n * @returns Code block observable\n */\nexport function watchCodeBlock(\n  el: HTMLElement\n): Observable<Overflow> {\n  return watchElementSize(el)\n    .pipe(\n      map(({ width }) => {\n        const content = getElementContentSize(el)\n        return {\n          scrollable: content.width > width\n        }\n      }),\n      distinctUntilKeyChanged(\"scrollable\")\n    )\n}\n\n/**\n * Mount code block\n *\n * This function ensures that an overflowing code block is focusable through\n * keyboard, so it can be scrolled without a mouse to improve on accessibility.\n * Furthermore, if code annotations are enabled, they are mounted if and only\n * if the code block is currently visible, e.g., not in a hidden content tab.\n *\n * Note that code blocks may be mounted eagerly or lazily. If they're mounted\n * lazily (on first visibility), code annotation anchor links will not work,\n * as they are evaluated on initial page load, and code annotations in general\n * might feel a little bumpier.\n *\n * @param el - Code block element\n * @param options - Options\n *\n * @returns Code block and annotation component observable\n */\nexport function mountCodeBlock(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<CodeBlock>> {\n  const { matches: hover } = matchMedia(\"(hover)\")\n\n  /* Defer mounting of code block - see https://bit.ly/3vHVoVD */\n  const factory$ = defer(() => {\n    const push$ = new Subject<Overflow>()\n    const done$ = push$.pipe(takeLast(1))\n    push$.subscribe(({ scrollable }) => {\n      if (scrollable && hover)\n        el.setAttribute(\"tabindex\", \"0\")\n      else\n        el.removeAttribute(\"tabindex\")\n    })\n\n    /* Render button for Clipboard.js integration */\n    const content$: Array<Observable<Component<CodeBlock>>> = []\n    if (ClipboardJS.isSupported()) {\n      if (el.closest(\".copy\") || (\n        feature(\"content.code.copy\") && !el.closest(\".no-copy\")\n      )) {\n        const parent = el.closest(\"pre\")!\n        parent.id = `__code_${sequence++}`\n\n        /* Mount tooltip, if enabled */\n        const button = renderClipboardButton(parent.id)\n        parent.insertBefore(button, el)\n        if (feature(\"content.tooltips\"))\n          content$.push(mountTooltip(button))\n      }\n    }\n\n    /* Handle code annotations */\n    const container = el.closest(\".highlight\")\n    if (container instanceof HTMLElement) {\n      const list = findCandidateList(container)\n\n      /* Mount code annotations, if enabled */\n      if (typeof list !== \"undefined\" && (\n        container.classList.contains(\"annotate\") ||\n        feature(\"content.code.annotate\")\n      )) {\n        const annotations$ = mountAnnotationList(list, el, options)\n        content$.push(\n          watchElementSize(container)\n            .pipe(\n              takeUntil(done$),\n              map(({ width, height }) => width && height),\n              distinctUntilChanged(),\n              switchMap(active => active ? annotations$ : EMPTY)\n            )\n        )\n      }\n    }\n\n    /* Create and return component */\n    return watchCodeBlock(el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state })),\n        mergeWith(...content$)\n      )\n  })\n\n  /* Mount code block lazily */\n  if (feature(\"content.lazy\"))\n    return watchElementVisibility(el)\n      .pipe(\n        filter(visible => visible),\n        take(1),\n        switchMap(() => factory$)\n      )\n\n  /* Mount code block */\n  return factory$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  defer,\n  filter,\n  finalize,\n  map,\n  merge,\n  tap\n} from \"rxjs\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Details\n */\nexport interface Details {\n  action: \"open\" | \"close\"             /* Details state */\n  reveal?: boolean                     /* Details is revealed */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch details\n *\n * @param el - Details element\n * @param options - Options\n *\n * @returns Details observable\n */\nexport function watchDetails(\n  el: HTMLDetailsElement, { target$, print$ }: WatchOptions\n): Observable<Details> {\n  let open = true\n  return merge(\n\n    /* Open and focus details on location target */\n    target$\n      .pipe(\n        map(target => target.closest(\"details:not([open])\")!),\n        filter(details => el === details),\n        map(() => ({\n          action: \"open\", reveal: true\n        }) as Details)\n      ),\n\n    /* Open details on print and close afterwards */\n    print$\n      .pipe(\n        filter(active => active || !open),\n        tap(() => open = el.open),\n        map(active => ({\n          action: active ? \"open\" : \"close\"\n        }) as Details)\n      )\n  )\n}\n\n/**\n * Mount details\n *\n * This function ensures that `details` tags are opened on anchor jumps and\n * prior to printing, so the whole content of the page is visible.\n *\n * @param el - Details element\n * @param options - Options\n *\n * @returns Details component observable\n */\nexport function mountDetails(\n  el: HTMLDetailsElement, options: MountOptions\n): Observable<Component<Details>> {\n  return defer(() => {\n    const push$ = new Subject<Details>()\n    push$.subscribe(({ action, reveal }) => {\n      el.toggleAttribute(\"open\", action === \"open\")\n      if (reveal)\n        el.scrollIntoView()\n    })\n\n    /* Create and return component */\n    return watchDetails(el, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", ".node circle,.node ellipse,.node path,.node polygon,.node rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}marker{fill:var(--md-mermaid-edge-color)!important}.edgeLabel .label rect{fill:#0000}.label{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.label foreignObject{line-height:normal;overflow:visible}.label div .edgeLabel{color:var(--md-mermaid-label-fg-color)}.edgeLabel,.edgeLabel rect,.label div .edgeLabel{background-color:var(--md-mermaid-label-bg-color)}.edgeLabel,.edgeLabel rect{fill:var(--md-mermaid-label-bg-color);color:var(--md-mermaid-edge-color)}.edgePath .path,.flowchart-link{stroke:var(--md-mermaid-edge-color);stroke-width:.05rem}.edgePath .arrowheadPath{fill:var(--md-mermaid-edge-color);stroke:none}.cluster rect{fill:var(--md-default-fg-color--lightest);stroke:var(--md-default-fg-color--lighter)}.cluster span{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}g #flowchart-circleEnd,g #flowchart-circleStart,g #flowchart-crossEnd,g #flowchart-crossStart,g #flowchart-pointEnd,g #flowchart-pointStart{stroke:none}g.classGroup line,g.classGroup rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}g.classGroup text{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.classLabel .box{fill:var(--md-mermaid-label-bg-color);background-color:var(--md-mermaid-label-bg-color);opacity:1}.classLabel .label{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.node .divider{stroke:var(--md-mermaid-node-fg-color)}.relation{stroke:var(--md-mermaid-edge-color)}.cardinality{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.cardinality text{fill:inherit!important}defs #classDiagram-compositionEnd,defs #classDiagram-compositionStart,defs #classDiagram-dependencyEnd,defs #classDiagram-dependencyStart,defs #classDiagram-extensionEnd,defs #classDiagram-extensionStart{fill:var(--md-mermaid-edge-color)!important;stroke:var(--md-mermaid-edge-color)!important}defs #classDiagram-aggregationEnd,defs #classDiagram-aggregationStart{fill:var(--md-mermaid-label-bg-color)!important;stroke:var(--md-mermaid-edge-color)!important}g.stateGroup rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}g.stateGroup .state-title{fill:var(--md-mermaid-label-fg-color)!important;font-family:var(--md-mermaid-font-family)}g.stateGroup .composit{fill:var(--md-mermaid-label-bg-color)}.nodeLabel{color:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.node circle.state-end,.node circle.state-start,.start-state{fill:var(--md-mermaid-edge-color);stroke:none}.end-state-inner,.end-state-outer{fill:var(--md-mermaid-edge-color)}.end-state-inner,.node circle.state-end{stroke:var(--md-mermaid-label-bg-color)}.transition{stroke:var(--md-mermaid-edge-color)}[id^=state-fork] rect,[id^=state-join] rect{fill:var(--md-mermaid-edge-color)!important;stroke:none!important}.statediagram-cluster.statediagram-cluster .inner{fill:var(--md-default-bg-color)}.statediagram-cluster rect{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}.statediagram-state rect.divider{fill:var(--md-default-fg-color--lightest);stroke:var(--md-default-fg-color--lighter)}defs #statediagram-barbEnd{stroke:var(--md-mermaid-edge-color)}.attributeBoxEven,.attributeBoxOdd{fill:var(--md-mermaid-node-bg-color);stroke:var(--md-mermaid-node-fg-color)}.entityBox{fill:var(--md-mermaid-label-bg-color);stroke:var(--md-mermaid-node-fg-color)}.entityLabel{fill:var(--md-mermaid-label-fg-color);font-family:var(--md-mermaid-font-family)}.relationshipLabelBox{fill:var(--md-mermaid-label-bg-color);fill-opacity:1;background-color:var(--md-mermaid-label-bg-color);opacity:1}.relationshipLabel{fill:var(--md-mermaid-label-fg-color)}.relationshipLine{stroke:var(--md-mermaid-edge-color)}defs #ONE_OR_MORE_END *,defs #ONE_OR_MORE_START *,defs #ONLY_ONE_END *,defs #ONLY_ONE_START *,defs #ZERO_OR_MORE_END *,defs #ZERO_OR_MORE_START *,defs #ZERO_OR_ONE_END *,defs #ZERO_OR_ONE_START *{stroke:var(--md-mermaid-edge-color)!important}defs #ZERO_OR_MORE_END circle,defs #ZERO_OR_MORE_START circle{fill:var(--md-mermaid-label-bg-color)}.actor{fill:var(--md-mermaid-sequence-actor-bg-color);stroke:var(--md-mermaid-sequence-actor-border-color)}text.actor>tspan{fill:var(--md-mermaid-sequence-actor-fg-color);font-family:var(--md-mermaid-font-family)}line{stroke:var(--md-mermaid-sequence-actor-line-color)}.actor-man circle,.actor-man line{fill:var(--md-mermaid-sequence-actorman-bg-color);stroke:var(--md-mermaid-sequence-actorman-line-color)}.messageLine0,.messageLine1{stroke:var(--md-mermaid-sequence-message-line-color)}.note{fill:var(--md-mermaid-sequence-note-bg-color);stroke:var(--md-mermaid-sequence-note-border-color)}.loopText,.loopText>tspan,.messageText,.noteText>tspan{stroke:none;font-family:var(--md-mermaid-font-family)!important}.messageText{fill:var(--md-mermaid-sequence-message-fg-color)}.loopText,.loopText>tspan{fill:var(--md-mermaid-sequence-loop-fg-color)}.noteText>tspan{fill:var(--md-mermaid-sequence-note-fg-color)}#arrowhead path{fill:var(--md-mermaid-sequence-message-line-color);stroke:none}.loopLine{fill:var(--md-mermaid-sequence-loop-bg-color);stroke:var(--md-mermaid-sequence-loop-border-color)}.labelBox{fill:var(--md-mermaid-sequence-label-bg-color);stroke:none}.labelText,.labelText>span{fill:var(--md-mermaid-sequence-label-fg-color);font-family:var(--md-mermaid-font-family)}.sequenceNumber{fill:var(--md-mermaid-sequence-number-fg-color)}rect.rect{fill:var(--md-mermaid-sequence-box-bg-color);stroke:none}rect.rect+text.text{fill:var(--md-mermaid-sequence-box-fg-color)}defs #sequencenumber{fill:var(--md-mermaid-sequence-number-bg-color)!important}", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  map,\n  of,\n  shareReplay,\n  tap\n} from \"rxjs\"\n\nimport { watchScript } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\nimport themeCSS from \"./index.css\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mermaid diagram\n */\nexport interface Mermaid {}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Mermaid instance observable\n */\nlet mermaid$: Observable<void>\n\n/**\n * Global sequence number for diagrams\n */\nlet sequence = 0\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch Mermaid script\n *\n * @returns Mermaid scripts observable\n */\nfunction fetchScripts(): Observable<void> {\n  return typeof mermaid === \"undefined\" || mermaid instanceof Element\n    ? watchScript(\"https://unpkg.com/mermaid@10.6.1/dist/mermaid.min.js\")\n    : of(undefined)\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount Mermaid diagram\n *\n * @param el - Code block element\n *\n * @returns Mermaid diagram component observable\n */\nexport function mountMermaid(\n  el: HTMLElement\n): Observable<Component<Mermaid>> {\n  el.classList.remove(\"mermaid\") // Hack: mitigate https://bit.ly/3CiN6Du\n  mermaid$ ||= fetchScripts()\n    .pipe(\n      tap(() => mermaid.initialize({\n        startOnLoad: false,\n        themeCSS,\n        sequence: {\n          actorFontSize: \"16px\", // Hack: mitigate https://bit.ly/3y0NEi3\n          messageFontSize: \"16px\",\n          noteFontSize: \"16px\"\n        }\n      })),\n      map(() => undefined),\n      shareReplay(1)\n    )\n\n  /* Render diagram */\n  mermaid$.subscribe(async () => {\n    el.classList.add(\"mermaid\") // Hack: mitigate https://bit.ly/3CiN6Du\n    const id = `__mermaid_${sequence++}`\n\n    /* Create host element to replace code block */\n    const host = h(\"div\", { class: \"mermaid\" })\n    const text = el.textContent\n\n    /* Render and inject diagram */\n    const { svg, fn } = await mermaid.render(id, text)\n\n    /* Create a shadow root and inject diagram */\n    const shadow = host.attachShadow({ mode: \"closed\" })\n    shadow.innerHTML = svg\n\n    /* Replace code block with diagram and bind functions */\n    el.replaceWith(host)\n    fn?.(shadow)\n  })\n\n  /* Create and return component */\n  return mermaid$\n    .pipe(\n      map(() => ({ ref: el }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Observable, of } from \"rxjs\"\n\nimport { renderTable } from \"~/templates\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Data table\n */\nexport interface DataTable {}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Sentinel for replacement\n */\nconst sentinel = h(\"table\")\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount data table\n *\n * This function wraps a data table in another scrollable container, so it can\n * be smoothly scrolled on smaller screen sizes and won't break the layout.\n *\n * @param el - Data table element\n *\n * @returns Data table component observable\n */\nexport function mountDataTable(\n  el: HTMLElement\n): Observable<Component<DataTable>> {\n  el.replaceWith(sentinel)\n  sentinel.replaceWith(renderTable(el))\n\n  /* Create and return component */\n  return of({ ref: el })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  animationFrameScheduler,\n  asyncScheduler,\n  auditTime,\n  combineLatest,\n  defer,\n  endWith,\n  filter,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  merge,\n  skip,\n  startWith,\n  subscribeOn,\n  takeUntil,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  getElement,\n  getElementContentOffset,\n  getElementContentSize,\n  getElementOffset,\n  getElementSize,\n  getElements,\n  watchElementContentOffset,\n  watchElementSize\n} from \"~/browser\"\nimport { renderTabbedControl } from \"~/templates\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Content tabs\n */\nexport interface ContentTabs {\n  active: HTMLLabelElement             /* Active tab label */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch content tabs\n *\n * @param inputs - Content tabs input elements\n *\n * @returns Content tabs observable\n */\nexport function watchContentTabs(\n  inputs: HTMLInputElement[]\n): Observable<ContentTabs> {\n  const initial = inputs.find(input => input.checked) || inputs[0]\n  return merge(...inputs.map(input => fromEvent(input, \"change\")\n    .pipe(\n      map(() => getElement<HTMLLabelElement>(`label[for=\"${input.id}\"]`))\n    )\n  ))\n    .pipe(\n      startWith(getElement<HTMLLabelElement>(`label[for=\"${initial.id}\"]`)),\n      map(active => ({ active }))\n    )\n}\n\n/**\n * Mount content tabs\n *\n * @param el - Content tabs element\n * @param options - Options\n *\n * @returns Content tabs component observable\n */\nexport function mountContentTabs(\n  el: HTMLElement, { viewport$, target$ }: MountOptions\n): Observable<Component<ContentTabs>> {\n  const container = getElement(\".tabbed-labels\", el)\n  const inputs = getElements<HTMLInputElement>(\":scope > input\", el)\n\n  /* Render content tab previous button for pagination */\n  const prev = renderTabbedControl(\"prev\")\n  el.append(prev)\n\n  /* Render content tab next button for pagination */\n  const next = renderTabbedControl(\"next\")\n  el.append(next)\n\n  /* Mount component on subscription */\n  return defer(() => {\n    const push$ = new Subject<ContentTabs>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    combineLatest([push$, watchElementSize(el)])\n      .pipe(\n        takeUntil(done$),\n        auditTime(1, animationFrameScheduler)\n      )\n        .subscribe({\n\n          /* Handle emission */\n          next([{ active }, size]) {\n            const offset = getElementOffset(active)\n            const { width } = getElementSize(active)\n\n            /* Set tab indicator offset and width */\n            el.style.setProperty(\"--md-indicator-x\", `${offset.x}px`)\n            el.style.setProperty(\"--md-indicator-width\", `${width}px`)\n\n            /* Scroll container to active content tab */\n            const content = getElementContentOffset(container)\n            if (\n              offset.x         < content.x              ||\n              offset.x + width > content.x + size.width\n            )\n              container.scrollTo({\n                left: Math.max(0, offset.x - 16),\n                behavior: \"smooth\"\n              })\n          },\n\n          /* Handle complete */\n          complete() {\n            el.style.removeProperty(\"--md-indicator-x\")\n            el.style.removeProperty(\"--md-indicator-width\")\n          }\n        })\n\n    /* Hide content tab buttons on borders */\n    combineLatest([\n      watchElementContentOffset(container),\n      watchElementSize(container)\n    ])\n      .pipe(\n        takeUntil(done$)\n      )\n        .subscribe(([offset, size]) => {\n          const content = getElementContentSize(container)\n          prev.hidden = offset.x < 16\n          next.hidden = offset.x > content.width - size.width - 16\n        })\n\n    /* Paginate content tab container on click */\n    merge(\n      fromEvent(prev, \"click\").pipe(map(() => -1)),\n      fromEvent(next, \"click\").pipe(map(() => +1))\n    )\n      .pipe(\n        takeUntil(done$)\n      )\n        .subscribe(direction => {\n          const { width } = getElementSize(container)\n          container.scrollBy({\n            left: width * direction,\n            behavior: \"smooth\"\n          })\n        })\n\n    /* Switch to content tab target */\n    target$\n      .pipe(\n        takeUntil(done$),\n        filter(input => inputs.includes(input as HTMLInputElement))\n      )\n        .subscribe(input => input.click())\n\n    /* Add link to each content tab label */\n    container.classList.add(\"tabbed-labels--linked\")\n    for (const input of inputs) {\n      const label = getElement<HTMLLabelElement>(`label[for=\"${input.id}\"]`)\n      label.replaceChildren(h(\"a\", {\n        href: `#${label.htmlFor}`,\n        tabIndex: -1\n      }, ...Array.from(label.childNodes)))\n\n      /* Allow to copy link without scrolling to anchor */\n      fromEvent<MouseEvent>(label.firstElementChild!, \"click\")\n        .pipe(\n          takeUntil(done$),\n          filter(ev => !(ev.metaKey || ev.ctrlKey)),\n          tap(ev => {\n            ev.preventDefault()\n            ev.stopPropagation()\n          })\n        )\n          // @todo we might need to remove the anchor link on complete\n          .subscribe(() => {\n            history.replaceState({}, \"\", `#${label.htmlFor}`)\n            label.click()\n          })\n    }\n\n    /* Set up linking of content tabs, if enabled */\n    if (feature(\"content.tabs.link\"))\n      push$.pipe(\n        skip(1),\n        withLatestFrom(viewport$)\n      )\n        .subscribe(([{ active }, { offset }]) => {\n          const tab = active.innerText.trim()\n          if (active.hasAttribute(\"data-md-switching\")) {\n            active.removeAttribute(\"data-md-switching\")\n\n          /* Determine viewport offset of active tab */\n          } else {\n            const y = el.offsetTop - offset.y\n\n            /* Passively activate other tabs */\n            for (const set of getElements(\"[data-tabs]\"))\n              for (const input of getElements<HTMLInputElement>(\n                \":scope > input\", set\n              )) {\n                const label = getElement(`label[for=\"${input.id}\"]`)\n                if (\n                  label !== active &&\n                  label.innerText.trim() === tab\n                ) {\n                  label.setAttribute(\"data-md-switching\", \"\")\n                  input.click()\n                  break\n                }\n              }\n\n            /* Bring active tab into view */\n            window.scrollTo({\n              top: el.offsetTop - y\n            })\n\n            /* Persist active tabs in local storage */\n            const tabs = __md_get<string[]>(\"__tabs\") || []\n            __md_set(\"__tabs\", [...new Set([tab, ...tabs])])\n          }\n        })\n\n    /* Pause media (audio, video) on switch - see https://bit.ly/3Bk6cel */\n    push$.pipe(takeUntil(done$))\n      .subscribe(() => {\n        for (const media of getElements<HTMLAudioElement>(\"audio, video\", el))\n          media.pause()\n      })\n\n    /* Create and return component */\n    return watchContentTabs(inputs)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n    .pipe(\n      subscribeOn(asyncScheduler)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Observable, merge } from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport { Viewport, getElements } from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport {\n  Tooltip,\n  mountTooltip\n} from \"../../tooltip\"\nimport {\n  Annotation,\n  mountAnnotationBlock\n} from \"../annotation\"\nimport {\n  CodeBlock,\n  mountCodeBlock\n} from \"../code\"\nimport {\n  Details,\n  mountDetails\n} from \"../details\"\nimport {\n  Mermaid,\n  mountMermaid\n} from \"../mermaid\"\nimport {\n  DataTable,\n  mountDataTable\n} from \"../table\"\nimport {\n  ContentTabs,\n  mountContentTabs\n} from \"../tabs\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Content\n */\nexport type Content =\n  | Annotation\n  | CodeBlock\n  | ContentTabs\n  | DataTable\n  | Details\n  | Mermaid\n  | Tooltip\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n  print$: Observable<boolean>          /* Media print observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount content\n *\n * This function mounts all components that are found in the content of the\n * actual article, including code blocks, data tables and details.\n *\n * @param el - Content element\n * @param options - Options\n *\n * @returns Content component observable\n */\nexport function mountContent(\n  el: HTMLElement, { viewport$, target$, print$ }: MountOptions\n): Observable<Component<Content>> {\n  return merge(\n\n    /* Annotations */\n    ...getElements(\".annotate:not(.highlight)\", el)\n      .map(child => mountAnnotationBlock(child, { target$, print$ })),\n\n    /* Code blocks */\n    ...getElements(\"pre:not(.mermaid) > code\", el)\n      .map(child => mountCodeBlock(child, { target$, print$ })),\n\n    /* Mermaid diagrams */\n    ...getElements(\"pre.mermaid\", el)\n      .map(child => mountMermaid(child)),\n\n    /* Data tables */\n    ...getElements(\"table:not([class])\", el)\n      .map(child => mountDataTable(child)),\n\n    /* Details */\n    ...getElements(\"details\", el)\n      .map(child => mountDetails(child, { target$, print$ })),\n\n    /* Content tabs */\n    ...getElements(\"[data-tabs]\", el)\n      .map(child => mountContentTabs(child, { viewport$, target$ })),\n\n    /* Tooltips */\n    ...getElements(\"[title]\", el)\n      .filter(() => feature(\"content.tooltips\"))\n      .map(child => mountTooltip(child))\n  )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  defer,\n  delay,\n  finalize,\n  map,\n  merge,\n  of,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { getElement } from \"~/browser\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Dialog\n */\nexport interface Dialog {\n  message: string                      /* Dialog message */\n  active: boolean                      /* Dialog is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  alert$: Subject<string>              /* Alert subject */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  alert$: Subject<string>              /* Alert subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch dialog\n *\n * @param _el - Dialog element\n * @param options - Options\n *\n * @returns Dialog observable\n */\nexport function watchDialog(\n  _el: HTMLElement, { alert$ }: WatchOptions\n): Observable<Dialog> {\n  return alert$\n    .pipe(\n      switchMap(message => merge(\n        of(true),\n        of(false).pipe(delay(2000))\n      )\n        .pipe(\n          map(active => ({ message, active }))\n        )\n      )\n    )\n}\n\n/**\n * Mount dialog\n *\n * This function reveals the dialog in the right corner when a new alert is\n * emitted through the subject that is passed as part of the options.\n *\n * @param el - Dialog element\n * @param options - Options\n *\n * @returns Dialog component observable\n */\nexport function mountDialog(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Dialog>> {\n  const inner = getElement(\".md-typeset\", el)\n  return defer(() => {\n    const push$ = new Subject<Dialog>()\n    push$.subscribe(({ message, active }) => {\n      el.classList.toggle(\"md-dialog--active\", active)\n      inner.textContent = message\n    })\n\n    /* Create and return component */\n    return watchDialog(el, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  bufferCount,\n  combineLatest,\n  combineLatestWith,\n  defer,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  filter,\n  from,\n  ignoreElements,\n  map,\n  mergeMap,\n  mergeWith,\n  of,\n  shareReplay,\n  startWith,\n  switchMap,\n  takeUntil\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  getElements,\n  watchElementSize,\n  watchToggle\n} from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { Main } from \"../../main\"\nimport {\n  Tooltip,\n  mountTooltip\n} from \"../../tooltip\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Header\n */\nexport interface Header {\n  height: number                       /* Header visible height */\n  hidden: boolean                      /* Header is hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Compute whether the header is hidden\n *\n * If the user scrolls past a certain threshold, the header can be hidden when\n * scrolling down, and shown when scrolling up.\n *\n * @param options - Options\n *\n * @returns Toggle observable\n */\nfunction isHidden({ viewport$ }: WatchOptions): Observable<boolean> {\n  if (!feature(\"header.autohide\"))\n    return of(false)\n\n  /* Compute direction and turning point */\n  const direction$ = viewport$\n    .pipe(\n      map(({ offset: { y } }) => y),\n      bufferCount(2, 1),\n      map(([a, b]) => [a < b, b] as const),\n      distinctUntilKeyChanged(0)\n    )\n\n  /* Compute whether header should be hidden */\n  const hidden$ = combineLatest([viewport$, direction$])\n    .pipe(\n      filter(([{ offset }, [, y]]) => Math.abs(y - offset.y) > 100),\n      map(([, [direction]]) => direction),\n      distinctUntilChanged()\n    )\n\n  /* Compute threshold for hiding */\n  const search$ = watchToggle(\"search\")\n  return combineLatest([viewport$, search$])\n    .pipe(\n      map(([{ offset }, search]) => offset.y > 400 && !search),\n      distinctUntilChanged(),\n      switchMap(active => active ? hidden$ : of(false)),\n      startWith(false)\n    )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch header\n *\n * @param el - Header element\n * @param options - Options\n *\n * @returns Header observable\n */\nexport function watchHeader(\n  el: HTMLElement, options: WatchOptions\n): Observable<Header> {\n  return defer(() => combineLatest([\n    watchElementSize(el),\n    isHidden(options)\n  ]))\n    .pipe(\n      map(([{ height }, hidden]) => ({\n        height,\n        hidden\n      })),\n      distinctUntilChanged((a, b) => (\n        a.height === b.height &&\n        a.hidden === b.hidden\n      )),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount header\n *\n * This function manages the different states of the header, i.e. whether it's\n * hidden or rendered with a shadow. This depends heavily on the main area.\n *\n * @param el - Header element\n * @param options - Options\n *\n * @returns Header component observable\n */\nexport function mountHeader(\n  el: HTMLElement, { header$, main$ }: MountOptions\n): Observable<Component<Header | Tooltip>> {\n  return defer(() => {\n    const push$ = new Subject<Main>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    push$\n      .pipe(\n        distinctUntilKeyChanged(\"active\"),\n        combineLatestWith(header$)\n      )\n        .subscribe(([{ active }, { hidden }]) => {\n          el.classList.toggle(\"md-header--shadow\", active && !hidden)\n          el.hidden = hidden\n        })\n\n    /* Mount tooltips, if enabled */\n    const tooltips = from(getElements(\"[title]\", el))\n      .pipe(\n        filter(() => feature(\"content.tooltips\")),\n        mergeMap(child => mountTooltip(child))\n      )\n\n    /* Link to main area */\n    main$.subscribe(push$)\n\n    /* Create and return component */\n    return header$\n      .pipe(\n        takeUntil(done$),\n        map(state => ({ ref: el, ...state })),\n        mergeWith(tooltips.pipe(takeUntil(done$)))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  defer,\n  distinctUntilKeyChanged,\n  finalize,\n  map,\n  tap\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  getElementSize,\n  getOptionalElement,\n  watchViewportAt\n} from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { Header } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Header\n */\nexport interface HeaderTitle {\n  active: boolean                      /* Header title is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch header title\n *\n * @param el - Heading element\n * @param options - Options\n *\n * @returns Header title observable\n */\nexport function watchHeaderTitle(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<HeaderTitle> {\n  return watchViewportAt(el, { viewport$, header$ })\n    .pipe(\n      map(({ offset: { y } }) => {\n        const { height } = getElementSize(el)\n        return {\n          active: y >= height\n        }\n      }),\n      distinctUntilKeyChanged(\"active\")\n    )\n}\n\n/**\n * Mount header title\n *\n * This function swaps the header title from the site title to the title of the\n * current page when the user scrolls past the first headline.\n *\n * @param el - Header title element\n * @param options - Options\n *\n * @returns Header title component observable\n */\nexport function mountHeaderTitle(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<HeaderTitle>> {\n  return defer(() => {\n    const push$ = new Subject<HeaderTitle>()\n    push$.subscribe({\n\n      /* Handle emission */\n      next({ active }) {\n        el.classList.toggle(\"md-header__title--active\", active)\n      },\n\n      /* Handle complete */\n      complete() {\n        el.classList.remove(\"md-header__title--active\")\n      }\n    })\n\n    /* Obtain headline, if any */\n    const heading = getOptionalElement(\".md-content h1\")\n    if (typeof heading === \"undefined\")\n      return EMPTY\n\n    /* Create and return component */\n    return watchHeaderTitle(heading, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  map,\n  switchMap\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  watchElementSize\n} from \"~/browser\"\n\nimport { Header } from \"../header\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Main area\n */\nexport interface Main {\n  offset: number                       /* Main area top offset */\n  height: number                       /* Main area visible height */\n  active: boolean                      /* Main area is active */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch main area\n *\n * This function returns an observable that computes the visual parameters of\n * the main area which depends on the viewport vertical offset and height, as\n * well as the height of the header element, if the header is fixed.\n *\n * @param el - Main area element\n * @param options - Options\n *\n * @returns Main area observable\n */\nexport function watchMain(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<Main> {\n\n  /* Compute necessary adjustment for header */\n  const adjust$ = header$\n    .pipe(\n      map(({ height }) => height),\n      distinctUntilChanged()\n    )\n\n  /* Compute the main area's top and bottom borders */\n  const border$ = adjust$\n    .pipe(\n      switchMap(() => watchElementSize(el)\n        .pipe(\n          map(({ height }) => ({\n            top:    el.offsetTop,\n            bottom: el.offsetTop + height\n          })),\n          distinctUntilKeyChanged(\"bottom\")\n        )\n      )\n    )\n\n  /* Compute the main area's offset, visible height and if we scrolled past */\n  return combineLatest([adjust$, border$, viewport$])\n    .pipe(\n      map(([header, { top, bottom }, { offset: { y }, size: { height } }]) => {\n        height = Math.max(0, height\n          - Math.max(0, top    - y,  header)\n          - Math.max(0, height + y - bottom)\n        )\n        return {\n          offset: top - header,\n          height,\n          active: top - header <= y\n        }\n      }),\n      distinctUntilChanged((a, b) => (\n        a.offset === b.offset &&\n        a.height === b.height &&\n        a.active === b.active\n      ))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  asyncScheduler,\n  defer,\n  finalize,\n  fromEvent,\n  map,\n  mergeMap,\n  observeOn,\n  of,\n  repeat,\n  shareReplay,\n  skip,\n  startWith,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport { getElements, watchMedia } from \"~/browser\"\nimport { h } from \"~/utilities\"\n\nimport {\n  Component,\n  getComponentElement\n} from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Palette colors\n */\nexport interface PaletteColor {\n  media?: string                       /* Media query */\n  scheme?: string                      /* Color scheme */\n  primary?: string                     /* Primary color */\n  accent?: string                      /* Accent color */\n}\n\n/**\n * Palette\n */\nexport interface Palette {\n  index: number                        /* Palette index */\n  color: PaletteColor                  /* Palette colors */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch color palette\n *\n * @param inputs - Color palette element\n *\n * @returns Color palette observable\n */\nexport function watchPalette(\n  inputs: HTMLInputElement[]\n): Observable<Palette> {\n  const current = __md_get<Palette>(\"__palette\") || {\n    index: inputs.findIndex(input => matchMedia(\n      input.getAttribute(\"data-md-color-media\")!\n    ).matches)\n  }\n\n  /* Emit changes in color palette */\n  return of(...inputs)\n    .pipe(\n      mergeMap(input => fromEvent(input, \"change\").pipe(map(() => input))),\n      startWith(inputs[Math.max(0, current.index)]),\n      map(input => ({\n        index: inputs.indexOf(input),\n        color: {\n          media:   input.getAttribute(\"data-md-color-media\"),\n          scheme:  input.getAttribute(\"data-md-color-scheme\"),\n          primary: input.getAttribute(\"data-md-color-primary\"),\n          accent:  input.getAttribute(\"data-md-color-accent\")\n        }\n      } as Palette)),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount color palette\n *\n * @param el - Color palette element\n *\n * @returns Color palette component observable\n */\nexport function mountPalette(\n  el: HTMLElement\n): Observable<Component<Palette>> {\n  const inputs = getElements<HTMLInputElement>(\"input\", el)\n  const meta = h(\"meta\", { name: \"theme-color\" })\n  document.head.appendChild(meta)\n\n  // Add color scheme meta tag\n  const scheme = h(\"meta\", { name: \"color-scheme\" })\n  document.head.appendChild(scheme)\n\n  /* Mount component on subscription */\n  const media$ = watchMedia(\"(prefers-color-scheme: light)\")\n  return defer(() => {\n    const push$ = new Subject<Palette>()\n    push$.subscribe(palette => {\n      document.body.setAttribute(\"data-md-color-switching\", \"\")\n\n      /* Retrieve color palette for system preference */\n      if (palette.color.media === \"(prefers-color-scheme)\") {\n        const media = matchMedia(\"(prefers-color-scheme: light)\")\n        const input = document.querySelector(media.matches\n          ? \"[data-md-color-media='(prefers-color-scheme: light)']\"\n          : \"[data-md-color-media='(prefers-color-scheme: dark)']\"\n        )!\n\n        /* Retrieve colors for system preference */\n        palette.color.scheme  = input.getAttribute(\"data-md-color-scheme\")!\n        palette.color.primary = input.getAttribute(\"data-md-color-primary\")!\n        palette.color.accent  = input.getAttribute(\"data-md-color-accent\")!\n      }\n\n      /* Set color palette */\n      for (const [key, value] of Object.entries(palette.color))\n        document.body.setAttribute(`data-md-color-${key}`, value)\n\n      /* Set toggle visibility */\n      for (let index = 0; index < inputs.length; index++) {\n        const label = inputs[index].nextElementSibling\n        if (label instanceof HTMLElement)\n          label.hidden = palette.index !== index\n      }\n\n      /* Persist preference in local storage */\n      __md_set(\"__palette\", palette)\n    })\n\n    /* Update theme-color meta tag */\n    push$\n      .pipe(\n        map(() => {\n          const header = getComponentElement(\"header\")\n          const style  = window.getComputedStyle(header)\n\n          // Set color scheme\n          scheme.content = style.colorScheme\n\n          /* Return color in hexadecimal format */\n          return style.backgroundColor.match(/\\d+/g)!\n            .map(value => (+value).toString(16).padStart(2, \"0\"))\n            .join(\"\")\n        })\n      )\n        .subscribe(color => meta.content = `#${color}`)\n\n    /* Revert transition durations after color switch */\n    push$.pipe(observeOn(asyncScheduler))\n      .subscribe(() => {\n        document.body.removeAttribute(\"data-md-color-switching\")\n      })\n\n    /* Create and return component */\n    return watchPalette(inputs)\n      .pipe(\n        takeUntil(media$.pipe(skip(1))),\n        repeat(),\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  defer,\n  finalize,\n  map,\n  tap\n} from \"rxjs\"\n\nimport { Component } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Progress indicator\n */\nexport interface Progress {\n  value: number                        // Progress value\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  progress$: Subject<number>           // Progress subject\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount progress indicator\n *\n * @param el - Progress indicator element\n * @param options - Options\n *\n * @returns Progress indicator component observable\n */\nexport function mountProgress(\n  el: HTMLElement, { progress$ }: MountOptions\n): Observable<Component<Progress>> {\n\n  // Mount component on subscription\n  return defer(() => {\n    const push$ = new Subject<Progress>()\n    push$.subscribe(({ value }) => {\n      el.style.setProperty(\"--md-progress-value\", `${value}`)\n    })\n\n    // Create and return component\n    return progress$\n      .pipe(\n        tap(value => push$.next({ value })),\n        finalize(() => push$.complete()),\n        map(value => ({ ref: el, value }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport ClipboardJS from \"clipboard\"\nimport {\n  Observable,\n  Subject,\n  map,\n  tap\n} from \"rxjs\"\n\nimport { translation } from \"~/_\"\nimport { getElement } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n  alert$: Subject<string>              /* Alert subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Extract text to copy\n *\n * @param el - HTML element\n *\n * @returns Extracted text\n */\nfunction extract(el: HTMLElement): string {\n  el.setAttribute(\"data-md-copying\", \"\")\n  const copy = el.closest(\"[data-copy]\")\n  const text = copy\n    ? copy.getAttribute(\"data-copy\")!\n    : el.innerText\n  el.removeAttribute(\"data-md-copying\")\n  return text.trimEnd()\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up Clipboard.js integration\n *\n * @param options - Options\n */\nexport function setupClipboardJS(\n  { alert$ }: SetupOptions\n): void {\n  if (ClipboardJS.isSupported()) {\n    new Observable<ClipboardJS.Event>(subscriber => {\n      new ClipboardJS(\"[data-clipboard-target], [data-clipboard-text]\", {\n        text: el => (\n          el.getAttribute(\"data-clipboard-text\")! ||\n          extract(getElement(\n            el.getAttribute(\"data-clipboard-target\")!\n          ))\n        )\n      })\n        .on(\"success\", ev => subscriber.next(ev))\n    })\n      .pipe(\n        tap(ev => {\n          const trigger = ev.trigger as HTMLElement\n          trigger.focus()\n        }),\n        map(() => translation(\"clipboard.copied\"))\n      )\n        .subscribe(alert$)\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  catchError,\n  defaultIfEmpty,\n  map,\n  of,\n  tap\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport { getElements, requestXML } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Sitemap, i.e. a list of URLs\n */\nexport type Sitemap = string[]\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Preprocess a list of URLs\n *\n * This function replaces the `site_url` in the sitemap with the actual base\n * URL, to allow instant navigation to work in occasions like Netlify previews.\n *\n * @param urls - URLs\n *\n * @returns URL path parts\n */\nfunction preprocess(urls: Sitemap): Sitemap {\n  if (urls.length < 2)\n    return [\"\"]\n\n  /* Take the first two URLs and remove everything after the last slash */\n  const [root, next] = [...urls]\n    .sort((a, b) => a.length - b.length)\n    .map(url => url.replace(/[^/]+$/, \"\"))\n\n  /* Compute common prefix */\n  let index = 0\n  if (root === next)\n    index = root.length\n  else\n    while (root.charCodeAt(index) === next.charCodeAt(index))\n      index++\n\n  /* Remove common prefix and return in original order */\n  return urls.map(url => url.replace(root.slice(0, index), \"\"))\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch the sitemap for the given base URL\n *\n * @param base - Base URL\n *\n * @returns Sitemap observable\n */\nexport function fetchSitemap(base?: URL): Observable<Sitemap> {\n  const cached = __md_get<Sitemap>(\"__sitemap\", sessionStorage, base)\n  if (cached) {\n    return of(cached)\n  } else {\n    const config = configuration()\n    return requestXML(new URL(\"sitemap.xml\", base || config.base))\n      .pipe(\n        map(sitemap => preprocess(getElements(\"loc\", sitemap)\n          .map(node => node.textContent!)\n        )),\n        catchError(() => EMPTY), // @todo refactor instant loading\n        defaultIfEmpty([]),\n        tap(sitemap => __md_set(\"__sitemap\", sitemap, sessionStorage, base))\n      )\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  bufferCount,\n  catchError,\n  concat,\n  debounceTime,\n  distinctUntilKeyChanged,\n  endWith,\n  filter,\n  fromEvent,\n  ignoreElements,\n  map,\n  of,\n  sample,\n  share,\n  skip,\n  startWith,\n  switchMap,\n  take,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { configuration, feature } from \"~/_\"\nimport {\n  Viewport,\n  getElements,\n  getLocation,\n  getOptionalElement,\n  request,\n  setLocation,\n  setLocationHash\n} from \"~/browser\"\nimport { getComponentElement } from \"~/components\"\n\nimport { fetchSitemap } from \"../sitemap\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n  location$: Subject<URL>              // Location subject\n  viewport$: Observable<Viewport>      // Viewport observable\n  progress$: Subject<number>           // Progress suject\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a map of head elements for lookup and replacement\n *\n * @param head - Document head\n *\n * @returns Element map\n */\nfunction lookup(head: HTMLHeadElement): Map<string, HTMLElement> {\n\n  // @todo When resolving URLs, we must make sure to use the correct base for\n  // resolution. The next time we refactor instant loading, we should use the\n  // location subject as a source, which is also used for anchor links tracking,\n  // but for now we just rely on canonical.\n  const canonical = getOptionalElement<HTMLLinkElement>(\"[rel=canonical]\", head)\n  if (typeof canonical !== \"undefined\")\n    canonical.href = canonical.href.replace(\"//localhost:\", \"//127.0.0.1:\")\n\n  // Create tag map and index elements in head\n  const tags = new Map<string, HTMLElement>()\n  for (const el of getElements(\":scope > *\", head)) {\n    let html = el.outerHTML\n\n    // If the current element is a style sheet or script, we must resolve the\n    // URL relative to the current location and make it absolute, so it's easy\n    // to deduplicate it later on by comparing the outer HTML of tags. We must\n    // keep identical style sheets and scripts without replacing them.\n    for (const key of [\"href\", \"src\"]) {\n      const value = el.getAttribute(key)!\n      if (value === null)\n        continue\n\n      // Resolve URL relative to current location\n      const url = new URL(value, canonical?.href)\n      const ref = el.cloneNode() as HTMLElement\n\n      // Set resolved URL and retrieve HTML for deduplication\n      ref.setAttribute(key, `${url}`)\n      html = ref.outerHTML\n      break\n    }\n\n    // Index element in tag map\n    tags.set(html, el)\n  }\n\n  // Return tag map\n  return tags\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up instant navigation\n *\n * This is a heavily orchestrated operation - see inline comments to learn how\n * this works with Material for MkDocs, and how you can hook into it.\n *\n * @param options - Options\n *\n * @returns Document observable\n */\nexport function setupInstantNavigation(\n  { location$, viewport$, progress$ }: SetupOptions\n): Observable<Document> {\n  const config = configuration()\n  if (location.protocol === \"file:\")\n    return EMPTY\n\n  // Load sitemap immediately, so we have it available when the user initiates\n  // the first instant navigation request, and canonicalize URLs to the current\n  // base URL. The base URL will remain stable in between loads, as it's only\n  // read at the first initialization of the application.\n  const sitemap$ = fetchSitemap()\n    .pipe(\n      map(paths => paths.map(path => `${new URL(path, config.base)}`))\n    )\n\n  // Intercept inter-site navigation - to keep the number of event listeners\n  // low we use the fact that uncaptured events bubble up to the body. This also\n  // has the nice property that we don't need to detach and then again attach\n  // event listeners when instant navigation occurs.\n  const instant$ = fromEvent<MouseEvent>(document.body, \"click\")\n    .pipe(\n      withLatestFrom(sitemap$),\n      switchMap(([ev, sitemap]) => {\n        if (!(ev.target instanceof Element))\n          return EMPTY\n\n        // Skip, as target is not within a link - clicks on non-link elements\n        // are also captured, which we need to exclude from processing\n        const el = ev.target.closest(\"a\")\n        if (el === null)\n          return EMPTY\n\n        // Skip, as link opens in new window - we now know we have captured a\n        // click on a link, but the link either has a `target` property defined,\n        // or the user pressed the `meta` or `ctrl` key to open it in a new\n        // window. Thus, we need to filter those events, too.\n        if (el.target || ev.metaKey || ev.ctrlKey)\n          return EMPTY\n\n        // Next, we must check if the URL is relevant for us, i.e., if it's an\n        // internal link to a page that is managed by MkDocs. Only then we can\n        // be sure that the structure of the page to be loaded adheres to the\n        // current document structure and can subsequently be injected into it\n        // without doing a full reload. For this reason, we must canonicalize\n        // the URL by removing all search parameters and hash fragments.\n        const url = new URL(el.href)\n        url.search = url.hash = \"\"\n\n        // Skip, if URL is not included in the sitemap - this could be the case\n        // when linking between versions or languages, or to another page that\n        // the author included as part of the build, but that is not managed by\n        // MkDocs. In that case we must not continue with instant navigation.\n        if (!sitemap.includes(`${url}`))\n          return EMPTY\n\n        // We now know that we have a link to an internal page, so we prevent\n        // the browser from navigation and emit the URL for instant navigation.\n        // Note that this also includes anchor links, which means we need to\n        // implement anchor positioning ourselves. The reason for this is that\n        // if we wouldn't manage anchor links as well, scroll restoration will\n        // not work correctly (e.g. following an anchor link and scrolling).\n        ev.preventDefault()\n        return of(new URL(el.href))\n      }),\n      share()\n    )\n\n  // Before fetching for the first time, resolve the absolute favicon position,\n  // as the browser will try to fetch the icon immediately\n  instant$.pipe(take(1))\n    .subscribe(() => {\n      const favicon = getOptionalElement<HTMLLinkElement>(\"link[rel=icon]\")\n      if (typeof favicon !== \"undefined\")\n        favicon.href = favicon.href\n    })\n\n  // Enable scroll restoration before window unloads - this is essential to\n  // ensure that full reloads (F5) restore the viewport offset correctly. If\n  // only popstate events wouldn't reset the scroll position prior to their\n  // emission, we could just reset this in popstate. Meh.\n  fromEvent(window, \"beforeunload\")\n    .subscribe(() => {\n      history.scrollRestoration = \"auto\"\n    })\n\n  // When an instant navigation event occurs, disable scroll restoration, since\n  // we must normalize and synchronize the behavior across all browsers. For\n  // instance, when the user clicks the back or forward button, the browser\n  // would immediately jump to the position of the previous document.\n  instant$.pipe(withLatestFrom(viewport$))\n    .subscribe(([url, { offset }]) => {\n      history.scrollRestoration = \"manual\"\n\n      // While it would be better UX to defer the history state change until the\n      // document was fully fetched and parsed, we must schedule it here, since\n      // popstate events are emitted when history state changes happen. Moreover\n      // we need to back up the current viewport offset, so we can restore it\n      // when popstate events occur, e.g., when the browser's back and forward\n      // buttons are used for navigation.\n      history.replaceState(offset, \"\")\n      history.pushState(null, \"\", url)\n    })\n\n  // Emit URL that should be fetched via instant navigation on location subject,\n  // which was passed into this function. Instant navigation can be intercepted\n  // by other parts of the application, which can synchronously back up or\n  // restore state before instant navigation happens.\n  instant$.subscribe(location$)\n\n  // Fetch document - when fetching, we could use `responseType: document`, but\n  // since all MkDocs links are relative, we need to make sure that the current\n  // location matches the document we just loaded. Otherwise any relative links\n  // in the document might use the old location. If the request fails for some\n  // reason, we fall back to regular navigation and set the location explicitly,\n  // which will force-load the page. Furthermore, we must pre-warm the buffer\n  // for the duplicate check, or the first click on an anchor link will also\n  // trigger an instant navigation event, which doesn't make sense.\n  const response$ = location$\n    .pipe(\n      startWith(getLocation()),\n      distinctUntilKeyChanged(\"pathname\"),\n      skip(1),\n      switchMap(url => request(url, { progress$ })\n        .pipe(\n          catchError(() => {\n            setLocation(url, true)\n            return EMPTY\n          })\n        )\n      )\n    )\n\n  // Initialize the DOM parser, parse the returned HTML, and replace selected\n  // components before handing control down to the application\n  const dom = new DOMParser()\n  const document$ = response$\n    .pipe(\n      switchMap(res => res.text()),\n      switchMap(res => {\n        const next = dom.parseFromString(res, \"text/html\")\n        for (const selector of [\n          \"[data-md-component=announce]\",\n          \"[data-md-component=container]\",\n          \"[data-md-component=header-topic]\",\n          \"[data-md-component=outdated]\",\n          \"[data-md-component=logo]\",\n          \"[data-md-component=skip]\",\n          ...feature(\"navigation.tabs.sticky\")\n            ? [\"[data-md-component=tabs]\"]\n            : []\n        ]) {\n          const source = getOptionalElement(selector)\n          const target = getOptionalElement(selector, next)\n          if (\n            typeof source !== \"undefined\" &&\n            typeof target !== \"undefined\"\n          ) {\n            source.replaceWith(target)\n          }\n        }\n\n        // Update meta tags\n        const source = lookup(document.head)\n        const target = lookup(next.head)\n        for (const [html, el] of target) {\n\n          // Hack: skip stylesheets and scripts until we manage to replace them\n          // entirely in order to omit flashes of white content @todo refactor\n          if (\n            el.getAttribute(\"rel\") === \"stylesheet\" ||\n            el.hasAttribute(\"src\")\n          )\n            continue\n\n          if (source.has(html)) {\n            source.delete(html)\n          } else {\n            document.head.appendChild(el)\n          }\n        }\n\n        // Remove meta tags that are not present in the new document\n        for (const el of source.values())\n\n          // Hack: skip stylesheets and scripts until we manage to replace them\n          // entirely in order to omit flashes of white content @todo refactor\n          if (\n            el.getAttribute(\"rel\") === \"stylesheet\" ||\n            el.hasAttribute(\"src\")\n          )\n            continue\n          else\n            el.remove()\n\n        // After components and meta tags were replaced, re-evaluate scripts\n        // that were provided by the author as part of Markdown files\n        const container = getComponentElement(\"container\")\n        return concat(getElements(\"script\", container))\n          .pipe(\n            switchMap(el => {\n              const script = next.createElement(\"script\")\n              if (el.src) {\n                for (const name of el.getAttributeNames())\n                  script.setAttribute(name, el.getAttribute(name)!)\n                el.replaceWith(script)\n\n                // Complete when script is loaded\n                return new Observable(observer => {\n                  script.onload = () => observer.complete()\n                })\n\n              // Complete immediately\n              } else {\n                script.textContent = el.textContent\n                el.replaceWith(script)\n                return EMPTY\n              }\n            }),\n            ignoreElements(),\n            endWith(next)\n          )\n      }),\n      share()\n    )\n\n  // Intercept popstate events, e.g. when using the browser's back and forward\n  // buttons, and emit new location for fetching and parsing\n  const popstate$ = fromEvent<PopStateEvent>(window, \"popstate\")\n  popstate$.pipe(map(getLocation))\n    .subscribe(location$)\n\n  // Intercept clicks on anchor links, and scroll document into position - as\n  // we disabled scroll restoration, we need to do this manually here\n  location$\n    .pipe(\n      startWith(getLocation()),\n      bufferCount(2, 1),\n      filter(([prev, next]) => (\n        prev.pathname === next.pathname &&\n        prev.hash     !== next.hash\n      )),\n      map(([, next]) => next)\n    )\n      .subscribe(url => {\n        if (history.state !== null || !url.hash) {\n          window.scrollTo(0, history.state?.y ?? 0)\n        } else {\n          history.scrollRestoration = \"auto\"\n          setLocationHash(url.hash)\n          history.scrollRestoration = \"manual\"\n        }\n      })\n\n  // Intercept clicks on the same anchor link - we must use a distinct pipeline\n  // for this, or we'd end up in a loop, setting the hash again and again\n  location$\n    .pipe(\n      sample(instant$),\n      startWith(getLocation()),\n      bufferCount(2, 1),\n      filter(([prev, next]) => (\n        prev.pathname === next.pathname &&\n        prev.hash     === next.hash\n      )),\n      map(([, next]) => next)\n    )\n      .subscribe(url => {\n        history.scrollRestoration = \"auto\"\n        setLocationHash(url.hash)\n        history.scrollRestoration = \"manual\"\n\n        // Hack: we need to make sure that we don't end up with multiple history\n        // entries for the same anchor link, so we just remove the last entry\n        history.back()\n      })\n\n  // After parsing the document, check if the current history entry has a state.\n  // This may happen when users press the back or forward button to visit a page\n  // that was already seen. If there's no state, it means a new page was visited\n  // and we should scroll to the top, unless an anchor is given.\n  document$.pipe(withLatestFrom(location$))\n    .subscribe(([, url]) => {\n      if (history.state !== null || !url.hash) {\n        window.scrollTo(0, history.state?.y ?? 0)\n      } else {\n        setLocationHash(url.hash)\n      }\n    })\n\n  // If the current history is not empty, register an event listener updating\n  // the current history state whenever the scroll position changes. This must\n  // be debounced and cannot be done in popstate, as popstate has already\n  // removed the entry from the history.\n  viewport$\n    .pipe(\n      distinctUntilKeyChanged(\"offset\"),\n      debounceTime(100)\n    )\n      .subscribe(({ offset }) => {\n        history.replaceState(offset, \"\")\n      })\n\n  // Return document\n  return document$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport escapeHTML from \"escape-html\"\n\nimport { SearchConfig } from \"../config\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search highlight function\n *\n * @param value - Value\n *\n * @returns Highlighted value\n */\nexport type SearchHighlightFn = (value: string) => string\n\n/**\n * Search highlight factory function\n *\n * @param query - Query value\n *\n * @returns Search highlight function\n */\nexport type SearchHighlightFactoryFn = (query: string) => SearchHighlightFn\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search highlighter\n *\n * @param config - Search configuration\n *\n * @returns Search highlight factory function\n */\nexport function setupSearchHighlighter(\n  config: SearchConfig\n): SearchHighlightFactoryFn {\n  // Hack: temporarily remove pure lookaheads and lookbehinds\n  const regex = config.separator.split(\"|\").map(term => {\n    const temp = term.replace(/(\\(\\?[!=<][^)]+\\))/g, \"\")\n    return temp.length === 0 ? \"\uFFFD\" : term\n  })\n    .join(\"|\")\n\n  const separator = new RegExp(regex, \"img\")\n  const highlight = (_: unknown, data: string, term: string) => {\n    return `${data}<mark data-md-highlight>${term}</mark>`\n  }\n\n  /* Return factory function */\n  return (query: string) => {\n    query = query\n      .replace(/[\\s*+\\-:~^]+/g, \" \")\n      .trim()\n\n    /* Create search term match expression */\n    const match = new RegExp(`(^|${config.separator}|)(${\n      query\n        .replace(/[|\\\\{}()[\\]^$+*?.-]/g, \"\\\\$&\")\n        .replace(separator, \"|\")\n    })`, \"img\")\n\n    /* Highlight string value */\n    return value => escapeHTML(value)\n      .replace(match, highlight)\n      .replace(/<\\/mark>(\\s+)<mark[^>]*>/img, \"$1\")\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { SearchResult } from \"../../_\"\nimport { SearchIndex } from \"../../config\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search message type\n */\nexport const enum SearchMessageType {\n  SETUP,                               /* Search index setup */\n  READY,                               /* Search index ready */\n  QUERY,                               /* Search query */\n  RESULT                               /* Search results */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Message containing the data necessary to setup the search index\n */\nexport interface SearchSetupMessage {\n  type: SearchMessageType.SETUP        /* Message type */\n  data: SearchIndex                    /* Message data */\n}\n\n/**\n * Message indicating the search index is ready\n */\nexport interface SearchReadyMessage {\n  type: SearchMessageType.READY        /* Message type */\n}\n\n/**\n * Message containing a search query\n */\nexport interface SearchQueryMessage {\n  type: SearchMessageType.QUERY        /* Message type */\n  data: string                         /* Message data */\n}\n\n/**\n * Message containing results for a search query\n */\nexport interface SearchResultMessage {\n  type: SearchMessageType.RESULT       /* Message type */\n  data: SearchResult                   /* Message data */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Message exchanged with the search worker\n */\nexport type SearchMessage =\n  | SearchSetupMessage\n  | SearchReadyMessage\n  | SearchQueryMessage\n  | SearchResultMessage\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Type guard for search ready messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchReadyMessage(\n  message: SearchMessage\n): message is SearchReadyMessage {\n  return message.type === SearchMessageType.READY\n}\n\n/**\n * Type guard for search result messages\n *\n * @param message - Search worker message\n *\n * @returns Test result\n */\nexport function isSearchResultMessage(\n  message: SearchMessage\n): message is SearchResultMessage {\n  return message.type === SearchMessageType.RESULT\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  ObservableInput,\n  Subject,\n  first,\n  merge,\n  of,\n  switchMap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport { watchToggle, watchWorker } from \"~/browser\"\n\nimport { SearchIndex } from \"../../config\"\nimport {\n  SearchMessage,\n  SearchMessageType\n} from \"../message\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up search worker\n *\n * This function creates and initializes a web worker that is used for search,\n * so that the user interface doesn't freeze. In general, the application does\n * not care how search is implemented, as long as the web worker conforms to\n * the format expected by the application as defined in `SearchMessage`. This\n * allows the author to implement custom search functionality, by providing a\n * custom web worker via configuration.\n *\n * Material for MkDocs' built-in search implementation makes use of Lunr.js, an\n * efficient and fast implementation for client-side search. Leveraging a tiny\n * iframe-based web worker shim, search is even supported for the `file://`\n * protocol, enabling search for local non-hosted builds.\n *\n * If the protocol is `file://`, search initialization is deferred to mitigate\n * freezing, as it's now synchronous by design - see https://bit.ly/3C521EO\n *\n * @see https://bit.ly/3igvtQv - How to implement custom search\n *\n * @param url - Worker URL\n * @param index$ - Search index observable input\n *\n * @returns Search worker\n */\nexport function setupSearchWorker(\n  url: string, index$: ObservableInput<SearchIndex>\n): Subject<SearchMessage> {\n  const worker$ = watchWorker<SearchMessage>(url)\n  merge(\n    of(location.protocol !== \"file:\"),\n    watchToggle(\"search\")\n  )\n    .pipe(\n      first(active => active),\n      switchMap(() => index$)\n    )\n      .subscribe(({ config, docs }) => worker$.next({\n        type: SearchMessageType.SETUP,\n        data: {\n          config,\n          docs,\n          options: {\n            suggest: feature(\"search.suggest\")\n          }\n        }\n      }))\n\n  /* Return search worker */\n  return worker$\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Subject,\n  catchError,\n  combineLatest,\n  filter,\n  fromEvent,\n  map,\n  of,\n  switchMap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport {\n  getElement,\n  getLocation,\n  requestJSON,\n  setLocation\n} from \"~/browser\"\nimport { getComponentElements } from \"~/components\"\nimport {\n  Version,\n  renderVersionSelector\n} from \"~/templates\"\n\nimport { fetchSitemap } from \"../sitemap\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Setup options\n */\ninterface SetupOptions {\n  document$: Subject<Document>         /* Document subject */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Set up version selector\n *\n * @param options - Options\n */\nexport function setupVersionSelector(\n  { document$ }: SetupOptions\n): void {\n  const config = configuration()\n  const versions$ = requestJSON<Version[]>(\n    new URL(\"../versions.json\", config.base)\n  )\n    .pipe(\n      catchError(() => EMPTY) // @todo refactor instant loading\n    )\n\n  /* Determine current version */\n  const current$ = versions$\n    .pipe(\n      map(versions => {\n        const [, current] = config.base.match(/([^/]+)\\/?$/)!\n        return versions.find(({ version, aliases }) => (\n          version === current || aliases.includes(current)\n        )) || versions[0]\n      })\n    )\n\n  /* Intercept inter-version navigation */\n  versions$\n    .pipe(\n      map(versions => new Map(versions.map(version => [\n        `${new URL(`../${version.version}/`, config.base)}`,\n        version\n      ]))),\n      switchMap(urls => fromEvent<MouseEvent>(document.body, \"click\")\n        .pipe(\n          filter(ev => !ev.metaKey && !ev.ctrlKey),\n          withLatestFrom(current$),\n          switchMap(([ev, current]) => {\n            if (ev.target instanceof Element) {\n              const el = ev.target.closest(\"a\")\n              if (el && !el.target && urls.has(el.href)) {\n                const url = el.href\n                // This is a temporary hack to detect if a version inside the\n                // version selector or on another part of the site was clicked.\n                // If we're inside the version selector, we definitely want to\n                // find the same page, as we might have different deployments\n                // due to aliases. However, if we're outside the version\n                // selector, we must abort here, because we might otherwise\n                // interfere with instant navigation. We need to refactor this\n                // at some point together with instant navigation.\n                //\n                // See https://github.com/squidfunk/mkdocs-material/issues/4012\n                if (!ev.target.closest(\".md-version\")) {\n                  const version = urls.get(url)!\n                  if (version === current)\n                    return EMPTY\n                }\n                ev.preventDefault()\n                return of(url)\n              }\n            }\n            return EMPTY\n          }),\n          switchMap(url => {\n            const { version } = urls.get(url)!\n            return fetchSitemap(new URL(url))\n              .pipe(\n                map(sitemap => {\n                  const location = getLocation()\n                  const path = location.href.replace(config.base, \"\")\n                  return sitemap.includes(path.split(\"#\")[0])\n                    ? new URL(`../${version}/${path}`, config.base)\n                    : new URL(url)\n                })\n              )\n          })\n        )\n      )\n    )\n      .subscribe(url => setLocation(url, true))\n\n  /* Render version selector and warning */\n  combineLatest([versions$, current$])\n    .subscribe(([versions, current]) => {\n      const topic = getElement(\".md-header__topic\")\n      topic.appendChild(renderVersionSelector(versions, current))\n    })\n\n  /* Integrate outdated version banner with instant navigation */\n  document$.pipe(switchMap(() => current$))\n    .subscribe(current => {\n\n      /* Check if version state was already determined */\n      let outdated = __md_get(\"__outdated\", sessionStorage)\n      if (outdated === null) {\n        outdated = true\n\n        /* Obtain and normalize default versions */\n        let ignored = config.version?.default || \"latest\"\n        if (!Array.isArray(ignored))\n          ignored = [ignored]\n\n        /* Check if version is considered a default */\n        main: for (const ignore of ignored)\n          for (const version of current.aliases.concat(current.version))\n            if (new RegExp(ignore, \"i\").test(version)) {\n              outdated = false\n              break main\n            }\n\n        /* Persist version state in session storage */\n        __md_set(\"__outdated\", outdated, sessionStorage)\n      }\n\n      /* Unhide outdated version banner */\n      if (outdated)\n        for (const warning of getComponentElements(\"outdated\"))\n          warning.hidden = false\n    })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  combineLatest,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  finalize,\n  first,\n  fromEvent,\n  ignoreElements,\n  map,\n  merge,\n  shareReplay,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport {\n  getElement,\n  getLocation,\n  setToggle,\n  watchElementFocus,\n  watchToggle\n} from \"~/browser\"\nimport {\n  SearchMessage,\n  SearchMessageType,\n  isSearchReadyMessage\n} from \"~/integrations\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search query\n */\nexport interface SearchQuery {\n  value: string                        /* Query value */\n  focus: boolean                       /* Query focus */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch search query\n *\n * Note that the focus event which triggers re-reading the current query value\n * is delayed by `1ms` so the input's empty state is allowed to propagate.\n *\n * @param el - Search query element\n * @param options - Options\n *\n * @returns Search query observable\n */\nexport function watchSearchQuery(\n  el: HTMLInputElement, { worker$ }: WatchOptions\n): Observable<SearchQuery> {\n\n  /* Support search deep linking */\n  const { searchParams } = getLocation()\n  if (searchParams.has(\"q\")) {\n    setToggle(\"search\", true)\n\n    /* Set query from parameter */\n    el.value = searchParams.get(\"q\")!\n    el.focus()\n\n    /* Remove query parameter on close */\n    watchToggle(\"search\")\n      .pipe(\n        first(active => !active)\n      )\n        .subscribe(() => {\n          const url = getLocation()\n          url.searchParams.delete(\"q\")\n          history.replaceState({}, \"\", `${url}`)\n        })\n  }\n\n  /* Intercept focus and input events */\n  const focus$ = watchElementFocus(el)\n  const value$ = merge(\n    worker$.pipe(first(isSearchReadyMessage)),\n    fromEvent(el, \"keyup\"),\n    focus$\n  )\n    .pipe(\n      map(() => el.value),\n      distinctUntilChanged()\n    )\n\n  /* Combine into single observable */\n  return combineLatest([value$, focus$])\n    .pipe(\n      map(([value, focus]) => ({ value, focus })),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount search query\n *\n * @param el - Search query element\n * @param options - Options\n *\n * @returns Search query component observable\n */\nexport function mountSearchQuery(\n  el: HTMLInputElement, { worker$ }: MountOptions\n): Observable<Component<SearchQuery, HTMLInputElement>> {\n  const push$ = new Subject<SearchQuery>()\n  const done$ = push$.pipe(ignoreElements(), endWith(true))\n\n  /* Handle value change */\n  combineLatest([\n    worker$.pipe(first(isSearchReadyMessage)),\n    push$\n  ], (_, query) => query)\n    .pipe(\n      distinctUntilKeyChanged(\"value\")\n    )\n      .subscribe(({ value }) => worker$.next({\n        type: SearchMessageType.QUERY,\n        data: value\n      }))\n\n  /* Handle focus change */\n  push$\n    .pipe(\n      distinctUntilKeyChanged(\"focus\")\n    )\n      .subscribe(({ focus }) => {\n        if (focus)\n          setToggle(\"search\", focus)\n      })\n\n  /* Handle reset */\n  fromEvent(el.form!, \"reset\")\n    .pipe(\n      takeUntil(done$)\n    )\n      .subscribe(() => el.focus())\n\n  // Focus search query on label click - note that this is necessary to bring\n  // up the keyboard on iOS and other mobile platforms, as the search dialog is\n  // not visible at first, and programatically focusing an input element must\n  // be triggered by a user interaction - see https://t.ly/Cb30n\n  const label = getElement(\"header [for=__search]\")\n  fromEvent(label, \"click\")\n    .subscribe(() => el.focus())\n\n  /* Create and return component */\n  return watchSearchQuery(el, { worker$ })\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state })),\n      shareReplay(1)\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  bufferCount,\n  filter,\n  finalize,\n  first,\n  fromEvent,\n  map,\n  merge,\n  mergeMap,\n  of,\n  share,\n  skipUntil,\n  switchMap,\n  takeUntil,\n  tap,\n  withLatestFrom,\n  zipWith\n} from \"rxjs\"\n\nimport { translation } from \"~/_\"\nimport {\n  getElement,\n  getOptionalElement,\n  watchElementBoundary,\n  watchToggle\n} from \"~/browser\"\nimport {\n  SearchMessage,\n  SearchResult,\n  isSearchReadyMessage,\n  isSearchResultMessage\n} from \"~/integrations\"\nimport { renderSearchResultItem } from \"~/templates\"\nimport { round } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\nimport { SearchQuery } from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  query$: Observable<SearchQuery>      /* Search query observable */\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search result list\n *\n * This function performs a lazy rendering of the search results, depending on\n * the vertical offset of the search result container.\n *\n * @param el - Search result list element\n * @param options - Options\n *\n * @returns Search result list component observable\n */\nexport function mountSearchResult(\n  el: HTMLElement, { worker$, query$ }: MountOptions\n): Observable<Component<SearchResult>> {\n  const push$ = new Subject<SearchResult>()\n  const boundary$ = watchElementBoundary(el.parentElement!)\n    .pipe(\n      filter(Boolean)\n    )\n\n  /* Retrieve container */\n  const container = el.parentElement!\n\n  /* Retrieve nested components */\n  const meta = getElement(\":scope > :first-child\", el)\n  const list = getElement(\":scope > :last-child\", el)\n\n  /* Reveal to accessibility tree \u2013 see https://bit.ly/3iAA7t8 */\n  watchToggle(\"search\")\n    .subscribe(active => list.setAttribute(\n      \"role\", active ? \"list\" : \"presentation\"\n    ))\n\n  /* Update search result metadata */\n  push$\n    .pipe(\n      withLatestFrom(query$),\n      skipUntil(worker$.pipe(first(isSearchReadyMessage)))\n    )\n      .subscribe(([{ items }, { value }]) => {\n        switch (items.length) {\n\n          /* No results */\n          case 0:\n            meta.textContent = value.length\n              ? translation(\"search.result.none\")\n              : translation(\"search.result.placeholder\")\n            break\n\n          /* One result */\n          case 1:\n            meta.textContent = translation(\"search.result.one\")\n            break\n\n          /* Multiple result */\n          default:\n            const count = round(items.length)\n            meta.textContent = translation(\"search.result.other\", count)\n        }\n      })\n\n  /* Render search result item */\n  const render$ = push$\n    .pipe(\n      tap(() => list.innerHTML = \"\"),\n      switchMap(({ items }) => merge(\n        of(...items.slice(0, 10)),\n        of(...items.slice(10))\n          .pipe(\n            bufferCount(4),\n            zipWith(boundary$),\n            switchMap(([chunk]) => chunk)\n          )\n      )),\n      map(renderSearchResultItem),\n      share()\n    )\n\n  /* Update search result list */\n  render$.subscribe(item => list.appendChild(item))\n  render$\n    .pipe(\n      mergeMap(item => {\n        const details = getOptionalElement(\"details\", item)\n        if (typeof details === \"undefined\")\n          return EMPTY\n\n        /* Keep position of details element stable */\n        return fromEvent(details, \"toggle\")\n          .pipe(\n            takeUntil(push$),\n            map(() => details)\n          )\n      })\n    )\n      .subscribe(details => {\n        if (\n          details.open === false &&\n          details.offsetTop <= container.scrollTop\n        )\n          container.scrollTo({ top: details.offsetTop })\n      })\n\n  /* Filter search result message */\n  const result$ = worker$\n    .pipe(\n      filter(isSearchResultMessage),\n      map(({ data }) => data)\n    )\n\n  /* Create and return component */\n  return result$\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  endWith,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport { getLocation } from \"~/browser\"\n\nimport { Component } from \"../../_\"\nimport { SearchQuery } from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search sharing\n */\nexport interface SearchShare {\n  url: URL                             /* Deep link for sharing */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  query$: Observable<SearchQuery>      /* Search query observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  query$: Observable<SearchQuery>      /* Search query observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search sharing\n *\n * @param _el - Search sharing element\n * @param options - Options\n *\n * @returns Search sharing observable\n */\nexport function watchSearchShare(\n  _el: HTMLElement, { query$ }: WatchOptions\n): Observable<SearchShare> {\n  return query$\n    .pipe(\n      map(({ value }) => {\n        const url = getLocation()\n        url.hash = \"\"\n\n        /* Compute readable query strings */\n        value = value\n          .replace(/\\s+/g, \"+\")        /* Collapse whitespace */\n          .replace(/&/g, \"%26\")        /* Escape '&' character */\n          .replace(/=/g, \"%3D\")        /* Escape '=' character */\n\n        /* Replace query string */\n        url.search = `q=${value}`\n        return { url }\n      })\n    )\n}\n\n/**\n * Mount search sharing\n *\n * @param el - Search sharing element\n * @param options - Options\n *\n * @returns Search sharing component observable\n */\nexport function mountSearchShare(\n  el: HTMLAnchorElement, options: MountOptions\n): Observable<Component<SearchShare>> {\n  const push$ = new Subject<SearchShare>()\n  const done$ = push$.pipe(ignoreElements(), endWith(true))\n  push$.subscribe(({ url }) => {\n    el.setAttribute(\"data-clipboard-text\", el.href)\n    el.href = `${url}`\n  })\n\n  /* Prevent following of link */\n  fromEvent(el, \"click\")\n    .pipe(\n      takeUntil(done$)\n    )\n      .subscribe(ev => ev.preventDefault())\n\n  /* Create and return component */\n  return watchSearchShare(el, options)\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  asyncScheduler,\n  combineLatestWith,\n  distinctUntilChanged,\n  filter,\n  finalize,\n  fromEvent,\n  map,\n  merge,\n  observeOn,\n  tap\n} from \"rxjs\"\n\nimport { Keyboard } from \"~/browser\"\nimport {\n  SearchMessage,\n  SearchResult,\n  isSearchResultMessage\n} from \"~/integrations\"\n\nimport { Component, getComponentElement } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search suggestions\n */\nexport interface SearchSuggest {}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  keyboard$: Observable<Keyboard>      /* Keyboard observable */\n  worker$: Subject<SearchMessage>      /* Search worker */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search suggestions\n *\n * This function will perform a lazy rendering of the search results, depending\n * on the vertical offset of the search result container.\n *\n * @param el - Search result list element\n * @param options - Options\n *\n * @returns Search result list component observable\n */\nexport function mountSearchSuggest(\n  el: HTMLElement, { worker$, keyboard$ }: MountOptions\n): Observable<Component<SearchSuggest>> {\n  const push$ = new Subject<SearchResult>()\n\n  /* Retrieve query component and track all changes */\n  const query  = getComponentElement(\"search-query\")\n  const query$ = merge(\n    fromEvent(query, \"keydown\"),\n    fromEvent(query, \"focus\")\n  )\n    .pipe(\n      observeOn(asyncScheduler),\n      map(() => query.value),\n      distinctUntilChanged(),\n    )\n\n  /* Update search suggestions */\n  push$\n    .pipe(\n      combineLatestWith(query$),\n      map(([{ suggest }, value]) => {\n        const words = value.split(/([\\s-]+)/)\n        if (suggest?.length && words[words.length - 1]) {\n          const last = suggest[suggest.length - 1]\n          if (last.startsWith(words[words.length - 1]))\n            words[words.length - 1] = last\n        } else {\n          words.length = 0\n        }\n        return words\n      })\n    )\n      .subscribe(words => el.innerHTML = words\n        .join(\"\")\n        .replace(/\\s/g, \"&nbsp;\")\n      )\n\n  /* Set up search keyboard handlers */\n  keyboard$\n    .pipe(\n      filter(({ mode }) => mode === \"search\")\n    )\n      .subscribe(key => {\n        switch (key.type) {\n\n          /* Right arrow: accept current suggestion */\n          case \"ArrowRight\":\n            if (\n              el.innerText.length &&\n              query.selectionStart === query.value.length\n            )\n              query.value = el.innerText\n            break\n        }\n      })\n\n  /* Filter search result message */\n  const result$ = worker$\n    .pipe(\n      filter(isSearchResultMessage),\n      map(({ data }) => data)\n    )\n\n  /* Create and return component */\n  return result$\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(() => ({ ref: el }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  NEVER,\n  Observable,\n  ObservableInput,\n  filter,\n  fromEvent,\n  merge,\n  mergeWith\n} from \"rxjs\"\n\nimport { configuration } from \"~/_\"\nimport {\n  Keyboard,\n  getActiveElement,\n  getElements,\n  setToggle\n} from \"~/browser\"\nimport {\n  SearchIndex,\n  SearchResult,\n  setupSearchWorker\n} from \"~/integrations\"\n\nimport {\n  Component,\n  getComponentElement,\n  getComponentElements\n} from \"../../_\"\nimport {\n  SearchQuery,\n  mountSearchQuery\n} from \"../query\"\nimport { mountSearchResult } from \"../result\"\nimport {\n  SearchShare,\n  mountSearchShare\n} from \"../share\"\nimport {\n  SearchSuggest,\n  mountSearchSuggest\n} from \"../suggest\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search\n */\nexport type Search =\n  | SearchQuery\n  | SearchResult\n  | SearchShare\n  | SearchSuggest\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  index$: ObservableInput<SearchIndex> /* Search index observable */\n  keyboard$: Observable<Keyboard>      /* Keyboard observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search\n *\n * This function sets up the search functionality, including the underlying\n * web worker and all keyboard bindings.\n *\n * @param el - Search element\n * @param options - Options\n *\n * @returns Search component observable\n */\nexport function mountSearch(\n  el: HTMLElement, { index$, keyboard$ }: MountOptions\n): Observable<Component<Search>> {\n  const config = configuration()\n  try {\n    const worker$ = setupSearchWorker(config.search, index$)\n\n    /* Retrieve query and result components */\n    const query  = getComponentElement(\"search-query\", el)\n    const result = getComponentElement(\"search-result\", el)\n\n    /* Always close search on result selection */\n    fromEvent<PointerEvent>(el, \"click\")\n      .pipe(\n        filter(({ target }) => (\n          target instanceof Element && !!target.closest(\"a\")\n        ))\n      )\n        .subscribe(() => setToggle(\"search\", false))\n\n    /* Set up search keyboard handlers */\n    keyboard$\n      .pipe(\n        filter(({ mode }) => mode === \"search\")\n      )\n        .subscribe(key => {\n          const active = getActiveElement()\n          switch (key.type) {\n\n            /* Enter: go to first (best) result */\n            case \"Enter\":\n              if (active === query) {\n                const anchors = new Map<HTMLAnchorElement, number>()\n                for (const anchor of getElements<HTMLAnchorElement>(\n                  \":first-child [href]\", result\n                )) {\n                  const article = anchor.firstElementChild!\n                  anchors.set(anchor, parseFloat(\n                    article.getAttribute(\"data-md-score\")!\n                  ))\n                }\n\n                /* Go to result with highest score, if any */\n                if (anchors.size) {\n                  const [[best]] = [...anchors].sort(([, a], [, b]) => b - a)\n                  best.click()\n                }\n\n                /* Otherwise omit form submission */\n                key.claim()\n              }\n              break\n\n            /* Escape or Tab: close search */\n            case \"Escape\":\n            case \"Tab\":\n              setToggle(\"search\", false)\n              query.blur()\n              break\n\n            /* Vertical arrows: select previous or next search result */\n            case \"ArrowUp\":\n            case \"ArrowDown\":\n              if (typeof active === \"undefined\") {\n                query.focus()\n              } else {\n                const els = [query, ...getElements(\n                  \":not(details) > [href], summary, details[open] [href]\",\n                  result\n                )]\n                const i = Math.max(0, (\n                  Math.max(0, els.indexOf(active)) + els.length + (\n                    key.type === \"ArrowUp\" ? -1 : +1\n                  )\n                ) % els.length)\n                els[i].focus()\n              }\n\n              /* Prevent scrolling of page */\n              key.claim()\n              break\n\n            /* All other keys: hand to search query */\n            default:\n              if (query !== getActiveElement())\n                query.focus()\n          }\n        })\n\n    /* Set up global keyboard handlers */\n    keyboard$\n      .pipe(\n        filter(({ mode }) => mode === \"global\")\n      )\n        .subscribe(key => {\n          switch (key.type) {\n\n            /* Open search and select query */\n            case \"f\":\n            case \"s\":\n            case \"/\":\n              query.focus()\n              query.select()\n\n              /* Prevent scrolling of page */\n              key.claim()\n              break\n          }\n        })\n\n    /* Create and return component */\n    const query$ = mountSearchQuery(query, { worker$ })\n    return merge(\n      query$,\n      mountSearchResult(result, { worker$, query$ })\n    )\n      .pipe(\n        mergeWith(\n\n          /* Search sharing */\n          ...getComponentElements(\"search-share\", el)\n            .map(child => mountSearchShare(child, { query$ })),\n\n          /* Search suggestions */\n          ...getComponentElements(\"search-suggest\", el)\n            .map(child => mountSearchSuggest(child, { worker$, keyboard$ }))\n        )\n      )\n\n  /* Gracefully handle broken search */\n  } catch (err) {\n    el.hidden = true\n    return NEVER\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  ObservableInput,\n  combineLatest,\n  filter,\n  map,\n  startWith\n} from \"rxjs\"\n\nimport { getLocation } from \"~/browser\"\nimport {\n  SearchIndex,\n  setupSearchHighlighter\n} from \"~/integrations\"\nimport { h } from \"~/utilities\"\n\nimport { Component } from \"../../_\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search highlighting\n */\nexport interface SearchHighlight {\n  nodes: Map<ChildNode, string>        /* Map of replacements */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  index$: ObservableInput<SearchIndex> /* Search index observable */\n  location$: Observable<URL>           /* Location observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Mount search highlighting\n *\n * @param el - Content element\n * @param options - Options\n *\n * @returns Search highlighting component observable\n */\nexport function mountSearchHiglight(\n  el: HTMLElement, { index$, location$ }: MountOptions\n): Observable<Component<SearchHighlight>> {\n  return combineLatest([\n    index$,\n    location$\n      .pipe(\n        startWith(getLocation()),\n        filter(url => !!url.searchParams.get(\"h\"))\n      )\n  ])\n    .pipe(\n      map(([index, url]) => setupSearchHighlighter(index.config)(\n        url.searchParams.get(\"h\")!\n      )),\n      map(fn => {\n        const nodes = new Map<ChildNode, string>()\n\n        /* Traverse text nodes and collect matches */\n        const it = document.createNodeIterator(el, NodeFilter.SHOW_TEXT)\n        for (let node = it.nextNode(); node; node = it.nextNode()) {\n          if (node.parentElement?.offsetHeight) {\n            const original = node.textContent!\n            const replaced = fn(original)\n            if (replaced.length > original.length)\n              nodes.set(node as ChildNode, replaced)\n          }\n        }\n\n        /* Replace original nodes with matches */\n        for (const [node, text] of nodes) {\n          const { childNodes } = h(\"span\", null, text)\n          node.replaceWith(...Array.from(childNodes))\n        }\n\n        /* Return component */\n        return { ref: el, nodes }\n      })\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  animationFrameScheduler,\n  asyncScheduler,\n  auditTime,\n  combineLatest,\n  defer,\n  distinctUntilChanged,\n  endWith,\n  finalize,\n  first,\n  from,\n  fromEvent,\n  ignoreElements,\n  map,\n  mergeMap,\n  observeOn,\n  takeUntil,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  getElement,\n  getElementOffset,\n  getElementSize,\n  getElements\n} from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Sidebar\n */\nexport interface Sidebar {\n  height: number                       /* Sidebar height */\n  locked: boolean                      /* Sidebar is locked */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  main$: Observable<Main>              /* Main area observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch sidebar\n *\n * This function returns an observable that computes the visual parameters of\n * the sidebar which depends on the vertical viewport offset, as well as the\n * height of the main area. When the page is scrolled beyond the header, the\n * sidebar is locked and fills the remaining space.\n *\n * @param el - Sidebar element\n * @param options - Options\n *\n * @returns Sidebar observable\n */\nexport function watchSidebar(\n  el: HTMLElement, { viewport$, main$ }: WatchOptions\n): Observable<Sidebar> {\n  const parent = el.closest<HTMLElement>(\".md-grid\")!\n  const adjust =\n    parent.offsetTop -\n    parent.parentElement!.offsetTop\n\n  /* Compute the sidebar's available height and if it should be locked */\n  return combineLatest([main$, viewport$])\n    .pipe(\n      map(([{ offset, height }, { offset: { y } }]) => {\n        height = height\n          + Math.min(adjust, Math.max(0, y - offset))\n          - adjust\n        return {\n          height,\n          locked: y >= offset + adjust\n        }\n      }),\n      distinctUntilChanged((a, b) => (\n        a.height === b.height &&\n        a.locked === b.locked\n      ))\n    )\n}\n\n/**\n * Mount sidebar\n *\n * This function doesn't set the height of the actual sidebar, but of its first\n * child \u2013 the `.md-sidebar__scrollwrap` element in order to mitigiate jittery\n * sidebars when the footer is scrolled into view. At some point we switched\n * from `absolute` / `fixed` positioning to `sticky` positioning, significantly\n * reducing jitter in some browsers (respectively Firefox and Safari) when\n * scrolling from the top. However, top-aligned sticky positioning means that\n * the sidebar snaps to the bottom when the end of the container is reached.\n * This is what leads to the mentioned jitter, as the sidebar's height may be\n * updated too slowly.\n *\n * This behaviour can be mitigiated by setting the height of the sidebar to `0`\n * while preserving the padding, and the height on its first element.\n *\n * @param el - Sidebar element\n * @param options - Options\n *\n * @returns Sidebar component observable\n */\nexport function mountSidebar(\n  el: HTMLElement, { header$, ...options }: MountOptions\n): Observable<Component<Sidebar>> {\n  const inner = getElement(\".md-sidebar__scrollwrap\", el)\n  const { y } = getElementOffset(inner)\n  return defer(() => {\n    const push$ = new Subject<Sidebar>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    const next$ = push$\n      .pipe(\n        auditTime(0, animationFrameScheduler)\n      )\n\n    /* Update sidebar height and offset */\n    next$.pipe(withLatestFrom(header$))\n      .subscribe({\n\n        /* Handle emission */\n        next([{ height }, { height: offset }]) {\n          inner.style.height = `${height - 2 * y}px`\n          el.style.top       = `${offset}px`\n        },\n\n        /* Handle complete */\n        complete() {\n          inner.style.height = \"\"\n          el.style.top       = \"\"\n        }\n      })\n\n    /* Bring active item into view on initial load */\n    next$.pipe(first())\n      .subscribe(() => {\n        for (const item of getElements(\".md-nav__link--active[href]\", el)) {\n          if (!item.clientHeight) // skip invisible toc in left sidebar\n            continue\n          const container = item.closest<HTMLElement>(\".md-sidebar__scrollwrap\")!\n          if (typeof container !== \"undefined\") {\n            const offset = item.offsetTop - container.offsetTop\n            const { height } = getElementSize(container)\n            container.scrollTo({\n              top: offset - height / 2\n            })\n          }\n        }\n      })\n\n    /* Handle accessibility for expandable items, see https://bit.ly/3jaod9p */\n    from(getElements<HTMLLabelElement>(\"label[tabindex]\", el))\n      .pipe(\n        mergeMap(label => fromEvent(label, \"click\")\n          .pipe(\n            observeOn(asyncScheduler),\n            map(() => label),\n            takeUntil(done$)\n          )\n        )\n      )\n        .subscribe(label => {\n          const input = getElement<HTMLInputElement>(`[id=\"${label.htmlFor}\"]`)\n          const nav = getElement(`[aria-labelledby=\"${label.id}\"]`)\n          nav.setAttribute(\"aria-expanded\", `${input.checked}`)\n        })\n\n    /* Create and return component */\n    return watchSidebar(el, options)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { Repo, User } from \"github-types\"\nimport {\n  EMPTY,\n  Observable,\n  catchError,\n  defaultIfEmpty,\n  map,\n  zip\n} from \"rxjs\"\n\nimport { requestJSON } from \"~/browser\"\n\nimport { SourceFacts } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * GitHub release (partial)\n */\ninterface Release {\n  tag_name: string                     /* Tag name */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch GitHub repository facts\n *\n * @param user - GitHub user or organization\n * @param repo - GitHub repository\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFactsFromGitHub(\n  user: string, repo?: string\n): Observable<SourceFacts> {\n  if (typeof repo !== \"undefined\") {\n    const url = `https://api.github.com/repos/${user}/${repo}`\n    return zip(\n\n      /* Fetch version */\n      requestJSON<Release>(`${url}/releases/latest`)\n        .pipe(\n          catchError(() => EMPTY), // @todo refactor instant loading\n          map(release => ({\n            version: release.tag_name\n          })),\n          defaultIfEmpty({})\n        ),\n\n      /* Fetch stars and forks */\n      requestJSON<Repo>(url)\n        .pipe(\n          catchError(() => EMPTY), // @todo refactor instant loading\n          map(info => ({\n            stars: info.stargazers_count,\n            forks: info.forks_count\n          })),\n          defaultIfEmpty({})\n        )\n    )\n      .pipe(\n        map(([release, info]) => ({ ...release, ...info }))\n      )\n\n  /* User or organization */\n  } else {\n    const url = `https://api.github.com/users/${user}`\n    return requestJSON<User>(url)\n      .pipe(\n        map(info => ({\n          repositories: info.public_repos\n        })),\n        defaultIfEmpty({})\n      )\n  }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { ProjectSchema } from \"gitlab\"\nimport {\n  EMPTY,\n  Observable,\n  catchError,\n  defaultIfEmpty,\n  map\n} from \"rxjs\"\n\nimport { requestJSON } from \"~/browser\"\n\nimport { SourceFacts } from \"../_\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch GitLab repository facts\n *\n * @param base - GitLab base\n * @param project - GitLab project\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFactsFromGitLab(\n  base: string, project: string\n): Observable<SourceFacts> {\n  const url = `https://${base}/api/v4/projects/${encodeURIComponent(project)}`\n  return requestJSON<ProjectSchema>(url)\n    .pipe(\n      catchError(() => EMPTY), // @todo refactor instant loading\n      map(({ star_count, forks_count }) => ({\n        stars: star_count,\n        forks: forks_count\n      })),\n      defaultIfEmpty({})\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { EMPTY, Observable } from \"rxjs\"\n\nimport { fetchSourceFactsFromGitHub } from \"../github\"\nimport { fetchSourceFactsFromGitLab } from \"../gitlab\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository facts for repositories\n */\nexport interface RepositoryFacts {\n  stars?: number                       /* Number of stars */\n  forks?: number                       /* Number of forks */\n  version?: string                     /* Latest version */\n}\n\n/**\n * Repository facts for organizations\n */\nexport interface OrganizationFacts {\n  repositories?: number                /* Number of repositories */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Repository facts\n */\nexport type SourceFacts =\n  | RepositoryFacts\n  | OrganizationFacts\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch repository facts\n *\n * @param url - Repository URL\n *\n * @returns Repository facts observable\n */\nexport function fetchSourceFacts(\n  url: string\n): Observable<SourceFacts> {\n\n  /* Try to match GitHub repository */\n  let match = url.match(/^.+github\\.com\\/([^/]+)\\/?([^/]+)?/i)\n  if (match) {\n    const [, user, repo] = match\n    return fetchSourceFactsFromGitHub(user, repo)\n  }\n\n  /* Try to match GitLab repository */\n  match = url.match(/^.+?([^/]*gitlab[^/]+)\\/(.+?)\\/?$/i)\n  if (match) {\n    const [, base, slug] = match\n    return fetchSourceFactsFromGitLab(base, slug)\n  }\n\n  /* Fallback */\n  return EMPTY\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  EMPTY,\n  Observable,\n  Subject,\n  catchError,\n  defer,\n  filter,\n  finalize,\n  map,\n  of,\n  shareReplay,\n  tap\n} from \"rxjs\"\n\nimport { getElement } from \"~/browser\"\nimport { ConsentDefaults } from \"~/components/consent\"\nimport { renderSourceFacts } from \"~/templates\"\n\nimport {\n  Component,\n  getComponentElements\n} from \"../../_\"\nimport {\n  SourceFacts,\n  fetchSourceFacts\n} from \"../facts\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository information\n */\nexport interface Source {\n  facts: SourceFacts                   /* Repository facts */\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Repository information observable\n */\nlet fetch$: Observable<Source>\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch repository information\n *\n * This function tries to read the repository facts from session storage, and\n * if unsuccessful, fetches them from the underlying provider.\n *\n * @param el - Repository information element\n *\n * @returns Repository information observable\n */\nexport function watchSource(\n  el: HTMLAnchorElement\n): Observable<Source> {\n  return fetch$ ||= defer(() => {\n    const cached = __md_get<SourceFacts>(\"__source\", sessionStorage)\n    if (cached) {\n      return of(cached)\n    } else {\n\n      /* Check if consent is configured and was given */\n      const els = getComponentElements(\"consent\")\n      if (els.length) {\n        const consent = __md_get<ConsentDefaults>(\"__consent\")\n        if (!(consent && consent.github))\n          return EMPTY\n      }\n\n      /* Fetch repository facts */\n      return fetchSourceFacts(el.href)\n        .pipe(\n          tap(facts => __md_set(\"__source\", facts, sessionStorage))\n        )\n    }\n  })\n    .pipe(\n      catchError(() => EMPTY),\n      filter(facts => Object.keys(facts).length > 0),\n      map(facts => ({ facts })),\n      shareReplay(1)\n    )\n}\n\n/**\n * Mount repository information\n *\n * @param el - Repository information element\n *\n * @returns Repository information component observable\n */\nexport function mountSource(\n  el: HTMLAnchorElement\n): Observable<Component<Source>> {\n  const inner = getElement(\":scope > :last-child\", el)\n  return defer(() => {\n    const push$ = new Subject<Source>()\n    push$.subscribe(({ facts }) => {\n      inner.appendChild(renderSourceFacts(facts))\n      inner.classList.add(\"md-source__repository--active\")\n    })\n\n    /* Create and return component */\n    return watchSource(el)\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  defer,\n  distinctUntilKeyChanged,\n  finalize,\n  map,\n  of,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  watchElementSize,\n  watchViewportAt\n} from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Navigation tabs\n */\nexport interface Tabs {\n  hidden: boolean                      /* Navigation tabs are hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch navigation tabs\n *\n * @param el - Navigation tabs element\n * @param options - Options\n *\n * @returns Navigation tabs observable\n */\nexport function watchTabs(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<Tabs> {\n  return watchElementSize(document.body)\n    .pipe(\n      switchMap(() => watchViewportAt(el, { header$, viewport$ })),\n      map(({ offset: { y } }) => {\n        return {\n          hidden: y >= 10\n        }\n      }),\n      distinctUntilKeyChanged(\"hidden\")\n    )\n}\n\n/**\n * Mount navigation tabs\n *\n * This function hides the navigation tabs when scrolling past the threshold\n * and makes them reappear in a nice CSS animation when scrolling back up.\n *\n * @param el - Navigation tabs element\n * @param options - Options\n *\n * @returns Navigation tabs component observable\n */\nexport function mountTabs(\n  el: HTMLElement, options: MountOptions\n): Observable<Component<Tabs>> {\n  return defer(() => {\n    const push$ = new Subject<Tabs>()\n    push$.subscribe({\n\n      /* Handle emission */\n      next({ hidden }) {\n        el.hidden = hidden\n      },\n\n      /* Handle complete */\n      complete() {\n        el.hidden = false\n      }\n    })\n\n    /* Create and return component */\n    return (\n      feature(\"navigation.tabs.sticky\")\n        ? of({ hidden: false })\n        : watchTabs(el, options)\n    )\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  asyncScheduler,\n  bufferCount,\n  combineLatestWith,\n  debounceTime,\n  defer,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  filter,\n  finalize,\n  ignoreElements,\n  map,\n  merge,\n  observeOn,\n  of,\n  repeat,\n  scan,\n  share,\n  skip,\n  startWith,\n  switchMap,\n  takeUntil,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { feature } from \"~/_\"\nimport {\n  Viewport,\n  getElement,\n  getElementContainer,\n  getElementSize,\n  getElements,\n  getLocation,\n  getOptionalElement,\n  watchElementSize\n} from \"~/browser\"\n\nimport {\n  Component,\n  getComponentElement\n} from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Table of contents\n */\nexport interface TableOfContents {\n  prev: HTMLAnchorElement[][]          /* Anchors (previous) */\n  next: HTMLAnchorElement[][]          /* Anchors (next) */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch table of contents\n *\n * This is effectively a scroll spy implementation which will account for the\n * fixed header and automatically re-calculate anchor offsets when the viewport\n * is resized. The returned observable will only emit if the table of contents\n * needs to be repainted.\n *\n * This implementation tracks an anchor element's entire path starting from its\n * level up to the top-most anchor element, e.g. `[h3, h2, h1]`. Although the\n * Material theme currently doesn't make use of this information, it enables\n * the styling of the entire hierarchy through customization.\n *\n * Note that the current anchor is the last item of the `prev` anchor list.\n *\n * @param el - Table of contents element\n * @param options - Options\n *\n * @returns Table of contents observable\n */\nexport function watchTableOfContents(\n  el: HTMLElement, { viewport$, header$ }: WatchOptions\n): Observable<TableOfContents> {\n  const table = new Map<HTMLAnchorElement, HTMLElement>()\n\n  /* Compute anchor-to-target mapping */\n  const anchors = getElements<HTMLAnchorElement>(\"[href^=\\\\#]\", el)\n  for (const anchor of anchors) {\n    const id = decodeURIComponent(anchor.hash.substring(1))\n    const target = getOptionalElement(`[id=\"${id}\"]`)\n    if (typeof target !== \"undefined\")\n      table.set(anchor, target)\n  }\n\n  /* Compute necessary adjustment for header */\n  const adjust$ = header$\n    .pipe(\n      distinctUntilKeyChanged(\"height\"),\n      map(({ height }) => {\n        const main = getComponentElement(\"main\")\n        const grid = getElement(\":scope > :first-child\", main)\n        return height + 0.8 * (\n          grid.offsetTop -\n          main.offsetTop\n        )\n      }),\n      share()\n    )\n\n  /* Compute partition of previous and next anchors */\n  const partition$ = watchElementSize(document.body)\n    .pipe(\n      distinctUntilKeyChanged(\"height\"),\n\n      /* Build index to map anchor paths to vertical offsets */\n      switchMap(body => defer(() => {\n        let path: HTMLAnchorElement[] = []\n        return of([...table].reduce((index, [anchor, target]) => {\n          while (path.length) {\n            const last = table.get(path[path.length - 1])!\n            if (last.tagName >= target.tagName) {\n              path.pop()\n            } else {\n              break\n            }\n          }\n\n          /* If the current anchor is hidden, continue with its parent */\n          let offset = target.offsetTop\n          while (!offset && target.parentElement) {\n            target = target.parentElement\n            offset = target.offsetTop\n          }\n\n          /* Fix anchor offsets in tables - see https://bit.ly/3CUFOcn */\n          let parent = target.offsetParent as HTMLElement\n          for (; parent; parent = parent.offsetParent as HTMLElement)\n            offset += parent.offsetTop\n\n          /* Map reversed anchor path to vertical offset */\n          return index.set(\n            [...path = [...path, anchor]].reverse(),\n            offset\n          )\n        }, new Map<HTMLAnchorElement[], number>()))\n      })\n        .pipe(\n\n          /* Sort index by vertical offset (see https://bit.ly/30z6QSO) */\n          map(index => new Map([...index].sort(([, a], [, b]) => a - b))),\n          combineLatestWith(adjust$),\n\n          /* Re-compute partition when viewport offset changes */\n          switchMap(([index, adjust]) => viewport$\n            .pipe(\n              scan(([prev, next], { offset: { y }, size }) => {\n                const last = y + size.height >= Math.floor(body.height)\n\n                /* Look forward */\n                while (next.length) {\n                  const [, offset] = next[0]\n                  if (offset - adjust < y || last) {\n                    prev = [...prev, next.shift()!]\n                  } else {\n                    break\n                  }\n                }\n\n                /* Look backward */\n                while (prev.length) {\n                  const [, offset] = prev[prev.length - 1]\n                  if (offset - adjust >= y && !last) {\n                    next = [prev.pop()!, ...next]\n                  } else {\n                    break\n                  }\n                }\n\n                /* Return partition */\n                return [prev, next]\n              }, [[], [...index]]),\n              distinctUntilChanged((a, b) => (\n                a[0] === b[0] &&\n                a[1] === b[1]\n              ))\n            )\n          )\n        )\n      )\n    )\n\n  /* Compute and return anchor list migrations */\n  return partition$\n    .pipe(\n      map(([prev, next]) => ({\n        prev: prev.map(([path]) => path),\n        next: next.map(([path]) => path)\n      })),\n\n      /* Extract anchor list migrations */\n      startWith({ prev: [], next: [] }),\n      bufferCount(2, 1),\n      map(([a, b]) => {\n\n        /* Moving down */\n        if (a.prev.length < b.prev.length) {\n          return {\n            prev: b.prev.slice(Math.max(0, a.prev.length - 1), b.prev.length),\n            next: []\n          }\n\n        /* Moving up */\n        } else {\n          return {\n            prev: b.prev.slice(-1),\n            next: b.next.slice(0, b.next.length - a.next.length)\n          }\n        }\n      })\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount table of contents\n *\n * @param el - Table of contents element\n * @param options - Options\n *\n * @returns Table of contents component observable\n */\nexport function mountTableOfContents(\n  el: HTMLElement, { viewport$, header$, main$, target$ }: MountOptions\n): Observable<Component<TableOfContents>> {\n  return defer(() => {\n    const push$ = new Subject<TableOfContents>()\n    const done$ = push$.pipe(ignoreElements(), endWith(true))\n    push$.subscribe(({ prev, next }) => {\n\n      /* Look forward */\n      for (const [anchor] of next) {\n        anchor.classList.remove(\"md-nav__link--passed\")\n        anchor.classList.remove(\"md-nav__link--active\")\n      }\n\n      /* Look backward */\n      for (const [index, [anchor]] of prev.entries()) {\n        anchor.classList.add(\"md-nav__link--passed\")\n        anchor.classList.toggle(\n          \"md-nav__link--active\",\n          index === prev.length - 1\n        )\n      }\n    })\n\n    /* Set up following, if enabled */\n    if (feature(\"toc.follow\")) {\n\n      /* Toggle smooth scrolling only for anchor clicks */\n      const smooth$ = merge(\n        viewport$.pipe(debounceTime(1), map(() => undefined)),\n        viewport$.pipe(debounceTime(250), map(() => \"smooth\" as const))\n      )\n\n      /* Bring active anchor into view */ // @todo: refactor\n      push$\n        .pipe(\n          filter(({ prev }) => prev.length > 0),\n          combineLatestWith(main$.pipe(observeOn(asyncScheduler))),\n          withLatestFrom(smooth$)\n        )\n          .subscribe(([[{ prev }], behavior]) => {\n            const [anchor] = prev[prev.length - 1]\n            if (anchor.offsetHeight) {\n\n              /* Retrieve overflowing container and scroll */\n              const container = getElementContainer(anchor)\n              if (typeof container !== \"undefined\") {\n                const offset = anchor.offsetTop - container.offsetTop\n                const { height } = getElementSize(container)\n                container.scrollTo({\n                  top: offset - height / 2,\n                  behavior\n                })\n              }\n            }\n          })\n    }\n\n    /* Set up anchor tracking, if enabled */\n    if (feature(\"navigation.tracking\"))\n      viewport$\n        .pipe(\n          takeUntil(done$),\n          distinctUntilKeyChanged(\"offset\"),\n          debounceTime(250),\n          skip(1),\n          takeUntil(target$.pipe(skip(1))),\n          repeat({ delay: 250 }),\n          withLatestFrom(push$)\n        )\n          .subscribe(([, { prev }]) => {\n            const url = getLocation()\n\n            /* Set hash fragment to active anchor */\n            const anchor = prev[prev.length - 1]\n            if (anchor && anchor.length) {\n              const [active] = anchor\n              const { hash } = new URL(active.href)\n              if (url.hash !== hash) {\n                url.hash = hash\n                history.replaceState({}, \"\", `${url}`)\n              }\n\n            /* Reset anchor when at the top */\n            } else {\n              url.hash = \"\"\n              history.replaceState({}, \"\", `${url}`)\n            }\n          })\n\n    /* Create and return component */\n    return watchTableOfContents(el, { viewport$, header$ })\n      .pipe(\n        tap(state => push$.next(state)),\n        finalize(() => push$.complete()),\n        map(state => ({ ref: el, ...state }))\n      )\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  Subject,\n  bufferCount,\n  combineLatest,\n  distinctUntilChanged,\n  distinctUntilKeyChanged,\n  endWith,\n  finalize,\n  fromEvent,\n  ignoreElements,\n  map,\n  repeat,\n  skip,\n  takeUntil,\n  tap\n} from \"rxjs\"\n\nimport { Viewport } from \"~/browser\"\n\nimport { Component } from \"../_\"\nimport { Header } from \"../header\"\nimport { Main } from \"../main\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Back-to-top button\n */\nexport interface BackToTop {\n  hidden: boolean                      /* Back-to-top button is hidden */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch options\n */\ninterface WatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  main$: Observable<Main>              /* Main area observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/**\n * Mount options\n */\ninterface MountOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  header$: Observable<Header>          /* Header observable */\n  main$: Observable<Main>              /* Main area observable */\n  target$: Observable<HTMLElement>     /* Location target observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Watch back-to-top\n *\n * @param _el - Back-to-top element\n * @param options - Options\n *\n * @returns Back-to-top observable\n */\nexport function watchBackToTop(\n  _el: HTMLElement, { viewport$, main$, target$ }: WatchOptions\n): Observable<BackToTop> {\n\n  /* Compute direction */\n  const direction$ = viewport$\n    .pipe(\n      map(({ offset: { y } }) => y),\n      bufferCount(2, 1),\n      map(([a, b]) => a > b && b > 0),\n      distinctUntilChanged()\n    )\n\n  /* Compute whether main area is active */\n  const active$ = main$\n    .pipe(\n      map(({ active }) => active)\n    )\n\n  /* Compute threshold for hiding */\n  return combineLatest([active$, direction$])\n    .pipe(\n      map(([active, direction]) => !(active && direction)),\n      distinctUntilChanged(),\n      takeUntil(target$.pipe(skip(1))),\n      endWith(true),\n      repeat({ delay: 250 }),\n      map(hidden => ({ hidden }))\n    )\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Mount back-to-top\n *\n * @param el - Back-to-top element\n * @param options - Options\n *\n * @returns Back-to-top component observable\n */\nexport function mountBackToTop(\n  el: HTMLElement, { viewport$, header$, main$, target$ }: MountOptions\n): Observable<Component<BackToTop>> {\n  const push$ = new Subject<BackToTop>()\n  const done$ = push$.pipe(ignoreElements(), endWith(true))\n  push$.subscribe({\n\n    /* Handle emission */\n    next({ hidden }) {\n      el.hidden = hidden\n      if (hidden) {\n        el.setAttribute(\"tabindex\", \"-1\")\n        el.blur()\n      } else {\n        el.removeAttribute(\"tabindex\")\n      }\n    },\n\n    /* Handle complete */\n    complete() {\n      el.style.top = \"\"\n      el.hidden = true\n      el.removeAttribute(\"tabindex\")\n    }\n  })\n\n  /* Watch header height */\n  header$\n    .pipe(\n      takeUntil(done$),\n      distinctUntilKeyChanged(\"height\")\n    )\n      .subscribe(({ height }) => {\n        el.style.top = `${height + 16}px`\n      })\n\n  /* Go back to top */\n  fromEvent(el, \"click\")\n    .subscribe(ev => {\n      ev.preventDefault()\n      window.scrollTo({ top: 0 })\n    })\n\n  /* Create and return component */\n  return watchBackToTop(el, { viewport$, main$, target$ })\n    .pipe(\n      tap(state => push$.next(state)),\n      finalize(() => push$.complete()),\n      map(state => ({ ref: el, ...state }))\n    )\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  filter,\n  finalize,\n  map,\n  mergeMap,\n  skip,\n  switchMap,\n  take,\n  takeUntil\n} from \"rxjs\"\n\nimport {\n  getElements,\n  watchElementVisibility\n} from \"~/browser\"\nimport { mountTooltip } from \"~/components\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n  document$: Observable<Document>      /* Document observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch ellipsis\n *\n * This function will fetch all elements that are shortened with ellipsis, and\n * filter those which are visible. Once they become visible, they stay in that\n * state, even though they may be hidden again. This optimization is necessary\n * to reduce pressure on the browser, with elements fading in and out of view.\n *\n * @param options - Options\n */\nexport function patchEllipsis(\n  { document$ }: PatchOptions\n): void {\n  document$\n    .pipe(\n      switchMap(() => getElements(\".md-ellipsis\")),\n      mergeMap(el => watchElementVisibility(el)\n        .pipe(\n          takeUntil(document$.pipe(skip(1))),\n          filter(visible => visible),\n          map(() => el),\n          take(1)\n        )\n      ),\n      filter(el => el.offsetWidth < el.scrollWidth),\n      mergeMap(el => {\n        const text = el.innerText\n        const host = el.closest(\"a\") || el\n        host.title = text\n\n        /* Mount tooltip */\n        return mountTooltip(host)\n          .pipe(\n            takeUntil(document$.pipe(skip(1))),\n            finalize(() => host.removeAttribute(\"title\"))\n          )\n      })\n    )\n      .subscribe()\n\n  // @todo move this outside of here and fix memleaks\n  document$\n    .pipe(\n      switchMap(() => getElements(\".md-status\")),\n      mergeMap(el => mountTooltip(el))\n    )\n      .subscribe()\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  fromEvent,\n  map,\n  mergeMap,\n  switchMap,\n  takeWhile,\n  tap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n  document$: Observable<Document>      /* Document observable */\n  tablet$: Observable<boolean>         /* Media tablet observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch indeterminate checkboxes\n *\n * This function replaces the indeterminate \"pseudo state\" with the actual\n * indeterminate state, which is used to keep navigation always expanded.\n *\n * @param options - Options\n */\nexport function patchIndeterminate(\n  { document$, tablet$ }: PatchOptions\n): void {\n  document$\n    .pipe(\n      switchMap(() => getElements<HTMLInputElement>(\n        \".md-toggle--indeterminate\"\n      )),\n      tap(el => {\n        el.indeterminate = true\n        el.checked = false\n      }),\n      mergeMap(el => fromEvent(el, \"change\")\n        .pipe(\n          takeWhile(() => el.classList.contains(\"md-toggle--indeterminate\")),\n          map(() => el)\n        )\n      ),\n      withLatestFrom(tablet$)\n    )\n      .subscribe(([el, tablet]) => {\n        el.classList.remove(\"md-toggle--indeterminate\")\n        if (tablet)\n          el.checked = false\n      })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  filter,\n  fromEvent,\n  map,\n  mergeMap,\n  switchMap,\n  tap\n} from \"rxjs\"\n\nimport { getElements } from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n  document$: Observable<Document>      /* Document observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Check whether the given device is an Apple device\n *\n * @returns Test result\n */\nfunction isAppleDevice(): boolean {\n  return /(iPad|iPhone|iPod)/.test(navigator.userAgent)\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch all elements with `data-md-scrollfix` attributes\n *\n * This is a year-old patch which ensures that overflow scrolling works at the\n * top and bottom of containers on iOS by ensuring a `1px` scroll offset upon\n * the start of a touch event.\n *\n * @see https://bit.ly/2SCtAOO - Original source\n *\n * @param options - Options\n */\nexport function patchScrollfix(\n  { document$ }: PatchOptions\n): void {\n  document$\n    .pipe(\n      switchMap(() => getElements(\"[data-md-scrollfix]\")),\n      tap(el => el.removeAttribute(\"data-md-scrollfix\")),\n      filter(isAppleDevice),\n      mergeMap(el => fromEvent(el, \"touchstart\")\n        .pipe(\n          map(() => el)\n        )\n      )\n    )\n      .subscribe(el => {\n        const top = el.scrollTop\n\n        /* We're at the top of the container */\n        if (top === 0) {\n          el.scrollTop = 1\n\n        /* We're at the bottom of the container */\n        } else if (top + el.offsetHeight === el.scrollHeight) {\n          el.scrollTop = top - 1\n        }\n      })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  Observable,\n  combineLatest,\n  delay,\n  map,\n  of,\n  switchMap,\n  withLatestFrom\n} from \"rxjs\"\n\nimport {\n  Viewport,\n  watchToggle\n} from \"~/browser\"\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch options\n */\ninterface PatchOptions {\n  viewport$: Observable<Viewport>      /* Viewport observable */\n  tablet$: Observable<boolean>         /* Media tablet observable */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Patch the document body to lock when search is open\n *\n * For mobile and tablet viewports, the search is rendered full screen, which\n * leads to scroll leaking when at the top or bottom of the search result. This\n * function locks the body when the search is in full screen mode, and restores\n * the scroll position when leaving.\n *\n * @param options - Options\n */\nexport function patchScrolllock(\n  { viewport$, tablet$ }: PatchOptions\n): void {\n  combineLatest([watchToggle(\"search\"), tablet$])\n    .pipe(\n      map(([active, tablet]) => active && !tablet),\n      switchMap(active => of(active)\n        .pipe(\n          delay(active ? 400 : 100)\n        )\n      ),\n      withLatestFrom(viewport$)\n    )\n      .subscribe(([active, { offset: { y }}]) => {\n        if (active) {\n          document.body.setAttribute(\"data-md-scrolllock\", \"\")\n          document.body.style.top = `-${y}px`\n        } else {\n          const value = -1 * parseInt(document.body.style.top, 10)\n          document.body.removeAttribute(\"data-md-scrolllock\")\n          document.body.style.top = \"\"\n          if (value)\n            window.scrollTo(0, value)\n        }\n      })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Polyfills\n * ------------------------------------------------------------------------- */\n\n/* Polyfill `Object.entries` */\nif (!Object.entries)\n  Object.entries = function (obj: object) {\n    const data: [string, string][] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push([key, obj[key]])\n\n    /* Return entries */\n    return data\n  }\n\n/* Polyfill `Object.values` */\nif (!Object.values)\n  Object.values = function (obj: object) {\n    const data: string[] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push(obj[key])\n\n    /* Return values */\n    return data\n  }\n\n/* ------------------------------------------------------------------------- */\n\n/* Polyfills for `Element` */\nif (typeof Element !== \"undefined\") {\n\n  /* Polyfill `Element.scrollTo` */\n  if (!Element.prototype.scrollTo)\n    Element.prototype.scrollTo = function (\n      x?: ScrollToOptions | number, y?: number\n    ): void {\n      if (typeof x === \"object\") {\n        this.scrollLeft = x.left!\n        this.scrollTop = x.top!\n      } else {\n        this.scrollLeft = x!\n        this.scrollTop = y!\n      }\n    }\n\n  /* Polyfill `Element.replaceWith` */\n  if (!Element.prototype.replaceWith)\n    Element.prototype.replaceWith = function (\n      ...nodes: Array<string | Node>\n    ): void {\n      const parent = this.parentNode\n      if (parent) {\n        if (nodes.length === 0)\n          parent.removeChild(this)\n\n        /* Replace children and create text nodes */\n        for (let i = nodes.length - 1; i >= 0; i--) {\n          let node = nodes[i]\n          if (typeof node === \"string\")\n            node = document.createTextNode(node)\n          else if (node.parentNode)\n            node.parentNode.removeChild(node)\n\n          /* Replace child or insert before previous sibling */\n          if (!i)\n            parent.replaceChild(node, this)\n          else\n            parent.insertBefore(this.previousSibling!, node)\n        }\n      }\n    }\n}\n"],
+  "mappings": 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+}
diff --git a/assets/javascripts/lunr/min/lunr.el.min.js b/assets/javascripts/lunr/min/lunr.el.min.js
new file mode 100644
index 000000000..ace017bd6
--- /dev/null
+++ b/assets/javascripts/lunr/min/lunr.el.min.js
@@ -0,0 +1 @@
+!function(e,t){"function"==typeof define&&define.amd?define(t):"object"==typeof exports?module.exports=t():t()(e.lunr)}(this,function(){return function(e){if(void 0===e)throw new Error("Lunr is not present. Please include / require Lunr before this script.");if(void 0===e.stemmerSupport)throw new Error("Lunr stemmer support is not present. Please include / require Lunr stemmer support before this script.");e.el=function(){this.pipeline.reset(),void 0===this.searchPipeline&&this.pipeline.add(e.el.trimmer,e.el.normilizer),this.pipeline.add(e.el.stopWordFilter,e.el.stemmer),this.searchPipeline&&(this.searchPipeline.reset(),this.searchPipeline.add(e.el.stemmer))},e.el.wordCharacters="A-Za-zΑαΒβΓγΔδΕεΖζΗηΘθΙιΚκΛλΜμΝνΞξΟοΠπΡρΣσςΤτΥυΦφΧχΨψΩωΆάΈέΉήΊίΌόΎύΏώΪΐΫΰΐΰ",e.el.trimmer=e.trimmerSupport.generateTrimmer(e.el.wordCharacters),e.Pipeline.registerFunction(e.el.trimmer,"trimmer-el"),e.el.stemmer=function(){function e(e){return s.test(e)}function t(e){return/[ΑΕΗΙΟΥΩ]$/.test(e)}function r(e){return/[ΑΕΗΙΟΩ]$/.test(e)}function n(n){var s=n;if(n.length<3)return s;if(!e(n))return s;if(i.indexOf(n)>=0)return s;var u=new RegExp("(.*)("+Object.keys(l).join("|")+")$"),o=u.exec(s);return null!==o&&(s=o[1]+l[o[2]]),null!==(o=/^(.+?)(ΑΔΕΣ|ΑΔΩΝ)$/.exec(s))&&(s=o[1],/(ΟΚ|ΜΑΜ|ΜΑΝ|ΜΠΑΜΠ|ΠΑΤΕΡ|ΓΙΑΓΙ|ΝΤΑΝΤ|ΚΥΡ|ΘΕΙ|ΠΕΘΕΡ|ΜΟΥΣΑΜ|ΚΑΠΛΑΜ|ΠΑΡ|ΨΑΡ|ΤΖΟΥΡ|ΤΑΜΠΟΥΡ|ΓΑΛΑΤ|ΦΑΦΛΑΤ)$/.test(o[1])||(s+="ΑΔ")),null!==(o=/^(.+?)(ΕΔΕΣ|ΕΔΩΝ)$/.exec(s))&&(s=o[1],/(ΟΠ|ΙΠ|ΕΜΠ|ΥΠ|ΓΗΠ|ΔΑΠ|ΚΡΑΣΠ|ΜΙΛ)$/.test(o[1])&&(s+="ΕΔ")),null!==(o=/^(.+?)(ΟΥΔΕΣ|ΟΥΔΩΝ)$/.exec(s))&&(s=o[1],/(ΑΡΚ|ΚΑΛΙΑΚ|ΠΕΤΑΛ|ΛΙΧ|ΠΛΕΞ|ΣΚ|Σ|ΦΛ|ΦΡ|ΒΕΛ|ΛΟΥΛ|ΧΝ|ΣΠ|ΤΡΑΓ|ΦΕ)$/.test(o[1])&&(s+="ΟΥΔ")),null!==(o=/^(.+?)(ΕΩΣ|ΕΩΝ|ΕΑΣ|ΕΑ)$/.exec(s))&&(s=o[1],/^(Θ|Δ|ΕΛ|ΓΑΛ|Ν|Π|ΙΔ|ΠΑΡ|ΣΤΕΡ|ΟΡΦ|ΑΝΔΡ|ΑΝΤΡ)$/.test(o[1])&&(s+="Ε")),null!==(o=/^(.+?)(ΕΙΟ|ΕΙΟΣ|ΕΙΟΙ|ΕΙΑ|ΕΙΑΣ|ΕΙΕΣ|ΕΙΟΥ|ΕΙΟΥΣ|ΕΙΩΝ)$/.exec(s))&&o[1].length>4&&(s=o[1]),null!==(o=/^(.+?)(ΙΟΥΣ|ΙΑΣ|ΙΕΣ|ΙΟΣ|ΙΟΥ|ΙΟΙ|ΙΩΝ|ΙΟΝ|ΙΑ|ΙΟ)$/.exec(s))&&(s=o[1],(t(s)||s.length<2||/^(ΑΓ|ΑΓΓΕΛ|ΑΓΡ|ΑΕΡ|ΑΘΛ|ΑΚΟΥΣ|ΑΞ|ΑΣ|Β|ΒΙΒΛ|ΒΥΤ|Γ|ΓΙΑΓ|ΓΩΝ|Δ|ΔΑΝ|ΔΗΛ|ΔΗΜ|ΔΟΚΙΜ|ΕΛ|ΖΑΧΑΡ|ΗΛ|ΗΠ|ΙΔ|ΙΣΚ|ΙΣΤ|ΙΟΝ|ΙΩΝ|ΚΙΜΩΛ|ΚΟΛΟΝ|ΚΟΡ|ΚΤΗΡ|ΚΥΡ|ΛΑΓ|ΛΟΓ|ΜΑΓ|ΜΠΑΝ|ΜΠΡ|ΝΑΥΤ|ΝΟΤ|ΟΠΑΛ|ΟΞ|ΟΡ|ΟΣ|ΠΑΝΑΓ|ΠΑΤΡ|ΠΗΛ|ΠΗΝ|ΠΛΑΙΣ|ΠΟΝΤ|ΡΑΔ|ΡΟΔ|ΣΚ|ΣΚΟΡΠ|ΣΟΥΝ|ΣΠΑΝ|ΣΤΑΔ|ΣΥΡ|ΤΗΛ|ΤΙΜ|ΤΟΚ|ΤΟΠ|ΤΡΟΧ|ΦΙΛ|ΦΩΤ|Χ|ΧΙΛ|ΧΡΩΜ|ΧΩΡ)$/.test(o[1]))&&(s+="Ι"),/^(ΠΑΛ)$/.test(o[1])&&(s+="ΑΙ")),null!==(o=/^(.+?)(ΙΚΟΣ|ΙΚΟΝ|ΙΚΕΙΣ|ΙΚΟΙ|ΙΚΕΣ|ΙΚΟΥΣ|ΙΚΗ|ΙΚΗΣ|ΙΚΟ|ΙΚΑ|ΙΚΟΥ|ΙΚΩΝ|ΙΚΩΣ)$/.exec(s))&&(s=o[1],(t(s)||/^(ΑΔ|ΑΛ|ΑΜΑΝ|ΑΜΕΡ|ΑΜΜΟΧΑΛ|ΑΝΗΘ|ΑΝΤΙΔ|ΑΠΛ|ΑΤΤ|ΑΦΡ|ΒΑΣ|ΒΡΩΜ|ΓΕΝ|ΓΕΡ|Δ|ΔΙΚΑΝ|ΔΥΤ|ΕΙΔ|ΕΝΔ|ΕΞΩΔ|ΗΘ|ΘΕΤ|ΚΑΛΛΙΝ|ΚΑΛΠ|ΚΑΤΑΔ|ΚΟΥΖΙΝ|ΚΡ|ΚΩΔ|ΛΟΓ|Μ|ΜΕΡ|ΜΟΝΑΔ|ΜΟΥΛ|ΜΟΥΣ|ΜΠΑΓΙΑΤ|ΜΠΑΝ|ΜΠΟΛ|ΜΠΟΣ|ΜΥΣΤ|Ν|ΝΙΤ|ΞΙΚ|ΟΠΤ|ΠΑΝ|ΠΕΤΣ|ΠΙΚΑΝΤ|ΠΙΤΣ|ΠΛΑΣΤ|ΠΛΙΑΤΣ|ΠΟΝΤ|ΠΟΣΤΕΛΝ|ΠΡΩΤΟΔ|ΣΕΡΤ|ΣΗΜΑΝΤ|ΣΤΑΤ|ΣΥΝΑΔ|ΣΥΝΟΜΗΛ|ΤΕΛ|ΤΕΧΝ|ΤΡΟΠ|ΤΣΑΜ|ΥΠΟΔ|Φ|ΦΙΛΟΝ|ΦΥΛΟΔ|ΦΥΣ|ΧΑΣ)$/.test(o[1])||/(ΦΟΙΝ)$/.test(o[1]))&&(s+="ΙΚ")),"ΑΓΑΜΕ"===s&&(s="ΑΓΑΜ"),null!==(o=/^(.+?)(ΑΓΑΜΕ|ΗΣΑΜΕ|ΟΥΣΑΜΕ|ΗΚΑΜΕ|ΗΘΗΚΑΜΕ)$/.exec(s))&&(s=o[1]),null!==(o=/^(.+?)(ΑΜΕ)$/.exec(s))&&(s=o[1],/^(ΑΝΑΠ|ΑΠΟΘ|ΑΠΟΚ|ΑΠΟΣΤ|ΒΟΥΒ|ΞΕΘ|ΟΥΛ|ΠΕΘ|ΠΙΚΡ|ΠΟΤ|ΣΙΧ|Χ)$/.test(o[1])&&(s+="ΑΜ")),null!==(o=/^(.+?)(ΑΓΑΝΕ|ΗΣΑΝΕ|ΟΥΣΑΝΕ|ΙΟΝΤΑΝΕ|ΙΟΤΑΝΕ|ΙΟΥΝΤΑΝΕ|ΟΝΤΑΝΕ|ΟΤΑΝΕ|ΟΥΝΤΑΝΕ|ΗΚΑΝΕ|ΗΘΗΚΑΝΕ)$/.exec(s))&&(s=o[1],/^(ΤΡ|ΤΣ)$/.test(o[1])&&(s+="ΑΓΑΝ")),null!==(o=/^(.+?)(ΑΝΕ)$/.exec(s))&&(s=o[1],(r(s)||/^(ΒΕΤΕΡ|ΒΟΥΛΚ|ΒΡΑΧΜ|Γ|ΔΡΑΔΟΥΜ|Θ|ΚΑΛΠΟΥΖ|ΚΑΣΤΕΛ|ΚΟΡΜΟΡ|ΛΑΟΠΛ|ΜΩΑΜΕΘ|Μ|ΜΟΥΣΟΥΛΜΑΝ|ΟΥΛ|Π|ΠΕΛΕΚ|ΠΛ|ΠΟΛΙΣ|ΠΟΡΤΟΛ|ΣΑΡΑΚΑΤΣ|ΣΟΥΛΤ|ΤΣΑΡΛΑΤ|ΟΡΦ|ΤΣΙΓΓ|ΤΣΟΠ|ΦΩΤΟΣΤΕΦ|Χ|ΨΥΧΟΠΛ|ΑΓ|ΟΡΦ|ΓΑΛ|ΓΕΡ|ΔΕΚ|ΔΙΠΛ|ΑΜΕΡΙΚΑΝ|ΟΥΡ|ΠΙΘ|ΠΟΥΡΙΤ|Σ|ΖΩΝΤ|ΙΚ|ΚΑΣΤ|ΚΟΠ|ΛΙΧ|ΛΟΥΘΗΡ|ΜΑΙΝΤ|ΜΕΛ|ΣΙΓ|ΣΠ|ΣΤΕΓ|ΤΡΑΓ|ΤΣΑΓ|Φ|ΕΡ|ΑΔΑΠ|ΑΘΙΓΓ|ΑΜΗΧ|ΑΝΙΚ|ΑΝΟΡΓ|ΑΠΗΓ|ΑΠΙΘ|ΑΤΣΙΓΓ|ΒΑΣ|ΒΑΣΚ|ΒΑΘΥΓΑΛ|ΒΙΟΜΗΧ|ΒΡΑΧΥΚ|ΔΙΑΤ|ΔΙΑΦ|ΕΝΟΡΓ|ΘΥΣ|ΚΑΠΝΟΒΙΟΜΗΧ|ΚΑΤΑΓΑΛ|ΚΛΙΒ|ΚΟΙΛΑΡΦ|ΛΙΒ|ΜΕΓΛΟΒΙΟΜΗΧ|ΜΙΚΡΟΒΙΟΜΗΧ|ΝΤΑΒ|ΞΗΡΟΚΛΙΒ|ΟΛΙΓΟΔΑΜ|ΟΛΟΓΑΛ|ΠΕΝΤΑΡΦ|ΠΕΡΗΦ|ΠΕΡΙΤΡ|ΠΛΑΤ|ΠΟΛΥΔΑΠ|ΠΟΛΥΜΗΧ|ΣΤΕΦ|ΤΑΒ|ΤΕΤ|ΥΠΕΡΗΦ|ΥΠΟΚΟΠ|ΧΑΜΗΛΟΔΑΠ|ΨΗΛΟΤΑΒ)$/.test(o[1]))&&(s+="ΑΝ")),null!==(o=/^(.+?)(ΗΣΕΤΕ)$/.exec(s))&&(s=o[1]),null!==(o=/^(.+?)(ΕΤΕ)$/.exec(s))&&(s=o[1],(r(s)||/(ΟΔ|ΑΙΡ|ΦΟΡ|ΤΑΘ|ΔΙΑΘ|ΣΧ|ΕΝΔ|ΕΥΡ|ΤΙΘ|ΥΠΕΡΘ|ΡΑΘ|ΕΝΘ|ΡΟΘ|ΣΘ|ΠΥΡ|ΑΙΝ|ΣΥΝΔ|ΣΥΝ|ΣΥΝΘ|ΧΩΡ|ΠΟΝ|ΒΡ|ΚΑΘ|ΕΥΘ|ΕΚΘ|ΝΕΤ|ΡΟΝ|ΑΡΚ|ΒΑΡ|ΒΟΛ|ΩΦΕΛ)$/.test(o[1])||/^(ΑΒΑΡ|ΒΕΝ|ΕΝΑΡ|ΑΒΡ|ΑΔ|ΑΘ|ΑΝ|ΑΠΛ|ΒΑΡΟΝ|ΝΤΡ|ΣΚ|ΚΟΠ|ΜΠΟΡ|ΝΙΦ|ΠΑΓ|ΠΑΡΑΚΑΛ|ΣΕΡΠ|ΣΚΕΛ|ΣΥΡΦ|ΤΟΚ|Υ|Δ|ΕΜ|ΘΑΡΡ|Θ)$/.test(o[1]))&&(s+="ΕΤ")),null!==(o=/^(.+?)(ΟΝΤΑΣ|ΩΝΤΑΣ)$/.exec(s))&&(s=o[1],/^ΑΡΧ$/.test(o[1])&&(s+="ΟΝΤ"),/ΚΡΕ$/.test(o[1])&&(s+="ΩΝΤ")),null!==(o=/^(.+?)(ΟΜΑΣΤΕ|ΙΟΜΑΣΤΕ)$/.exec(s))&&(s=o[1],/^ΟΝ$/.test(o[1])&&(s+="ΟΜΑΣΤ")),null!==(o=/^(.+?)(ΙΕΣΤΕ)$/.exec(s))&&(s=o[1],/^(Π|ΑΠ|ΣΥΜΠ|ΑΣΥΜΠ|ΑΚΑΤΑΠ|ΑΜΕΤΑΜΦ)$/.test(o[1])&&(s+="ΙΕΣΤ")),null!==(o=/^(.+?)(ΕΣΤΕ)$/.exec(s))&&(s=o[1],/^(ΑΛ|ΑΡ|ΕΚΤΕΛ|Ζ|Μ|Ξ|ΠΑΡΑΚΑΛ|ΠΡΟ|ΝΙΣ)$/.test(o[1])&&(s+="ΕΣΤ")),null!==(o=/^(.+?)(ΗΘΗΚΑ|ΗΘΗΚΕΣ|ΗΘΗΚΕ)$/.exec(s))&&(s=o[1]),null!==(o=/^(.+?)(ΗΚΑ|ΗΚΕΣ|ΗΚΕ)$/.exec(s))&&(s=o[1],(/(ΣΚΩΛ|ΣΚΟΥΛ|ΝΑΡΘ|ΣΦ|ΟΘ|ΠΙΘ)$/.test(o[1])||/^(ΔΙΑΘ|Θ|ΠΑΡΑΚΑΤΑΘ|ΠΡΟΣΘ|ΣΥΝΘ)$/.test(o[1]))&&(s+="ΗΚ")),null!==(o=/^(.+?)(ΟΥΣΑ|ΟΥΣΕΣ|ΟΥΣΕ)$/.exec(s))&&(s=o[1],(t(s)||/^(ΦΑΡΜΑΚ|ΧΑΔ|ΑΓΚ|ΑΝΑΡΡ|ΒΡΟΜ|ΕΚΛΙΠ|ΛΑΜΠΙΔ|ΛΕΧ|Μ|ΠΑΤ|Ρ|Λ|ΜΕΔ|ΜΕΣΑΖ|ΥΠΟΤΕΙΝ|ΑΜ|ΑΙΘ|ΑΝΗΚ|ΔΕΣΠΟΖ|ΕΝΔΙΑΦΕΡ)$/.test(o[1])||/(ΠΟΔΑΡ|ΒΛΕΠ|ΠΑΝΤΑΧ|ΦΡΥΔ|ΜΑΝΤΙΛ|ΜΑΛΛ|ΚΥΜΑΤ|ΛΑΧ|ΛΗΓ|ΦΑΓ|ΟΜ|ΠΡΩΤ)$/.test(o[1]))&&(s+="ΟΥΣ")),null!==(o=/^(.+?)(ΑΓΑ|ΑΓΕΣ|ΑΓΕ)$/.exec(s))&&(s=o[1],(/^(ΑΒΑΣΤ|ΠΟΛΥΦ|ΑΔΗΦ|ΠΑΜΦ|Ρ|ΑΣΠ|ΑΦ|ΑΜΑΛ|ΑΜΑΛΛΙ|ΑΝΥΣΤ|ΑΠΕΡ|ΑΣΠΑΡ|ΑΧΑΡ|ΔΕΡΒΕΝ|ΔΡΟΣΟΠ|ΞΕΦ|ΝΕΟΠ|ΝΟΜΟΤ|ΟΛΟΠ|ΟΜΟΤ|ΠΡΟΣΤ|ΠΡΟΣΩΠΟΠ|ΣΥΜΠ|ΣΥΝΤ|Τ|ΥΠΟΤ|ΧΑΡ|ΑΕΙΠ|ΑΙΜΟΣΤ|ΑΝΥΠ|ΑΠΟΤ|ΑΡΤΙΠ|ΔΙΑΤ|ΕΝ|ΕΠΙΤ|ΚΡΟΚΑΛΟΠ|ΣΙΔΗΡΟΠ|Λ|ΝΑΥ|ΟΥΛΑΜ|ΟΥΡ|Π|ΤΡ|Μ)$/.test(o[1])||/(ΟΦ|ΠΕΛ|ΧΟΡΤ|ΛΛ|ΣΦ|ΡΠ|ΦΡ|ΠΡ|ΛΟΧ|ΣΜΗΝ)$/.test(o[1])&&!/^(ΨΟΦ|ΝΑΥΛΟΧ)$/.test(o[1])||/(ΚΟΛΛ)$/.test(o[1]))&&(s+="ΑΓ")),null!==(o=/^(.+?)(ΗΣΕ|ΗΣΟΥ|ΗΣΑ)$/.exec(s))&&(s=o[1],/^(Ν|ΧΕΡΣΟΝ|ΔΩΔΕΚΑΝ|ΕΡΗΜΟΝ|ΜΕΓΑΛΟΝ|ΕΠΤΑΝ|Ι)$/.test(o[1])&&(s+="ΗΣ")),null!==(o=/^(.+?)(ΗΣΤΕ)$/.exec(s))&&(s=o[1],/^(ΑΣΒ|ΣΒ|ΑΧΡ|ΧΡ|ΑΠΛ|ΑΕΙΜΝ|ΔΥΣΧΡ|ΕΥΧΡ|ΚΟΙΝΟΧΡ|ΠΑΛΙΜΨ)$/.test(o[1])&&(s+="ΗΣΤ")),null!==(o=/^(.+?)(ΟΥΝΕ|ΗΣΟΥΝΕ|ΗΘΟΥΝΕ)$/.exec(s))&&(s=o[1],/^(Ν|Ρ|ΣΠΙ|ΣΤΡΑΒΟΜΟΥΤΣ|ΚΑΚΟΜΟΥΤΣ|ΕΞΩΝ)$/.test(o[1])&&(s+="ΟΥΝ")),null!==(o=/^(.+?)(ΟΥΜΕ|ΗΣΟΥΜΕ|ΗΘΟΥΜΕ)$/.exec(s))&&(s=o[1],/^(ΠΑΡΑΣΟΥΣ|Φ|Χ|ΩΡΙΟΠΛ|ΑΖ|ΑΛΛΟΣΟΥΣ|ΑΣΟΥΣ)$/.test(o[1])&&(s+="ΟΥΜ")),null!=(o=/^(.+?)(ΜΑΤΟΙ|ΜΑΤΟΥΣ|ΜΑΤΟ|ΜΑΤΑ|ΜΑΤΩΣ|ΜΑΤΩΝ|ΜΑΤΟΣ|ΜΑΤΕΣ|ΜΑΤΗ|ΜΑΤΗΣ|ΜΑΤΟΥ)$/.exec(s))&&(s=o[1]+"Μ",/^(ΓΡΑΜ)$/.test(o[1])?s+="Α":/^(ΓΕ|ΣΤΑ)$/.test(o[1])&&(s+="ΑΤ")),null!==(o=/^(.+?)(ΟΥΑ)$/.exec(s))&&(s=o[1]+"ΟΥ"),n.length===s.length&&null!==(o=/^(.+?)(Α|ΑΓΑΤΕ|ΑΓΑΝ|ΑΕΙ|ΑΜΑΙ|ΑΝ|ΑΣ|ΑΣΑΙ|ΑΤΑΙ|ΑΩ|Ε|ΕΙ|ΕΙΣ|ΕΙΤΕ|ΕΣΑΙ|ΕΣ|ΕΤΑΙ|Ι|ΙΕΜΑΙ|ΙΕΜΑΣΤΕ|ΙΕΤΑΙ|ΙΕΣΑΙ|ΙΕΣΑΣΤΕ|ΙΟΜΑΣΤΑΝ|ΙΟΜΟΥΝ|ΙΟΜΟΥΝΑ|ΙΟΝΤΑΝ|ΙΟΝΤΟΥΣΑΝ|ΙΟΣΑΣΤΑΝ|ΙΟΣΑΣΤΕ|ΙΟΣΟΥΝ|ΙΟΣΟΥΝΑ|ΙΟΤΑΝ|ΙΟΥΜΑ|ΙΟΥΜΑΣΤΕ|ΙΟΥΝΤΑΙ|ΙΟΥΝΤΑΝ|Η|ΗΔΕΣ|ΗΔΩΝ|ΗΘΕΙ|ΗΘΕΙΣ|ΗΘΕΙΤΕ|ΗΘΗΚΑΤΕ|ΗΘΗΚΑΝ|ΗΘΟΥΝ|ΗΘΩ|ΗΚΑΤΕ|ΗΚΑΝ|ΗΣ|ΗΣΑΝ|ΗΣΑΤΕ|ΗΣΕΙ|ΗΣΕΣ|ΗΣΟΥΝ|ΗΣΩ|Ο|ΟΙ|ΟΜΑΙ|ΟΜΑΣΤΑΝ|ΟΜΟΥΝ|ΟΜΟΥΝΑ|ΟΝΤΑΙ|ΟΝΤΑΝ|ΟΝΤΟΥΣΑΝ|ΟΣ|ΟΣΑΣΤΑΝ|ΟΣΑΣΤΕ|ΟΣΟΥΝ|ΟΣΟΥΝΑ|ΟΤΑΝ|ΟΥ|ΟΥΜΑΙ|ΟΥΜΑΣΤΕ|ΟΥΝ|ΟΥΝΤΑΙ|ΟΥΝΤΑΝ|ΟΥΣ|ΟΥΣΑΝ|ΟΥΣΑΤΕ|Υ||ΥΑ|ΥΣ|Ω|ΩΝ|ΟΙΣ)$/.exec(s))&&(s=o[1]),null!=(o=/^(.+?)(ΕΣΤΕΡ|ΕΣΤΑΤ|ΟΤΕΡ|ΟΤΑΤ|ΥΤΕΡ|ΥΤΑΤ|ΩΤΕΡ|ΩΤΑΤ)$/.exec(s))&&(/^(ΕΞ|ΕΣ|ΑΝ|ΚΑΤ|Κ|ΠΡ)$/.test(o[1])||(s=o[1]),/^(ΚΑ|Μ|ΕΛΕ|ΛΕ|ΔΕ)$/.test(o[1])&&(s+="ΥΤ")),s}var 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\ No newline at end of file
diff --git a/assets/javascripts/workers/search.a264c092.min.js b/assets/javascripts/workers/search.f886a092.min.js
similarity index 75%
rename from assets/javascripts/workers/search.a264c092.min.js
rename to assets/javascripts/workers/search.f886a092.min.js
index dd7649e29..283b855c3 100644
--- a/assets/javascripts/workers/search.a264c092.min.js
+++ b/assets/javascripts/workers/search.f886a092.min.js
@@ -11,18 +11,18 @@
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  * Copyright (C) 2020 Oliver Nightingale
- */t.tokenizer=function(e,r){if(e==null||e==null)return[];if(Array.isArray(e))return e.map(function(g){return new t.Token(t.utils.asString(g).toLowerCase(),t.utils.clone(r))});for(var n=e.toString().toLowerCase(),i=n.length,s=[],o=0,a=0;o<=i;o++){var u=n.charAt(o),c=o-a;if(u.match(t.tokenizer.separator)||o==i){if(c>0){var l=t.utils.clone(r)||{};l.position=[a,c],l.index=s.length,s.push(new t.Token(n.slice(a,o),l))}a=o+1}}return s},t.tokenizer.separator=/[\s\-]+/;/*!
+ */t.tokenizer=function(e,r){if(e==null||e==null)return[];if(Array.isArray(e))return e.map(function(g){return new t.Token(t.utils.asString(g).toLowerCase(),t.utils.clone(r))});for(var n=e.toString().toLowerCase(),i=n.length,s=[],o=0,a=0;o<=i;o++){var u=n.charAt(o),c=o-a;if(u.match(t.tokenizer.separator)||o==i){if(c>0){var f=t.utils.clone(r)||{};f.position=[a,c],f.index=s.length,s.push(new t.Token(n.slice(a,o),f))}a=o+1}}return s},t.tokenizer.separator=/[\s\-]+/;/*!
  * lunr.Pipeline
  * Copyright (C) 2020 Oliver Nightingale
  */t.Pipeline=function(){this._stack=[]},t.Pipeline.registeredFunctions=Object.create(null),t.Pipeline.registerFunction=function(e,r){r in this.registeredFunctions&&t.utils.warn("Overwriting existing registered function: "+r),e.label=r,t.Pipeline.registeredFunctions[e.label]=e},t.Pipeline.warnIfFunctionNotRegistered=function(e){var r=e.label&&e.label in this.registeredFunctions;r||t.utils.warn(`Function is not registered with pipeline. This may cause problems when serialising the index.
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  * Copyright (C) 2020 Oliver Nightingale
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+ */t.Vector=function(e){this._magnitude=0,this.elements=e||[]},t.Vector.prototype.positionForIndex=function(e){if(this.elements.length==0)return 0;for(var r=0,n=this.elements.length/2,i=n-r,s=Math.floor(i/2),o=this.elements[s*2];i>1&&(o<e&&(r=s),o>e&&(n=s),o!=e);)i=n-r,s=r+Math.floor(i/2),o=this.elements[s*2];if(o==e||o>e)return s*2;if(o<e)return(s+1)*2},t.Vector.prototype.insert=function(e,r){this.upsert(e,r,function(){throw"duplicate index"})},t.Vector.prototype.upsert=function(e,r,n){this._magnitude=0;var i=this.positionForIndex(e);this.elements[i]==e?this.elements[i+1]=n(this.elements[i+1],r):this.elements.splice(i,0,e,r)},t.Vector.prototype.magnitude=function(){if(this._magnitude)return this._magnitude;for(var e=0,r=this.elements.length,n=1;n<r;n+=2){var i=this.elements[n];e+=i*i}return this._magnitude=Math.sqrt(e)},t.Vector.prototype.dot=function(e){for(var r=0,n=this.elements,i=e.elements,s=n.length,o=i.length,a=0,u=0,c=0,f=0;c<s&&f<o;)a=n[c],u=i[f],a<u?c+=2:a>u?f+=2:a==u&&(r+=n[c+1]*i[f+1],c+=2,f+=2);return r},t.Vector.prototype.similarity=function(e){return this.dot(e)/this.magnitude()||0},t.Vector.prototype.toArray=function(){for(var e=new Array(this.elements.length/2),r=1,n=0;r<this.elements.length;r+=2,n++)e[n]=this.elements[r];return e},t.Vector.prototype.toJSON=function(){return this.elements};/*!
  * lunr.stemmer
  * Copyright (C) 2020 Oliver Nightingale
  * Includes code from - http://tartarus.org/~martin/PorterStemmer/js.txt
- */t.stemmer=function(){var e={ational:"ate",tional:"tion",enci:"ence",anci:"ance",izer:"ize",bli:"ble",alli:"al",entli:"ent",eli:"e",ousli:"ous",ization:"ize",ation:"ate",ator:"ate",alism:"al",iveness:"ive",fulness:"ful",ousness:"ous",aliti:"al",iviti:"ive",biliti:"ble",logi:"log"},r={icate:"ic",ative:"",alize:"al",iciti:"ic",ical:"ic",ful:"",ness:""},n="[^aeiou]",i="[aeiouy]",s=n+"[^aeiouy]*",o=i+"[aeiou]*",a="^("+s+")?"+o+s,u="^("+s+")?"+o+s+"("+o+")?$",c="^("+s+")?"+o+s+o+s,l="^("+s+")?"+i,g=new RegExp(a),f=new RegExp(c),v=new RegExp(u),m=new RegExp(l),x=/^(.+?)(ss|i)es$/,d=/^(.+?)([^s])s$/,y=/^(.+?)eed$/,b=/^(.+?)(ed|ing)$/,E=/.$/,w=/(at|bl|iz)$/,R=new RegExp("([^aeiouylsz])\\1$"),j=new RegExp("^"+s+i+"[^aeiouwxy]$"),_=/^(.+?[^aeiou])y$/,D=/^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/,N=/^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/,C=/^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/,V=/^(.+?)(s|t)(ion)$/,P=/^(.+?)e$/,z=/ll$/,$=new RegExp("^"+s+i+"[^aeiouwxy]$"),M=function(h){var S,k,L,p,T,O,F;if(h.length<3)return h;if(L=h.substr(0,1),L=="y"&&(h=L.toUpperCase()+h.substr(1)),p=x,T=d,p.test(h)?h=h.replace(p,"$1$2"):T.test(h)&&(h=h.replace(T,"$1$2")),p=y,T=b,p.test(h)){var Q=p.exec(h);p=g,p.test(Q[1])&&(p=E,h=h.replace(p,""))}else if(T.test(h)){var Q=T.exec(h);S=Q[1],T=m,T.test(S)&&(h=S,T=w,O=R,F=j,T.test(h)?h=h+"e":O.test(h)?(p=E,h=h.replace(p,"")):F.test(h)&&(h=h+"e"))}if(p=_,p.test(h)){var Q=p.exec(h);S=Q[1],h=S+"i"}if(p=D,p.test(h)){var Q=p.exec(h);S=Q[1],k=Q[2],p=g,p.test(S)&&(h=S+e[k])}if(p=N,p.test(h)){var Q=p.exec(h);S=Q[1],k=Q[2],p=g,p.test(S)&&(h=S+r[k])}if(p=C,T=V,p.test(h)){var Q=p.exec(h);S=Q[1],p=f,p.test(S)&&(h=S)}else if(T.test(h)){var Q=T.exec(h);S=Q[1]+Q[2],T=f,T.test(S)&&(h=S)}if(p=P,p.test(h)){var Q=p.exec(h);S=Q[1],p=f,T=v,O=$,(p.test(S)||T.test(S)&&!O.test(S))&&(h=S)}return p=z,T=f,p.test(h)&&T.test(h)&&(p=E,h=h.replace(p,"")),L=="y"&&(h=L.toLowerCase()+h.substr(1)),h};return function(I){return I.update(M)}}(),t.Pipeline.registerFunction(t.stemmer,"stemmer");/*!
+ */t.stemmer=function(){var e={ational:"ate",tional:"tion",enci:"ence",anci:"ance",izer:"ize",bli:"ble",alli:"al",entli:"ent",eli:"e",ousli:"ous",ization:"ize",ation:"ate",ator:"ate",alism:"al",iveness:"ive",fulness:"ful",ousness:"ous",aliti:"al",iviti:"ive",biliti:"ble",logi:"log"},r={icate:"ic",ative:"",alize:"al",iciti:"ic",ical:"ic",ful:"",ness:""},n="[^aeiou]",i="[aeiouy]",s=n+"[^aeiouy]*",o=i+"[aeiou]*",a="^("+s+")?"+o+s,u="^("+s+")?"+o+s+"("+o+")?$",c="^("+s+")?"+o+s+o+s,f="^("+s+")?"+i,g=new RegExp(a),l=new RegExp(c),m=new RegExp(u),x=new RegExp(f),v=/^(.+?)(ss|i)es$/,d=/^(.+?)([^s])s$/,y=/^(.+?)eed$/,b=/^(.+?)(ed|ing)$/,E=/.$/,w=/(at|bl|iz)$/,R=new RegExp("([^aeiouylsz])\\1$"),j=new RegExp("^"+s+i+"[^aeiouwxy]$"),_=/^(.+?[^aeiou])y$/,D=/^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/,N=/^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/,C=/^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/,V=/^(.+?)(s|t)(ion)$/,P=/^(.+?)e$/,z=/ll$/,$=new RegExp("^"+s+i+"[^aeiouwxy]$"),M=function(h){var S,k,L,p,T,O,F;if(h.length<3)return h;if(L=h.substr(0,1),L=="y"&&(h=L.toUpperCase()+h.substr(1)),p=v,T=d,p.test(h)?h=h.replace(p,"$1$2"):T.test(h)&&(h=h.replace(T,"$1$2")),p=y,T=b,p.test(h)){var Q=p.exec(h);p=g,p.test(Q[1])&&(p=E,h=h.replace(p,""))}else if(T.test(h)){var Q=T.exec(h);S=Q[1],T=x,T.test(S)&&(h=S,T=w,O=R,F=j,T.test(h)?h=h+"e":O.test(h)?(p=E,h=h.replace(p,"")):F.test(h)&&(h=h+"e"))}if(p=_,p.test(h)){var Q=p.exec(h);S=Q[1],h=S+"i"}if(p=D,p.test(h)){var Q=p.exec(h);S=Q[1],k=Q[2],p=g,p.test(S)&&(h=S+e[k])}if(p=N,p.test(h)){var Q=p.exec(h);S=Q[1],k=Q[2],p=g,p.test(S)&&(h=S+r[k])}if(p=C,T=V,p.test(h)){var Q=p.exec(h);S=Q[1],p=l,p.test(S)&&(h=S)}else if(T.test(h)){var Q=T.exec(h);S=Q[1]+Q[2],T=l,T.test(S)&&(h=S)}if(p=P,p.test(h)){var Q=p.exec(h);S=Q[1],p=l,T=m,O=$,(p.test(S)||T.test(S)&&!O.test(S))&&(h=S)}return p=z,T=l,p.test(h)&&T.test(h)&&(p=E,h=h.replace(p,"")),L=="y"&&(h=L.toLowerCase()+h.substr(1)),h};return function(I){return I.update(M)}}(),t.Pipeline.registerFunction(t.stemmer,"stemmer");/*!
  * lunr.stopWordFilter
  * Copyright (C) 2020 Oliver Nightingale
  */t.generateStopWordFilter=function(e){var r=e.reduce(function(n,i){return n[i]=i,n},{});return function(n){if(n&&r[n.toString()]!==n.toString())return n}},t.stopWordFilter=t.generateStopWordFilter(["a","able","about","across","after","all","almost","also","am","among","an","and","any","are","as","at","be","because","been","but","by","can","cannot","could","dear","did","do","does","either","else","ever","every","for","from","get","got","had","has","have","he","her","hers","him","his","how","however","i","if","in","into","is","it","its","just","least","let","like","likely","may","me","might","most","must","my","neither","no","nor","not","of","off","often","on","only","or","other","our","own","rather","said","say","says","she","should","since","so","some","than","that","the","their","them","then","there","these","they","this","tis","to","too","twas","us","wants","was","we","were","what","when","where","which","while","who","whom","why","will","with","would","yet","you","your"]),t.Pipeline.registerFunction(t.stopWordFilter,"stopWordFilter");/*!
@@ -31,12 +31,12 @@
  */t.trimmer=function(e){return e.update(function(r){return r.replace(/^\W+/,"").replace(/\W+$/,"")})},t.Pipeline.registerFunction(t.trimmer,"trimmer");/*!
  * lunr.TokenSet
  * Copyright (C) 2020 Oliver Nightingale
- */t.TokenSet=function(){this.final=!1,this.edges={},this.id=t.TokenSet._nextId,t.TokenSet._nextId+=1},t.TokenSet._nextId=1,t.TokenSet.fromArray=function(e){for(var r=new t.TokenSet.Builder,n=0,i=e.length;n<i;n++)r.insert(e[n]);return r.finish(),r.root},t.TokenSet.fromClause=function(e){return"editDistance"in e?t.TokenSet.fromFuzzyString(e.term,e.editDistance):t.TokenSet.fromString(e.term)},t.TokenSet.fromFuzzyString=function(e,r){for(var n=new t.TokenSet,i=[{node:n,editsRemaining:r,str:e}];i.length;){var s=i.pop();if(s.str.length>0){var o=s.str.charAt(0),a;o in s.node.edges?a=s.node.edges[o]:(a=new t.TokenSet,s.node.edges[o]=a),s.str.length==1&&(a.final=!0),i.push({node:a,editsRemaining:s.editsRemaining,str:s.str.slice(1)})}if(s.editsRemaining!=0){if("*"in s.node.edges)var u=s.node.edges["*"];else{var u=new t.TokenSet;s.node.edges["*"]=u}if(s.str.length==0&&(u.final=!0),i.push({node:u,editsRemaining:s.editsRemaining-1,str:s.str}),s.str.length>1&&i.push({node:s.node,editsRemaining:s.editsRemaining-1,str:s.str.slice(1)}),s.str.length==1&&(s.node.final=!0),s.str.length>=1){if("*"in s.node.edges)var c=s.node.edges["*"];else{var c=new t.TokenSet;s.node.edges["*"]=c}s.str.length==1&&(c.final=!0),i.push({node:c,editsRemaining:s.editsRemaining-1,str:s.str.slice(1)})}if(s.str.length>1){var l=s.str.charAt(0),g=s.str.charAt(1),f;g in s.node.edges?f=s.node.edges[g]:(f=new t.TokenSet,s.node.edges[g]=f),s.str.length==1&&(f.final=!0),i.push({node:f,editsRemaining:s.editsRemaining-1,str:l+s.str.slice(2)})}}}return n},t.TokenSet.fromString=function(e){for(var r=new t.TokenSet,n=r,i=0,s=e.length;i<s;i++){var o=e[i],a=i==s-1;if(o=="*")r.edges[o]=r,r.final=a;else{var u=new t.TokenSet;u.final=a,r.edges[o]=u,r=u}}return n},t.TokenSet.prototype.toArray=function(){for(var e=[],r=[{prefix:"",node:this}];r.length;){var n=r.pop(),i=Object.keys(n.node.edges),s=i.length;n.node.final&&(n.prefix.charAt(0),e.push(n.prefix));for(var o=0;o<s;o++){var a=i[o];r.push({prefix:n.prefix.concat(a),node:n.node.edges[a]})}}return e},t.TokenSet.prototype.toString=function(){if(this._str)return this._str;for(var e=this.final?"1":"0",r=Object.keys(this.edges).sort(),n=r.length,i=0;i<n;i++){var s=r[i],o=this.edges[s];e=e+s+o.id}return e},t.TokenSet.prototype.intersect=function(e){for(var r=new t.TokenSet,n=void 0,i=[{qNode:e,output:r,node:this}];i.length;){n=i.pop();for(var s=Object.keys(n.qNode.edges),o=s.length,a=Object.keys(n.node.edges),u=a.length,c=0;c<o;c++)for(var l=s[c],g=0;g<u;g++){var f=a[g];if(f==l||l=="*"){var v=n.node.edges[f],m=n.qNode.edges[l],x=v.final&&m.final,d=void 0;f in n.output.edges?(d=n.output.edges[f],d.final=d.final||x):(d=new t.TokenSet,d.final=x,n.output.edges[f]=d),i.push({qNode:m,output:d,node:v})}}}return r},t.TokenSet.Builder=function(){this.previousWord="",this.root=new t.TokenSet,this.uncheckedNodes=[],this.minimizedNodes={}},t.TokenSet.Builder.prototype.insert=function(e){var r,n=0;if(e<this.previousWord)throw new Error("Out of order word insertion");for(var i=0;i<e.length&&i<this.previousWord.length&&e[i]==this.previousWord[i];i++)n++;this.minimize(n),this.uncheckedNodes.length==0?r=this.root:r=this.uncheckedNodes[this.uncheckedNodes.length-1].child;for(var i=n;i<e.length;i++){var s=new t.TokenSet,o=e[i];r.edges[o]=s,this.uncheckedNodes.push({parent:r,char:o,child:s}),r=s}r.final=!0,this.previousWord=e},t.TokenSet.Builder.prototype.finish=function(){this.minimize(0)},t.TokenSet.Builder.prototype.minimize=function(e){for(var r=this.uncheckedNodes.length-1;r>=e;r--){var n=this.uncheckedNodes[r],i=n.child.toString();i in this.minimizedNodes?n.parent.edges[n.char]=this.minimizedNodes[i]:(n.child._str=i,this.minimizedNodes[i]=n.child),this.uncheckedNodes.pop()}};/*!
+ */t.TokenSet=function(){this.final=!1,this.edges={},this.id=t.TokenSet._nextId,t.TokenSet._nextId+=1},t.TokenSet._nextId=1,t.TokenSet.fromArray=function(e){for(var r=new t.TokenSet.Builder,n=0,i=e.length;n<i;n++)r.insert(e[n]);return r.finish(),r.root},t.TokenSet.fromClause=function(e){return"editDistance"in e?t.TokenSet.fromFuzzyString(e.term,e.editDistance):t.TokenSet.fromString(e.term)},t.TokenSet.fromFuzzyString=function(e,r){for(var n=new t.TokenSet,i=[{node:n,editsRemaining:r,str:e}];i.length;){var s=i.pop();if(s.str.length>0){var o=s.str.charAt(0),a;o in s.node.edges?a=s.node.edges[o]:(a=new t.TokenSet,s.node.edges[o]=a),s.str.length==1&&(a.final=!0),i.push({node:a,editsRemaining:s.editsRemaining,str:s.str.slice(1)})}if(s.editsRemaining!=0){if("*"in s.node.edges)var u=s.node.edges["*"];else{var u=new t.TokenSet;s.node.edges["*"]=u}if(s.str.length==0&&(u.final=!0),i.push({node:u,editsRemaining:s.editsRemaining-1,str:s.str}),s.str.length>1&&i.push({node:s.node,editsRemaining:s.editsRemaining-1,str:s.str.slice(1)}),s.str.length==1&&(s.node.final=!0),s.str.length>=1){if("*"in s.node.edges)var c=s.node.edges["*"];else{var c=new t.TokenSet;s.node.edges["*"]=c}s.str.length==1&&(c.final=!0),i.push({node:c,editsRemaining:s.editsRemaining-1,str:s.str.slice(1)})}if(s.str.length>1){var f=s.str.charAt(0),g=s.str.charAt(1),l;g in s.node.edges?l=s.node.edges[g]:(l=new t.TokenSet,s.node.edges[g]=l),s.str.length==1&&(l.final=!0),i.push({node:l,editsRemaining:s.editsRemaining-1,str:f+s.str.slice(2)})}}}return n},t.TokenSet.fromString=function(e){for(var r=new t.TokenSet,n=r,i=0,s=e.length;i<s;i++){var o=e[i],a=i==s-1;if(o=="*")r.edges[o]=r,r.final=a;else{var u=new t.TokenSet;u.final=a,r.edges[o]=u,r=u}}return n},t.TokenSet.prototype.toArray=function(){for(var e=[],r=[{prefix:"",node:this}];r.length;){var n=r.pop(),i=Object.keys(n.node.edges),s=i.length;n.node.final&&(n.prefix.charAt(0),e.push(n.prefix));for(var o=0;o<s;o++){var a=i[o];r.push({prefix:n.prefix.concat(a),node:n.node.edges[a]})}}return e},t.TokenSet.prototype.toString=function(){if(this._str)return this._str;for(var e=this.final?"1":"0",r=Object.keys(this.edges).sort(),n=r.length,i=0;i<n;i++){var s=r[i],o=this.edges[s];e=e+s+o.id}return e},t.TokenSet.prototype.intersect=function(e){for(var r=new t.TokenSet,n=void 0,i=[{qNode:e,output:r,node:this}];i.length;){n=i.pop();for(var s=Object.keys(n.qNode.edges),o=s.length,a=Object.keys(n.node.edges),u=a.length,c=0;c<o;c++)for(var f=s[c],g=0;g<u;g++){var l=a[g];if(l==f||f=="*"){var m=n.node.edges[l],x=n.qNode.edges[f],v=m.final&&x.final,d=void 0;l in n.output.edges?(d=n.output.edges[l],d.final=d.final||v):(d=new t.TokenSet,d.final=v,n.output.edges[l]=d),i.push({qNode:x,output:d,node:m})}}}return r},t.TokenSet.Builder=function(){this.previousWord="",this.root=new t.TokenSet,this.uncheckedNodes=[],this.minimizedNodes={}},t.TokenSet.Builder.prototype.insert=function(e){var r,n=0;if(e<this.previousWord)throw new Error("Out of order word insertion");for(var i=0;i<e.length&&i<this.previousWord.length&&e[i]==this.previousWord[i];i++)n++;this.minimize(n),this.uncheckedNodes.length==0?r=this.root:r=this.uncheckedNodes[this.uncheckedNodes.length-1].child;for(var i=n;i<e.length;i++){var s=new t.TokenSet,o=e[i];r.edges[o]=s,this.uncheckedNodes.push({parent:r,char:o,child:s}),r=s}r.final=!0,this.previousWord=e},t.TokenSet.Builder.prototype.finish=function(){this.minimize(0)},t.TokenSet.Builder.prototype.minimize=function(e){for(var r=this.uncheckedNodes.length-1;r>=e;r--){var n=this.uncheckedNodes[r],i=n.child.toString();i in this.minimizedNodes?n.parent.edges[n.char]=this.minimizedNodes[i]:(n.child._str=i,this.minimizedNodes[i]=n.child),this.uncheckedNodes.pop()}};/*!
  * lunr.Index
  * Copyright (C) 2020 Oliver Nightingale
- */t.Index=function(e){this.invertedIndex=e.invertedIndex,this.fieldVectors=e.fieldVectors,this.tokenSet=e.tokenSet,this.fields=e.fields,this.pipeline=e.pipeline},t.Index.prototype.search=function(e){return this.query(function(r){var n=new t.QueryParser(e,r);n.parse()})},t.Index.prototype.query=function(e){for(var r=new t.Query(this.fields),n=Object.create(null),i=Object.create(null),s=Object.create(null),o=Object.create(null),a=Object.create(null),u=0;u<this.fields.length;u++)i[this.fields[u]]=new t.Vector;e.call(r,r);for(var u=0;u<r.clauses.length;u++){var c=r.clauses[u],l=null,g=t.Set.empty;c.usePipeline?l=this.pipeline.runString(c.term,{fields:c.fields}):l=[c.term];for(var f=0;f<l.length;f++){var v=l[f];c.term=v;var m=t.TokenSet.fromClause(c),x=this.tokenSet.intersect(m).toArray();if(x.length===0&&c.presence===t.Query.presence.REQUIRED){for(var d=0;d<c.fields.length;d++){var y=c.fields[d];o[y]=t.Set.empty}break}for(var b=0;b<x.length;b++)for(var E=x[b],w=this.invertedIndex[E],R=w._index,d=0;d<c.fields.length;d++){var y=c.fields[d],j=w[y],_=Object.keys(j),D=E+"/"+y,N=new t.Set(_);if(c.presence==t.Query.presence.REQUIRED&&(g=g.union(N),o[y]===void 0&&(o[y]=t.Set.complete)),c.presence==t.Query.presence.PROHIBITED){a[y]===void 0&&(a[y]=t.Set.empty),a[y]=a[y].union(N);continue}if(i[y].upsert(R,c.boost,function(ye,me){return ye+me}),!s[D]){for(var C=0;C<_.length;C++){var V=_[C],P=new t.FieldRef(V,y),z=j[V],$;($=n[P])===void 0?n[P]=new t.MatchData(E,y,z):$.add(E,y,z)}s[D]=!0}}}if(c.presence===t.Query.presence.REQUIRED)for(var d=0;d<c.fields.length;d++){var y=c.fields[d];o[y]=o[y].intersect(g)}}for(var M=t.Set.complete,I=t.Set.empty,u=0;u<this.fields.length;u++){var y=this.fields[u];o[y]&&(M=M.intersect(o[y])),a[y]&&(I=I.union(a[y]))}var h=Object.keys(n),S=[],k=Object.create(null);if(r.isNegated()){h=Object.keys(this.fieldVectors);for(var u=0;u<h.length;u++){var P=h[u],L=t.FieldRef.fromString(P);n[P]=new t.MatchData}}for(var u=0;u<h.length;u++){var L=t.FieldRef.fromString(h[u]),p=L.docRef;if(M.contains(p)&&!I.contains(p)){var T=this.fieldVectors[L],O=i[L.fieldName].similarity(T),F;if((F=k[p])!==void 0)F.score+=O,F.matchData.combine(n[L]);else{var Q={ref:p,score:O,matchData:n[L]};k[p]=Q,S.push(Q)}}}return S.sort(function(pe,ge){return ge.score-pe.score})},t.Index.prototype.toJSON=function(){var e=Object.keys(this.invertedIndex).sort().map(function(n){return[n,this.invertedIndex[n]]},this),r=Object.keys(this.fieldVectors).map(function(n){return[n,this.fieldVectors[n].toJSON()]},this);return{version:t.version,fields:this.fields,fieldVectors:r,invertedIndex:e,pipeline:this.pipeline.toJSON()}},t.Index.load=function(e){var r={},n={},i=e.fieldVectors,s=Object.create(null),o=e.invertedIndex,a=new t.TokenSet.Builder,u=t.Pipeline.load(e.pipeline);e.version!=t.version&&t.utils.warn("Version mismatch when loading serialised index. Current version of lunr '"+t.version+"' does not match serialized index '"+e.version+"'");for(var c=0;c<i.length;c++){var l=i[c],g=l[0],f=l[1];n[g]=new t.Vector(f)}for(var c=0;c<o.length;c++){var l=o[c],v=l[0],m=l[1];a.insert(v),s[v]=m}return a.finish(),r.fields=e.fields,r.fieldVectors=n,r.invertedIndex=s,r.tokenSet=a.root,r.pipeline=u,new t.Index(r)};/*!
+ */t.Index=function(e){this.invertedIndex=e.invertedIndex,this.fieldVectors=e.fieldVectors,this.tokenSet=e.tokenSet,this.fields=e.fields,this.pipeline=e.pipeline},t.Index.prototype.search=function(e){return this.query(function(r){var n=new t.QueryParser(e,r);n.parse()})},t.Index.prototype.query=function(e){for(var r=new t.Query(this.fields),n=Object.create(null),i=Object.create(null),s=Object.create(null),o=Object.create(null),a=Object.create(null),u=0;u<this.fields.length;u++)i[this.fields[u]]=new t.Vector;e.call(r,r);for(var u=0;u<r.clauses.length;u++){var c=r.clauses[u],f=null,g=t.Set.empty;c.usePipeline?f=this.pipeline.runString(c.term,{fields:c.fields}):f=[c.term];for(var l=0;l<f.length;l++){var m=f[l];c.term=m;var x=t.TokenSet.fromClause(c),v=this.tokenSet.intersect(x).toArray();if(v.length===0&&c.presence===t.Query.presence.REQUIRED){for(var d=0;d<c.fields.length;d++){var y=c.fields[d];o[y]=t.Set.empty}break}for(var b=0;b<v.length;b++)for(var E=v[b],w=this.invertedIndex[E],R=w._index,d=0;d<c.fields.length;d++){var y=c.fields[d],j=w[y],_=Object.keys(j),D=E+"/"+y,N=new t.Set(_);if(c.presence==t.Query.presence.REQUIRED&&(g=g.union(N),o[y]===void 0&&(o[y]=t.Set.complete)),c.presence==t.Query.presence.PROHIBITED){a[y]===void 0&&(a[y]=t.Set.empty),a[y]=a[y].union(N);continue}if(i[y].upsert(R,c.boost,function(ye,me){return ye+me}),!s[D]){for(var C=0;C<_.length;C++){var V=_[C],P=new t.FieldRef(V,y),z=j[V],$;($=n[P])===void 0?n[P]=new t.MatchData(E,y,z):$.add(E,y,z)}s[D]=!0}}}if(c.presence===t.Query.presence.REQUIRED)for(var d=0;d<c.fields.length;d++){var y=c.fields[d];o[y]=o[y].intersect(g)}}for(var M=t.Set.complete,I=t.Set.empty,u=0;u<this.fields.length;u++){var y=this.fields[u];o[y]&&(M=M.intersect(o[y])),a[y]&&(I=I.union(a[y]))}var h=Object.keys(n),S=[],k=Object.create(null);if(r.isNegated()){h=Object.keys(this.fieldVectors);for(var u=0;u<h.length;u++){var P=h[u],L=t.FieldRef.fromString(P);n[P]=new t.MatchData}}for(var u=0;u<h.length;u++){var L=t.FieldRef.fromString(h[u]),p=L.docRef;if(M.contains(p)&&!I.contains(p)){var T=this.fieldVectors[L],O=i[L.fieldName].similarity(T),F;if((F=k[p])!==void 0)F.score+=O,F.matchData.combine(n[L]);else{var Q={ref:p,score:O,matchData:n[L]};k[p]=Q,S.push(Q)}}}return S.sort(function(pe,ge){return ge.score-pe.score})},t.Index.prototype.toJSON=function(){var e=Object.keys(this.invertedIndex).sort().map(function(n){return[n,this.invertedIndex[n]]},this),r=Object.keys(this.fieldVectors).map(function(n){return[n,this.fieldVectors[n].toJSON()]},this);return{version:t.version,fields:this.fields,fieldVectors:r,invertedIndex:e,pipeline:this.pipeline.toJSON()}},t.Index.load=function(e){var r={},n={},i=e.fieldVectors,s=Object.create(null),o=e.invertedIndex,a=new t.TokenSet.Builder,u=t.Pipeline.load(e.pipeline);e.version!=t.version&&t.utils.warn("Version mismatch when loading serialised index. Current version of lunr '"+t.version+"' does not match serialized index '"+e.version+"'");for(var c=0;c<i.length;c++){var f=i[c],g=f[0],l=f[1];n[g]=new t.Vector(l)}for(var c=0;c<o.length;c++){var f=o[c],m=f[0],x=f[1];a.insert(m),s[m]=x}return a.finish(),r.fields=e.fields,r.fieldVectors=n,r.invertedIndex=s,r.tokenSet=a.root,r.pipeline=u,new t.Index(r)};/*!
  * lunr.Builder
  * Copyright (C) 2020 Oliver Nightingale
- */t.Builder=function(){this._ref="id",this._fields=Object.create(null),this._documents=Object.create(null),this.invertedIndex=Object.create(null),this.fieldTermFrequencies={},this.fieldLengths={},this.tokenizer=t.tokenizer,this.pipeline=new t.Pipeline,this.searchPipeline=new t.Pipeline,this.documentCount=0,this._b=.75,this._k1=1.2,this.termIndex=0,this.metadataWhitelist=[]},t.Builder.prototype.ref=function(e){this._ref=e},t.Builder.prototype.field=function(e,r){if(/\//.test(e))throw new RangeError("Field '"+e+"' contains illegal character '/'");this._fields[e]=r||{}},t.Builder.prototype.b=function(e){e<0?this._b=0:e>1?this._b=1:this._b=e},t.Builder.prototype.k1=function(e){this._k1=e},t.Builder.prototype.add=function(e,r){var n=e[this._ref],i=Object.keys(this._fields);this._documents[n]=r||{},this.documentCount+=1;for(var s=0;s<i.length;s++){var o=i[s],a=this._fields[o].extractor,u=a?a(e):e[o],c=this.tokenizer(u,{fields:[o]}),l=this.pipeline.run(c),g=new t.FieldRef(n,o),f=Object.create(null);this.fieldTermFrequencies[g]=f,this.fieldLengths[g]=0,this.fieldLengths[g]+=l.length;for(var v=0;v<l.length;v++){var m=l[v];if(f[m]==null&&(f[m]=0),f[m]+=1,this.invertedIndex[m]==null){var x=Object.create(null);x._index=this.termIndex,this.termIndex+=1;for(var d=0;d<i.length;d++)x[i[d]]=Object.create(null);this.invertedIndex[m]=x}this.invertedIndex[m][o][n]==null&&(this.invertedIndex[m][o][n]=Object.create(null));for(var y=0;y<this.metadataWhitelist.length;y++){var b=this.metadataWhitelist[y],E=m.metadata[b];this.invertedIndex[m][o][n][b]==null&&(this.invertedIndex[m][o][n][b]=[]),this.invertedIndex[m][o][n][b].push(E)}}}},t.Builder.prototype.calculateAverageFieldLengths=function(){for(var e=Object.keys(this.fieldLengths),r=e.length,n={},i={},s=0;s<r;s++){var o=t.FieldRef.fromString(e[s]),a=o.fieldName;i[a]||(i[a]=0),i[a]+=1,n[a]||(n[a]=0),n[a]+=this.fieldLengths[o]}for(var u=Object.keys(this._fields),s=0;s<u.length;s++){var c=u[s];n[c]=n[c]/i[c]}this.averageFieldLength=n},t.Builder.prototype.createFieldVectors=function(){for(var e={},r=Object.keys(this.fieldTermFrequencies),n=r.length,i=Object.create(null),s=0;s<n;s++){for(var o=t.FieldRef.fromString(r[s]),a=o.fieldName,u=this.fieldLengths[o],c=new t.Vector,l=this.fieldTermFrequencies[o],g=Object.keys(l),f=g.length,v=this._fields[a].boost||1,m=this._documents[o.docRef].boost||1,x=0;x<f;x++){var d=g[x],y=l[d],b=this.invertedIndex[d]._index,E,w,R;i[d]===void 0?(E=t.idf(this.invertedIndex[d],this.documentCount),i[d]=E):E=i[d],w=E*((this._k1+1)*y)/(this._k1*(1-this._b+this._b*(u/this.averageFieldLength[a]))+y),w*=v,w*=m,R=Math.round(w*1e3)/1e3,c.insert(b,R)}e[o]=c}this.fieldVectors=e},t.Builder.prototype.createTokenSet=function(){this.tokenSet=t.TokenSet.fromArray(Object.keys(this.invertedIndex).sort())},t.Builder.prototype.build=function(){return this.calculateAverageFieldLengths(),this.createFieldVectors(),this.createTokenSet(),new t.Index({invertedIndex:this.invertedIndex,fieldVectors:this.fieldVectors,tokenSet:this.tokenSet,fields:Object.keys(this._fields),pipeline:this.searchPipeline})},t.Builder.prototype.use=function(e){var r=Array.prototype.slice.call(arguments,1);r.unshift(this),e.apply(this,r)},t.MatchData=function(e,r,n){for(var i=Object.create(null),s=Object.keys(n||{}),o=0;o<s.length;o++){var a=s[o];i[a]=n[a].slice()}this.metadata=Object.create(null),e!==void 0&&(this.metadata[e]=Object.create(null),this.metadata[e][r]=i)},t.MatchData.prototype.combine=function(e){for(var r=Object.keys(e.metadata),n=0;n<r.length;n++){var i=r[n],s=Object.keys(e.metadata[i]);this.metadata[i]==null&&(this.metadata[i]=Object.create(null));for(var o=0;o<s.length;o++){var a=s[o],u=Object.keys(e.metadata[i][a]);this.metadata[i][a]==null&&(this.metadata[i][a]=Object.create(null));for(var c=0;c<u.length;c++){var l=u[c];this.metadata[i][a][l]==null?this.metadata[i][a][l]=e.metadata[i][a][l]:this.metadata[i][a][l]=this.metadata[i][a][l].concat(e.metadata[i][a][l])}}}},t.MatchData.prototype.add=function(e,r,n){if(!(e in this.metadata)){this.metadata[e]=Object.create(null),this.metadata[e][r]=n;return}if(!(r in this.metadata[e])){this.metadata[e][r]=n;return}for(var i=Object.keys(n),s=0;s<i.length;s++){var o=i[s];o in this.metadata[e][r]?this.metadata[e][r][o]=this.metadata[e][r][o].concat(n[o]):this.metadata[e][r][o]=n[o]}},t.Query=function(e){this.clauses=[],this.allFields=e},t.Query.wildcard=new String("*"),t.Query.wildcard.NONE=0,t.Query.wildcard.LEADING=1,t.Query.wildcard.TRAILING=2,t.Query.presence={OPTIONAL:1,REQUIRED:2,PROHIBITED:3},t.Query.prototype.clause=function(e){return"fields"in e||(e.fields=this.allFields),"boost"in e||(e.boost=1),"usePipeline"in e||(e.usePipeline=!0),"wildcard"in e||(e.wildcard=t.Query.wildcard.NONE),e.wildcard&t.Query.wildcard.LEADING&&e.term.charAt(0)!=t.Query.wildcard&&(e.term="*"+e.term),e.wildcard&t.Query.wildcard.TRAILING&&e.term.slice(-1)!=t.Query.wildcard&&(e.term=""+e.term+"*"),"presence"in e||(e.presence=t.Query.presence.OPTIONAL),this.clauses.push(e),this},t.Query.prototype.isNegated=function(){for(var e=0;e<this.clauses.length;e++)if(this.clauses[e].presence!=t.Query.presence.PROHIBITED)return!1;return!0},t.Query.prototype.term=function(e,r){if(Array.isArray(e))return e.forEach(function(i){this.term(i,t.utils.clone(r))},this),this;var n=r||{};return n.term=e.toString(),this.clause(n),this},t.QueryParseError=function(e,r,n){this.name="QueryParseError",this.message=e,this.start=r,this.end=n},t.QueryParseError.prototype=new Error,t.QueryLexer=function(e){this.lexemes=[],this.str=e,this.length=e.length,this.pos=0,this.start=0,this.escapeCharPositions=[]},t.QueryLexer.prototype.run=function(){for(var e=t.QueryLexer.lexText;e;)e=e(this)},t.QueryLexer.prototype.sliceString=function(){for(var e=[],r=this.start,n=this.pos,i=0;i<this.escapeCharPositions.length;i++)n=this.escapeCharPositions[i],e.push(this.str.slice(r,n)),r=n+1;return e.push(this.str.slice(r,this.pos)),this.escapeCharPositions.length=0,e.join("")},t.QueryLexer.prototype.emit=function(e){this.lexemes.push({type:e,str:this.sliceString(),start:this.start,end:this.pos}),this.start=this.pos},t.QueryLexer.prototype.escapeCharacter=function(){this.escapeCharPositions.push(this.pos-1),this.pos+=1},t.QueryLexer.prototype.next=function(){if(this.pos>=this.length)return t.QueryLexer.EOS;var e=this.str.charAt(this.pos);return this.pos+=1,e},t.QueryLexer.prototype.width=function(){return this.pos-this.start},t.QueryLexer.prototype.ignore=function(){this.start==this.pos&&(this.pos+=1),this.start=this.pos},t.QueryLexer.prototype.backup=function(){this.pos-=1},t.QueryLexer.prototype.acceptDigitRun=function(){var e,r;do e=this.next(),r=e.charCodeAt(0);while(r>47&&r<58);e!=t.QueryLexer.EOS&&this.backup()},t.QueryLexer.prototype.more=function(){return this.pos<this.length},t.QueryLexer.EOS="EOS",t.QueryLexer.FIELD="FIELD",t.QueryLexer.TERM="TERM",t.QueryLexer.EDIT_DISTANCE="EDIT_DISTANCE",t.QueryLexer.BOOST="BOOST",t.QueryLexer.PRESENCE="PRESENCE",t.QueryLexer.lexField=function(e){return e.backup(),e.emit(t.QueryLexer.FIELD),e.ignore(),t.QueryLexer.lexText},t.QueryLexer.lexTerm=function(e){if(e.width()>1&&(e.backup(),e.emit(t.QueryLexer.TERM)),e.ignore(),e.more())return t.QueryLexer.lexText},t.QueryLexer.lexEditDistance=function(e){return e.ignore(),e.acceptDigitRun(),e.emit(t.QueryLexer.EDIT_DISTANCE),t.QueryLexer.lexText},t.QueryLexer.lexBoost=function(e){return e.ignore(),e.acceptDigitRun(),e.emit(t.QueryLexer.BOOST),t.QueryLexer.lexText},t.QueryLexer.lexEOS=function(e){e.width()>0&&e.emit(t.QueryLexer.TERM)},t.QueryLexer.termSeparator=t.tokenizer.separator,t.QueryLexer.lexText=function(e){for(;;){var r=e.next();if(r==t.QueryLexer.EOS)return t.QueryLexer.lexEOS;if(r.charCodeAt(0)==92){e.escapeCharacter();continue}if(r==":")return t.QueryLexer.lexField;if(r=="~")return e.backup(),e.width()>0&&e.emit(t.QueryLexer.TERM),t.QueryLexer.lexEditDistance;if(r=="^")return e.backup(),e.width()>0&&e.emit(t.QueryLexer.TERM),t.QueryLexer.lexBoost;if(r=="+"&&e.width()===1||r=="-"&&e.width()===1)return e.emit(t.QueryLexer.PRESENCE),t.QueryLexer.lexText;if(r.match(t.QueryLexer.termSeparator))return t.QueryLexer.lexTerm}},t.QueryParser=function(e,r){this.lexer=new t.QueryLexer(e),this.query=r,this.currentClause={},this.lexemeIdx=0},t.QueryParser.prototype.parse=function(){this.lexer.run(),this.lexemes=this.lexer.lexemes;for(var e=t.QueryParser.parseClause;e;)e=e(this);return this.query},t.QueryParser.prototype.peekLexeme=function(){return this.lexemes[this.lexemeIdx]},t.QueryParser.prototype.consumeLexeme=function(){var e=this.peekLexeme();return this.lexemeIdx+=1,e},t.QueryParser.prototype.nextClause=function(){var e=this.currentClause;this.query.clause(e),this.currentClause={}},t.QueryParser.parseClause=function(e){var r=e.peekLexeme();if(r!=null)switch(r.type){case t.QueryLexer.PRESENCE:return t.QueryParser.parsePresence;case t.QueryLexer.FIELD:return t.QueryParser.parseField;case t.QueryLexer.TERM:return t.QueryParser.parseTerm;default:var n="expected either a field or a term, found "+r.type;throw r.str.length>=1&&(n+=" with value '"+r.str+"'"),new t.QueryParseError(n,r.start,r.end)}},t.QueryParser.parsePresence=function(e){var r=e.consumeLexeme();if(r!=null){switch(r.str){case"-":e.currentClause.presence=t.Query.presence.PROHIBITED;break;case"+":e.currentClause.presence=t.Query.presence.REQUIRED;break;default:var n="unrecognised presence operator'"+r.str+"'";throw new t.QueryParseError(n,r.start,r.end)}var i=e.peekLexeme();if(i==null){var n="expecting term or field, found nothing";throw new t.QueryParseError(n,r.start,r.end)}switch(i.type){case t.QueryLexer.FIELD:return t.QueryParser.parseField;case t.QueryLexer.TERM:return t.QueryParser.parseTerm;default:var n="expecting term or field, found '"+i.type+"'";throw new t.QueryParseError(n,i.start,i.end)}}},t.QueryParser.parseField=function(e){var r=e.consumeLexeme();if(r!=null){if(e.query.allFields.indexOf(r.str)==-1){var n=e.query.allFields.map(function(o){return"'"+o+"'"}).join(", "),i="unrecognised field '"+r.str+"', possible fields: "+n;throw new t.QueryParseError(i,r.start,r.end)}e.currentClause.fields=[r.str];var s=e.peekLexeme();if(s==null){var i="expecting term, found nothing";throw new t.QueryParseError(i,r.start,r.end)}switch(s.type){case t.QueryLexer.TERM:return t.QueryParser.parseTerm;default:var i="expecting term, found '"+s.type+"'";throw new t.QueryParseError(i,s.start,s.end)}}},t.QueryParser.parseTerm=function(e){var r=e.consumeLexeme();if(r!=null){e.currentClause.term=r.str.toLowerCase(),r.str.indexOf("*")!=-1&&(e.currentClause.usePipeline=!1);var n=e.peekLexeme();if(n==null){e.nextClause();return}switch(n.type){case t.QueryLexer.TERM:return e.nextClause(),t.QueryParser.parseTerm;case t.QueryLexer.FIELD:return e.nextClause(),t.QueryParser.parseField;case t.QueryLexer.EDIT_DISTANCE:return t.QueryParser.parseEditDistance;case t.QueryLexer.BOOST:return t.QueryParser.parseBoost;case t.QueryLexer.PRESENCE:return e.nextClause(),t.QueryParser.parsePresence;default:var i="Unexpected lexeme type '"+n.type+"'";throw new t.QueryParseError(i,n.start,n.end)}}},t.QueryParser.parseEditDistance=function(e){var r=e.consumeLexeme();if(r!=null){var n=parseInt(r.str,10);if(isNaN(n)){var i="edit distance must be numeric";throw new t.QueryParseError(i,r.start,r.end)}e.currentClause.editDistance=n;var s=e.peekLexeme();if(s==null){e.nextClause();return}switch(s.type){case t.QueryLexer.TERM:return e.nextClause(),t.QueryParser.parseTerm;case t.QueryLexer.FIELD:return e.nextClause(),t.QueryParser.parseField;case t.QueryLexer.EDIT_DISTANCE:return t.QueryParser.parseEditDistance;case t.QueryLexer.BOOST:return t.QueryParser.parseBoost;case t.QueryLexer.PRESENCE:return e.nextClause(),t.QueryParser.parsePresence;default:var i="Unexpected lexeme type '"+s.type+"'";throw new t.QueryParseError(i,s.start,s.end)}}},t.QueryParser.parseBoost=function(e){var r=e.consumeLexeme();if(r!=null){var n=parseInt(r.str,10);if(isNaN(n)){var i="boost must be numeric";throw new t.QueryParseError(i,r.start,r.end)}e.currentClause.boost=n;var s=e.peekLexeme();if(s==null){e.nextClause();return}switch(s.type){case 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t.FieldRef(n,o),l=Object.create(null);this.fieldTermFrequencies[g]=l,this.fieldLengths[g]=0,this.fieldLengths[g]+=f.length;for(var m=0;m<f.length;m++){var x=f[m];if(l[x]==null&&(l[x]=0),l[x]+=1,this.invertedIndex[x]==null){var v=Object.create(null);v._index=this.termIndex,this.termIndex+=1;for(var d=0;d<i.length;d++)v[i[d]]=Object.create(null);this.invertedIndex[x]=v}this.invertedIndex[x][o][n]==null&&(this.invertedIndex[x][o][n]=Object.create(null));for(var y=0;y<this.metadataWhitelist.length;y++){var b=this.metadataWhitelist[y],E=x.metadata[b];this.invertedIndex[x][o][n][b]==null&&(this.invertedIndex[x][o][n][b]=[]),this.invertedIndex[x][o][n][b].push(E)}}}},t.Builder.prototype.calculateAverageFieldLengths=function(){for(var e=Object.keys(this.fieldLengths),r=e.length,n={},i={},s=0;s<r;s++){var o=t.FieldRef.fromString(e[s]),a=o.fieldName;i[a]||(i[a]=0),i[a]+=1,n[a]||(n[a]=0),n[a]+=this.fieldLengths[o]}for(var u=Object.keys(this._fields),s=0;s<u.length;s++){var c=u[s];n[c]=n[c]/i[c]}this.averageFieldLength=n},t.Builder.prototype.createFieldVectors=function(){for(var e={},r=Object.keys(this.fieldTermFrequencies),n=r.length,i=Object.create(null),s=0;s<n;s++){for(var o=t.FieldRef.fromString(r[s]),a=o.fieldName,u=this.fieldLengths[o],c=new t.Vector,f=this.fieldTermFrequencies[o],g=Object.keys(f),l=g.length,m=this._fields[a].boost||1,x=this._documents[o.docRef].boost||1,v=0;v<l;v++){var d=g[v],y=f[d],b=this.invertedIndex[d]._index,E,w,R;i[d]===void 0?(E=t.idf(this.invertedIndex[d],this.documentCount),i[d]=E):E=i[d],w=E*((this._k1+1)*y)/(this._k1*(1-this._b+this._b*(u/this.averageFieldLength[a]))+y),w*=m,w*=x,R=Math.round(w*1e3)/1e3,c.insert(b,R)}e[o]=c}this.fieldVectors=e},t.Builder.prototype.createTokenSet=function(){this.tokenSet=t.TokenSet.fromArray(Object.keys(this.invertedIndex).sort())},t.Builder.prototype.build=function(){return this.calculateAverageFieldLengths(),this.createFieldVectors(),this.createTokenSet(),new t.Index({invertedIndex:this.invertedIndex,fieldVectors:this.fieldVectors,tokenSet:this.tokenSet,fields:Object.keys(this._fields),pipeline:this.searchPipeline})},t.Builder.prototype.use=function(e){var r=Array.prototype.slice.call(arguments,1);r.unshift(this),e.apply(this,r)},t.MatchData=function(e,r,n){for(var i=Object.create(null),s=Object.keys(n||{}),o=0;o<s.length;o++){var a=s[o];i[a]=n[a].slice()}this.metadata=Object.create(null),e!==void 0&&(this.metadata[e]=Object.create(null),this.metadata[e][r]=i)},t.MatchData.prototype.combine=function(e){for(var r=Object.keys(e.metadata),n=0;n<r.length;n++){var i=r[n],s=Object.keys(e.metadata[i]);this.metadata[i]==null&&(this.metadata[i]=Object.create(null));for(var o=0;o<s.length;o++){var a=s[o],u=Object.keys(e.metadata[i][a]);this.metadata[i][a]==null&&(this.metadata[i][a]=Object.create(null));for(var c=0;c<u.length;c++){var f=u[c];this.metadata[i][a][f]==null?this.metadata[i][a][f]=e.metadata[i][a][f]:this.metadata[i][a][f]=this.metadata[i][a][f].concat(e.metadata[i][a][f])}}}},t.MatchData.prototype.add=function(e,r,n){if(!(e in this.metadata)){this.metadata[e]=Object.create(null),this.metadata[e][r]=n;return}if(!(r in this.metadata[e])){this.metadata[e][r]=n;return}for(var i=Object.keys(n),s=0;s<i.length;s++){var o=i[s];o in this.metadata[e][r]?this.metadata[e][r][o]=this.metadata[e][r][o].concat(n[o]):this.metadata[e][r][o]=n[o]}},t.Query=function(e){this.clauses=[],this.allFields=e},t.Query.wildcard=new String("*"),t.Query.wildcard.NONE=0,t.Query.wildcard.LEADING=1,t.Query.wildcard.TRAILING=2,t.Query.presence={OPTIONAL:1,REQUIRED:2,PROHIBITED:3},t.Query.prototype.clause=function(e){return"fields"in e||(e.fields=this.allFields),"boost"in e||(e.boost=1),"usePipeline"in e||(e.usePipeline=!0),"wildcard"in e||(e.wildcard=t.Query.wildcard.NONE),e.wildcard&t.Query.wildcard.LEADING&&e.term.charAt(0)!=t.Query.wildcard&&(e.term="*"+e.term),e.wildcard&t.Query.wildcard.TRAILING&&e.term.slice(-1)!=t.Query.wildcard&&(e.term=""+e.term+"*"),"presence"in e||(e.presence=t.Query.presence.OPTIONAL),this.clauses.push(e),this},t.Query.prototype.isNegated=function(){for(var e=0;e<this.clauses.length;e++)if(this.clauses[e].presence!=t.Query.presence.PROHIBITED)return!1;return!0},t.Query.prototype.term=function(e,r){if(Array.isArray(e))return e.forEach(function(i){this.term(i,t.utils.clone(r))},this),this;var n=r||{};return n.term=e.toString(),this.clause(n),this},t.QueryParseError=function(e,r,n){this.name="QueryParseError",this.message=e,this.start=r,this.end=n},t.QueryParseError.prototype=new Error,t.QueryLexer=function(e){this.lexemes=[],this.str=e,this.length=e.length,this.pos=0,this.start=0,this.escapeCharPositions=[]},t.QueryLexer.prototype.run=function(){for(var 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i;if(this.options.suggest){let s=this.index.query(o=>{for(let a of r)o.term(a.term,{fields:["title"],presence:lunr.Query.presence.REQUIRED,wildcard:lunr.Query.wildcard.TRAILING})});i=s.length?Object.keys(s[0].matchData.metadata):[]}return A({items:[...n.values()]},typeof i!="undefined"&&{suggest:i})}};var fe;function Ie(t){return B(this,null,function*(){let e="../lunr";if(typeof parent!="undefined"&&"IFrameWorker"in parent){let n=re("script[src]"),[i]=n.src.split("/worker");e=e.replace("..",i)}let r=[];for(let n of t.lang){switch(n){case"ja":r.push(`${e}/tinyseg.js`);break;case"hi":case"th":r.push(`${e}/wordcut.js`);break}n!=="en"&&r.push(`${e}/min/lunr.${n}.min.js`)}t.lang.length>1&&r.push(`${e}/min/lunr.multi.min.js`),r.length&&(yield importScripts(`${e}/min/lunr.stemmer.support.min.js`,...r))})}function Fe(t){return B(this,null,function*(){switch(t.type){case 0:return yield Ie(t.data.config),fe=new H(t.data),{type:1};case 2:let e=t.data;try{return{type:3,data:fe.search(e)}}catch(r){return console.warn(`Invalid query: ${e} \u2013 see https://bit.ly/2s3ChXG`),console.warn(r),{type:3,data:{items:[]}}}default:throw new TypeError("Invalid message type")}})}self.lunr=de.default;addEventListener("message",t=>B(void 0,null,function*(){postMessage(yield Fe(t.data))}));})();
+//# sourceMappingURL=search.f886a092.min.js.map
 
diff --git a/assets/javascripts/workers/search.a264c092.min.js.map b/assets/javascripts/workers/search.f886a092.min.js.map
similarity index 80%
rename from assets/javascripts/workers/search.a264c092.min.js.map
rename to assets/javascripts/workers/search.f886a092.min.js.map
index dc00979a0..9115984a7 100644
--- a/assets/javascripts/workers/search.a264c092.min.js.map
+++ b/assets/javascripts/workers/search.f886a092.min.js.map
@@ -1,8 +1,7 @@
 {
   "version": 3,
   "sources": ["node_modules/lunr/lunr.js", "src/templates/assets/javascripts/integrations/search/worker/main/index.ts", "src/templates/assets/javascripts/browser/element/_/index.ts", "src/templates/assets/javascripts/polyfills/index.ts", "src/templates/assets/javascripts/integrations/search/config/index.ts", "src/templates/assets/javascripts/integrations/search/internal/_/index.ts", "src/templates/assets/javascripts/integrations/search/internal/extract/index.ts", "src/templates/assets/javascripts/integrations/search/internal/highlight/index.ts", "src/templates/assets/javascripts/integrations/search/internal/tokenize/index.ts", "src/templates/assets/javascripts/integrations/search/query/transform/index.ts", "src/templates/assets/javascripts/integrations/search/query/_/index.ts", "src/templates/assets/javascripts/integrations/search/query/segment/index.ts", "src/templates/assets/javascripts/integrations/search/_/index.ts"],
-  "sourceRoot": "../../../../..",
-  "sourcesContent": ["/**\n * lunr - http://lunrjs.com - A bit like Solr, but much smaller and not as bright - 2.3.9\n * Copyright (C) 2020 Oliver Nightingale\n * @license MIT\n */\n\n;(function(){\n\n/**\n * A convenience function for configuring and constructing\n * a new lunr Index.\n *\n * A lunr.Builder instance is created and the pipeline setup\n * with a trimmer, stop word filter and stemmer.\n *\n * This builder object is yielded to the configuration function\n * that is passed as a parameter, allowing the list of fields\n * and other builder parameters to be customised.\n *\n * All documents _must_ be added within the passed config function.\n *\n * @example\n * var idx = lunr(function () {\n *   this.field('title')\n *   this.field('body')\n *   this.ref('id')\n *\n *   documents.forEach(function (doc) {\n *     this.add(doc)\n *   }, this)\n * })\n *\n * @see {@link lunr.Builder}\n * @see {@link lunr.Pipeline}\n * @see {@link lunr.trimmer}\n * @see {@link lunr.stopWordFilter}\n * @see {@link lunr.stemmer}\n * @namespace {function} lunr\n */\nvar lunr = function (config) {\n  var builder = new lunr.Builder\n\n  builder.pipeline.add(\n    lunr.trimmer,\n    lunr.stopWordFilter,\n    lunr.stemmer\n  )\n\n  builder.searchPipeline.add(\n    lunr.stemmer\n  )\n\n  config.call(builder, builder)\n  return builder.build()\n}\n\nlunr.version = \"2.3.9\"\n/*!\n * lunr.utils\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A namespace containing utils for the rest of the lunr library\n * @namespace lunr.utils\n */\nlunr.utils = {}\n\n/**\n * Print a warning message to the console.\n *\n * @param {String} message The message to be printed.\n * @memberOf lunr.utils\n * @function\n */\nlunr.utils.warn = (function (global) {\n  /* eslint-disable no-console */\n  return function (message) {\n    if (global.console && console.warn) {\n      console.warn(message)\n    }\n  }\n  /* eslint-enable no-console */\n})(this)\n\n/**\n * Convert an object to a string.\n *\n * In the case of `null` and `undefined` the function returns\n * the empty string, in all other cases the result of calling\n * `toString` on the passed object is returned.\n *\n * @param {Any} obj The object to convert to a string.\n * @return {String} string representation of the passed object.\n * @memberOf lunr.utils\n */\nlunr.utils.asString = function (obj) {\n  if (obj === void 0 || obj === null) {\n    return \"\"\n  } else {\n    return obj.toString()\n  }\n}\n\n/**\n * Clones an object.\n *\n * Will create a copy of an existing object such that any mutations\n * on the copy cannot affect the original.\n *\n * Only shallow objects are supported, passing a nested object to this\n * function will cause a TypeError.\n *\n * Objects with primitives, and arrays of primitives are supported.\n *\n * @param {Object} obj The object to clone.\n * @return {Object} a clone of the passed object.\n * @throws {TypeError} when a nested object is passed.\n * @memberOf Utils\n */\nlunr.utils.clone = function (obj) {\n  if (obj === null || obj === undefined) {\n    return obj\n  }\n\n  var clone = Object.create(null),\n      keys = Object.keys(obj)\n\n  for (var i = 0; i < keys.length; i++) {\n    var key = keys[i],\n        val = obj[key]\n\n    if (Array.isArray(val)) {\n      clone[key] = val.slice()\n      continue\n    }\n\n    if (typeof val === 'string' ||\n        typeof val === 'number' ||\n        typeof val === 'boolean') {\n      clone[key] = val\n      continue\n    }\n\n    throw new TypeError(\"clone is not deep and does not support nested objects\")\n  }\n\n  return clone\n}\nlunr.FieldRef = function (docRef, fieldName, stringValue) {\n  this.docRef = docRef\n  this.fieldName = fieldName\n  this._stringValue = stringValue\n}\n\nlunr.FieldRef.joiner = \"/\"\n\nlunr.FieldRef.fromString = function (s) {\n  var n = s.indexOf(lunr.FieldRef.joiner)\n\n  if (n === -1) {\n    throw \"malformed field ref string\"\n  }\n\n  var fieldRef = s.slice(0, n),\n      docRef = s.slice(n + 1)\n\n  return new lunr.FieldRef (docRef, fieldRef, s)\n}\n\nlunr.FieldRef.prototype.toString = function () {\n  if (this._stringValue == undefined) {\n    this._stringValue = this.fieldName + lunr.FieldRef.joiner + this.docRef\n  }\n\n  return this._stringValue\n}\n/*!\n * lunr.Set\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A lunr set.\n *\n * @constructor\n */\nlunr.Set = function (elements) {\n  this.elements = Object.create(null)\n\n  if (elements) {\n    this.length = elements.length\n\n    for (var i = 0; i < this.length; i++) {\n      this.elements[elements[i]] = true\n    }\n  } else {\n    this.length = 0\n  }\n}\n\n/**\n * A complete set that contains all elements.\n *\n * @static\n * @readonly\n * @type {lunr.Set}\n */\nlunr.Set.complete = {\n  intersect: function (other) {\n    return other\n  },\n\n  union: function () {\n    return this\n  },\n\n  contains: function () {\n    return true\n  }\n}\n\n/**\n * An empty set that contains no elements.\n *\n * @static\n * @readonly\n * @type {lunr.Set}\n */\nlunr.Set.empty = {\n  intersect: function () {\n    return this\n  },\n\n  union: function (other) {\n    return other\n  },\n\n  contains: function () {\n    return false\n  }\n}\n\n/**\n * Returns true if this set contains the specified object.\n *\n * @param {object} object - Object whose presence in this set is to be tested.\n * @returns {boolean} - True if this set contains the specified object.\n */\nlunr.Set.prototype.contains = function (object) {\n  return !!this.elements[object]\n}\n\n/**\n * Returns a new set containing only the elements that are present in both\n * this set and the specified set.\n *\n * @param {lunr.Set} other - set to intersect with this set.\n * @returns {lunr.Set} a new set that is the intersection of this and the specified set.\n */\n\nlunr.Set.prototype.intersect = function (other) {\n  var a, b, elements, intersection = []\n\n  if (other === lunr.Set.complete) {\n    return this\n  }\n\n  if (other === lunr.Set.empty) {\n    return other\n  }\n\n  if (this.length < other.length) {\n    a = this\n    b = other\n  } else {\n    a = other\n    b = this\n  }\n\n  elements = Object.keys(a.elements)\n\n  for (var i = 0; i < elements.length; i++) {\n    var element = elements[i]\n    if (element in b.elements) {\n      intersection.push(element)\n    }\n  }\n\n  return new lunr.Set (intersection)\n}\n\n/**\n * Returns a new set combining the elements of this and the specified set.\n *\n * @param {lunr.Set} other - set to union with this set.\n * @return {lunr.Set} a new set that is the union of this and the specified set.\n */\n\nlunr.Set.prototype.union = function (other) {\n  if (other === lunr.Set.complete) {\n    return lunr.Set.complete\n  }\n\n  if (other === lunr.Set.empty) {\n    return this\n  }\n\n  return new lunr.Set(Object.keys(this.elements).concat(Object.keys(other.elements)))\n}\n/**\n * A function to calculate the inverse document frequency for\n * a posting. This is shared between the builder and the index\n *\n * @private\n * @param {object} posting - The posting for a given term\n * @param {number} documentCount - The total number of documents.\n */\nlunr.idf = function (posting, documentCount) {\n  var documentsWithTerm = 0\n\n  for (var fieldName in posting) {\n    if (fieldName == '_index') continue // Ignore the term index, its not a field\n    documentsWithTerm += Object.keys(posting[fieldName]).length\n  }\n\n  var x = (documentCount - documentsWithTerm + 0.5) / (documentsWithTerm + 0.5)\n\n  return Math.log(1 + Math.abs(x))\n}\n\n/**\n * A token wraps a string representation of a token\n * as it is passed through the text processing pipeline.\n *\n * @constructor\n * @param {string} [str=''] - The string token being wrapped.\n * @param {object} [metadata={}] - Metadata associated with this token.\n */\nlunr.Token = function (str, metadata) {\n  this.str = str || \"\"\n  this.metadata = metadata || {}\n}\n\n/**\n * Returns the token string that is being wrapped by this object.\n *\n * @returns {string}\n */\nlunr.Token.prototype.toString = function () {\n  return this.str\n}\n\n/**\n * A token update function is used when updating or optionally\n * when cloning a token.\n *\n * @callback lunr.Token~updateFunction\n * @param {string} str - The string representation of the token.\n * @param {Object} metadata - All metadata associated with this token.\n */\n\n/**\n * Applies the given function to the wrapped string token.\n *\n * @example\n * token.update(function (str, metadata) {\n *   return str.toUpperCase()\n * })\n *\n * @param {lunr.Token~updateFunction} fn - A function to apply to the token string.\n * @returns {lunr.Token}\n */\nlunr.Token.prototype.update = function (fn) {\n  this.str = fn(this.str, this.metadata)\n  return this\n}\n\n/**\n * Creates a clone of this token. Optionally a function can be\n * applied to the cloned token.\n *\n * @param {lunr.Token~updateFunction} [fn] - An optional function to apply to the cloned token.\n * @returns {lunr.Token}\n */\nlunr.Token.prototype.clone = function (fn) {\n  fn = fn || function (s) { return s }\n  return new lunr.Token (fn(this.str, this.metadata), this.metadata)\n}\n/*!\n * lunr.tokenizer\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A function for splitting a string into tokens ready to be inserted into\n * the search index. Uses `lunr.tokenizer.separator` to split strings, change\n * the value of this property to change how strings are split into tokens.\n *\n * This tokenizer will convert its parameter to a string by calling `toString` and\n * then will split this string on the character in `lunr.tokenizer.separator`.\n * Arrays will have their elements converted to strings and wrapped in a lunr.Token.\n *\n * Optional metadata can be passed to the tokenizer, this metadata will be cloned and\n * added as metadata to every token that is created from the object to be tokenized.\n *\n * @static\n * @param {?(string|object|object[])} obj - The object to convert into tokens\n * @param {?object} metadata - Optional metadata to associate with every token\n * @returns {lunr.Token[]}\n * @see {@link lunr.Pipeline}\n */\nlunr.tokenizer = function (obj, metadata) {\n  if (obj == null || obj == undefined) {\n    return []\n  }\n\n  if (Array.isArray(obj)) {\n    return obj.map(function (t) {\n      return new lunr.Token(\n        lunr.utils.asString(t).toLowerCase(),\n        lunr.utils.clone(metadata)\n      )\n    })\n  }\n\n  var str = obj.toString().toLowerCase(),\n      len = str.length,\n      tokens = []\n\n  for (var sliceEnd = 0, sliceStart = 0; sliceEnd <= len; sliceEnd++) {\n    var char = str.charAt(sliceEnd),\n        sliceLength = sliceEnd - sliceStart\n\n    if ((char.match(lunr.tokenizer.separator) || sliceEnd == len)) {\n\n      if (sliceLength > 0) {\n        var tokenMetadata = lunr.utils.clone(metadata) || {}\n        tokenMetadata[\"position\"] = [sliceStart, sliceLength]\n        tokenMetadata[\"index\"] = tokens.length\n\n        tokens.push(\n          new lunr.Token (\n            str.slice(sliceStart, sliceEnd),\n            tokenMetadata\n          )\n        )\n      }\n\n      sliceStart = sliceEnd + 1\n    }\n\n  }\n\n  return tokens\n}\n\n/**\n * The separator used to split a string into tokens. Override this property to change the behaviour of\n * `lunr.tokenizer` behaviour when tokenizing strings. By default this splits on whitespace and hyphens.\n *\n * @static\n * @see lunr.tokenizer\n */\nlunr.tokenizer.separator = /[\\s\\-]+/\n/*!\n * lunr.Pipeline\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.Pipelines maintain an ordered list of functions to be applied to all\n * tokens in documents entering the search index and queries being ran against\n * the index.\n *\n * An instance of lunr.Index created with the lunr shortcut will contain a\n * pipeline with a stop word filter and an English language stemmer. Extra\n * functions can be added before or after either of these functions or these\n * default functions can be removed.\n *\n * When run the pipeline will call each function in turn, passing a token, the\n * index of that token in the original list of all tokens and finally a list of\n * all the original tokens.\n *\n * The output of functions in the pipeline will be passed to the next function\n * in the pipeline. To exclude a token from entering the index the function\n * should return undefined, the rest of the pipeline will not be called with\n * this token.\n *\n * For serialisation of pipelines to work, all functions used in an instance of\n * a pipeline should be registered with lunr.Pipeline. Registered functions can\n * then be loaded. If trying to load a serialised pipeline that uses functions\n * that are not registered an error will be thrown.\n *\n * If not planning on serialising the pipeline then registering pipeline functions\n * is not necessary.\n *\n * @constructor\n */\nlunr.Pipeline = function () {\n  this._stack = []\n}\n\nlunr.Pipeline.registeredFunctions = Object.create(null)\n\n/**\n * A pipeline function maps lunr.Token to lunr.Token. A lunr.Token contains the token\n * string as well as all known metadata. A pipeline function can mutate the token string\n * or mutate (or add) metadata for a given token.\n *\n * A pipeline function can indicate that the passed token should be discarded by returning\n * null, undefined or an empty string. This token will not be passed to any downstream pipeline\n * functions and will not be added to the index.\n *\n * Multiple tokens can be returned by returning an array of tokens. Each token will be passed\n * to any downstream pipeline functions and all will returned tokens will be added to the index.\n *\n * Any number of pipeline functions may be chained together using a lunr.Pipeline.\n *\n * @interface lunr.PipelineFunction\n * @param {lunr.Token} token - A token from the document being processed.\n * @param {number} i - The index of this token in the complete list of tokens for this document/field.\n * @param {lunr.Token[]} tokens - All tokens for this document/field.\n * @returns {(?lunr.Token|lunr.Token[])}\n */\n\n/**\n * Register a function with the pipeline.\n *\n * Functions that are used in the pipeline should be registered if the pipeline\n * needs to be serialised, or a serialised pipeline needs to be loaded.\n *\n * Registering a function does not add it to a pipeline, functions must still be\n * added to instances of the pipeline for them to be used when running a pipeline.\n *\n * @param {lunr.PipelineFunction} fn - The function to check for.\n * @param {String} label - The label to register this function with\n */\nlunr.Pipeline.registerFunction = function (fn, label) {\n  if (label in this.registeredFunctions) {\n    lunr.utils.warn('Overwriting existing registered function: ' + label)\n  }\n\n  fn.label = label\n  lunr.Pipeline.registeredFunctions[fn.label] = fn\n}\n\n/**\n * Warns if the function is not registered as a Pipeline function.\n *\n * @param {lunr.PipelineFunction} fn - The function to check for.\n * @private\n */\nlunr.Pipeline.warnIfFunctionNotRegistered = function (fn) {\n  var isRegistered = fn.label && (fn.label in this.registeredFunctions)\n\n  if (!isRegistered) {\n    lunr.utils.warn('Function is not registered with pipeline. This may cause problems when serialising the index.\\n', fn)\n  }\n}\n\n/**\n * Loads a previously serialised pipeline.\n *\n * All functions to be loaded must already be registered with lunr.Pipeline.\n * If any function from the serialised data has not been registered then an\n * error will be thrown.\n *\n * @param {Object} serialised - The serialised pipeline to load.\n * @returns {lunr.Pipeline}\n */\nlunr.Pipeline.load = function (serialised) {\n  var pipeline = new lunr.Pipeline\n\n  serialised.forEach(function (fnName) {\n    var fn = lunr.Pipeline.registeredFunctions[fnName]\n\n    if (fn) {\n      pipeline.add(fn)\n    } else {\n      throw new Error('Cannot load unregistered function: ' + fnName)\n    }\n  })\n\n  return pipeline\n}\n\n/**\n * Adds new functions to the end of the pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction[]} functions - Any number of functions to add to the pipeline.\n */\nlunr.Pipeline.prototype.add = function () {\n  var fns = Array.prototype.slice.call(arguments)\n\n  fns.forEach(function (fn) {\n    lunr.Pipeline.warnIfFunctionNotRegistered(fn)\n    this._stack.push(fn)\n  }, this)\n}\n\n/**\n * Adds a single function after a function that already exists in the\n * pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline.\n * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline.\n */\nlunr.Pipeline.prototype.after = function (existingFn, newFn) {\n  lunr.Pipeline.warnIfFunctionNotRegistered(newFn)\n\n  var pos = this._stack.indexOf(existingFn)\n  if (pos == -1) {\n    throw new Error('Cannot find existingFn')\n  }\n\n  pos = pos + 1\n  this._stack.splice(pos, 0, newFn)\n}\n\n/**\n * Adds a single function before a function that already exists in the\n * pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline.\n * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline.\n */\nlunr.Pipeline.prototype.before = function (existingFn, newFn) {\n  lunr.Pipeline.warnIfFunctionNotRegistered(newFn)\n\n  var pos = this._stack.indexOf(existingFn)\n  if (pos == -1) {\n    throw new Error('Cannot find existingFn')\n  }\n\n  this._stack.splice(pos, 0, newFn)\n}\n\n/**\n * Removes a function from the pipeline.\n *\n * @param {lunr.PipelineFunction} fn The function to remove from the pipeline.\n */\nlunr.Pipeline.prototype.remove = function (fn) {\n  var pos = this._stack.indexOf(fn)\n  if (pos == -1) {\n    return\n  }\n\n  this._stack.splice(pos, 1)\n}\n\n/**\n * Runs the current list of functions that make up the pipeline against the\n * passed tokens.\n *\n * @param {Array} tokens The tokens to run through the pipeline.\n * @returns {Array}\n */\nlunr.Pipeline.prototype.run = function (tokens) {\n  var stackLength = this._stack.length\n\n  for (var i = 0; i < stackLength; i++) {\n    var fn = this._stack[i]\n    var memo = []\n\n    for (var j = 0; j < tokens.length; j++) {\n      var result = fn(tokens[j], j, tokens)\n\n      if (result === null || result === void 0 || result === '') continue\n\n      if (Array.isArray(result)) {\n        for (var k = 0; k < result.length; k++) {\n          memo.push(result[k])\n        }\n      } else {\n        memo.push(result)\n      }\n    }\n\n    tokens = memo\n  }\n\n  return tokens\n}\n\n/**\n * Convenience method for passing a string through a pipeline and getting\n * strings out. This method takes care of wrapping the passed string in a\n * token and mapping the resulting tokens back to strings.\n *\n * @param {string} str - The string to pass through the pipeline.\n * @param {?object} metadata - Optional metadata to associate with the token\n * passed to the pipeline.\n * @returns {string[]}\n */\nlunr.Pipeline.prototype.runString = function (str, metadata) {\n  var token = new lunr.Token (str, metadata)\n\n  return this.run([token]).map(function (t) {\n    return t.toString()\n  })\n}\n\n/**\n * Resets the pipeline by removing any existing processors.\n *\n */\nlunr.Pipeline.prototype.reset = function () {\n  this._stack = []\n}\n\n/**\n * Returns a representation of the pipeline ready for serialisation.\n *\n * Logs a warning if the function has not been registered.\n *\n * @returns {Array}\n */\nlunr.Pipeline.prototype.toJSON = function () {\n  return this._stack.map(function (fn) {\n    lunr.Pipeline.warnIfFunctionNotRegistered(fn)\n\n    return fn.label\n  })\n}\n/*!\n * lunr.Vector\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A vector is used to construct the vector space of documents and queries. These\n * vectors support operations to determine the similarity between two documents or\n * a document and a query.\n *\n * Normally no parameters are required for initializing a vector, but in the case of\n * loading a previously dumped vector the raw elements can be provided to the constructor.\n *\n * For performance reasons vectors are implemented with a flat array, where an elements\n * index is immediately followed by its value. E.g. [index, value, index, value]. This\n * allows the underlying array to be as sparse as possible and still offer decent\n * performance when being used for vector calculations.\n *\n * @constructor\n * @param {Number[]} [elements] - The flat list of element index and element value pairs.\n */\nlunr.Vector = function (elements) {\n  this._magnitude = 0\n  this.elements = elements || []\n}\n\n\n/**\n * Calculates the position within the vector to insert a given index.\n *\n * This is used internally by insert and upsert. If there are duplicate indexes then\n * the position is returned as if the value for that index were to be updated, but it\n * is the callers responsibility to check whether there is a duplicate at that index\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @returns {Number}\n */\nlunr.Vector.prototype.positionForIndex = function (index) {\n  // For an empty vector the tuple can be inserted at the beginning\n  if (this.elements.length == 0) {\n    return 0\n  }\n\n  var start = 0,\n      end = this.elements.length / 2,\n      sliceLength = end - start,\n      pivotPoint = Math.floor(sliceLength / 2),\n      pivotIndex = this.elements[pivotPoint * 2]\n\n  while (sliceLength > 1) {\n    if (pivotIndex < index) {\n      start = pivotPoint\n    }\n\n    if (pivotIndex > index) {\n      end = pivotPoint\n    }\n\n    if (pivotIndex == index) {\n      break\n    }\n\n    sliceLength = end - start\n    pivotPoint = start + Math.floor(sliceLength / 2)\n    pivotIndex = this.elements[pivotPoint * 2]\n  }\n\n  if (pivotIndex == index) {\n    return pivotPoint * 2\n  }\n\n  if (pivotIndex > index) {\n    return pivotPoint * 2\n  }\n\n  if (pivotIndex < index) {\n    return (pivotPoint + 1) * 2\n  }\n}\n\n/**\n * Inserts an element at an index within the vector.\n *\n * Does not allow duplicates, will throw an error if there is already an entry\n * for this index.\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @param {Number} val - The value to be inserted into the vector.\n */\nlunr.Vector.prototype.insert = function (insertIdx, val) {\n  this.upsert(insertIdx, val, function () {\n    throw \"duplicate index\"\n  })\n}\n\n/**\n * Inserts or updates an existing index within the vector.\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @param {Number} val - The value to be inserted into the vector.\n * @param {function} fn - A function that is called for updates, the existing value and the\n * requested value are passed as arguments\n */\nlunr.Vector.prototype.upsert = function (insertIdx, val, fn) {\n  this._magnitude = 0\n  var position = this.positionForIndex(insertIdx)\n\n  if (this.elements[position] == insertIdx) {\n    this.elements[position + 1] = fn(this.elements[position + 1], val)\n  } else {\n    this.elements.splice(position, 0, insertIdx, val)\n  }\n}\n\n/**\n * Calculates the magnitude of this vector.\n *\n * @returns {Number}\n */\nlunr.Vector.prototype.magnitude = function () {\n  if (this._magnitude) return this._magnitude\n\n  var sumOfSquares = 0,\n      elementsLength = this.elements.length\n\n  for (var i = 1; i < elementsLength; i += 2) {\n    var val = this.elements[i]\n    sumOfSquares += val * val\n  }\n\n  return this._magnitude = Math.sqrt(sumOfSquares)\n}\n\n/**\n * Calculates the dot product of this vector and another vector.\n *\n * @param {lunr.Vector} otherVector - The vector to compute the dot product with.\n * @returns {Number}\n */\nlunr.Vector.prototype.dot = function (otherVector) {\n  var dotProduct = 0,\n      a = this.elements, b = otherVector.elements,\n      aLen = a.length, bLen = b.length,\n      aVal = 0, bVal = 0,\n      i = 0, j = 0\n\n  while (i < aLen && j < bLen) {\n    aVal = a[i], bVal = b[j]\n    if (aVal < bVal) {\n      i += 2\n    } else if (aVal > bVal) {\n      j += 2\n    } else if (aVal == bVal) {\n      dotProduct += a[i + 1] * b[j + 1]\n      i += 2\n      j += 2\n    }\n  }\n\n  return dotProduct\n}\n\n/**\n * Calculates the similarity between this vector and another vector.\n *\n * @param {lunr.Vector} otherVector - The other vector to calculate the\n * similarity with.\n * @returns {Number}\n */\nlunr.Vector.prototype.similarity = function (otherVector) {\n  return this.dot(otherVector) / this.magnitude() || 0\n}\n\n/**\n * Converts the vector to an array of the elements within the vector.\n *\n * @returns {Number[]}\n */\nlunr.Vector.prototype.toArray = function () {\n  var output = new Array (this.elements.length / 2)\n\n  for (var i = 1, j = 0; i < this.elements.length; i += 2, j++) {\n    output[j] = this.elements[i]\n  }\n\n  return output\n}\n\n/**\n * A JSON serializable representation of the vector.\n *\n * @returns {Number[]}\n */\nlunr.Vector.prototype.toJSON = function () {\n  return this.elements\n}\n/* eslint-disable */\n/*!\n * lunr.stemmer\n * Copyright (C) 2020 Oliver Nightingale\n * Includes code from - http://tartarus.org/~martin/PorterStemmer/js.txt\n */\n\n/**\n * lunr.stemmer is an english language stemmer, this is a JavaScript\n * implementation of the PorterStemmer taken from http://tartarus.org/~martin\n *\n * @static\n * @implements {lunr.PipelineFunction}\n * @param {lunr.Token} token - The string to stem\n * @returns {lunr.Token}\n * @see {@link lunr.Pipeline}\n * @function\n */\nlunr.stemmer = (function(){\n  var step2list = {\n      \"ational\" : \"ate\",\n      \"tional\" : \"tion\",\n      \"enci\" : \"ence\",\n      \"anci\" : \"ance\",\n      \"izer\" : \"ize\",\n      \"bli\" : \"ble\",\n      \"alli\" : \"al\",\n      \"entli\" : \"ent\",\n      \"eli\" : \"e\",\n      \"ousli\" : \"ous\",\n      \"ization\" : \"ize\",\n      \"ation\" : \"ate\",\n      \"ator\" : \"ate\",\n      \"alism\" : \"al\",\n      \"iveness\" : \"ive\",\n      \"fulness\" : \"ful\",\n      \"ousness\" : \"ous\",\n      \"aliti\" : \"al\",\n      \"iviti\" : \"ive\",\n      \"biliti\" : \"ble\",\n      \"logi\" : \"log\"\n    },\n\n    step3list = {\n      \"icate\" : \"ic\",\n      \"ative\" : \"\",\n      \"alize\" : \"al\",\n      \"iciti\" : \"ic\",\n      \"ical\" : \"ic\",\n      \"ful\" : \"\",\n      \"ness\" : \"\"\n    },\n\n    c = \"[^aeiou]\",          // consonant\n    v = \"[aeiouy]\",          // vowel\n    C = c + \"[^aeiouy]*\",    // consonant sequence\n    V = v + \"[aeiou]*\",      // vowel sequence\n\n    mgr0 = \"^(\" + C + \")?\" + V + C,               // [C]VC... is m>0\n    meq1 = \"^(\" + C + \")?\" + V + C + \"(\" + V + \")?$\",  // [C]VC[V] is m=1\n    mgr1 = \"^(\" + C + \")?\" + V + C + V + C,       // [C]VCVC... is m>1\n    s_v = \"^(\" + C + \")?\" + v;                   // vowel in stem\n\n  var re_mgr0 = new RegExp(mgr0);\n  var re_mgr1 = new RegExp(mgr1);\n  var re_meq1 = new RegExp(meq1);\n  var re_s_v = new RegExp(s_v);\n\n  var re_1a = /^(.+?)(ss|i)es$/;\n  var re2_1a = /^(.+?)([^s])s$/;\n  var re_1b = /^(.+?)eed$/;\n  var re2_1b = /^(.+?)(ed|ing)$/;\n  var re_1b_2 = /.$/;\n  var re2_1b_2 = /(at|bl|iz)$/;\n  var re3_1b_2 = new RegExp(\"([^aeiouylsz])\\\\1$\");\n  var re4_1b_2 = new RegExp(\"^\" + C + v + \"[^aeiouwxy]$\");\n\n  var re_1c = /^(.+?[^aeiou])y$/;\n  var re_2 = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;\n\n  var re_3 = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;\n\n  var re_4 = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;\n  var re2_4 = /^(.+?)(s|t)(ion)$/;\n\n  var re_5 = /^(.+?)e$/;\n  var re_5_1 = /ll$/;\n  var re3_5 = new RegExp(\"^\" + C + v + \"[^aeiouwxy]$\");\n\n  var porterStemmer = function porterStemmer(w) {\n    var stem,\n      suffix,\n      firstch,\n      re,\n      re2,\n      re3,\n      re4;\n\n    if (w.length < 3) { return w; }\n\n    firstch = w.substr(0,1);\n    if (firstch == \"y\") {\n      w = firstch.toUpperCase() + w.substr(1);\n    }\n\n    // Step 1a\n    re = re_1a\n    re2 = re2_1a;\n\n    if (re.test(w)) { w = w.replace(re,\"$1$2\"); }\n    else if (re2.test(w)) { w = w.replace(re2,\"$1$2\"); }\n\n    // Step 1b\n    re = re_1b;\n    re2 = re2_1b;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      re = re_mgr0;\n      if (re.test(fp[1])) {\n        re = re_1b_2;\n        w = w.replace(re,\"\");\n      }\n    } else if (re2.test(w)) {\n      var fp = re2.exec(w);\n      stem = fp[1];\n      re2 = re_s_v;\n      if (re2.test(stem)) {\n        w = stem;\n        re2 = re2_1b_2;\n        re3 = re3_1b_2;\n        re4 = re4_1b_2;\n        if (re2.test(w)) { w = w + \"e\"; }\n        else if (re3.test(w)) { re = re_1b_2; w = w.replace(re,\"\"); }\n        else if (re4.test(w)) { w = w + \"e\"; }\n      }\n    }\n\n    // Step 1c - replace suffix y or Y by i if preceded by a non-vowel which is not the first letter of the word (so cry -> cri, by -> by, say -> say)\n    re = re_1c;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      w = stem + \"i\";\n    }\n\n    // Step 2\n    re = re_2;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      suffix = fp[2];\n      re = re_mgr0;\n      if (re.test(stem)) {\n        w = stem + step2list[suffix];\n      }\n    }\n\n    // Step 3\n    re = re_3;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      suffix = fp[2];\n      re = re_mgr0;\n      if (re.test(stem)) {\n        w = stem + step3list[suffix];\n      }\n    }\n\n    // Step 4\n    re = re_4;\n    re2 = re2_4;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      re = re_mgr1;\n      if (re.test(stem)) {\n        w = stem;\n      }\n    } else if (re2.test(w)) {\n      var fp = re2.exec(w);\n      stem = fp[1] + fp[2];\n      re2 = re_mgr1;\n      if (re2.test(stem)) {\n        w = stem;\n      }\n    }\n\n    // Step 5\n    re = re_5;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      re = re_mgr1;\n      re2 = re_meq1;\n      re3 = re3_5;\n      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem)))) {\n        w = stem;\n      }\n    }\n\n    re = re_5_1;\n    re2 = re_mgr1;\n    if (re.test(w) && re2.test(w)) {\n      re = re_1b_2;\n      w = w.replace(re,\"\");\n    }\n\n    // and turn initial Y back to y\n\n    if (firstch == \"y\") {\n      w = firstch.toLowerCase() + w.substr(1);\n    }\n\n    return w;\n  };\n\n  return function (token) {\n    return token.update(porterStemmer);\n  }\n})();\n\nlunr.Pipeline.registerFunction(lunr.stemmer, 'stemmer')\n/*!\n * lunr.stopWordFilter\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.generateStopWordFilter builds a stopWordFilter function from the provided\n * list of stop words.\n *\n * The built in lunr.stopWordFilter is built using this generator and can be used\n * to generate custom stopWordFilters for applications or non English languages.\n *\n * @function\n * @param {Array} token The token to pass through the filter\n * @returns {lunr.PipelineFunction}\n * @see lunr.Pipeline\n * @see lunr.stopWordFilter\n */\nlunr.generateStopWordFilter = function (stopWords) {\n  var words = stopWords.reduce(function (memo, stopWord) {\n    memo[stopWord] = stopWord\n    return memo\n  }, {})\n\n  return function (token) {\n    if (token && words[token.toString()] !== token.toString()) return token\n  }\n}\n\n/**\n * lunr.stopWordFilter is an English language stop word list filter, any words\n * contained in the list will not be passed through the filter.\n *\n * This is intended to be used in the Pipeline. If the token does not pass the\n * filter then undefined will be returned.\n *\n * @function\n * @implements {lunr.PipelineFunction}\n * @params {lunr.Token} token - A token to check for being a stop word.\n * @returns {lunr.Token}\n * @see {@link lunr.Pipeline}\n */\nlunr.stopWordFilter = lunr.generateStopWordFilter([\n  'a',\n  'able',\n  'about',\n  'across',\n  'after',\n  'all',\n  'almost',\n  'also',\n  'am',\n  'among',\n  'an',\n  'and',\n  'any',\n  'are',\n  'as',\n  'at',\n  'be',\n  'because',\n  'been',\n  'but',\n  'by',\n  'can',\n  'cannot',\n  'could',\n  'dear',\n  'did',\n  'do',\n  'does',\n  'either',\n  'else',\n  'ever',\n  'every',\n  'for',\n  'from',\n  'get',\n  'got',\n  'had',\n  'has',\n  'have',\n  'he',\n  'her',\n  'hers',\n  'him',\n  'his',\n  'how',\n  'however',\n  'i',\n  'if',\n  'in',\n  'into',\n  'is',\n  'it',\n  'its',\n  'just',\n  'least',\n  'let',\n  'like',\n  'likely',\n  'may',\n  'me',\n  'might',\n  'most',\n  'must',\n  'my',\n  'neither',\n  'no',\n  'nor',\n  'not',\n  'of',\n  'off',\n  'often',\n  'on',\n  'only',\n  'or',\n  'other',\n  'our',\n  'own',\n  'rather',\n  'said',\n  'say',\n  'says',\n  'she',\n  'should',\n  'since',\n  'so',\n  'some',\n  'than',\n  'that',\n  'the',\n  'their',\n  'them',\n  'then',\n  'there',\n  'these',\n  'they',\n  'this',\n  'tis',\n  'to',\n  'too',\n  'twas',\n  'us',\n  'wants',\n  'was',\n  'we',\n  'were',\n  'what',\n  'when',\n  'where',\n  'which',\n  'while',\n  'who',\n  'whom',\n  'why',\n  'will',\n  'with',\n  'would',\n  'yet',\n  'you',\n  'your'\n])\n\nlunr.Pipeline.registerFunction(lunr.stopWordFilter, 'stopWordFilter')\n/*!\n * lunr.trimmer\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.trimmer is a pipeline function for trimming non word\n * characters from the beginning and end of tokens before they\n * enter the index.\n *\n * This implementation may not work correctly for non latin\n * characters and should either be removed or adapted for use\n * with languages with non-latin characters.\n *\n * @static\n * @implements {lunr.PipelineFunction}\n * @param {lunr.Token} token The token to pass through the filter\n * @returns {lunr.Token}\n * @see lunr.Pipeline\n */\nlunr.trimmer = function (token) {\n  return token.update(function (s) {\n    return s.replace(/^\\W+/, '').replace(/\\W+$/, '')\n  })\n}\n\nlunr.Pipeline.registerFunction(lunr.trimmer, 'trimmer')\n/*!\n * lunr.TokenSet\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A token set is used to store the unique list of all tokens\n * within an index. Token sets are also used to represent an\n * incoming query to the index, this query token set and index\n * token set are then intersected to find which tokens to look\n * up in the inverted index.\n *\n * A token set can hold multiple tokens, as in the case of the\n * index token set, or it can hold a single token as in the\n * case of a simple query token set.\n *\n * Additionally token sets are used to perform wildcard matching.\n * Leading, contained and trailing wildcards are supported, and\n * from this edit distance matching can also be provided.\n *\n * Token sets are implemented as a minimal finite state automata,\n * where both common prefixes and suffixes are shared between tokens.\n * This helps to reduce the space used for storing the token set.\n *\n * @constructor\n */\nlunr.TokenSet = function () {\n  this.final = false\n  this.edges = {}\n  this.id = lunr.TokenSet._nextId\n  lunr.TokenSet._nextId += 1\n}\n\n/**\n * Keeps track of the next, auto increment, identifier to assign\n * to a new tokenSet.\n *\n * TokenSets require a unique identifier to be correctly minimised.\n *\n * @private\n */\nlunr.TokenSet._nextId = 1\n\n/**\n * Creates a TokenSet instance from the given sorted array of words.\n *\n * @param {String[]} arr - A sorted array of strings to create the set from.\n * @returns {lunr.TokenSet}\n * @throws Will throw an error if the input array is not sorted.\n */\nlunr.TokenSet.fromArray = function (arr) {\n  var builder = new lunr.TokenSet.Builder\n\n  for (var i = 0, len = arr.length; i < len; i++) {\n    builder.insert(arr[i])\n  }\n\n  builder.finish()\n  return builder.root\n}\n\n/**\n * Creates a token set from a query clause.\n *\n * @private\n * @param {Object} clause - A single clause from lunr.Query.\n * @param {string} clause.term - The query clause term.\n * @param {number} [clause.editDistance] - The optional edit distance for the term.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.fromClause = function (clause) {\n  if ('editDistance' in clause) {\n    return lunr.TokenSet.fromFuzzyString(clause.term, clause.editDistance)\n  } else {\n    return lunr.TokenSet.fromString(clause.term)\n  }\n}\n\n/**\n * Creates a token set representing a single string with a specified\n * edit distance.\n *\n * Insertions, deletions, substitutions and transpositions are each\n * treated as an edit distance of 1.\n *\n * Increasing the allowed edit distance will have a dramatic impact\n * on the performance of both creating and intersecting these TokenSets.\n * It is advised to keep the edit distance less than 3.\n *\n * @param {string} str - The string to create the token set from.\n * @param {number} editDistance - The allowed edit distance to match.\n * @returns {lunr.Vector}\n */\nlunr.TokenSet.fromFuzzyString = function (str, editDistance) {\n  var root = new lunr.TokenSet\n\n  var stack = [{\n    node: root,\n    editsRemaining: editDistance,\n    str: str\n  }]\n\n  while (stack.length) {\n    var frame = stack.pop()\n\n    // no edit\n    if (frame.str.length > 0) {\n      var char = frame.str.charAt(0),\n          noEditNode\n\n      if (char in frame.node.edges) {\n        noEditNode = frame.node.edges[char]\n      } else {\n        noEditNode = new lunr.TokenSet\n        frame.node.edges[char] = noEditNode\n      }\n\n      if (frame.str.length == 1) {\n        noEditNode.final = true\n      }\n\n      stack.push({\n        node: noEditNode,\n        editsRemaining: frame.editsRemaining,\n        str: frame.str.slice(1)\n      })\n    }\n\n    if (frame.editsRemaining == 0) {\n      continue\n    }\n\n    // insertion\n    if (\"*\" in frame.node.edges) {\n      var insertionNode = frame.node.edges[\"*\"]\n    } else {\n      var insertionNode = new lunr.TokenSet\n      frame.node.edges[\"*\"] = insertionNode\n    }\n\n    if (frame.str.length == 0) {\n      insertionNode.final = true\n    }\n\n    stack.push({\n      node: insertionNode,\n      editsRemaining: frame.editsRemaining - 1,\n      str: frame.str\n    })\n\n    // deletion\n    // can only do a deletion if we have enough edits remaining\n    // and if there are characters left to delete in the string\n    if (frame.str.length > 1) {\n      stack.push({\n        node: frame.node,\n        editsRemaining: frame.editsRemaining - 1,\n        str: frame.str.slice(1)\n      })\n    }\n\n    // deletion\n    // just removing the last character from the str\n    if (frame.str.length == 1) {\n      frame.node.final = true\n    }\n\n    // substitution\n    // can only do a substitution if we have enough edits remaining\n    // and if there are characters left to substitute\n    if (frame.str.length >= 1) {\n      if (\"*\" in frame.node.edges) {\n        var substitutionNode = frame.node.edges[\"*\"]\n      } else {\n        var substitutionNode = new lunr.TokenSet\n        frame.node.edges[\"*\"] = substitutionNode\n      }\n\n      if (frame.str.length == 1) {\n        substitutionNode.final = true\n      }\n\n      stack.push({\n        node: substitutionNode,\n        editsRemaining: frame.editsRemaining - 1,\n        str: frame.str.slice(1)\n      })\n    }\n\n    // transposition\n    // can only do a transposition if there are edits remaining\n    // and there are enough characters to transpose\n    if (frame.str.length > 1) {\n      var charA = frame.str.charAt(0),\n          charB = frame.str.charAt(1),\n          transposeNode\n\n      if (charB in frame.node.edges) {\n        transposeNode = frame.node.edges[charB]\n      } else {\n        transposeNode = new lunr.TokenSet\n        frame.node.edges[charB] = transposeNode\n      }\n\n      if (frame.str.length == 1) {\n        transposeNode.final = true\n      }\n\n      stack.push({\n        node: transposeNode,\n        editsRemaining: frame.editsRemaining - 1,\n        str: charA + frame.str.slice(2)\n      })\n    }\n  }\n\n  return root\n}\n\n/**\n * Creates a TokenSet from a string.\n *\n * The string may contain one or more wildcard characters (*)\n * that will allow wildcard matching when intersecting with\n * another TokenSet.\n *\n * @param {string} str - The string to create a TokenSet from.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.fromString = function (str) {\n  var node = new lunr.TokenSet,\n      root = node\n\n  /*\n   * Iterates through all characters within the passed string\n   * appending a node for each character.\n   *\n   * When a wildcard character is found then a self\n   * referencing edge is introduced to continually match\n   * any number of any characters.\n   */\n  for (var i = 0, len = str.length; i < len; i++) {\n    var char = str[i],\n        final = (i == len - 1)\n\n    if (char == \"*\") {\n      node.edges[char] = node\n      node.final = final\n\n    } else {\n      var next = new lunr.TokenSet\n      next.final = final\n\n      node.edges[char] = next\n      node = next\n    }\n  }\n\n  return root\n}\n\n/**\n * Converts this TokenSet into an array of strings\n * contained within the TokenSet.\n *\n * This is not intended to be used on a TokenSet that\n * contains wildcards, in these cases the results are\n * undefined and are likely to cause an infinite loop.\n *\n * @returns {string[]}\n */\nlunr.TokenSet.prototype.toArray = function () {\n  var words = []\n\n  var stack = [{\n    prefix: \"\",\n    node: this\n  }]\n\n  while (stack.length) {\n    var frame = stack.pop(),\n        edges = Object.keys(frame.node.edges),\n        len = edges.length\n\n    if (frame.node.final) {\n      /* In Safari, at this point the prefix is sometimes corrupted, see:\n       * https://github.com/olivernn/lunr.js/issues/279 Calling any\n       * String.prototype method forces Safari to \"cast\" this string to what\n       * it's supposed to be, fixing the bug. */\n      frame.prefix.charAt(0)\n      words.push(frame.prefix)\n    }\n\n    for (var i = 0; i < len; i++) {\n      var edge = edges[i]\n\n      stack.push({\n        prefix: frame.prefix.concat(edge),\n        node: frame.node.edges[edge]\n      })\n    }\n  }\n\n  return words\n}\n\n/**\n * Generates a string representation of a TokenSet.\n *\n * This is intended to allow TokenSets to be used as keys\n * in objects, largely to aid the construction and minimisation\n * of a TokenSet. As such it is not designed to be a human\n * friendly representation of the TokenSet.\n *\n * @returns {string}\n */\nlunr.TokenSet.prototype.toString = function () {\n  // NOTE: Using Object.keys here as this.edges is very likely\n  // to enter 'hash-mode' with many keys being added\n  //\n  // avoiding a for-in loop here as it leads to the function\n  // being de-optimised (at least in V8). From some simple\n  // benchmarks the performance is comparable, but allowing\n  // V8 to optimize may mean easy performance wins in the future.\n\n  if (this._str) {\n    return this._str\n  }\n\n  var str = this.final ? '1' : '0',\n      labels = Object.keys(this.edges).sort(),\n      len = labels.length\n\n  for (var i = 0; i < len; i++) {\n    var label = labels[i],\n        node = this.edges[label]\n\n    str = str + label + node.id\n  }\n\n  return str\n}\n\n/**\n * Returns a new TokenSet that is the intersection of\n * this TokenSet and the passed TokenSet.\n *\n * This intersection will take into account any wildcards\n * contained within the TokenSet.\n *\n * @param {lunr.TokenSet} b - An other TokenSet to intersect with.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.prototype.intersect = function (b) {\n  var output = new lunr.TokenSet,\n      frame = undefined\n\n  var stack = [{\n    qNode: b,\n    output: output,\n    node: this\n  }]\n\n  while (stack.length) {\n    frame = stack.pop()\n\n    // NOTE: As with the #toString method, we are using\n    // Object.keys and a for loop instead of a for-in loop\n    // as both of these objects enter 'hash' mode, causing\n    // the function to be de-optimised in V8\n    var qEdges = Object.keys(frame.qNode.edges),\n        qLen = qEdges.length,\n        nEdges = Object.keys(frame.node.edges),\n        nLen = nEdges.length\n\n    for (var q = 0; q < qLen; q++) {\n      var qEdge = qEdges[q]\n\n      for (var n = 0; n < nLen; n++) {\n        var nEdge = nEdges[n]\n\n        if (nEdge == qEdge || qEdge == '*') {\n          var node = frame.node.edges[nEdge],\n              qNode = frame.qNode.edges[qEdge],\n              final = node.final && qNode.final,\n              next = undefined\n\n          if (nEdge in frame.output.edges) {\n            // an edge already exists for this character\n            // no need to create a new node, just set the finality\n            // bit unless this node is already final\n            next = frame.output.edges[nEdge]\n            next.final = next.final || final\n\n          } else {\n            // no edge exists yet, must create one\n            // set the finality bit and insert it\n            // into the output\n            next = new lunr.TokenSet\n            next.final = final\n            frame.output.edges[nEdge] = next\n          }\n\n          stack.push({\n            qNode: qNode,\n            output: next,\n            node: node\n          })\n        }\n      }\n    }\n  }\n\n  return output\n}\nlunr.TokenSet.Builder = function () {\n  this.previousWord = \"\"\n  this.root = new lunr.TokenSet\n  this.uncheckedNodes = []\n  this.minimizedNodes = {}\n}\n\nlunr.TokenSet.Builder.prototype.insert = function (word) {\n  var node,\n      commonPrefix = 0\n\n  if (word < this.previousWord) {\n    throw new Error (\"Out of order word insertion\")\n  }\n\n  for (var i = 0; i < word.length && i < this.previousWord.length; i++) {\n    if (word[i] != this.previousWord[i]) break\n    commonPrefix++\n  }\n\n  this.minimize(commonPrefix)\n\n  if (this.uncheckedNodes.length == 0) {\n    node = this.root\n  } else {\n    node = this.uncheckedNodes[this.uncheckedNodes.length - 1].child\n  }\n\n  for (var i = commonPrefix; i < word.length; i++) {\n    var nextNode = new lunr.TokenSet,\n        char = word[i]\n\n    node.edges[char] = nextNode\n\n    this.uncheckedNodes.push({\n      parent: node,\n      char: char,\n      child: nextNode\n    })\n\n    node = nextNode\n  }\n\n  node.final = true\n  this.previousWord = word\n}\n\nlunr.TokenSet.Builder.prototype.finish = function () {\n  this.minimize(0)\n}\n\nlunr.TokenSet.Builder.prototype.minimize = function (downTo) {\n  for (var i = this.uncheckedNodes.length - 1; i >= downTo; i--) {\n    var node = this.uncheckedNodes[i],\n        childKey = node.child.toString()\n\n    if (childKey in this.minimizedNodes) {\n      node.parent.edges[node.char] = this.minimizedNodes[childKey]\n    } else {\n      // Cache the key for this node since\n      // we know it can't change anymore\n      node.child._str = childKey\n\n      this.minimizedNodes[childKey] = node.child\n    }\n\n    this.uncheckedNodes.pop()\n  }\n}\n/*!\n * lunr.Index\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * An index contains the built index of all documents and provides a query interface\n * to the index.\n *\n * Usually instances of lunr.Index will not be created using this constructor, instead\n * lunr.Builder should be used to construct new indexes, or lunr.Index.load should be\n * used to load previously built and serialized indexes.\n *\n * @constructor\n * @param {Object} attrs - The attributes of the built search index.\n * @param {Object} attrs.invertedIndex - An index of term/field to document reference.\n * @param {Object<string, lunr.Vector>} attrs.fieldVectors - Field vectors\n * @param {lunr.TokenSet} attrs.tokenSet - An set of all corpus tokens.\n * @param {string[]} attrs.fields - The names of indexed document fields.\n * @param {lunr.Pipeline} attrs.pipeline - The pipeline to use for search terms.\n */\nlunr.Index = function (attrs) {\n  this.invertedIndex = attrs.invertedIndex\n  this.fieldVectors = attrs.fieldVectors\n  this.tokenSet = attrs.tokenSet\n  this.fields = attrs.fields\n  this.pipeline = attrs.pipeline\n}\n\n/**\n * A result contains details of a document matching a search query.\n * @typedef {Object} lunr.Index~Result\n * @property {string} ref - The reference of the document this result represents.\n * @property {number} score - A number between 0 and 1 representing how similar this document is to the query.\n * @property {lunr.MatchData} matchData - Contains metadata about this match including which term(s) caused the match.\n */\n\n/**\n * Although lunr provides the ability to create queries using lunr.Query, it also provides a simple\n * query language which itself is parsed into an instance of lunr.Query.\n *\n * For programmatically building queries it is advised to directly use lunr.Query, the query language\n * is best used for human entered text rather than program generated text.\n *\n * At its simplest queries can just be a single term, e.g. `hello`, multiple terms are also supported\n * and will be combined with OR, e.g `hello world` will match documents that contain either 'hello'\n * or 'world', though those that contain both will rank higher in the results.\n *\n * Wildcards can be included in terms to match one or more unspecified characters, these wildcards can\n * be inserted anywhere within the term, and more than one wildcard can exist in a single term. Adding\n * wildcards will increase the number of documents that will be found but can also have a negative\n * impact on query performance, especially with wildcards at the beginning of a term.\n *\n * Terms can be restricted to specific fields, e.g. `title:hello`, only documents with the term\n * hello in the title field will match this query. Using a field not present in the index will lead\n * to an error being thrown.\n *\n * Modifiers can also be added to terms, lunr supports edit distance and boost modifiers on terms. A term\n * boost will make documents matching that term score higher, e.g. `foo^5`. Edit distance is also supported\n * to provide fuzzy matching, e.g. 'hello~2' will match documents with hello with an edit distance of 2.\n * Avoid large values for edit distance to improve query performance.\n *\n * Each term also supports a presence modifier. By default a term's presence in document is optional, however\n * this can be changed to either required or prohibited. For a term's presence to be required in a document the\n * term should be prefixed with a '+', e.g. `+foo bar` is a search for documents that must contain 'foo' and\n * optionally contain 'bar'. Conversely a leading '-' sets the terms presence to prohibited, i.e. it must not\n * appear in a document, e.g. `-foo bar` is a search for documents that do not contain 'foo' but may contain 'bar'.\n *\n * To escape special characters the backslash character '\\' can be used, this allows searches to include\n * characters that would normally be considered modifiers, e.g. `foo\\~2` will search for a term \"foo~2\" instead\n * of attempting to apply a boost of 2 to the search term \"foo\".\n *\n * @typedef {string} lunr.Index~QueryString\n * @example <caption>Simple single term query</caption>\n * hello\n * @example <caption>Multiple term query</caption>\n * hello world\n * @example <caption>term scoped to a field</caption>\n * title:hello\n * @example <caption>term with a boost of 10</caption>\n * hello^10\n * @example <caption>term with an edit distance of 2</caption>\n * hello~2\n * @example <caption>terms with presence modifiers</caption>\n * -foo +bar baz\n */\n\n/**\n * Performs a search against the index using lunr query syntax.\n *\n * Results will be returned sorted by their score, the most relevant results\n * will be returned first.  For details on how the score is calculated, please see\n * the {@link https://lunrjs.com/guides/searching.html#scoring|guide}.\n *\n * For more programmatic querying use lunr.Index#query.\n *\n * @param {lunr.Index~QueryString} queryString - A string containing a lunr query.\n * @throws {lunr.QueryParseError} If the passed query string cannot be parsed.\n * @returns {lunr.Index~Result[]}\n */\nlunr.Index.prototype.search = function (queryString) {\n  return this.query(function (query) {\n    var parser = new lunr.QueryParser(queryString, query)\n    parser.parse()\n  })\n}\n\n/**\n * A query builder callback provides a query object to be used to express\n * the query to perform on the index.\n *\n * @callback lunr.Index~queryBuilder\n * @param {lunr.Query} query - The query object to build up.\n * @this lunr.Query\n */\n\n/**\n * Performs a query against the index using the yielded lunr.Query object.\n *\n * If performing programmatic queries against the index, this method is preferred\n * over lunr.Index#search so as to avoid the additional query parsing overhead.\n *\n * A query object is yielded to the supplied function which should be used to\n * express the query to be run against the index.\n *\n * Note that although this function takes a callback parameter it is _not_ an\n * asynchronous operation, the callback is just yielded a query object to be\n * customized.\n *\n * @param {lunr.Index~queryBuilder} fn - A function that is used to build the query.\n * @returns {lunr.Index~Result[]}\n */\nlunr.Index.prototype.query = function (fn) {\n  // for each query clause\n  // * process terms\n  // * expand terms from token set\n  // * find matching documents and metadata\n  // * get document vectors\n  // * score documents\n\n  var query = new lunr.Query(this.fields),\n      matchingFields = Object.create(null),\n      queryVectors = Object.create(null),\n      termFieldCache = Object.create(null),\n      requiredMatches = Object.create(null),\n      prohibitedMatches = Object.create(null)\n\n  /*\n   * To support field level boosts a query vector is created per\n   * field. An empty vector is eagerly created to support negated\n   * queries.\n   */\n  for (var i = 0; i < this.fields.length; i++) {\n    queryVectors[this.fields[i]] = new lunr.Vector\n  }\n\n  fn.call(query, query)\n\n  for (var i = 0; i < query.clauses.length; i++) {\n    /*\n     * Unless the pipeline has been disabled for this term, which is\n     * the case for terms with wildcards, we need to pass the clause\n     * term through the search pipeline. A pipeline returns an array\n     * of processed terms. Pipeline functions may expand the passed\n     * term, which means we may end up performing multiple index lookups\n     * for a single query term.\n     */\n    var clause = query.clauses[i],\n        terms = null,\n        clauseMatches = lunr.Set.empty\n\n    if (clause.usePipeline) {\n      terms = this.pipeline.runString(clause.term, {\n        fields: clause.fields\n      })\n    } else {\n      terms = [clause.term]\n    }\n\n    for (var m = 0; m < terms.length; m++) {\n      var term = terms[m]\n\n      /*\n       * Each term returned from the pipeline needs to use the same query\n       * clause object, e.g. the same boost and or edit distance. The\n       * simplest way to do this is to re-use the clause object but mutate\n       * its term property.\n       */\n      clause.term = term\n\n      /*\n       * From the term in the clause we create a token set which will then\n       * be used to intersect the indexes token set to get a list of terms\n       * to lookup in the inverted index\n       */\n      var termTokenSet = lunr.TokenSet.fromClause(clause),\n          expandedTerms = this.tokenSet.intersect(termTokenSet).toArray()\n\n      /*\n       * If a term marked as required does not exist in the tokenSet it is\n       * impossible for the search to return any matches. We set all the field\n       * scoped required matches set to empty and stop examining any further\n       * clauses.\n       */\n      if (expandedTerms.length === 0 && clause.presence === lunr.Query.presence.REQUIRED) {\n        for (var k = 0; k < clause.fields.length; k++) {\n          var field = clause.fields[k]\n          requiredMatches[field] = lunr.Set.empty\n        }\n\n        break\n      }\n\n      for (var j = 0; j < expandedTerms.length; j++) {\n        /*\n         * For each term get the posting and termIndex, this is required for\n         * building the query vector.\n         */\n        var expandedTerm = expandedTerms[j],\n            posting = this.invertedIndex[expandedTerm],\n            termIndex = posting._index\n\n        for (var k = 0; k < clause.fields.length; k++) {\n          /*\n           * For each field that this query term is scoped by (by default\n           * all fields are in scope) we need to get all the document refs\n           * that have this term in that field.\n           *\n           * The posting is the entry in the invertedIndex for the matching\n           * term from above.\n           */\n          var field = clause.fields[k],\n              fieldPosting = posting[field],\n              matchingDocumentRefs = Object.keys(fieldPosting),\n              termField = expandedTerm + \"/\" + field,\n              matchingDocumentsSet = new lunr.Set(matchingDocumentRefs)\n\n          /*\n           * if the presence of this term is required ensure that the matching\n           * documents are added to the set of required matches for this clause.\n           *\n           */\n          if (clause.presence == lunr.Query.presence.REQUIRED) {\n            clauseMatches = clauseMatches.union(matchingDocumentsSet)\n\n            if (requiredMatches[field] === undefined) {\n              requiredMatches[field] = lunr.Set.complete\n            }\n          }\n\n          /*\n           * if the presence of this term is prohibited ensure that the matching\n           * documents are added to the set of prohibited matches for this field,\n           * creating that set if it does not yet exist.\n           */\n          if (clause.presence == lunr.Query.presence.PROHIBITED) {\n            if (prohibitedMatches[field] === undefined) {\n              prohibitedMatches[field] = lunr.Set.empty\n            }\n\n            prohibitedMatches[field] = prohibitedMatches[field].union(matchingDocumentsSet)\n\n            /*\n             * Prohibited matches should not be part of the query vector used for\n             * similarity scoring and no metadata should be extracted so we continue\n             * to the next field\n             */\n            continue\n          }\n\n          /*\n           * The query field vector is populated using the termIndex found for\n           * the term and a unit value with the appropriate boost applied.\n           * Using upsert because there could already be an entry in the vector\n           * for the term we are working with. In that case we just add the scores\n           * together.\n           */\n          queryVectors[field].upsert(termIndex, clause.boost, function (a, b) { return a + b })\n\n          /**\n           * If we've already seen this term, field combo then we've already collected\n           * the matching documents and metadata, no need to go through all that again\n           */\n          if (termFieldCache[termField]) {\n            continue\n          }\n\n          for (var l = 0; l < matchingDocumentRefs.length; l++) {\n            /*\n             * All metadata for this term/field/document triple\n             * are then extracted and collected into an instance\n             * of lunr.MatchData ready to be returned in the query\n             * results\n             */\n            var matchingDocumentRef = matchingDocumentRefs[l],\n                matchingFieldRef = new lunr.FieldRef (matchingDocumentRef, field),\n                metadata = fieldPosting[matchingDocumentRef],\n                fieldMatch\n\n            if ((fieldMatch = matchingFields[matchingFieldRef]) === undefined) {\n              matchingFields[matchingFieldRef] = new lunr.MatchData (expandedTerm, field, metadata)\n            } else {\n              fieldMatch.add(expandedTerm, field, metadata)\n            }\n\n          }\n\n          termFieldCache[termField] = true\n        }\n      }\n    }\n\n    /**\n     * If the presence was required we need to update the requiredMatches field sets.\n     * We do this after all fields for the term have collected their matches because\n     * the clause terms presence is required in _any_ of the fields not _all_ of the\n     * fields.\n     */\n    if (clause.presence === lunr.Query.presence.REQUIRED) {\n      for (var k = 0; k < clause.fields.length; k++) {\n        var field = clause.fields[k]\n        requiredMatches[field] = requiredMatches[field].intersect(clauseMatches)\n      }\n    }\n  }\n\n  /**\n   * Need to combine the field scoped required and prohibited\n   * matching documents into a global set of required and prohibited\n   * matches\n   */\n  var allRequiredMatches = lunr.Set.complete,\n      allProhibitedMatches = lunr.Set.empty\n\n  for (var i = 0; i < this.fields.length; i++) {\n    var field = this.fields[i]\n\n    if (requiredMatches[field]) {\n      allRequiredMatches = allRequiredMatches.intersect(requiredMatches[field])\n    }\n\n    if (prohibitedMatches[field]) {\n      allProhibitedMatches = allProhibitedMatches.union(prohibitedMatches[field])\n    }\n  }\n\n  var matchingFieldRefs = Object.keys(matchingFields),\n      results = [],\n      matches = Object.create(null)\n\n  /*\n   * If the query is negated (contains only prohibited terms)\n   * we need to get _all_ fieldRefs currently existing in the\n   * index. This is only done when we know that the query is\n   * entirely prohibited terms to avoid any cost of getting all\n   * fieldRefs unnecessarily.\n   *\n   * Additionally, blank MatchData must be created to correctly\n   * populate the results.\n   */\n  if (query.isNegated()) {\n    matchingFieldRefs = Object.keys(this.fieldVectors)\n\n    for (var i = 0; i < matchingFieldRefs.length; i++) {\n      var matchingFieldRef = matchingFieldRefs[i]\n      var fieldRef = lunr.FieldRef.fromString(matchingFieldRef)\n      matchingFields[matchingFieldRef] = new lunr.MatchData\n    }\n  }\n\n  for (var i = 0; i < matchingFieldRefs.length; i++) {\n    /*\n     * Currently we have document fields that match the query, but we\n     * need to return documents. The matchData and scores are combined\n     * from multiple fields belonging to the same document.\n     *\n     * Scores are calculated by field, using the query vectors created\n     * above, and combined into a final document score using addition.\n     */\n    var fieldRef = lunr.FieldRef.fromString(matchingFieldRefs[i]),\n        docRef = fieldRef.docRef\n\n    if (!allRequiredMatches.contains(docRef)) {\n      continue\n    }\n\n    if (allProhibitedMatches.contains(docRef)) {\n      continue\n    }\n\n    var fieldVector = this.fieldVectors[fieldRef],\n        score = queryVectors[fieldRef.fieldName].similarity(fieldVector),\n        docMatch\n\n    if ((docMatch = matches[docRef]) !== undefined) {\n      docMatch.score += score\n      docMatch.matchData.combine(matchingFields[fieldRef])\n    } else {\n      var match = {\n        ref: docRef,\n        score: score,\n        matchData: matchingFields[fieldRef]\n      }\n      matches[docRef] = match\n      results.push(match)\n    }\n  }\n\n  /*\n   * Sort the results objects by score, highest first.\n   */\n  return results.sort(function (a, b) {\n    return b.score - a.score\n  })\n}\n\n/**\n * Prepares the index for JSON serialization.\n *\n * The schema for this JSON blob will be described in a\n * separate JSON schema file.\n *\n * @returns {Object}\n */\nlunr.Index.prototype.toJSON = function () {\n  var invertedIndex = Object.keys(this.invertedIndex)\n    .sort()\n    .map(function (term) {\n      return [term, this.invertedIndex[term]]\n    }, this)\n\n  var fieldVectors = Object.keys(this.fieldVectors)\n    .map(function (ref) {\n      return [ref, this.fieldVectors[ref].toJSON()]\n    }, this)\n\n  return {\n    version: lunr.version,\n    fields: this.fields,\n    fieldVectors: fieldVectors,\n    invertedIndex: invertedIndex,\n    pipeline: this.pipeline.toJSON()\n  }\n}\n\n/**\n * Loads a previously serialized lunr.Index\n *\n * @param {Object} serializedIndex - A previously serialized lunr.Index\n * @returns {lunr.Index}\n */\nlunr.Index.load = function (serializedIndex) {\n  var attrs = {},\n      fieldVectors = {},\n      serializedVectors = serializedIndex.fieldVectors,\n      invertedIndex = Object.create(null),\n      serializedInvertedIndex = serializedIndex.invertedIndex,\n      tokenSetBuilder = new lunr.TokenSet.Builder,\n      pipeline = lunr.Pipeline.load(serializedIndex.pipeline)\n\n  if (serializedIndex.version != lunr.version) {\n    lunr.utils.warn(\"Version mismatch when loading serialised index. Current version of lunr '\" + lunr.version + \"' does not match serialized index '\" + serializedIndex.version + \"'\")\n  }\n\n  for (var i = 0; i < serializedVectors.length; i++) {\n    var tuple = serializedVectors[i],\n        ref = tuple[0],\n        elements = tuple[1]\n\n    fieldVectors[ref] = new lunr.Vector(elements)\n  }\n\n  for (var i = 0; i < serializedInvertedIndex.length; i++) {\n    var tuple = serializedInvertedIndex[i],\n        term = tuple[0],\n        posting = tuple[1]\n\n    tokenSetBuilder.insert(term)\n    invertedIndex[term] = posting\n  }\n\n  tokenSetBuilder.finish()\n\n  attrs.fields = serializedIndex.fields\n\n  attrs.fieldVectors = fieldVectors\n  attrs.invertedIndex = invertedIndex\n  attrs.tokenSet = tokenSetBuilder.root\n  attrs.pipeline = pipeline\n\n  return new lunr.Index(attrs)\n}\n/*!\n * lunr.Builder\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.Builder performs indexing on a set of documents and\n * returns instances of lunr.Index ready for querying.\n *\n * All configuration of the index is done via the builder, the\n * fields to index, the document reference, the text processing\n * pipeline and document scoring parameters are all set on the\n * builder before indexing.\n *\n * @constructor\n * @property {string} _ref - Internal reference to the document reference field.\n * @property {string[]} _fields - Internal reference to the document fields to index.\n * @property {object} invertedIndex - The inverted index maps terms to document fields.\n * @property {object} documentTermFrequencies - Keeps track of document term frequencies.\n * @property {object} documentLengths - Keeps track of the length of documents added to the index.\n * @property {lunr.tokenizer} tokenizer - Function for splitting strings into tokens for indexing.\n * @property {lunr.Pipeline} pipeline - The pipeline performs text processing on tokens before indexing.\n * @property {lunr.Pipeline} searchPipeline - A pipeline for processing search terms before querying the index.\n * @property {number} documentCount - Keeps track of the total number of documents indexed.\n * @property {number} _b - A parameter to control field length normalization, setting this to 0 disabled normalization, 1 fully normalizes field lengths, the default value is 0.75.\n * @property {number} _k1 - A parameter to control how quickly an increase in term frequency results in term frequency saturation, the default value is 1.2.\n * @property {number} termIndex - A counter incremented for each unique term, used to identify a terms position in the vector space.\n * @property {array} metadataWhitelist - A list of metadata keys that have been whitelisted for entry in the index.\n */\nlunr.Builder = function () {\n  this._ref = \"id\"\n  this._fields = Object.create(null)\n  this._documents = Object.create(null)\n  this.invertedIndex = Object.create(null)\n  this.fieldTermFrequencies = {}\n  this.fieldLengths = {}\n  this.tokenizer = lunr.tokenizer\n  this.pipeline = new lunr.Pipeline\n  this.searchPipeline = new lunr.Pipeline\n  this.documentCount = 0\n  this._b = 0.75\n  this._k1 = 1.2\n  this.termIndex = 0\n  this.metadataWhitelist = []\n}\n\n/**\n * Sets the document field used as the document reference. Every document must have this field.\n * The type of this field in the document should be a string, if it is not a string it will be\n * coerced into a string by calling toString.\n *\n * The default ref is 'id'.\n *\n * The ref should _not_ be changed during indexing, it should be set before any documents are\n * added to the index. Changing it during indexing can lead to inconsistent results.\n *\n * @param {string} ref - The name of the reference field in the document.\n */\nlunr.Builder.prototype.ref = function (ref) {\n  this._ref = ref\n}\n\n/**\n * A function that is used to extract a field from a document.\n *\n * Lunr expects a field to be at the top level of a document, if however the field\n * is deeply nested within a document an extractor function can be used to extract\n * the right field for indexing.\n *\n * @callback fieldExtractor\n * @param {object} doc - The document being added to the index.\n * @returns {?(string|object|object[])} obj - The object that will be indexed for this field.\n * @example <caption>Extracting a nested field</caption>\n * function (doc) { return doc.nested.field }\n */\n\n/**\n * Adds a field to the list of document fields that will be indexed. Every document being\n * indexed should have this field. Null values for this field in indexed documents will\n * not cause errors but will limit the chance of that document being retrieved by searches.\n *\n * All fields should be added before adding documents to the index. Adding fields after\n * a document has been indexed will have no effect on already indexed documents.\n *\n * Fields can be boosted at build time. This allows terms within that field to have more\n * importance when ranking search results. Use a field boost to specify that matches within\n * one field are more important than other fields.\n *\n * @param {string} fieldName - The name of a field to index in all documents.\n * @param {object} attributes - Optional attributes associated with this field.\n * @param {number} [attributes.boost=1] - Boost applied to all terms within this field.\n * @param {fieldExtractor} [attributes.extractor] - Function to extract a field from a document.\n * @throws {RangeError} fieldName cannot contain unsupported characters '/'\n */\nlunr.Builder.prototype.field = function (fieldName, attributes) {\n  if (/\\//.test(fieldName)) {\n    throw new RangeError (\"Field '\" + fieldName + \"' contains illegal character '/'\")\n  }\n\n  this._fields[fieldName] = attributes || {}\n}\n\n/**\n * A parameter to tune the amount of field length normalisation that is applied when\n * calculating relevance scores. A value of 0 will completely disable any normalisation\n * and a value of 1 will fully normalise field lengths. The default is 0.75. Values of b\n * will be clamped to the range 0 - 1.\n *\n * @param {number} number - The value to set for this tuning parameter.\n */\nlunr.Builder.prototype.b = function (number) {\n  if (number < 0) {\n    this._b = 0\n  } else if (number > 1) {\n    this._b = 1\n  } else {\n    this._b = number\n  }\n}\n\n/**\n * A parameter that controls the speed at which a rise in term frequency results in term\n * frequency saturation. The default value is 1.2. Setting this to a higher value will give\n * slower saturation levels, a lower value will result in quicker saturation.\n *\n * @param {number} number - The value to set for this tuning parameter.\n */\nlunr.Builder.prototype.k1 = function (number) {\n  this._k1 = number\n}\n\n/**\n * Adds a document to the index.\n *\n * Before adding fields to the index the index should have been fully setup, with the document\n * ref and all fields to index already having been specified.\n *\n * The document must have a field name as specified by the ref (by default this is 'id') and\n * it should have all fields defined for indexing, though null or undefined values will not\n * cause errors.\n *\n * Entire documents can be boosted at build time. Applying a boost to a document indicates that\n * this document should rank higher in search results than other documents.\n *\n * @param {object} doc - The document to add to the index.\n * @param {object} attributes - Optional attributes associated with this document.\n * @param {number} [attributes.boost=1] - Boost applied to all terms within this document.\n */\nlunr.Builder.prototype.add = function (doc, attributes) {\n  var docRef = doc[this._ref],\n      fields = Object.keys(this._fields)\n\n  this._documents[docRef] = attributes || {}\n  this.documentCount += 1\n\n  for (var i = 0; i < fields.length; i++) {\n    var fieldName = fields[i],\n        extractor = this._fields[fieldName].extractor,\n        field = extractor ? extractor(doc) : doc[fieldName],\n        tokens = this.tokenizer(field, {\n          fields: [fieldName]\n        }),\n        terms = this.pipeline.run(tokens),\n        fieldRef = new lunr.FieldRef (docRef, fieldName),\n        fieldTerms = Object.create(null)\n\n    this.fieldTermFrequencies[fieldRef] = fieldTerms\n    this.fieldLengths[fieldRef] = 0\n\n    // store the length of this field for this document\n    this.fieldLengths[fieldRef] += terms.length\n\n    // calculate term frequencies for this field\n    for (var j = 0; j < terms.length; j++) {\n      var term = terms[j]\n\n      if (fieldTerms[term] == undefined) {\n        fieldTerms[term] = 0\n      }\n\n      fieldTerms[term] += 1\n\n      // add to inverted index\n      // create an initial posting if one doesn't exist\n      if (this.invertedIndex[term] == undefined) {\n        var posting = Object.create(null)\n        posting[\"_index\"] = this.termIndex\n        this.termIndex += 1\n\n        for (var k = 0; k < fields.length; k++) {\n          posting[fields[k]] = Object.create(null)\n        }\n\n        this.invertedIndex[term] = posting\n      }\n\n      // add an entry for this term/fieldName/docRef to the invertedIndex\n      if (this.invertedIndex[term][fieldName][docRef] == undefined) {\n        this.invertedIndex[term][fieldName][docRef] = Object.create(null)\n      }\n\n      // store all whitelisted metadata about this token in the\n      // inverted index\n      for (var l = 0; l < this.metadataWhitelist.length; l++) {\n        var metadataKey = this.metadataWhitelist[l],\n            metadata = term.metadata[metadataKey]\n\n        if (this.invertedIndex[term][fieldName][docRef][metadataKey] == undefined) {\n          this.invertedIndex[term][fieldName][docRef][metadataKey] = []\n        }\n\n        this.invertedIndex[term][fieldName][docRef][metadataKey].push(metadata)\n      }\n    }\n\n  }\n}\n\n/**\n * Calculates the average document length for this index\n *\n * @private\n */\nlunr.Builder.prototype.calculateAverageFieldLengths = function () {\n\n  var fieldRefs = Object.keys(this.fieldLengths),\n      numberOfFields = fieldRefs.length,\n      accumulator = {},\n      documentsWithField = {}\n\n  for (var i = 0; i < numberOfFields; i++) {\n    var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]),\n        field = fieldRef.fieldName\n\n    documentsWithField[field] || (documentsWithField[field] = 0)\n    documentsWithField[field] += 1\n\n    accumulator[field] || (accumulator[field] = 0)\n    accumulator[field] += this.fieldLengths[fieldRef]\n  }\n\n  var fields = Object.keys(this._fields)\n\n  for (var i = 0; i < fields.length; i++) {\n    var fieldName = fields[i]\n    accumulator[fieldName] = accumulator[fieldName] / documentsWithField[fieldName]\n  }\n\n  this.averageFieldLength = accumulator\n}\n\n/**\n * Builds a vector space model of every document using lunr.Vector\n *\n * @private\n */\nlunr.Builder.prototype.createFieldVectors = function () {\n  var fieldVectors = {},\n      fieldRefs = Object.keys(this.fieldTermFrequencies),\n      fieldRefsLength = fieldRefs.length,\n      termIdfCache = Object.create(null)\n\n  for (var i = 0; i < fieldRefsLength; i++) {\n    var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]),\n        fieldName = fieldRef.fieldName,\n        fieldLength = this.fieldLengths[fieldRef],\n        fieldVector = new lunr.Vector,\n        termFrequencies = this.fieldTermFrequencies[fieldRef],\n        terms = Object.keys(termFrequencies),\n        termsLength = terms.length\n\n\n    var fieldBoost = this._fields[fieldName].boost || 1,\n        docBoost = this._documents[fieldRef.docRef].boost || 1\n\n    for (var j = 0; j < termsLength; j++) {\n      var term = terms[j],\n          tf = termFrequencies[term],\n          termIndex = this.invertedIndex[term]._index,\n          idf, score, scoreWithPrecision\n\n      if (termIdfCache[term] === undefined) {\n        idf = lunr.idf(this.invertedIndex[term], this.documentCount)\n        termIdfCache[term] = idf\n      } else {\n        idf = termIdfCache[term]\n      }\n\n      score = idf * ((this._k1 + 1) * tf) / (this._k1 * (1 - this._b + this._b * (fieldLength / this.averageFieldLength[fieldName])) + tf)\n      score *= fieldBoost\n      score *= docBoost\n      scoreWithPrecision = Math.round(score * 1000) / 1000\n      // Converts 1.23456789 to 1.234.\n      // Reducing the precision so that the vectors take up less\n      // space when serialised. Doing it now so that they behave\n      // the same before and after serialisation. Also, this is\n      // the fastest approach to reducing a number's precision in\n      // JavaScript.\n\n      fieldVector.insert(termIndex, scoreWithPrecision)\n    }\n\n    fieldVectors[fieldRef] = fieldVector\n  }\n\n  this.fieldVectors = fieldVectors\n}\n\n/**\n * Creates a token set of all tokens in the index using lunr.TokenSet\n *\n * @private\n */\nlunr.Builder.prototype.createTokenSet = function () {\n  this.tokenSet = lunr.TokenSet.fromArray(\n    Object.keys(this.invertedIndex).sort()\n  )\n}\n\n/**\n * Builds the index, creating an instance of lunr.Index.\n *\n * This completes the indexing process and should only be called\n * once all documents have been added to the index.\n *\n * @returns {lunr.Index}\n */\nlunr.Builder.prototype.build = function () {\n  this.calculateAverageFieldLengths()\n  this.createFieldVectors()\n  this.createTokenSet()\n\n  return new lunr.Index({\n    invertedIndex: this.invertedIndex,\n    fieldVectors: this.fieldVectors,\n    tokenSet: this.tokenSet,\n    fields: Object.keys(this._fields),\n    pipeline: this.searchPipeline\n  })\n}\n\n/**\n * Applies a plugin to the index builder.\n *\n * A plugin is a function that is called with the index builder as its context.\n * Plugins can be used to customise or extend the behaviour of the index\n * in some way. A plugin is just a function, that encapsulated the custom\n * behaviour that should be applied when building the index.\n *\n * The plugin function will be called with the index builder as its argument, additional\n * arguments can also be passed when calling use. The function will be called\n * with the index builder as its context.\n *\n * @param {Function} plugin The plugin to apply.\n */\nlunr.Builder.prototype.use = function (fn) {\n  var args = Array.prototype.slice.call(arguments, 1)\n  args.unshift(this)\n  fn.apply(this, args)\n}\n/**\n * Contains and collects metadata about a matching document.\n * A single instance of lunr.MatchData is returned as part of every\n * lunr.Index~Result.\n *\n * @constructor\n * @param {string} term - The term this match data is associated with\n * @param {string} field - The field in which the term was found\n * @param {object} metadata - The metadata recorded about this term in this field\n * @property {object} metadata - A cloned collection of metadata associated with this document.\n * @see {@link lunr.Index~Result}\n */\nlunr.MatchData = function (term, field, metadata) {\n  var clonedMetadata = Object.create(null),\n      metadataKeys = Object.keys(metadata || {})\n\n  // Cloning the metadata to prevent the original\n  // being mutated during match data combination.\n  // Metadata is kept in an array within the inverted\n  // index so cloning the data can be done with\n  // Array#slice\n  for (var i = 0; i < metadataKeys.length; i++) {\n    var key = metadataKeys[i]\n    clonedMetadata[key] = metadata[key].slice()\n  }\n\n  this.metadata = Object.create(null)\n\n  if (term !== undefined) {\n    this.metadata[term] = Object.create(null)\n    this.metadata[term][field] = clonedMetadata\n  }\n}\n\n/**\n * An instance of lunr.MatchData will be created for every term that matches a\n * document. However only one instance is required in a lunr.Index~Result. This\n * method combines metadata from another instance of lunr.MatchData with this\n * objects metadata.\n *\n * @param {lunr.MatchData} otherMatchData - Another instance of match data to merge with this one.\n * @see {@link lunr.Index~Result}\n */\nlunr.MatchData.prototype.combine = function (otherMatchData) {\n  var terms = Object.keys(otherMatchData.metadata)\n\n  for (var i = 0; i < terms.length; i++) {\n    var term = terms[i],\n        fields = Object.keys(otherMatchData.metadata[term])\n\n    if (this.metadata[term] == undefined) {\n      this.metadata[term] = Object.create(null)\n    }\n\n    for (var j = 0; j < fields.length; j++) {\n      var field = fields[j],\n          keys = Object.keys(otherMatchData.metadata[term][field])\n\n      if (this.metadata[term][field] == undefined) {\n        this.metadata[term][field] = Object.create(null)\n      }\n\n      for (var k = 0; k < keys.length; k++) {\n        var key = keys[k]\n\n        if (this.metadata[term][field][key] == undefined) {\n          this.metadata[term][field][key] = otherMatchData.metadata[term][field][key]\n        } else {\n          this.metadata[term][field][key] = this.metadata[term][field][key].concat(otherMatchData.metadata[term][field][key])\n        }\n\n      }\n    }\n  }\n}\n\n/**\n * Add metadata for a term/field pair to this instance of match data.\n *\n * @param {string} term - The term this match data is associated with\n * @param {string} field - The field in which the term was found\n * @param {object} metadata - The metadata recorded about this term in this field\n */\nlunr.MatchData.prototype.add = function (term, field, metadata) {\n  if (!(term in this.metadata)) {\n    this.metadata[term] = Object.create(null)\n    this.metadata[term][field] = metadata\n    return\n  }\n\n  if (!(field in this.metadata[term])) {\n    this.metadata[term][field] = metadata\n    return\n  }\n\n  var metadataKeys = Object.keys(metadata)\n\n  for (var i = 0; i < metadataKeys.length; i++) {\n    var key = metadataKeys[i]\n\n    if (key in this.metadata[term][field]) {\n      this.metadata[term][field][key] = this.metadata[term][field][key].concat(metadata[key])\n    } else {\n      this.metadata[term][field][key] = metadata[key]\n    }\n  }\n}\n/**\n * A lunr.Query provides a programmatic way of defining queries to be performed\n * against a {@link lunr.Index}.\n *\n * Prefer constructing a lunr.Query using the {@link lunr.Index#query} method\n * so the query object is pre-initialized with the right index fields.\n *\n * @constructor\n * @property {lunr.Query~Clause[]} clauses - An array of query clauses.\n * @property {string[]} allFields - An array of all available fields in a lunr.Index.\n */\nlunr.Query = function (allFields) {\n  this.clauses = []\n  this.allFields = allFields\n}\n\n/**\n * Constants for indicating what kind of automatic wildcard insertion will be used when constructing a query clause.\n *\n * This allows wildcards to be added to the beginning and end of a term without having to manually do any string\n * concatenation.\n *\n * The wildcard constants can be bitwise combined to select both leading and trailing wildcards.\n *\n * @constant\n * @default\n * @property {number} wildcard.NONE - The term will have no wildcards inserted, this is the default behaviour\n * @property {number} wildcard.LEADING - Prepend the term with a wildcard, unless a leading wildcard already exists\n * @property {number} wildcard.TRAILING - Append a wildcard to the term, unless a trailing wildcard already exists\n * @see lunr.Query~Clause\n * @see lunr.Query#clause\n * @see lunr.Query#term\n * @example <caption>query term with trailing wildcard</caption>\n * query.term('foo', { wildcard: lunr.Query.wildcard.TRAILING })\n * @example <caption>query term with leading and trailing wildcard</caption>\n * query.term('foo', {\n *   wildcard: lunr.Query.wildcard.LEADING | lunr.Query.wildcard.TRAILING\n * })\n */\n\nlunr.Query.wildcard = new String (\"*\")\nlunr.Query.wildcard.NONE = 0\nlunr.Query.wildcard.LEADING = 1\nlunr.Query.wildcard.TRAILING = 2\n\n/**\n * Constants for indicating what kind of presence a term must have in matching documents.\n *\n * @constant\n * @enum {number}\n * @see lunr.Query~Clause\n * @see lunr.Query#clause\n * @see lunr.Query#term\n * @example <caption>query term with required presence</caption>\n * query.term('foo', { presence: lunr.Query.presence.REQUIRED })\n */\nlunr.Query.presence = {\n  /**\n   * Term's presence in a document is optional, this is the default value.\n   */\n  OPTIONAL: 1,\n\n  /**\n   * Term's presence in a document is required, documents that do not contain\n   * this term will not be returned.\n   */\n  REQUIRED: 2,\n\n  /**\n   * Term's presence in a document is prohibited, documents that do contain\n   * this term will not be returned.\n   */\n  PROHIBITED: 3\n}\n\n/**\n * A single clause in a {@link lunr.Query} contains a term and details on how to\n * match that term against a {@link lunr.Index}.\n *\n * @typedef {Object} lunr.Query~Clause\n * @property {string[]} fields - The fields in an index this clause should be matched against.\n * @property {number} [boost=1] - Any boost that should be applied when matching this clause.\n * @property {number} [editDistance] - Whether the term should have fuzzy matching applied, and how fuzzy the match should be.\n * @property {boolean} [usePipeline] - Whether the term should be passed through the search pipeline.\n * @property {number} [wildcard=lunr.Query.wildcard.NONE] - Whether the term should have wildcards appended or prepended.\n * @property {number} [presence=lunr.Query.presence.OPTIONAL] - The terms presence in any matching documents.\n */\n\n/**\n * Adds a {@link lunr.Query~Clause} to this query.\n *\n * Unless the clause contains the fields to be matched all fields will be matched. In addition\n * a default boost of 1 is applied to the clause.\n *\n * @param {lunr.Query~Clause} clause - The clause to add to this query.\n * @see lunr.Query~Clause\n * @returns {lunr.Query}\n */\nlunr.Query.prototype.clause = function (clause) {\n  if (!('fields' in clause)) {\n    clause.fields = this.allFields\n  }\n\n  if (!('boost' in clause)) {\n    clause.boost = 1\n  }\n\n  if (!('usePipeline' in clause)) {\n    clause.usePipeline = true\n  }\n\n  if (!('wildcard' in clause)) {\n    clause.wildcard = lunr.Query.wildcard.NONE\n  }\n\n  if ((clause.wildcard & lunr.Query.wildcard.LEADING) && (clause.term.charAt(0) != lunr.Query.wildcard)) {\n    clause.term = \"*\" + clause.term\n  }\n\n  if ((clause.wildcard & lunr.Query.wildcard.TRAILING) && (clause.term.slice(-1) != lunr.Query.wildcard)) {\n    clause.term = \"\" + clause.term + \"*\"\n  }\n\n  if (!('presence' in clause)) {\n    clause.presence = lunr.Query.presence.OPTIONAL\n  }\n\n  this.clauses.push(clause)\n\n  return this\n}\n\n/**\n * A negated query is one in which every clause has a presence of\n * prohibited. These queries require some special processing to return\n * the expected results.\n *\n * @returns boolean\n */\nlunr.Query.prototype.isNegated = function () {\n  for (var i = 0; i < this.clauses.length; i++) {\n    if (this.clauses[i].presence != lunr.Query.presence.PROHIBITED) {\n      return false\n    }\n  }\n\n  return true\n}\n\n/**\n * Adds a term to the current query, under the covers this will create a {@link lunr.Query~Clause}\n * to the list of clauses that make up this query.\n *\n * The term is used as is, i.e. no tokenization will be performed by this method. Instead conversion\n * to a token or token-like string should be done before calling this method.\n *\n * The term will be converted to a string by calling `toString`. Multiple terms can be passed as an\n * array, each term in the array will share the same options.\n *\n * @param {object|object[]} term - The term(s) to add to the query.\n * @param {object} [options] - Any additional properties to add to the query clause.\n * @returns {lunr.Query}\n * @see lunr.Query#clause\n * @see lunr.Query~Clause\n * @example <caption>adding a single term to a query</caption>\n * query.term(\"foo\")\n * @example <caption>adding a single term to a query and specifying search fields, term boost and automatic trailing wildcard</caption>\n * query.term(\"foo\", {\n *   fields: [\"title\"],\n *   boost: 10,\n *   wildcard: lunr.Query.wildcard.TRAILING\n * })\n * @example <caption>using lunr.tokenizer to convert a string to tokens before using them as terms</caption>\n * query.term(lunr.tokenizer(\"foo bar\"))\n */\nlunr.Query.prototype.term = function (term, options) {\n  if (Array.isArray(term)) {\n    term.forEach(function (t) { this.term(t, lunr.utils.clone(options)) }, this)\n    return this\n  }\n\n  var clause = options || {}\n  clause.term = term.toString()\n\n  this.clause(clause)\n\n  return this\n}\nlunr.QueryParseError = function (message, start, end) {\n  this.name = \"QueryParseError\"\n  this.message = message\n  this.start = start\n  this.end = end\n}\n\nlunr.QueryParseError.prototype = new Error\nlunr.QueryLexer = function (str) {\n  this.lexemes = []\n  this.str = str\n  this.length = str.length\n  this.pos = 0\n  this.start = 0\n  this.escapeCharPositions = []\n}\n\nlunr.QueryLexer.prototype.run = function () {\n  var state = lunr.QueryLexer.lexText\n\n  while (state) {\n    state = state(this)\n  }\n}\n\nlunr.QueryLexer.prototype.sliceString = function () {\n  var subSlices = [],\n      sliceStart = this.start,\n      sliceEnd = this.pos\n\n  for (var i = 0; i < this.escapeCharPositions.length; i++) {\n    sliceEnd = this.escapeCharPositions[i]\n    subSlices.push(this.str.slice(sliceStart, sliceEnd))\n    sliceStart = sliceEnd + 1\n  }\n\n  subSlices.push(this.str.slice(sliceStart, this.pos))\n  this.escapeCharPositions.length = 0\n\n  return subSlices.join('')\n}\n\nlunr.QueryLexer.prototype.emit = function (type) {\n  this.lexemes.push({\n    type: type,\n    str: this.sliceString(),\n    start: this.start,\n    end: this.pos\n  })\n\n  this.start = this.pos\n}\n\nlunr.QueryLexer.prototype.escapeCharacter = function () {\n  this.escapeCharPositions.push(this.pos - 1)\n  this.pos += 1\n}\n\nlunr.QueryLexer.prototype.next = function () {\n  if (this.pos >= this.length) {\n    return lunr.QueryLexer.EOS\n  }\n\n  var char = this.str.charAt(this.pos)\n  this.pos += 1\n  return char\n}\n\nlunr.QueryLexer.prototype.width = function () {\n  return this.pos - this.start\n}\n\nlunr.QueryLexer.prototype.ignore = function () {\n  if (this.start == this.pos) {\n    this.pos += 1\n  }\n\n  this.start = this.pos\n}\n\nlunr.QueryLexer.prototype.backup = function () {\n  this.pos -= 1\n}\n\nlunr.QueryLexer.prototype.acceptDigitRun = function () {\n  var char, charCode\n\n  do {\n    char = this.next()\n    charCode = char.charCodeAt(0)\n  } while (charCode > 47 && charCode < 58)\n\n  if (char != lunr.QueryLexer.EOS) {\n    this.backup()\n  }\n}\n\nlunr.QueryLexer.prototype.more = function () {\n  return this.pos < this.length\n}\n\nlunr.QueryLexer.EOS = 'EOS'\nlunr.QueryLexer.FIELD = 'FIELD'\nlunr.QueryLexer.TERM = 'TERM'\nlunr.QueryLexer.EDIT_DISTANCE = 'EDIT_DISTANCE'\nlunr.QueryLexer.BOOST = 'BOOST'\nlunr.QueryLexer.PRESENCE = 'PRESENCE'\n\nlunr.QueryLexer.lexField = function (lexer) {\n  lexer.backup()\n  lexer.emit(lunr.QueryLexer.FIELD)\n  lexer.ignore()\n  return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexTerm = function (lexer) {\n  if (lexer.width() > 1) {\n    lexer.backup()\n    lexer.emit(lunr.QueryLexer.TERM)\n  }\n\n  lexer.ignore()\n\n  if (lexer.more()) {\n    return lunr.QueryLexer.lexText\n  }\n}\n\nlunr.QueryLexer.lexEditDistance = function (lexer) {\n  lexer.ignore()\n  lexer.acceptDigitRun()\n  lexer.emit(lunr.QueryLexer.EDIT_DISTANCE)\n  return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexBoost = function (lexer) {\n  lexer.ignore()\n  lexer.acceptDigitRun()\n  lexer.emit(lunr.QueryLexer.BOOST)\n  return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexEOS = function (lexer) {\n  if (lexer.width() > 0) {\n    lexer.emit(lunr.QueryLexer.TERM)\n  }\n}\n\n// This matches the separator used when tokenising fields\n// within a document. These should match otherwise it is\n// not possible to search for some tokens within a document.\n//\n// It is possible for the user to change the separator on the\n// tokenizer so it _might_ clash with any other of the special\n// characters already used within the search string, e.g. :.\n//\n// This means that it is possible to change the separator in\n// such a way that makes some words unsearchable using a search\n// string.\nlunr.QueryLexer.termSeparator = lunr.tokenizer.separator\n\nlunr.QueryLexer.lexText = function (lexer) {\n  while (true) {\n    var char = lexer.next()\n\n    if (char == lunr.QueryLexer.EOS) {\n      return lunr.QueryLexer.lexEOS\n    }\n\n    // Escape character is '\\'\n    if (char.charCodeAt(0) == 92) {\n      lexer.escapeCharacter()\n      continue\n    }\n\n    if (char == \":\") {\n      return lunr.QueryLexer.lexField\n    }\n\n    if (char == \"~\") {\n      lexer.backup()\n      if (lexer.width() > 0) {\n        lexer.emit(lunr.QueryLexer.TERM)\n      }\n      return lunr.QueryLexer.lexEditDistance\n    }\n\n    if (char == \"^\") {\n      lexer.backup()\n      if (lexer.width() > 0) {\n        lexer.emit(lunr.QueryLexer.TERM)\n      }\n      return lunr.QueryLexer.lexBoost\n    }\n\n    // \"+\" indicates term presence is required\n    // checking for length to ensure that only\n    // leading \"+\" are considered\n    if (char == \"+\" && lexer.width() === 1) {\n      lexer.emit(lunr.QueryLexer.PRESENCE)\n      return lunr.QueryLexer.lexText\n    }\n\n    // \"-\" indicates term presence is prohibited\n    // checking for length to ensure that only\n    // leading \"-\" are considered\n    if (char == \"-\" && lexer.width() === 1) {\n      lexer.emit(lunr.QueryLexer.PRESENCE)\n      return lunr.QueryLexer.lexText\n    }\n\n    if (char.match(lunr.QueryLexer.termSeparator)) {\n      return lunr.QueryLexer.lexTerm\n    }\n  }\n}\n\nlunr.QueryParser = function (str, query) {\n  this.lexer = new lunr.QueryLexer (str)\n  this.query = query\n  this.currentClause = {}\n  this.lexemeIdx = 0\n}\n\nlunr.QueryParser.prototype.parse = function () {\n  this.lexer.run()\n  this.lexemes = this.lexer.lexemes\n\n  var state = lunr.QueryParser.parseClause\n\n  while (state) {\n    state = state(this)\n  }\n\n  return this.query\n}\n\nlunr.QueryParser.prototype.peekLexeme = function () {\n  return this.lexemes[this.lexemeIdx]\n}\n\nlunr.QueryParser.prototype.consumeLexeme = function () {\n  var lexeme = this.peekLexeme()\n  this.lexemeIdx += 1\n  return lexeme\n}\n\nlunr.QueryParser.prototype.nextClause = function () {\n  var completedClause = this.currentClause\n  this.query.clause(completedClause)\n  this.currentClause = {}\n}\n\nlunr.QueryParser.parseClause = function (parser) {\n  var lexeme = parser.peekLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  switch (lexeme.type) {\n    case lunr.QueryLexer.PRESENCE:\n      return lunr.QueryParser.parsePresence\n    case lunr.QueryLexer.FIELD:\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.TERM:\n      return lunr.QueryParser.parseTerm\n    default:\n      var errorMessage = \"expected either a field or a term, found \" + lexeme.type\n\n      if (lexeme.str.length >= 1) {\n        errorMessage += \" with value '\" + lexeme.str + \"'\"\n      }\n\n      throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n}\n\nlunr.QueryParser.parsePresence = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  switch (lexeme.str) {\n    case \"-\":\n      parser.currentClause.presence = lunr.Query.presence.PROHIBITED\n      break\n    case \"+\":\n      parser.currentClause.presence = lunr.Query.presence.REQUIRED\n      break\n    default:\n      var errorMessage = \"unrecognised presence operator'\" + lexeme.str + \"'\"\n      throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    var errorMessage = \"expecting term or field, found nothing\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.FIELD:\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.TERM:\n      return lunr.QueryParser.parseTerm\n    default:\n      var errorMessage = \"expecting term or field, found '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseField = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  if (parser.query.allFields.indexOf(lexeme.str) == -1) {\n    var possibleFields = parser.query.allFields.map(function (f) { return \"'\" + f + \"'\" }).join(', '),\n        errorMessage = \"unrecognised field '\" + lexeme.str + \"', possible fields: \" + possibleFields\n\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  parser.currentClause.fields = [lexeme.str]\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    var errorMessage = \"expecting term, found nothing\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      return lunr.QueryParser.parseTerm\n    default:\n      var errorMessage = \"expecting term, found '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseTerm = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  parser.currentClause.term = lexeme.str.toLowerCase()\n\n  if (lexeme.str.indexOf(\"*\") != -1) {\n    parser.currentClause.usePipeline = false\n  }\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    parser.nextClause()\n    return\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      parser.nextClause()\n      return lunr.QueryParser.parseTerm\n    case lunr.QueryLexer.FIELD:\n      parser.nextClause()\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.EDIT_DISTANCE:\n      return lunr.QueryParser.parseEditDistance\n    case lunr.QueryLexer.BOOST:\n      return lunr.QueryParser.parseBoost\n    case lunr.QueryLexer.PRESENCE:\n      parser.nextClause()\n      return lunr.QueryParser.parsePresence\n    default:\n      var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseEditDistance = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  var editDistance = parseInt(lexeme.str, 10)\n\n  if (isNaN(editDistance)) {\n    var errorMessage = \"edit distance must be numeric\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  parser.currentClause.editDistance = editDistance\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    parser.nextClause()\n    return\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      parser.nextClause()\n      return lunr.QueryParser.parseTerm\n    case lunr.QueryLexer.FIELD:\n      parser.nextClause()\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.EDIT_DISTANCE:\n      return lunr.QueryParser.parseEditDistance\n    case lunr.QueryLexer.BOOST:\n      return lunr.QueryParser.parseBoost\n    case lunr.QueryLexer.PRESENCE:\n      parser.nextClause()\n      return lunr.QueryParser.parsePresence\n    default:\n      var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseBoost = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  var boost = parseInt(lexeme.str, 10)\n\n  if (isNaN(boost)) {\n    var errorMessage = \"boost must be numeric\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  parser.currentClause.boost = boost\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    parser.nextClause()\n    return\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      parser.nextClause()\n      return lunr.QueryParser.parseTerm\n    case lunr.QueryLexer.FIELD:\n      parser.nextClause()\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.EDIT_DISTANCE:\n      return lunr.QueryParser.parseEditDistance\n    case lunr.QueryLexer.BOOST:\n      return lunr.QueryParser.parseBoost\n    case lunr.QueryLexer.PRESENCE:\n      parser.nextClause()\n      return lunr.QueryParser.parsePresence\n    default:\n      var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\n  /**\n   * export the module via AMD, CommonJS or as a browser global\n   * Export code from https://github.com/umdjs/umd/blob/master/returnExports.js\n   */\n  ;(function (root, factory) {\n    if (typeof define === 'function' && define.amd) {\n      // AMD. Register as an anonymous module.\n      define(factory)\n    } else if (typeof exports === 'object') {\n      /**\n       * Node. Does not work with strict CommonJS, but\n       * only CommonJS-like enviroments that support module.exports,\n       * like Node.\n       */\n      module.exports = factory()\n    } else {\n      // Browser globals (root is window)\n      root.lunr = factory()\n    }\n  }(this, function () {\n    /**\n     * Just return a value to define the module export.\n     * This example returns an object, but the module\n     * can return a function as the exported value.\n     */\n    return lunr\n  }))\n})();\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport lunr from \"lunr\"\n\nimport { getElement } from \"~/browser/element/_\"\nimport \"~/polyfills\"\n\nimport { Search } from \"../../_\"\nimport { SearchConfig } from \"../../config\"\nimport {\n  SearchMessage,\n  SearchMessageType\n} from \"../message\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Add support for `iframe-worker` shim\n *\n * While `importScripts` is synchronous when executed inside of a web worker,\n * it's not possible to provide a synchronous shim implementation. The cool\n * thing is that awaiting a non-Promise will convert it into a Promise, so\n * extending the type definition to return a `Promise` shouldn't break anything.\n *\n * @see https://bit.ly/2PjDnXi - GitHub comment\n *\n * @param urls - Scripts to load\n *\n * @returns Promise resolving with no result\n */\ndeclare global {\n  function importScripts(...urls: string[]): Promise<void> | void\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nlet index: Search\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch (= import) multi-language support through `lunr-languages`\n *\n * This function automatically imports the stemmers necessary to process the\n * languages which are defined as part of the search configuration.\n *\n * If the worker runs inside of an `iframe` (when using `iframe-worker` as\n * a shim), the base URL for the stemmers to be loaded must be determined by\n * searching for the first `script` element with a `src` attribute, which will\n * contain the contents of this script.\n *\n * @param config - Search configuration\n *\n * @returns Promise resolving with no result\n */\nasync function setupSearchLanguages(\n  config: SearchConfig\n): Promise<void> {\n  let base = \"../lunr\"\n\n  /* Detect `iframe-worker` and fix base URL */\n  if (typeof parent !== \"undefined\" && \"IFrameWorker\" in parent) {\n    const worker = getElement<HTMLScriptElement>(\"script[src]\")!\n    const [path] = worker.src.split(\"/worker\")\n\n    /* Prefix base with path */\n    base = base.replace(\"..\", path)\n  }\n\n  /* Add scripts for languages */\n  const scripts = []\n  for (const lang of config.lang) {\n    switch (lang) {\n\n      /* Add segmenter for Japanese */\n      case \"ja\":\n        scripts.push(`${base}/tinyseg.js`)\n        break\n\n      /* Add segmenter for Hindi and Thai */\n      case \"hi\":\n      case \"th\":\n        scripts.push(`${base}/wordcut.js`)\n        break\n    }\n\n    /* Add language support */\n    if (lang !== \"en\")\n      scripts.push(`${base}/min/lunr.${lang}.min.js`)\n  }\n\n  /* Add multi-language support */\n  if (config.lang.length > 1)\n    scripts.push(`${base}/min/lunr.multi.min.js`)\n\n  /* Load scripts synchronously */\n  if (scripts.length)\n    await importScripts(\n      `${base}/min/lunr.stemmer.support.min.js`,\n      ...scripts\n    )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Message handler\n *\n * @param message - Source message\n *\n * @returns Target message\n */\nexport async function handler(\n  message: SearchMessage\n): Promise<SearchMessage> {\n  switch (message.type) {\n\n    /* Search setup message */\n    case SearchMessageType.SETUP:\n      await setupSearchLanguages(message.data.config)\n      index = new Search(message.data)\n      return {\n        type: SearchMessageType.READY\n      }\n\n    /* Search query message */\n    case SearchMessageType.QUERY:\n      const query = message.data\n      try {\n        return {\n          type: SearchMessageType.RESULT,\n          data: index.search(query)\n        }\n\n      /* Return empty result in case of error */\n      } catch (err) {\n        console.warn(`Invalid query: ${query} \u2013 see https://bit.ly/2s3ChXG`)\n        console.warn(err)\n        return {\n          type: SearchMessageType.RESULT,\n          data: { items: [] }\n        }\n      }\n\n    /* All other messages */\n    default:\n      throw new TypeError(\"Invalid message type\")\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Worker\n * ------------------------------------------------------------------------- */\n\n/* Expose Lunr.js in global scope, or stemmers won't work */\nself.lunr = lunr\n\n/* Handle messages */\naddEventListener(\"message\", async ev => {\n  postMessage(await handler(ev.data))\n})\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve all elements matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getElements<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T][]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T[]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T[] {\n  return Array.from(node.querySelectorAll<T>(selector))\n}\n\n/**\n * Retrieve an element matching a query selector or throw a reference error\n *\n * Note that this function assumes that the element is present. If unsure if an\n * element is existent, use the `getOptionalElement` function instead.\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T]\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T {\n  const el = getOptionalElement<T>(selector, node)\n  if (typeof el === \"undefined\")\n    throw new ReferenceError(\n      `Missing element: expected \"${selector}\" to be present`\n    )\n\n  /* Return element */\n  return el\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Retrieve an optional element matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element or nothing\n */\nexport function getOptionalElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T] | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T | undefined {\n  return node.querySelector<T>(selector) || undefined\n}\n\n/**\n * Retrieve the currently active element\n *\n * @returns Element or nothing\n */\nexport function getActiveElement(): HTMLElement | undefined {\n  return document.activeElement instanceof HTMLElement\n    ? document.activeElement || undefined\n    : undefined\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Polyfills\n * ------------------------------------------------------------------------- */\n\n/* Polyfill `Object.entries` */\nif (!Object.entries)\n  Object.entries = function (obj: object) {\n    const data: [string, string][] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push([key, obj[key]])\n\n    /* Return entries */\n    return data\n  }\n\n/* Polyfill `Object.values` */\nif (!Object.values)\n  Object.values = function (obj: object) {\n    const data: string[] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push(obj[key])\n\n    /* Return values */\n    return data\n  }\n\n/* ------------------------------------------------------------------------- */\n\n/* Polyfills for `Element` */\nif (typeof Element !== \"undefined\") {\n\n  /* Polyfill `Element.scrollTo` */\n  if (!Element.prototype.scrollTo)\n    Element.prototype.scrollTo = function (\n      x?: ScrollToOptions | number, y?: number\n    ): void {\n      if (typeof x === \"object\") {\n        this.scrollLeft = x.left!\n        this.scrollTop = x.top!\n      } else {\n        this.scrollLeft = x!\n        this.scrollTop = y!\n      }\n    }\n\n  /* Polyfill `Element.replaceWith` */\n  if (!Element.prototype.replaceWith)\n    Element.prototype.replaceWith = function (\n      ...nodes: Array<string | Node>\n    ): void {\n      const parent = this.parentNode\n      if (parent) {\n        if (nodes.length === 0)\n          parent.removeChild(this)\n\n        /* Replace children and create text nodes */\n        for (let i = nodes.length - 1; i >= 0; i--) {\n          let node = nodes[i]\n          if (typeof node === \"string\")\n            node = document.createTextNode(node)\n          else if (node.parentNode)\n            node.parentNode.removeChild(node)\n\n          /* Replace child or insert before previous sibling */\n          if (!i)\n            parent.replaceChild(node, this)\n          else\n            parent.insertBefore(this.previousSibling!, node)\n        }\n      }\n    }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search configuration\n */\nexport interface SearchConfig {\n  lang: string[]                       /* Search languages */\n  separator: string                    /* Search separator */\n  pipeline: SearchPipelineFn[]         /* Search pipeline */\n}\n\n/**\n * Search document\n */\nexport interface SearchDocument {\n  location: string                     /* Document location */\n  title: string                        /* Document title */\n  text: string                         /* Document text */\n  tags?: string[]                      /* Document tags */\n  boost?: number                       /* Document boost */\n  parent?: SearchDocument              /* Document parent */\n}\n\n/**\n * Search options\n */\nexport interface SearchOptions {\n  suggest: boolean                     /* Search suggestions */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nexport interface SearchIndex {\n  config: SearchConfig                 /* Search configuration */\n  docs: SearchDocument[]               /* Search documents */\n  options: SearchOptions               /* Search options */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search pipeline function\n */\ntype SearchPipelineFn =\n  | \"trimmer\"                          /* Trimmer */\n  | \"stopWordFilter\"                   /* Stop word filter */\n  | \"stemmer\"                          /* Stemmer */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search document map\n *\n * This function creates a mapping of URLs (including anchors) to the actual\n * articles and sections. It relies on the invariant that the search index is\n * ordered with the main article appearing before all sections with anchors.\n * If this is not the case, the logic music be changed.\n *\n * @param docs - Search documents\n *\n * @returns Search document map\n */\nexport function setupSearchDocumentMap(\n  docs: SearchDocument[]\n): Map<string, SearchDocument> {\n  const map = new Map<string, SearchDocument>()\n  for (const doc of docs) {\n    const [path] = doc.location.split(\"#\")\n\n    /* Add document article */\n    const article = map.get(path)\n    if (typeof article === \"undefined\") {\n      map.set(path, doc)\n\n      /* Add document section */\n    } else {\n      map.set(doc.location, doc)\n      doc.parent = article\n    }\n  }\n\n  /* Return search document map */\n  return map\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Visitor function\n *\n * @param start - Start offset\n * @param end - End offset\n */\ntype VisitorFn = (\n  start: number, end: number\n) => void\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Split a string using the given separator\n *\n * @param input - Input value\n * @param separator - Separator\n * @param fn - Visitor function\n */\nexport function split(\n  input: string, separator: RegExp, fn: VisitorFn\n): void {\n  separator = new RegExp(separator, \"g\")\n\n  /* Split string using separator */\n  let match: RegExpExecArray | null\n  let index = 0\n  do {\n    match = separator.exec(input)\n\n    /* Emit non-empty range */\n    const until = match?.index ?? input.length\n    if (index < until)\n      fn(index, until)\n\n    /* Update last index */\n    if (match) {\n      const [term] = match\n      index = match.index + term.length\n\n      /* Support zero-length lookaheads */\n      if (term.length === 0)\n        separator.lastIndex = match.index + 1\n    }\n  } while (match)\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Extraction type\n *\n * This type defines the possible values that are encoded into the first two\n * bits of a section that is part of the blocks of a tokenization table. There\n * are three types of interest: HTML opening and closing tags, as well as the\n * actual text content we need to extract for indexing.\n */\nexport const enum Extract {\n  TAG_OPEN  = 0,                       /* HTML opening tag */\n  TEXT      = 1,                       /* Text content */\n  TAG_CLOSE = 2                        /* HTML closing tag */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Visitor function\n *\n * @param block - Block index\n * @param type - Extraction type\n * @param start - Start offset\n * @param end - End offset\n */\ntype VisitorFn = (\n  block: number, type: Extract, start: number, end: number\n) => void\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Split a string into markup and text sections\n *\n * This function scans a string and divides it up into sections of markup and\n * text. For each section, it invokes the given visitor function with the block\n * index, extraction type, as well as start and end offsets. Using a visitor\n * function (= streaming data) is ideal for minimizing pressure on the GC.\n *\n * @param input - Input value\n * @param fn - Visitor function\n */\nexport function extract(\n  input: string, fn: VisitorFn\n): void {\n\n  let block = 0                        /* Current block */\n  let start = 0                        /* Current start offset */\n  let end = 0                          /* Current end offset */\n\n  /* Split string into sections */\n  for (let stack = 0; end < input.length; end++) {\n\n    /* Opening tag after non-empty section */\n    if (input.charAt(end) === \"<\" && end > start) {\n      fn(block, Extract.TEXT, start, start = end)\n\n    /* Closing tag */\n    } else if (input.charAt(end) === \">\") {\n      if (input.charAt(start + 1) === \"/\") {\n        if (--stack === 0)\n          fn(block++, Extract.TAG_CLOSE, start, end + 1)\n\n      /* Tag is not self-closing */\n      } else if (input.charAt(end - 1) !== \"/\") {\n        if (stack++ === 0)\n          fn(block, Extract.TAG_OPEN, start, end + 1)\n      }\n\n      /* New section */\n      start = end + 1\n    }\n  }\n\n  /* Add trailing section */\n  if (end > start)\n    fn(block, Extract.TEXT, start, end)\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Position table\n */\nexport type PositionTable = number[][]\n\n/**\n * Position\n */\nexport type Position = number\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Highlight all occurrences in a string\n *\n * This function receives a field's value (e.g. like `title` or `text`), it's\n * position table that was generated during indexing, and the positions found\n * when executing the query. It then highlights all occurrences, and returns\n * their concatenation. In case of multiple blocks, two are returned.\n *\n * @param input - Input value\n * @param table - Table for indexing\n * @param positions - Occurrences\n *\n * @returns Highlighted string value\n */\nexport function highlight(\n  input: string, table: PositionTable, positions: Position[]\n): string {\n  return highlightAll([input], table, positions).pop()!\n}\n\n/**\n * Highlight all occurrences in a set of strings\n *\n * @param inputs - Input values\n * @param table - Table for indexing\n * @param positions - Occurrences\n *\n * @returns Highlighted string values\n */\nexport function highlightAll(\n  inputs: string[], table: PositionTable, positions: Position[]\n): string[] {\n\n  /* Map blocks to input values */\n  const mapping = [0]\n  for (let t = 1; t < table.length; t++) {\n    const prev = table[t - 1]\n    const next = table[t]\n\n    /* Check if table points to new block */\n    const p = prev[prev.length - 1] >>> 2 & 0x3FF\n    const q = next[0]               >>> 12\n\n    /* Add block to mapping */\n    mapping.push(+(p > q) + mapping[mapping.length - 1])\n  }\n\n  /* Highlight strings one after another */\n  return inputs.map((input, i) => {\n\n    /* Map occurrences to blocks */\n    const blocks = new Map<number, number[]>()\n    for (const p of positions.sort((a, b) => a - b)) {\n      const index = p & 0xFFFFF\n      const block = p >>> 20\n      if (mapping[block] !== i)\n        continue\n\n      /* Ensure presence of block group */\n      let group = blocks.get(block)\n      if (typeof group === \"undefined\")\n        blocks.set(block, group = [])\n\n      /* Add index to group */\n      group.push(index)\n    }\n\n    /* Just return string, if no occurrences */\n    if (blocks.size === 0)\n      return input\n\n    /* Compute slices */\n    const slices: string[] = []\n    for (const [block, indexes] of blocks) {\n      const t = table[block]\n\n      /* Extract positions and length */\n      const start  = t[0]            >>> 12\n      const end    = t[t.length - 1] >>> 12\n      const length = t[t.length - 1] >>> 2 & 0x3FF\n\n      /* Extract and highlight slice */\n      let slice = input.slice(start, end + length)\n      for (const j of indexes.sort((a, b) => b - a)) {\n\n        /* Retrieve offset and length of match */\n        const p = (t[j] >>> 12) - start\n        const q = (t[j] >>> 2 & 0x3FF) + p\n\n        /* Wrap occurrence */\n        slice = [\n          slice.slice(0, p),\n          \"<mark>\",\n          slice.slice(p, q),\n          \"</mark>\",\n          slice.slice(q)\n        ].join(\"\")\n      }\n\n      /* Append slice and abort if we have two */\n      if (slices.push(slice) === 2)\n        break\n    }\n\n    /* Return highlighted slices */\n    return slices.join(\"\")\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { split } from \"../_\"\nimport {\n  Extract,\n  extract\n} from \"../extract\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Split a string or set of strings into tokens\n *\n * This tokenizer supersedes the default tokenizer that is provided by Lunr.js,\n * as it is aware of HTML tags and allows for multi-character splitting.\n *\n * It takes the given inputs, splits each of them into markup and text sections,\n * tokenizes and segments (if necessary) each of them, and then indexes them in\n * a table by using a compact bit representation. Bitwise techniques are used\n * to write and read from the table during indexing and querying.\n *\n * @see https://bit.ly/3W3Xw4J - Search: better, faster, smaller\n *\n * @param input - Input value(s)\n *\n * @returns Tokens\n */\nexport function tokenize(\n  input?: string | string[]\n): lunr.Token[] {\n  const tokens: lunr.Token[] = []\n  if (typeof input === \"undefined\")\n    return tokens\n\n  /* Tokenize strings one after another */\n  const inputs = Array.isArray(input) ? input : [input]\n  for (let i = 0; i < inputs.length; i++) {\n    const table = lunr.tokenizer.table\n    const total = table.length\n\n    /* Split string into sections and tokenize content blocks */\n    extract(inputs[i], (block, type, start, end) => {\n      table[block += total] ||= []\n      switch (type) {\n\n        /* Handle markup */\n        case Extract.TAG_OPEN:\n        case Extract.TAG_CLOSE:\n          table[block].push(\n            start       << 12 |\n            end - start <<  2 |\n            type\n          )\n          break\n\n        /* Handle text content */\n        case Extract.TEXT:\n          const section = inputs[i].slice(start, end)\n          split(section, lunr.tokenizer.separator, (index, until) => {\n\n            /**\n             * Apply segmenter after tokenization. Note that the segmenter will\n             * also split words at word boundaries, which is not what we want,\n             * so we need to check if we can somehow mitigate this behavior.\n             */\n            if (typeof lunr.segmenter !== \"undefined\") {\n              const subsection = section.slice(index, until)\n              if (/^[MHIK]$/.test(lunr.segmenter.ctype_(subsection))) {\n                const segments = lunr.segmenter.segment(subsection)\n                for (let s = 0, l = 0; s < segments.length; s++) {\n\n                  /* Add block to section */\n                  table[block] ||= []\n                  table[block].push(\n                    start + index + l  << 12 |\n                    segments[s].length <<  2 |\n                    type\n                  )\n\n                  /* Add token with position */\n                  tokens.push(new lunr.Token(\n                    segments[s].toLowerCase(), {\n                      position: block << 20 | table[block].length - 1\n                    }\n                  ))\n\n                  /* Keep track of length */\n                  l += segments[s].length\n                }\n                return\n              }\n            }\n\n            /* Add block to section */\n            table[block].push(\n              start + index << 12 |\n              until - index <<  2 |\n              type\n            )\n\n            /* Add token with position */\n            tokens.push(new lunr.Token(\n              section.slice(index, until).toLowerCase(), {\n                position: block << 20 | table[block].length - 1\n              }\n            ))\n          })\n      }\n    })\n  }\n\n  /* Return tokens */\n  return tokens\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Visitor function\n *\n * @param value - String value\n *\n * @returns String term(s)\n */\ntype VisitorFn = (\n  value: string\n) => string | string[]\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Default transformation function\n *\n * 1. Trim excess whitespace from left and right.\n *\n * 2. Search for parts in quotation marks and prepend a `+` modifier to denote\n *    that the resulting document must contain all parts, converting the query\n *    to an `AND` query (as opposed to the default `OR` behavior). While users\n *    may expect parts enclosed in quotation marks to map to span queries, i.e.\n *    for which order is important, Lunr.js doesn't support them, so the best\n *    we can do is to convert the parts to an `AND` query.\n *\n * 3. Replace control characters which are not located at the beginning of the\n *    query or preceded by white space, or are not followed by a non-whitespace\n *    character or are at the end of the query string. Furthermore, filter\n *    unmatched quotation marks.\n *\n * 4. Split the query string at whitespace, then pass each part to the visitor\n *    function for tokenization, and append a wildcard to every resulting term\n *    that is not explicitly marked with a `+`, `-`, `~` or `^` modifier, since\n *    it ensures consistent and stable ranking when multiple terms are entered.\n *    Also, if a fuzzy or boost modifier are given, but no numeric value has\n *    been entered, default to 1 to not induce a query error.\n *\n * @param query - Query value\n * @param fn - Visitor function\n *\n * @returns Transformed query value\n */\nexport function transform(\n  query: string, fn: VisitorFn = term => term\n): string {\n  return query\n\n    /* => 1 */\n    .trim()\n\n    /* => 2 */\n    .split(/\"([^\"]+)\"/g)\n      .map((parts, index) => index & 1\n        ? parts.replace(/^\\b|^(?![^\\x00-\\x7F]|$)|\\s+/g, \" +\")\n        : parts\n      )\n      .join(\"\")\n\n    /* => 3 */\n    .replace(/\"|(?:^|\\s+)[*+\\-:^~]+(?=\\s+|$)/g, \"\")\n\n    /* => 4 */\n    .split(/\\s+/g)\n      .reduce((prev, term) => {\n        const next = fn(term)\n        return [...prev, ...Array.isArray(next) ? next : [next]]\n      }, [] as string[])\n      .map(term => /([~^]$)/.test(term) ? `${term}1` : term)\n      .map(term => /(^[+-]|[~^]\\d+$)/.test(term) ? term : `${term}*`)\n      .join(\" \")\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { split } from \"../../internal\"\nimport { transform } from \"../transform\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search query clause\n */\nexport interface SearchQueryClause {\n  presence: lunr.Query.presence        /* Clause presence */\n  term: string                         /* Clause term */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search query terms\n */\nexport type SearchQueryTerms = Record<string, boolean>\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Transform search query\n *\n * This function lexes the given search query and applies the transformation\n * function to each term, preserving markup like `+` and `-` modifiers.\n *\n * @param query - Search query\n *\n * @returns Search query\n */\nexport function transformSearchQuery(\n  query: string\n): string {\n\n  /* Split query terms with tokenizer */\n  return transform(query, part => {\n    const terms: string[] = []\n\n    /* Initialize lexer and analyze part */\n    const lexer = new lunr.QueryLexer(part)\n    lexer.run()\n\n    /* Extract and tokenize term from lexeme */\n    for (const { type, str: term, start, end } of lexer.lexemes)\n      switch (type) {\n\n        /* Hack: remove colon - see https://bit.ly/3wD3T3I */\n        case \"FIELD\":\n          if (![\"title\", \"text\", \"tags\"].includes(term))\n            part = [\n              part.slice(0, end),\n              \" \",\n              part.slice(end + 1)\n            ].join(\"\")\n          break\n\n        /* Tokenize term */\n        case \"TERM\":\n          split(term, lunr.tokenizer.separator, (...range) => {\n            terms.push([\n              part.slice(0, start),\n              term.slice(...range),\n              part.slice(end)\n            ].join(\"\"))\n          })\n      }\n\n    /* Return terms */\n    return terms\n  })\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Parse a search query for analysis\n *\n * Lunr.js itself has a bug where it doesn't detect or remove wildcards for\n * query clauses, so we must do this here.\n *\n * @see https://bit.ly/3DpTGtz - GitHub issue\n *\n * @param value - Query value\n *\n * @returns Search query clauses\n */\nexport function parseSearchQuery(\n  value: string\n): SearchQueryClause[] {\n  const query  = new lunr.Query([\"title\", \"text\", \"tags\"])\n  const parser = new lunr.QueryParser(value, query)\n\n  /* Parse Search query */\n  parser.parse()\n  for (const clause of query.clauses) {\n    clause.usePipeline = true\n\n    /* Handle leading wildcard */\n    if (clause.term.startsWith(\"*\")) {\n      clause.wildcard = lunr.Query.wildcard.LEADING\n      clause.term = clause.term.slice(1)\n    }\n\n    /* Handle trailing wildcard */\n    if (clause.term.endsWith(\"*\")) {\n      clause.wildcard = lunr.Query.wildcard.TRAILING\n      clause.term = clause.term.slice(0, -1)\n    }\n  }\n\n  /* Return query clauses */\n  return query.clauses\n}\n\n/**\n * Analyze the search query clauses in regard to the search terms found\n *\n * @param query - Search query clauses\n * @param terms - Search terms\n *\n * @returns Search query terms\n */\nexport function getSearchQueryTerms(\n  query: SearchQueryClause[], terms: string[]\n): SearchQueryTerms {\n  const clauses = new Set<SearchQueryClause>(query)\n\n  /* Match query clauses against terms */\n  const result: SearchQueryTerms = {}\n  for (let t = 0; t < terms.length; t++)\n    for (const clause of clauses)\n      if (terms[t].startsWith(clause.term)) {\n        result[clause.term] = true\n        clauses.delete(clause)\n      }\n\n  /* Annotate unmatched non-stopword query clauses */\n  for (const clause of clauses)\n    if (lunr.stopWordFilter?.(clause.term))\n      result[clause.term] = false\n\n  /* Return query terms */\n  return result\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Segment a search query using the inverted index\n *\n * This function implements a clever approach to text segmentation for Asian\n * languages, as it used the information already available in the search index.\n * The idea is to greedily segment the search query based on the tokens that are\n * already part of the index, as described in the linked issue.\n *\n * @see https://bit.ly/3lwjrk7 - GitHub issue\n *\n * @param query - Query value\n * @param index - Inverted index\n *\n * @returns Segmented query value\n */\nexport function segment(\n  query: string, index: object\n): Iterable<string> {\n  const segments = new Set<string>()\n\n  /* Segment search query */\n  const wordcuts = new Uint16Array(query.length)\n  for (let i = 0; i < query.length; i++)\n    for (let j = i + 1; j < query.length; j++) {\n      const value = query.slice(i, j)\n      if (value in index)\n        wordcuts[i] = j - i\n    }\n\n  /* Compute longest matches with minimum overlap */\n  const stack = [0]\n  for (let s = stack.length; s > 0;) {\n    const p = stack[--s]\n    for (let q = 1; q < wordcuts[p]; q++)\n      if (wordcuts[p + q] > wordcuts[p] - q) {\n        segments.add(query.slice(p, p + q))\n        stack[s++] = p + q\n      }\n\n    /* Continue at end of query string */\n    const q = p + wordcuts[p]\n    if (wordcuts[q] && q < query.length - 1)\n      stack[s++] = q\n\n    /* Add current segment */\n    segments.add(query.slice(p, q))\n  }\n\n  // @todo fix this case in the code block above, this is a hotfix\n  if (segments.has(\"\"))\n    return new Set([query])\n\n  /* Return segmented query value */\n  return segments\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  SearchDocument,\n  SearchIndex,\n  SearchOptions,\n  setupSearchDocumentMap\n} from \"../config\"\nimport {\n  Position,\n  PositionTable,\n  highlight,\n  highlightAll,\n  tokenize\n} from \"../internal\"\nimport {\n  SearchQueryTerms,\n  getSearchQueryTerms,\n  parseSearchQuery,\n  segment,\n  transformSearchQuery\n} from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search item\n */\nexport interface SearchItem\n  extends SearchDocument\n{\n  score: number                        /* Score (relevance) */\n  terms: SearchQueryTerms              /* Search query terms */\n}\n\n/**\n * Search result\n */\nexport interface SearchResult {\n  items: SearchItem[][]                /* Search items */\n  suggest?: string[]                   /* Search suggestions */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create field extractor factory\n *\n * @param table - Position table map\n *\n * @returns Extractor factory\n */\nfunction extractor(table: Map<string, PositionTable>) {\n  return (name: keyof SearchDocument) => {\n    return (doc: SearchDocument) => {\n      if (typeof doc[name] === \"undefined\")\n        return undefined\n\n      /* Compute identifier and initialize table */\n      const id = [doc.location, name].join(\":\")\n      table.set(id, lunr.tokenizer.table = [])\n\n      /* Return field value */\n      return doc[name]\n    }\n  }\n}\n\n/**\n * Compute the difference of two lists of strings\n *\n * @param a - 1st list of strings\n * @param b - 2nd list of strings\n *\n * @returns Difference\n */\nfunction difference(a: string[], b: string[]): string[] {\n  const [x, y] = [new Set(a), new Set(b)]\n  return [\n    ...new Set([...x].filter(value => !y.has(value)))\n  ]\n}\n\n/* ----------------------------------------------------------------------------\n * Class\n * ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nexport class Search {\n\n  /**\n   * Search document map\n   */\n  protected map: Map<string, SearchDocument>\n\n  /**\n   * Search options\n   */\n  protected options: SearchOptions\n\n  /**\n   * The underlying Lunr.js search index\n   */\n  protected index: lunr.Index\n\n  /**\n   * Internal position table map\n   */\n  protected table: Map<string, PositionTable>\n\n  /**\n   * Create the search integration\n   *\n   * @param data - Search index\n   */\n  public constructor({ config, docs, options }: SearchIndex) {\n    const field = extractor(this.table = new Map())\n\n    /* Set up document map and options */\n    this.map = setupSearchDocumentMap(docs)\n    this.options = options\n\n    /* Set up document index */\n    this.index = lunr(function () {\n      this.metadataWhitelist = [\"position\"]\n      this.b(0)\n\n      /* Set up (multi-)language support */\n      if (config.lang.length === 1 && config.lang[0] !== \"en\") {\n        // @ts-expect-error - namespace indexing not supported\n        this.use(lunr[config.lang[0]])\n      } else if (config.lang.length > 1) {\n        this.use(lunr.multiLanguage(...config.lang))\n      }\n\n      /* Set up custom tokenizer (must be after language setup) */\n      this.tokenizer = tokenize as typeof lunr.tokenizer\n      lunr.tokenizer.separator = new RegExp(config.separator)\n\n      /* Set up custom segmenter, if loaded */\n      lunr.segmenter = \"TinySegmenter\" in lunr\n        ? new lunr.TinySegmenter()\n        : undefined\n\n      /* Compute functions to be removed from the pipeline */\n      const fns = difference([\n        \"trimmer\", \"stopWordFilter\", \"stemmer\"\n      ], config.pipeline)\n\n      /* Remove functions from the pipeline for registered languages */\n      for (const lang of config.lang.map(language => (\n        // @ts-expect-error - namespace indexing not supported\n        language === \"en\" ? lunr : lunr[language]\n      )))\n        for (const fn of fns) {\n          this.pipeline.remove(lang[fn])\n          this.searchPipeline.remove(lang[fn])\n        }\n\n      /* Set up index reference */\n      this.ref(\"location\")\n\n      /* Set up index fields */\n      this.field(\"title\", { boost: 1e3, extractor: field(\"title\") })\n      this.field(\"text\",  { boost: 1e0, extractor: field(\"text\") })\n      this.field(\"tags\",  { boost: 1e6, extractor: field(\"tags\") })\n\n      /* Add documents to index */\n      for (const doc of docs)\n        this.add(doc, { boost: doc.boost })\n    })\n  }\n\n  /**\n   * Search for matching documents\n   *\n   * @param query - Search query\n   *\n   * @returns Search result\n   */\n  public search(query: string): SearchResult {\n\n    // Experimental Chinese segmentation\n    query = query.replace(/\\p{sc=Han}+/gu, value => {\n      return [...segment(value, this.index.invertedIndex)]\n        .join(\"* \")\n    })\n\n    // @todo: move segmenter (above) into transformSearchQuery\n    query = transformSearchQuery(query)\n    if (!query)\n      return { items: [] }\n\n    /* Parse query to extract clauses for analysis */\n    const clauses = parseSearchQuery(query)\n      .filter(clause => (\n        clause.presence !== lunr.Query.presence.PROHIBITED\n      ))\n\n    /* Perform search and post-process results */\n    const groups = this.index.search(query)\n\n      /* Apply post-query boosts based on title and search query terms */\n      .reduce<SearchItem[]>((item, { ref, score, matchData }) => {\n        let doc = this.map.get(ref)\n        if (typeof doc !== \"undefined\") {\n\n          /* Shallow copy document */\n          doc = { ...doc }\n          if (doc.tags)\n            doc.tags = [...doc.tags]\n\n          /* Compute and analyze search query terms */\n          const terms = getSearchQueryTerms(\n            clauses,\n            Object.keys(matchData.metadata)\n          )\n\n          /* Highlight matches in fields */\n          for (const field of this.index.fields) {\n            if (typeof doc[field] === \"undefined\")\n              continue\n\n            /* Collect positions from matches */\n            const positions: Position[] = []\n            for (const match of Object.values(matchData.metadata))\n              if (typeof match[field] !== \"undefined\")\n                positions.push(...match[field].position)\n\n            /* Skip highlighting, if no positions were collected */\n            if (!positions.length)\n              continue\n\n            /* Load table and determine highlighting method */\n            const table = this.table.get([doc.location, field].join(\":\"))!\n            const fn = Array.isArray(doc[field])\n              ? highlightAll\n              : highlight\n\n            // @ts-expect-error - stop moaning, TypeScript!\n            doc[field] = fn(doc[field], table, positions)\n          }\n\n          /* Highlight title and text and apply post-query boosts */\n          const boost = +!doc.parent +\n            Object.values(terms)\n              .filter(t => t).length /\n            Object.keys(terms).length\n\n          /* Append item */\n          item.push({\n            ...doc,\n            score: score * (1 + boost ** 2),\n            terms\n          })\n        }\n        return item\n      }, [])\n\n      /* Sort search results again after applying boosts */\n      .sort((a, b) => b.score - a.score)\n\n      /* Group search results by article */\n      .reduce((items, result) => {\n        const doc = this.map.get(result.location)\n        if (typeof doc !== \"undefined\") {\n          const ref = doc.parent\n            ? doc.parent.location\n            : doc.location\n          items.set(ref, [...items.get(ref) || [], result])\n        }\n        return items\n      }, new Map<string, SearchItem[]>())\n\n    /* Ensure that every item set has an article */\n    for (const [ref, items] of groups)\n      if (!items.find(item => item.location === ref)) {\n        const doc = this.map.get(ref)!\n        items.push({ ...doc, score: 0, terms: {} })\n      }\n\n    /* Generate search suggestions, if desired */\n    let suggest: string[] | undefined\n    if (this.options.suggest) {\n      const titles = this.index.query(builder => {\n        for (const clause of clauses)\n          builder.term(clause.term, {\n            fields: [\"title\"],\n            presence: lunr.Query.presence.REQUIRED,\n            wildcard: lunr.Query.wildcard.TRAILING\n          })\n      })\n\n      /* Retrieve suggestions for best match */\n      suggest = titles.length\n        ? Object.keys(titles[0].matchData.metadata)\n        : []\n    }\n\n    /* Return search result */\n    return {\n      items: [...groups.values()],\n      ...typeof suggest !== \"undefined\" && { suggest }\n    }\n  }\n}\n"],
-  "mappings": 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+  "sourcesContent": ["/**\n * lunr - http://lunrjs.com - A bit like Solr, but much smaller and not as bright - 2.3.9\n * Copyright (C) 2020 Oliver Nightingale\n * @license MIT\n */\n\n;(function(){\n\n/**\n * A convenience function for configuring and constructing\n * a new lunr Index.\n *\n * A lunr.Builder instance is created and the pipeline setup\n * with a trimmer, stop word filter and stemmer.\n *\n * This builder object is yielded to the configuration function\n * that is passed as a parameter, allowing the list of fields\n * and other builder parameters to be customised.\n *\n * All documents _must_ be added within the passed config function.\n *\n * @example\n * var idx = lunr(function () {\n *   this.field('title')\n *   this.field('body')\n *   this.ref('id')\n *\n *   documents.forEach(function (doc) {\n *     this.add(doc)\n *   }, this)\n * })\n *\n * @see {@link lunr.Builder}\n * @see {@link lunr.Pipeline}\n * @see {@link lunr.trimmer}\n * @see {@link lunr.stopWordFilter}\n * @see {@link lunr.stemmer}\n * @namespace {function} lunr\n */\nvar lunr = function (config) {\n  var builder = new lunr.Builder\n\n  builder.pipeline.add(\n    lunr.trimmer,\n    lunr.stopWordFilter,\n    lunr.stemmer\n  )\n\n  builder.searchPipeline.add(\n    lunr.stemmer\n  )\n\n  config.call(builder, builder)\n  return builder.build()\n}\n\nlunr.version = \"2.3.9\"\n/*!\n * lunr.utils\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A namespace containing utils for the rest of the lunr library\n * @namespace lunr.utils\n */\nlunr.utils = {}\n\n/**\n * Print a warning message to the console.\n *\n * @param {String} message The message to be printed.\n * @memberOf lunr.utils\n * @function\n */\nlunr.utils.warn = (function (global) {\n  /* eslint-disable no-console */\n  return function (message) {\n    if (global.console && console.warn) {\n      console.warn(message)\n    }\n  }\n  /* eslint-enable no-console */\n})(this)\n\n/**\n * Convert an object to a string.\n *\n * In the case of `null` and `undefined` the function returns\n * the empty string, in all other cases the result of calling\n * `toString` on the passed object is returned.\n *\n * @param {Any} obj The object to convert to a string.\n * @return {String} string representation of the passed object.\n * @memberOf lunr.utils\n */\nlunr.utils.asString = function (obj) {\n  if (obj === void 0 || obj === null) {\n    return \"\"\n  } else {\n    return obj.toString()\n  }\n}\n\n/**\n * Clones an object.\n *\n * Will create a copy of an existing object such that any mutations\n * on the copy cannot affect the original.\n *\n * Only shallow objects are supported, passing a nested object to this\n * function will cause a TypeError.\n *\n * Objects with primitives, and arrays of primitives are supported.\n *\n * @param {Object} obj The object to clone.\n * @return {Object} a clone of the passed object.\n * @throws {TypeError} when a nested object is passed.\n * @memberOf Utils\n */\nlunr.utils.clone = function (obj) {\n  if (obj === null || obj === undefined) {\n    return obj\n  }\n\n  var clone = Object.create(null),\n      keys = Object.keys(obj)\n\n  for (var i = 0; i < keys.length; i++) {\n    var key = keys[i],\n        val = obj[key]\n\n    if (Array.isArray(val)) {\n      clone[key] = val.slice()\n      continue\n    }\n\n    if (typeof val === 'string' ||\n        typeof val === 'number' ||\n        typeof val === 'boolean') {\n      clone[key] = val\n      continue\n    }\n\n    throw new TypeError(\"clone is not deep and does not support nested objects\")\n  }\n\n  return clone\n}\nlunr.FieldRef = function (docRef, fieldName, stringValue) {\n  this.docRef = docRef\n  this.fieldName = fieldName\n  this._stringValue = stringValue\n}\n\nlunr.FieldRef.joiner = \"/\"\n\nlunr.FieldRef.fromString = function (s) {\n  var n = s.indexOf(lunr.FieldRef.joiner)\n\n  if (n === -1) {\n    throw \"malformed field ref string\"\n  }\n\n  var fieldRef = s.slice(0, n),\n      docRef = s.slice(n + 1)\n\n  return new lunr.FieldRef (docRef, fieldRef, s)\n}\n\nlunr.FieldRef.prototype.toString = function () {\n  if (this._stringValue == undefined) {\n    this._stringValue = this.fieldName + lunr.FieldRef.joiner + this.docRef\n  }\n\n  return this._stringValue\n}\n/*!\n * lunr.Set\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A lunr set.\n *\n * @constructor\n */\nlunr.Set = function (elements) {\n  this.elements = Object.create(null)\n\n  if (elements) {\n    this.length = elements.length\n\n    for (var i = 0; i < this.length; i++) {\n      this.elements[elements[i]] = true\n    }\n  } else {\n    this.length = 0\n  }\n}\n\n/**\n * A complete set that contains all elements.\n *\n * @static\n * @readonly\n * @type {lunr.Set}\n */\nlunr.Set.complete = {\n  intersect: function (other) {\n    return other\n  },\n\n  union: function () {\n    return this\n  },\n\n  contains: function () {\n    return true\n  }\n}\n\n/**\n * An empty set that contains no elements.\n *\n * @static\n * @readonly\n * @type {lunr.Set}\n */\nlunr.Set.empty = {\n  intersect: function () {\n    return this\n  },\n\n  union: function (other) {\n    return other\n  },\n\n  contains: function () {\n    return false\n  }\n}\n\n/**\n * Returns true if this set contains the specified object.\n *\n * @param {object} object - Object whose presence in this set is to be tested.\n * @returns {boolean} - True if this set contains the specified object.\n */\nlunr.Set.prototype.contains = function (object) {\n  return !!this.elements[object]\n}\n\n/**\n * Returns a new set containing only the elements that are present in both\n * this set and the specified set.\n *\n * @param {lunr.Set} other - set to intersect with this set.\n * @returns {lunr.Set} a new set that is the intersection of this and the specified set.\n */\n\nlunr.Set.prototype.intersect = function (other) {\n  var a, b, elements, intersection = []\n\n  if (other === lunr.Set.complete) {\n    return this\n  }\n\n  if (other === lunr.Set.empty) {\n    return other\n  }\n\n  if (this.length < other.length) {\n    a = this\n    b = other\n  } else {\n    a = other\n    b = this\n  }\n\n  elements = Object.keys(a.elements)\n\n  for (var i = 0; i < elements.length; i++) {\n    var element = elements[i]\n    if (element in b.elements) {\n      intersection.push(element)\n    }\n  }\n\n  return new lunr.Set (intersection)\n}\n\n/**\n * Returns a new set combining the elements of this and the specified set.\n *\n * @param {lunr.Set} other - set to union with this set.\n * @return {lunr.Set} a new set that is the union of this and the specified set.\n */\n\nlunr.Set.prototype.union = function (other) {\n  if (other === lunr.Set.complete) {\n    return lunr.Set.complete\n  }\n\n  if (other === lunr.Set.empty) {\n    return this\n  }\n\n  return new lunr.Set(Object.keys(this.elements).concat(Object.keys(other.elements)))\n}\n/**\n * A function to calculate the inverse document frequency for\n * a posting. This is shared between the builder and the index\n *\n * @private\n * @param {object} posting - The posting for a given term\n * @param {number} documentCount - The total number of documents.\n */\nlunr.idf = function (posting, documentCount) {\n  var documentsWithTerm = 0\n\n  for (var fieldName in posting) {\n    if (fieldName == '_index') continue // Ignore the term index, its not a field\n    documentsWithTerm += Object.keys(posting[fieldName]).length\n  }\n\n  var x = (documentCount - documentsWithTerm + 0.5) / (documentsWithTerm + 0.5)\n\n  return Math.log(1 + Math.abs(x))\n}\n\n/**\n * A token wraps a string representation of a token\n * as it is passed through the text processing pipeline.\n *\n * @constructor\n * @param {string} [str=''] - The string token being wrapped.\n * @param {object} [metadata={}] - Metadata associated with this token.\n */\nlunr.Token = function (str, metadata) {\n  this.str = str || \"\"\n  this.metadata = metadata || {}\n}\n\n/**\n * Returns the token string that is being wrapped by this object.\n *\n * @returns {string}\n */\nlunr.Token.prototype.toString = function () {\n  return this.str\n}\n\n/**\n * A token update function is used when updating or optionally\n * when cloning a token.\n *\n * @callback lunr.Token~updateFunction\n * @param {string} str - The string representation of the token.\n * @param {Object} metadata - All metadata associated with this token.\n */\n\n/**\n * Applies the given function to the wrapped string token.\n *\n * @example\n * token.update(function (str, metadata) {\n *   return str.toUpperCase()\n * })\n *\n * @param {lunr.Token~updateFunction} fn - A function to apply to the token string.\n * @returns {lunr.Token}\n */\nlunr.Token.prototype.update = function (fn) {\n  this.str = fn(this.str, this.metadata)\n  return this\n}\n\n/**\n * Creates a clone of this token. Optionally a function can be\n * applied to the cloned token.\n *\n * @param {lunr.Token~updateFunction} [fn] - An optional function to apply to the cloned token.\n * @returns {lunr.Token}\n */\nlunr.Token.prototype.clone = function (fn) {\n  fn = fn || function (s) { return s }\n  return new lunr.Token (fn(this.str, this.metadata), this.metadata)\n}\n/*!\n * lunr.tokenizer\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A function for splitting a string into tokens ready to be inserted into\n * the search index. Uses `lunr.tokenizer.separator` to split strings, change\n * the value of this property to change how strings are split into tokens.\n *\n * This tokenizer will convert its parameter to a string by calling `toString` and\n * then will split this string on the character in `lunr.tokenizer.separator`.\n * Arrays will have their elements converted to strings and wrapped in a lunr.Token.\n *\n * Optional metadata can be passed to the tokenizer, this metadata will be cloned and\n * added as metadata to every token that is created from the object to be tokenized.\n *\n * @static\n * @param {?(string|object|object[])} obj - The object to convert into tokens\n * @param {?object} metadata - Optional metadata to associate with every token\n * @returns {lunr.Token[]}\n * @see {@link lunr.Pipeline}\n */\nlunr.tokenizer = function (obj, metadata) {\n  if (obj == null || obj == undefined) {\n    return []\n  }\n\n  if (Array.isArray(obj)) {\n    return obj.map(function (t) {\n      return new lunr.Token(\n        lunr.utils.asString(t).toLowerCase(),\n        lunr.utils.clone(metadata)\n      )\n    })\n  }\n\n  var str = obj.toString().toLowerCase(),\n      len = str.length,\n      tokens = []\n\n  for (var sliceEnd = 0, sliceStart = 0; sliceEnd <= len; sliceEnd++) {\n    var char = str.charAt(sliceEnd),\n        sliceLength = sliceEnd - sliceStart\n\n    if ((char.match(lunr.tokenizer.separator) || sliceEnd == len)) {\n\n      if (sliceLength > 0) {\n        var tokenMetadata = lunr.utils.clone(metadata) || {}\n        tokenMetadata[\"position\"] = [sliceStart, sliceLength]\n        tokenMetadata[\"index\"] = tokens.length\n\n        tokens.push(\n          new lunr.Token (\n            str.slice(sliceStart, sliceEnd),\n            tokenMetadata\n          )\n        )\n      }\n\n      sliceStart = sliceEnd + 1\n    }\n\n  }\n\n  return tokens\n}\n\n/**\n * The separator used to split a string into tokens. Override this property to change the behaviour of\n * `lunr.tokenizer` behaviour when tokenizing strings. By default this splits on whitespace and hyphens.\n *\n * @static\n * @see lunr.tokenizer\n */\nlunr.tokenizer.separator = /[\\s\\-]+/\n/*!\n * lunr.Pipeline\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.Pipelines maintain an ordered list of functions to be applied to all\n * tokens in documents entering the search index and queries being ran against\n * the index.\n *\n * An instance of lunr.Index created with the lunr shortcut will contain a\n * pipeline with a stop word filter and an English language stemmer. Extra\n * functions can be added before or after either of these functions or these\n * default functions can be removed.\n *\n * When run the pipeline will call each function in turn, passing a token, the\n * index of that token in the original list of all tokens and finally a list of\n * all the original tokens.\n *\n * The output of functions in the pipeline will be passed to the next function\n * in the pipeline. To exclude a token from entering the index the function\n * should return undefined, the rest of the pipeline will not be called with\n * this token.\n *\n * For serialisation of pipelines to work, all functions used in an instance of\n * a pipeline should be registered with lunr.Pipeline. Registered functions can\n * then be loaded. If trying to load a serialised pipeline that uses functions\n * that are not registered an error will be thrown.\n *\n * If not planning on serialising the pipeline then registering pipeline functions\n * is not necessary.\n *\n * @constructor\n */\nlunr.Pipeline = function () {\n  this._stack = []\n}\n\nlunr.Pipeline.registeredFunctions = Object.create(null)\n\n/**\n * A pipeline function maps lunr.Token to lunr.Token. A lunr.Token contains the token\n * string as well as all known metadata. A pipeline function can mutate the token string\n * or mutate (or add) metadata for a given token.\n *\n * A pipeline function can indicate that the passed token should be discarded by returning\n * null, undefined or an empty string. This token will not be passed to any downstream pipeline\n * functions and will not be added to the index.\n *\n * Multiple tokens can be returned by returning an array of tokens. Each token will be passed\n * to any downstream pipeline functions and all will returned tokens will be added to the index.\n *\n * Any number of pipeline functions may be chained together using a lunr.Pipeline.\n *\n * @interface lunr.PipelineFunction\n * @param {lunr.Token} token - A token from the document being processed.\n * @param {number} i - The index of this token in the complete list of tokens for this document/field.\n * @param {lunr.Token[]} tokens - All tokens for this document/field.\n * @returns {(?lunr.Token|lunr.Token[])}\n */\n\n/**\n * Register a function with the pipeline.\n *\n * Functions that are used in the pipeline should be registered if the pipeline\n * needs to be serialised, or a serialised pipeline needs to be loaded.\n *\n * Registering a function does not add it to a pipeline, functions must still be\n * added to instances of the pipeline for them to be used when running a pipeline.\n *\n * @param {lunr.PipelineFunction} fn - The function to check for.\n * @param {String} label - The label to register this function with\n */\nlunr.Pipeline.registerFunction = function (fn, label) {\n  if (label in this.registeredFunctions) {\n    lunr.utils.warn('Overwriting existing registered function: ' + label)\n  }\n\n  fn.label = label\n  lunr.Pipeline.registeredFunctions[fn.label] = fn\n}\n\n/**\n * Warns if the function is not registered as a Pipeline function.\n *\n * @param {lunr.PipelineFunction} fn - The function to check for.\n * @private\n */\nlunr.Pipeline.warnIfFunctionNotRegistered = function (fn) {\n  var isRegistered = fn.label && (fn.label in this.registeredFunctions)\n\n  if (!isRegistered) {\n    lunr.utils.warn('Function is not registered with pipeline. This may cause problems when serialising the index.\\n', fn)\n  }\n}\n\n/**\n * Loads a previously serialised pipeline.\n *\n * All functions to be loaded must already be registered with lunr.Pipeline.\n * If any function from the serialised data has not been registered then an\n * error will be thrown.\n *\n * @param {Object} serialised - The serialised pipeline to load.\n * @returns {lunr.Pipeline}\n */\nlunr.Pipeline.load = function (serialised) {\n  var pipeline = new lunr.Pipeline\n\n  serialised.forEach(function (fnName) {\n    var fn = lunr.Pipeline.registeredFunctions[fnName]\n\n    if (fn) {\n      pipeline.add(fn)\n    } else {\n      throw new Error('Cannot load unregistered function: ' + fnName)\n    }\n  })\n\n  return pipeline\n}\n\n/**\n * Adds new functions to the end of the pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction[]} functions - Any number of functions to add to the pipeline.\n */\nlunr.Pipeline.prototype.add = function () {\n  var fns = Array.prototype.slice.call(arguments)\n\n  fns.forEach(function (fn) {\n    lunr.Pipeline.warnIfFunctionNotRegistered(fn)\n    this._stack.push(fn)\n  }, this)\n}\n\n/**\n * Adds a single function after a function that already exists in the\n * pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline.\n * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline.\n */\nlunr.Pipeline.prototype.after = function (existingFn, newFn) {\n  lunr.Pipeline.warnIfFunctionNotRegistered(newFn)\n\n  var pos = this._stack.indexOf(existingFn)\n  if (pos == -1) {\n    throw new Error('Cannot find existingFn')\n  }\n\n  pos = pos + 1\n  this._stack.splice(pos, 0, newFn)\n}\n\n/**\n * Adds a single function before a function that already exists in the\n * pipeline.\n *\n * Logs a warning if the function has not been registered.\n *\n * @param {lunr.PipelineFunction} existingFn - A function that already exists in the pipeline.\n * @param {lunr.PipelineFunction} newFn - The new function to add to the pipeline.\n */\nlunr.Pipeline.prototype.before = function (existingFn, newFn) {\n  lunr.Pipeline.warnIfFunctionNotRegistered(newFn)\n\n  var pos = this._stack.indexOf(existingFn)\n  if (pos == -1) {\n    throw new Error('Cannot find existingFn')\n  }\n\n  this._stack.splice(pos, 0, newFn)\n}\n\n/**\n * Removes a function from the pipeline.\n *\n * @param {lunr.PipelineFunction} fn The function to remove from the pipeline.\n */\nlunr.Pipeline.prototype.remove = function (fn) {\n  var pos = this._stack.indexOf(fn)\n  if (pos == -1) {\n    return\n  }\n\n  this._stack.splice(pos, 1)\n}\n\n/**\n * Runs the current list of functions that make up the pipeline against the\n * passed tokens.\n *\n * @param {Array} tokens The tokens to run through the pipeline.\n * @returns {Array}\n */\nlunr.Pipeline.prototype.run = function (tokens) {\n  var stackLength = this._stack.length\n\n  for (var i = 0; i < stackLength; i++) {\n    var fn = this._stack[i]\n    var memo = []\n\n    for (var j = 0; j < tokens.length; j++) {\n      var result = fn(tokens[j], j, tokens)\n\n      if (result === null || result === void 0 || result === '') continue\n\n      if (Array.isArray(result)) {\n        for (var k = 0; k < result.length; k++) {\n          memo.push(result[k])\n        }\n      } else {\n        memo.push(result)\n      }\n    }\n\n    tokens = memo\n  }\n\n  return tokens\n}\n\n/**\n * Convenience method for passing a string through a pipeline and getting\n * strings out. This method takes care of wrapping the passed string in a\n * token and mapping the resulting tokens back to strings.\n *\n * @param {string} str - The string to pass through the pipeline.\n * @param {?object} metadata - Optional metadata to associate with the token\n * passed to the pipeline.\n * @returns {string[]}\n */\nlunr.Pipeline.prototype.runString = function (str, metadata) {\n  var token = new lunr.Token (str, metadata)\n\n  return this.run([token]).map(function (t) {\n    return t.toString()\n  })\n}\n\n/**\n * Resets the pipeline by removing any existing processors.\n *\n */\nlunr.Pipeline.prototype.reset = function () {\n  this._stack = []\n}\n\n/**\n * Returns a representation of the pipeline ready for serialisation.\n *\n * Logs a warning if the function has not been registered.\n *\n * @returns {Array}\n */\nlunr.Pipeline.prototype.toJSON = function () {\n  return this._stack.map(function (fn) {\n    lunr.Pipeline.warnIfFunctionNotRegistered(fn)\n\n    return fn.label\n  })\n}\n/*!\n * lunr.Vector\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A vector is used to construct the vector space of documents and queries. These\n * vectors support operations to determine the similarity between two documents or\n * a document and a query.\n *\n * Normally no parameters are required for initializing a vector, but in the case of\n * loading a previously dumped vector the raw elements can be provided to the constructor.\n *\n * For performance reasons vectors are implemented with a flat array, where an elements\n * index is immediately followed by its value. E.g. [index, value, index, value]. This\n * allows the underlying array to be as sparse as possible and still offer decent\n * performance when being used for vector calculations.\n *\n * @constructor\n * @param {Number[]} [elements] - The flat list of element index and element value pairs.\n */\nlunr.Vector = function (elements) {\n  this._magnitude = 0\n  this.elements = elements || []\n}\n\n\n/**\n * Calculates the position within the vector to insert a given index.\n *\n * This is used internally by insert and upsert. If there are duplicate indexes then\n * the position is returned as if the value for that index were to be updated, but it\n * is the callers responsibility to check whether there is a duplicate at that index\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @returns {Number}\n */\nlunr.Vector.prototype.positionForIndex = function (index) {\n  // For an empty vector the tuple can be inserted at the beginning\n  if (this.elements.length == 0) {\n    return 0\n  }\n\n  var start = 0,\n      end = this.elements.length / 2,\n      sliceLength = end - start,\n      pivotPoint = Math.floor(sliceLength / 2),\n      pivotIndex = this.elements[pivotPoint * 2]\n\n  while (sliceLength > 1) {\n    if (pivotIndex < index) {\n      start = pivotPoint\n    }\n\n    if (pivotIndex > index) {\n      end = pivotPoint\n    }\n\n    if (pivotIndex == index) {\n      break\n    }\n\n    sliceLength = end - start\n    pivotPoint = start + Math.floor(sliceLength / 2)\n    pivotIndex = this.elements[pivotPoint * 2]\n  }\n\n  if (pivotIndex == index) {\n    return pivotPoint * 2\n  }\n\n  if (pivotIndex > index) {\n    return pivotPoint * 2\n  }\n\n  if (pivotIndex < index) {\n    return (pivotPoint + 1) * 2\n  }\n}\n\n/**\n * Inserts an element at an index within the vector.\n *\n * Does not allow duplicates, will throw an error if there is already an entry\n * for this index.\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @param {Number} val - The value to be inserted into the vector.\n */\nlunr.Vector.prototype.insert = function (insertIdx, val) {\n  this.upsert(insertIdx, val, function () {\n    throw \"duplicate index\"\n  })\n}\n\n/**\n * Inserts or updates an existing index within the vector.\n *\n * @param {Number} insertIdx - The index at which the element should be inserted.\n * @param {Number} val - The value to be inserted into the vector.\n * @param {function} fn - A function that is called for updates, the existing value and the\n * requested value are passed as arguments\n */\nlunr.Vector.prototype.upsert = function (insertIdx, val, fn) {\n  this._magnitude = 0\n  var position = this.positionForIndex(insertIdx)\n\n  if (this.elements[position] == insertIdx) {\n    this.elements[position + 1] = fn(this.elements[position + 1], val)\n  } else {\n    this.elements.splice(position, 0, insertIdx, val)\n  }\n}\n\n/**\n * Calculates the magnitude of this vector.\n *\n * @returns {Number}\n */\nlunr.Vector.prototype.magnitude = function () {\n  if (this._magnitude) return this._magnitude\n\n  var sumOfSquares = 0,\n      elementsLength = this.elements.length\n\n  for (var i = 1; i < elementsLength; i += 2) {\n    var val = this.elements[i]\n    sumOfSquares += val * val\n  }\n\n  return this._magnitude = Math.sqrt(sumOfSquares)\n}\n\n/**\n * Calculates the dot product of this vector and another vector.\n *\n * @param {lunr.Vector} otherVector - The vector to compute the dot product with.\n * @returns {Number}\n */\nlunr.Vector.prototype.dot = function (otherVector) {\n  var dotProduct = 0,\n      a = this.elements, b = otherVector.elements,\n      aLen = a.length, bLen = b.length,\n      aVal = 0, bVal = 0,\n      i = 0, j = 0\n\n  while (i < aLen && j < bLen) {\n    aVal = a[i], bVal = b[j]\n    if (aVal < bVal) {\n      i += 2\n    } else if (aVal > bVal) {\n      j += 2\n    } else if (aVal == bVal) {\n      dotProduct += a[i + 1] * b[j + 1]\n      i += 2\n      j += 2\n    }\n  }\n\n  return dotProduct\n}\n\n/**\n * Calculates the similarity between this vector and another vector.\n *\n * @param {lunr.Vector} otherVector - The other vector to calculate the\n * similarity with.\n * @returns {Number}\n */\nlunr.Vector.prototype.similarity = function (otherVector) {\n  return this.dot(otherVector) / this.magnitude() || 0\n}\n\n/**\n * Converts the vector to an array of the elements within the vector.\n *\n * @returns {Number[]}\n */\nlunr.Vector.prototype.toArray = function () {\n  var output = new Array (this.elements.length / 2)\n\n  for (var i = 1, j = 0; i < this.elements.length; i += 2, j++) {\n    output[j] = this.elements[i]\n  }\n\n  return output\n}\n\n/**\n * A JSON serializable representation of the vector.\n *\n * @returns {Number[]}\n */\nlunr.Vector.prototype.toJSON = function () {\n  return this.elements\n}\n/* eslint-disable */\n/*!\n * lunr.stemmer\n * Copyright (C) 2020 Oliver Nightingale\n * Includes code from - http://tartarus.org/~martin/PorterStemmer/js.txt\n */\n\n/**\n * lunr.stemmer is an english language stemmer, this is a JavaScript\n * implementation of the PorterStemmer taken from http://tartarus.org/~martin\n *\n * @static\n * @implements {lunr.PipelineFunction}\n * @param {lunr.Token} token - The string to stem\n * @returns {lunr.Token}\n * @see {@link lunr.Pipeline}\n * @function\n */\nlunr.stemmer = (function(){\n  var step2list = {\n      \"ational\" : \"ate\",\n      \"tional\" : \"tion\",\n      \"enci\" : \"ence\",\n      \"anci\" : \"ance\",\n      \"izer\" : \"ize\",\n      \"bli\" : \"ble\",\n      \"alli\" : \"al\",\n      \"entli\" : \"ent\",\n      \"eli\" : \"e\",\n      \"ousli\" : \"ous\",\n      \"ization\" : \"ize\",\n      \"ation\" : \"ate\",\n      \"ator\" : \"ate\",\n      \"alism\" : \"al\",\n      \"iveness\" : \"ive\",\n      \"fulness\" : \"ful\",\n      \"ousness\" : \"ous\",\n      \"aliti\" : \"al\",\n      \"iviti\" : \"ive\",\n      \"biliti\" : \"ble\",\n      \"logi\" : \"log\"\n    },\n\n    step3list = {\n      \"icate\" : \"ic\",\n      \"ative\" : \"\",\n      \"alize\" : \"al\",\n      \"iciti\" : \"ic\",\n      \"ical\" : \"ic\",\n      \"ful\" : \"\",\n      \"ness\" : \"\"\n    },\n\n    c = \"[^aeiou]\",          // consonant\n    v = \"[aeiouy]\",          // vowel\n    C = c + \"[^aeiouy]*\",    // consonant sequence\n    V = v + \"[aeiou]*\",      // vowel sequence\n\n    mgr0 = \"^(\" + C + \")?\" + V + C,               // [C]VC... is m>0\n    meq1 = \"^(\" + C + \")?\" + V + C + \"(\" + V + \")?$\",  // [C]VC[V] is m=1\n    mgr1 = \"^(\" + C + \")?\" + V + C + V + C,       // [C]VCVC... is m>1\n    s_v = \"^(\" + C + \")?\" + v;                   // vowel in stem\n\n  var re_mgr0 = new RegExp(mgr0);\n  var re_mgr1 = new RegExp(mgr1);\n  var re_meq1 = new RegExp(meq1);\n  var re_s_v = new RegExp(s_v);\n\n  var re_1a = /^(.+?)(ss|i)es$/;\n  var re2_1a = /^(.+?)([^s])s$/;\n  var re_1b = /^(.+?)eed$/;\n  var re2_1b = /^(.+?)(ed|ing)$/;\n  var re_1b_2 = /.$/;\n  var re2_1b_2 = /(at|bl|iz)$/;\n  var re3_1b_2 = new RegExp(\"([^aeiouylsz])\\\\1$\");\n  var re4_1b_2 = new RegExp(\"^\" + C + v + \"[^aeiouwxy]$\");\n\n  var re_1c = /^(.+?[^aeiou])y$/;\n  var re_2 = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;\n\n  var re_3 = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;\n\n  var re_4 = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;\n  var re2_4 = /^(.+?)(s|t)(ion)$/;\n\n  var re_5 = /^(.+?)e$/;\n  var re_5_1 = /ll$/;\n  var re3_5 = new RegExp(\"^\" + C + v + \"[^aeiouwxy]$\");\n\n  var porterStemmer = function porterStemmer(w) {\n    var stem,\n      suffix,\n      firstch,\n      re,\n      re2,\n      re3,\n      re4;\n\n    if (w.length < 3) { return w; }\n\n    firstch = w.substr(0,1);\n    if (firstch == \"y\") {\n      w = firstch.toUpperCase() + w.substr(1);\n    }\n\n    // Step 1a\n    re = re_1a\n    re2 = re2_1a;\n\n    if (re.test(w)) { w = w.replace(re,\"$1$2\"); }\n    else if (re2.test(w)) { w = w.replace(re2,\"$1$2\"); }\n\n    // Step 1b\n    re = re_1b;\n    re2 = re2_1b;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      re = re_mgr0;\n      if (re.test(fp[1])) {\n        re = re_1b_2;\n        w = w.replace(re,\"\");\n      }\n    } else if (re2.test(w)) {\n      var fp = re2.exec(w);\n      stem = fp[1];\n      re2 = re_s_v;\n      if (re2.test(stem)) {\n        w = stem;\n        re2 = re2_1b_2;\n        re3 = re3_1b_2;\n        re4 = re4_1b_2;\n        if (re2.test(w)) { w = w + \"e\"; }\n        else if (re3.test(w)) { re = re_1b_2; w = w.replace(re,\"\"); }\n        else if (re4.test(w)) { w = w + \"e\"; }\n      }\n    }\n\n    // Step 1c - replace suffix y or Y by i if preceded by a non-vowel which is not the first letter of the word (so cry -> cri, by -> by, say -> say)\n    re = re_1c;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      w = stem + \"i\";\n    }\n\n    // Step 2\n    re = re_2;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      suffix = fp[2];\n      re = re_mgr0;\n      if (re.test(stem)) {\n        w = stem + step2list[suffix];\n      }\n    }\n\n    // Step 3\n    re = re_3;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      suffix = fp[2];\n      re = re_mgr0;\n      if (re.test(stem)) {\n        w = stem + step3list[suffix];\n      }\n    }\n\n    // Step 4\n    re = re_4;\n    re2 = re2_4;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      re = re_mgr1;\n      if (re.test(stem)) {\n        w = stem;\n      }\n    } else if (re2.test(w)) {\n      var fp = re2.exec(w);\n      stem = fp[1] + fp[2];\n      re2 = re_mgr1;\n      if (re2.test(stem)) {\n        w = stem;\n      }\n    }\n\n    // Step 5\n    re = re_5;\n    if (re.test(w)) {\n      var fp = re.exec(w);\n      stem = fp[1];\n      re = re_mgr1;\n      re2 = re_meq1;\n      re3 = re3_5;\n      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem)))) {\n        w = stem;\n      }\n    }\n\n    re = re_5_1;\n    re2 = re_mgr1;\n    if (re.test(w) && re2.test(w)) {\n      re = re_1b_2;\n      w = w.replace(re,\"\");\n    }\n\n    // and turn initial Y back to y\n\n    if (firstch == \"y\") {\n      w = firstch.toLowerCase() + w.substr(1);\n    }\n\n    return w;\n  };\n\n  return function (token) {\n    return token.update(porterStemmer);\n  }\n})();\n\nlunr.Pipeline.registerFunction(lunr.stemmer, 'stemmer')\n/*!\n * lunr.stopWordFilter\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.generateStopWordFilter builds a stopWordFilter function from the provided\n * list of stop words.\n *\n * The built in lunr.stopWordFilter is built using this generator and can be used\n * to generate custom stopWordFilters for applications or non English languages.\n *\n * @function\n * @param {Array} token The token to pass through the filter\n * @returns {lunr.PipelineFunction}\n * @see lunr.Pipeline\n * @see lunr.stopWordFilter\n */\nlunr.generateStopWordFilter = function (stopWords) {\n  var words = stopWords.reduce(function (memo, stopWord) {\n    memo[stopWord] = stopWord\n    return memo\n  }, {})\n\n  return function (token) {\n    if (token && words[token.toString()] !== token.toString()) return token\n  }\n}\n\n/**\n * lunr.stopWordFilter is an English language stop word list filter, any words\n * contained in the list will not be passed through the filter.\n *\n * This is intended to be used in the Pipeline. If the token does not pass the\n * filter then undefined will be returned.\n *\n * @function\n * @implements {lunr.PipelineFunction}\n * @params {lunr.Token} token - A token to check for being a stop word.\n * @returns {lunr.Token}\n * @see {@link lunr.Pipeline}\n */\nlunr.stopWordFilter = lunr.generateStopWordFilter([\n  'a',\n  'able',\n  'about',\n  'across',\n  'after',\n  'all',\n  'almost',\n  'also',\n  'am',\n  'among',\n  'an',\n  'and',\n  'any',\n  'are',\n  'as',\n  'at',\n  'be',\n  'because',\n  'been',\n  'but',\n  'by',\n  'can',\n  'cannot',\n  'could',\n  'dear',\n  'did',\n  'do',\n  'does',\n  'either',\n  'else',\n  'ever',\n  'every',\n  'for',\n  'from',\n  'get',\n  'got',\n  'had',\n  'has',\n  'have',\n  'he',\n  'her',\n  'hers',\n  'him',\n  'his',\n  'how',\n  'however',\n  'i',\n  'if',\n  'in',\n  'into',\n  'is',\n  'it',\n  'its',\n  'just',\n  'least',\n  'let',\n  'like',\n  'likely',\n  'may',\n  'me',\n  'might',\n  'most',\n  'must',\n  'my',\n  'neither',\n  'no',\n  'nor',\n  'not',\n  'of',\n  'off',\n  'often',\n  'on',\n  'only',\n  'or',\n  'other',\n  'our',\n  'own',\n  'rather',\n  'said',\n  'say',\n  'says',\n  'she',\n  'should',\n  'since',\n  'so',\n  'some',\n  'than',\n  'that',\n  'the',\n  'their',\n  'them',\n  'then',\n  'there',\n  'these',\n  'they',\n  'this',\n  'tis',\n  'to',\n  'too',\n  'twas',\n  'us',\n  'wants',\n  'was',\n  'we',\n  'were',\n  'what',\n  'when',\n  'where',\n  'which',\n  'while',\n  'who',\n  'whom',\n  'why',\n  'will',\n  'with',\n  'would',\n  'yet',\n  'you',\n  'your'\n])\n\nlunr.Pipeline.registerFunction(lunr.stopWordFilter, 'stopWordFilter')\n/*!\n * lunr.trimmer\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.trimmer is a pipeline function for trimming non word\n * characters from the beginning and end of tokens before they\n * enter the index.\n *\n * This implementation may not work correctly for non latin\n * characters and should either be removed or adapted for use\n * with languages with non-latin characters.\n *\n * @static\n * @implements {lunr.PipelineFunction}\n * @param {lunr.Token} token The token to pass through the filter\n * @returns {lunr.Token}\n * @see lunr.Pipeline\n */\nlunr.trimmer = function (token) {\n  return token.update(function (s) {\n    return s.replace(/^\\W+/, '').replace(/\\W+$/, '')\n  })\n}\n\nlunr.Pipeline.registerFunction(lunr.trimmer, 'trimmer')\n/*!\n * lunr.TokenSet\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * A token set is used to store the unique list of all tokens\n * within an index. Token sets are also used to represent an\n * incoming query to the index, this query token set and index\n * token set are then intersected to find which tokens to look\n * up in the inverted index.\n *\n * A token set can hold multiple tokens, as in the case of the\n * index token set, or it can hold a single token as in the\n * case of a simple query token set.\n *\n * Additionally token sets are used to perform wildcard matching.\n * Leading, contained and trailing wildcards are supported, and\n * from this edit distance matching can also be provided.\n *\n * Token sets are implemented as a minimal finite state automata,\n * where both common prefixes and suffixes are shared between tokens.\n * This helps to reduce the space used for storing the token set.\n *\n * @constructor\n */\nlunr.TokenSet = function () {\n  this.final = false\n  this.edges = {}\n  this.id = lunr.TokenSet._nextId\n  lunr.TokenSet._nextId += 1\n}\n\n/**\n * Keeps track of the next, auto increment, identifier to assign\n * to a new tokenSet.\n *\n * TokenSets require a unique identifier to be correctly minimised.\n *\n * @private\n */\nlunr.TokenSet._nextId = 1\n\n/**\n * Creates a TokenSet instance from the given sorted array of words.\n *\n * @param {String[]} arr - A sorted array of strings to create the set from.\n * @returns {lunr.TokenSet}\n * @throws Will throw an error if the input array is not sorted.\n */\nlunr.TokenSet.fromArray = function (arr) {\n  var builder = new lunr.TokenSet.Builder\n\n  for (var i = 0, len = arr.length; i < len; i++) {\n    builder.insert(arr[i])\n  }\n\n  builder.finish()\n  return builder.root\n}\n\n/**\n * Creates a token set from a query clause.\n *\n * @private\n * @param {Object} clause - A single clause from lunr.Query.\n * @param {string} clause.term - The query clause term.\n * @param {number} [clause.editDistance] - The optional edit distance for the term.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.fromClause = function (clause) {\n  if ('editDistance' in clause) {\n    return lunr.TokenSet.fromFuzzyString(clause.term, clause.editDistance)\n  } else {\n    return lunr.TokenSet.fromString(clause.term)\n  }\n}\n\n/**\n * Creates a token set representing a single string with a specified\n * edit distance.\n *\n * Insertions, deletions, substitutions and transpositions are each\n * treated as an edit distance of 1.\n *\n * Increasing the allowed edit distance will have a dramatic impact\n * on the performance of both creating and intersecting these TokenSets.\n * It is advised to keep the edit distance less than 3.\n *\n * @param {string} str - The string to create the token set from.\n * @param {number} editDistance - The allowed edit distance to match.\n * @returns {lunr.Vector}\n */\nlunr.TokenSet.fromFuzzyString = function (str, editDistance) {\n  var root = new lunr.TokenSet\n\n  var stack = [{\n    node: root,\n    editsRemaining: editDistance,\n    str: str\n  }]\n\n  while (stack.length) {\n    var frame = stack.pop()\n\n    // no edit\n    if (frame.str.length > 0) {\n      var char = frame.str.charAt(0),\n          noEditNode\n\n      if (char in frame.node.edges) {\n        noEditNode = frame.node.edges[char]\n      } else {\n        noEditNode = new lunr.TokenSet\n        frame.node.edges[char] = noEditNode\n      }\n\n      if (frame.str.length == 1) {\n        noEditNode.final = true\n      }\n\n      stack.push({\n        node: noEditNode,\n        editsRemaining: frame.editsRemaining,\n        str: frame.str.slice(1)\n      })\n    }\n\n    if (frame.editsRemaining == 0) {\n      continue\n    }\n\n    // insertion\n    if (\"*\" in frame.node.edges) {\n      var insertionNode = frame.node.edges[\"*\"]\n    } else {\n      var insertionNode = new lunr.TokenSet\n      frame.node.edges[\"*\"] = insertionNode\n    }\n\n    if (frame.str.length == 0) {\n      insertionNode.final = true\n    }\n\n    stack.push({\n      node: insertionNode,\n      editsRemaining: frame.editsRemaining - 1,\n      str: frame.str\n    })\n\n    // deletion\n    // can only do a deletion if we have enough edits remaining\n    // and if there are characters left to delete in the string\n    if (frame.str.length > 1) {\n      stack.push({\n        node: frame.node,\n        editsRemaining: frame.editsRemaining - 1,\n        str: frame.str.slice(1)\n      })\n    }\n\n    // deletion\n    // just removing the last character from the str\n    if (frame.str.length == 1) {\n      frame.node.final = true\n    }\n\n    // substitution\n    // can only do a substitution if we have enough edits remaining\n    // and if there are characters left to substitute\n    if (frame.str.length >= 1) {\n      if (\"*\" in frame.node.edges) {\n        var substitutionNode = frame.node.edges[\"*\"]\n      } else {\n        var substitutionNode = new lunr.TokenSet\n        frame.node.edges[\"*\"] = substitutionNode\n      }\n\n      if (frame.str.length == 1) {\n        substitutionNode.final = true\n      }\n\n      stack.push({\n        node: substitutionNode,\n        editsRemaining: frame.editsRemaining - 1,\n        str: frame.str.slice(1)\n      })\n    }\n\n    // transposition\n    // can only do a transposition if there are edits remaining\n    // and there are enough characters to transpose\n    if (frame.str.length > 1) {\n      var charA = frame.str.charAt(0),\n          charB = frame.str.charAt(1),\n          transposeNode\n\n      if (charB in frame.node.edges) {\n        transposeNode = frame.node.edges[charB]\n      } else {\n        transposeNode = new lunr.TokenSet\n        frame.node.edges[charB] = transposeNode\n      }\n\n      if (frame.str.length == 1) {\n        transposeNode.final = true\n      }\n\n      stack.push({\n        node: transposeNode,\n        editsRemaining: frame.editsRemaining - 1,\n        str: charA + frame.str.slice(2)\n      })\n    }\n  }\n\n  return root\n}\n\n/**\n * Creates a TokenSet from a string.\n *\n * The string may contain one or more wildcard characters (*)\n * that will allow wildcard matching when intersecting with\n * another TokenSet.\n *\n * @param {string} str - The string to create a TokenSet from.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.fromString = function (str) {\n  var node = new lunr.TokenSet,\n      root = node\n\n  /*\n   * Iterates through all characters within the passed string\n   * appending a node for each character.\n   *\n   * When a wildcard character is found then a self\n   * referencing edge is introduced to continually match\n   * any number of any characters.\n   */\n  for (var i = 0, len = str.length; i < len; i++) {\n    var char = str[i],\n        final = (i == len - 1)\n\n    if (char == \"*\") {\n      node.edges[char] = node\n      node.final = final\n\n    } else {\n      var next = new lunr.TokenSet\n      next.final = final\n\n      node.edges[char] = next\n      node = next\n    }\n  }\n\n  return root\n}\n\n/**\n * Converts this TokenSet into an array of strings\n * contained within the TokenSet.\n *\n * This is not intended to be used on a TokenSet that\n * contains wildcards, in these cases the results are\n * undefined and are likely to cause an infinite loop.\n *\n * @returns {string[]}\n */\nlunr.TokenSet.prototype.toArray = function () {\n  var words = []\n\n  var stack = [{\n    prefix: \"\",\n    node: this\n  }]\n\n  while (stack.length) {\n    var frame = stack.pop(),\n        edges = Object.keys(frame.node.edges),\n        len = edges.length\n\n    if (frame.node.final) {\n      /* In Safari, at this point the prefix is sometimes corrupted, see:\n       * https://github.com/olivernn/lunr.js/issues/279 Calling any\n       * String.prototype method forces Safari to \"cast\" this string to what\n       * it's supposed to be, fixing the bug. */\n      frame.prefix.charAt(0)\n      words.push(frame.prefix)\n    }\n\n    for (var i = 0; i < len; i++) {\n      var edge = edges[i]\n\n      stack.push({\n        prefix: frame.prefix.concat(edge),\n        node: frame.node.edges[edge]\n      })\n    }\n  }\n\n  return words\n}\n\n/**\n * Generates a string representation of a TokenSet.\n *\n * This is intended to allow TokenSets to be used as keys\n * in objects, largely to aid the construction and minimisation\n * of a TokenSet. As such it is not designed to be a human\n * friendly representation of the TokenSet.\n *\n * @returns {string}\n */\nlunr.TokenSet.prototype.toString = function () {\n  // NOTE: Using Object.keys here as this.edges is very likely\n  // to enter 'hash-mode' with many keys being added\n  //\n  // avoiding a for-in loop here as it leads to the function\n  // being de-optimised (at least in V8). From some simple\n  // benchmarks the performance is comparable, but allowing\n  // V8 to optimize may mean easy performance wins in the future.\n\n  if (this._str) {\n    return this._str\n  }\n\n  var str = this.final ? '1' : '0',\n      labels = Object.keys(this.edges).sort(),\n      len = labels.length\n\n  for (var i = 0; i < len; i++) {\n    var label = labels[i],\n        node = this.edges[label]\n\n    str = str + label + node.id\n  }\n\n  return str\n}\n\n/**\n * Returns a new TokenSet that is the intersection of\n * this TokenSet and the passed TokenSet.\n *\n * This intersection will take into account any wildcards\n * contained within the TokenSet.\n *\n * @param {lunr.TokenSet} b - An other TokenSet to intersect with.\n * @returns {lunr.TokenSet}\n */\nlunr.TokenSet.prototype.intersect = function (b) {\n  var output = new lunr.TokenSet,\n      frame = undefined\n\n  var stack = [{\n    qNode: b,\n    output: output,\n    node: this\n  }]\n\n  while (stack.length) {\n    frame = stack.pop()\n\n    // NOTE: As with the #toString method, we are using\n    // Object.keys and a for loop instead of a for-in loop\n    // as both of these objects enter 'hash' mode, causing\n    // the function to be de-optimised in V8\n    var qEdges = Object.keys(frame.qNode.edges),\n        qLen = qEdges.length,\n        nEdges = Object.keys(frame.node.edges),\n        nLen = nEdges.length\n\n    for (var q = 0; q < qLen; q++) {\n      var qEdge = qEdges[q]\n\n      for (var n = 0; n < nLen; n++) {\n        var nEdge = nEdges[n]\n\n        if (nEdge == qEdge || qEdge == '*') {\n          var node = frame.node.edges[nEdge],\n              qNode = frame.qNode.edges[qEdge],\n              final = node.final && qNode.final,\n              next = undefined\n\n          if (nEdge in frame.output.edges) {\n            // an edge already exists for this character\n            // no need to create a new node, just set the finality\n            // bit unless this node is already final\n            next = frame.output.edges[nEdge]\n            next.final = next.final || final\n\n          } else {\n            // no edge exists yet, must create one\n            // set the finality bit and insert it\n            // into the output\n            next = new lunr.TokenSet\n            next.final = final\n            frame.output.edges[nEdge] = next\n          }\n\n          stack.push({\n            qNode: qNode,\n            output: next,\n            node: node\n          })\n        }\n      }\n    }\n  }\n\n  return output\n}\nlunr.TokenSet.Builder = function () {\n  this.previousWord = \"\"\n  this.root = new lunr.TokenSet\n  this.uncheckedNodes = []\n  this.minimizedNodes = {}\n}\n\nlunr.TokenSet.Builder.prototype.insert = function (word) {\n  var node,\n      commonPrefix = 0\n\n  if (word < this.previousWord) {\n    throw new Error (\"Out of order word insertion\")\n  }\n\n  for (var i = 0; i < word.length && i < this.previousWord.length; i++) {\n    if (word[i] != this.previousWord[i]) break\n    commonPrefix++\n  }\n\n  this.minimize(commonPrefix)\n\n  if (this.uncheckedNodes.length == 0) {\n    node = this.root\n  } else {\n    node = this.uncheckedNodes[this.uncheckedNodes.length - 1].child\n  }\n\n  for (var i = commonPrefix; i < word.length; i++) {\n    var nextNode = new lunr.TokenSet,\n        char = word[i]\n\n    node.edges[char] = nextNode\n\n    this.uncheckedNodes.push({\n      parent: node,\n      char: char,\n      child: nextNode\n    })\n\n    node = nextNode\n  }\n\n  node.final = true\n  this.previousWord = word\n}\n\nlunr.TokenSet.Builder.prototype.finish = function () {\n  this.minimize(0)\n}\n\nlunr.TokenSet.Builder.prototype.minimize = function (downTo) {\n  for (var i = this.uncheckedNodes.length - 1; i >= downTo; i--) {\n    var node = this.uncheckedNodes[i],\n        childKey = node.child.toString()\n\n    if (childKey in this.minimizedNodes) {\n      node.parent.edges[node.char] = this.minimizedNodes[childKey]\n    } else {\n      // Cache the key for this node since\n      // we know it can't change anymore\n      node.child._str = childKey\n\n      this.minimizedNodes[childKey] = node.child\n    }\n\n    this.uncheckedNodes.pop()\n  }\n}\n/*!\n * lunr.Index\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * An index contains the built index of all documents and provides a query interface\n * to the index.\n *\n * Usually instances of lunr.Index will not be created using this constructor, instead\n * lunr.Builder should be used to construct new indexes, or lunr.Index.load should be\n * used to load previously built and serialized indexes.\n *\n * @constructor\n * @param {Object} attrs - The attributes of the built search index.\n * @param {Object} attrs.invertedIndex - An index of term/field to document reference.\n * @param {Object<string, lunr.Vector>} attrs.fieldVectors - Field vectors\n * @param {lunr.TokenSet} attrs.tokenSet - An set of all corpus tokens.\n * @param {string[]} attrs.fields - The names of indexed document fields.\n * @param {lunr.Pipeline} attrs.pipeline - The pipeline to use for search terms.\n */\nlunr.Index = function (attrs) {\n  this.invertedIndex = attrs.invertedIndex\n  this.fieldVectors = attrs.fieldVectors\n  this.tokenSet = attrs.tokenSet\n  this.fields = attrs.fields\n  this.pipeline = attrs.pipeline\n}\n\n/**\n * A result contains details of a document matching a search query.\n * @typedef {Object} lunr.Index~Result\n * @property {string} ref - The reference of the document this result represents.\n * @property {number} score - A number between 0 and 1 representing how similar this document is to the query.\n * @property {lunr.MatchData} matchData - Contains metadata about this match including which term(s) caused the match.\n */\n\n/**\n * Although lunr provides the ability to create queries using lunr.Query, it also provides a simple\n * query language which itself is parsed into an instance of lunr.Query.\n *\n * For programmatically building queries it is advised to directly use lunr.Query, the query language\n * is best used for human entered text rather than program generated text.\n *\n * At its simplest queries can just be a single term, e.g. `hello`, multiple terms are also supported\n * and will be combined with OR, e.g `hello world` will match documents that contain either 'hello'\n * or 'world', though those that contain both will rank higher in the results.\n *\n * Wildcards can be included in terms to match one or more unspecified characters, these wildcards can\n * be inserted anywhere within the term, and more than one wildcard can exist in a single term. Adding\n * wildcards will increase the number of documents that will be found but can also have a negative\n * impact on query performance, especially with wildcards at the beginning of a term.\n *\n * Terms can be restricted to specific fields, e.g. `title:hello`, only documents with the term\n * hello in the title field will match this query. Using a field not present in the index will lead\n * to an error being thrown.\n *\n * Modifiers can also be added to terms, lunr supports edit distance and boost modifiers on terms. A term\n * boost will make documents matching that term score higher, e.g. `foo^5`. Edit distance is also supported\n * to provide fuzzy matching, e.g. 'hello~2' will match documents with hello with an edit distance of 2.\n * Avoid large values for edit distance to improve query performance.\n *\n * Each term also supports a presence modifier. By default a term's presence in document is optional, however\n * this can be changed to either required or prohibited. For a term's presence to be required in a document the\n * term should be prefixed with a '+', e.g. `+foo bar` is a search for documents that must contain 'foo' and\n * optionally contain 'bar'. Conversely a leading '-' sets the terms presence to prohibited, i.e. it must not\n * appear in a document, e.g. `-foo bar` is a search for documents that do not contain 'foo' but may contain 'bar'.\n *\n * To escape special characters the backslash character '\\' can be used, this allows searches to include\n * characters that would normally be considered modifiers, e.g. `foo\\~2` will search for a term \"foo~2\" instead\n * of attempting to apply a boost of 2 to the search term \"foo\".\n *\n * @typedef {string} lunr.Index~QueryString\n * @example <caption>Simple single term query</caption>\n * hello\n * @example <caption>Multiple term query</caption>\n * hello world\n * @example <caption>term scoped to a field</caption>\n * title:hello\n * @example <caption>term with a boost of 10</caption>\n * hello^10\n * @example <caption>term with an edit distance of 2</caption>\n * hello~2\n * @example <caption>terms with presence modifiers</caption>\n * -foo +bar baz\n */\n\n/**\n * Performs a search against the index using lunr query syntax.\n *\n * Results will be returned sorted by their score, the most relevant results\n * will be returned first.  For details on how the score is calculated, please see\n * the {@link https://lunrjs.com/guides/searching.html#scoring|guide}.\n *\n * For more programmatic querying use lunr.Index#query.\n *\n * @param {lunr.Index~QueryString} queryString - A string containing a lunr query.\n * @throws {lunr.QueryParseError} If the passed query string cannot be parsed.\n * @returns {lunr.Index~Result[]}\n */\nlunr.Index.prototype.search = function (queryString) {\n  return this.query(function (query) {\n    var parser = new lunr.QueryParser(queryString, query)\n    parser.parse()\n  })\n}\n\n/**\n * A query builder callback provides a query object to be used to express\n * the query to perform on the index.\n *\n * @callback lunr.Index~queryBuilder\n * @param {lunr.Query} query - The query object to build up.\n * @this lunr.Query\n */\n\n/**\n * Performs a query against the index using the yielded lunr.Query object.\n *\n * If performing programmatic queries against the index, this method is preferred\n * over lunr.Index#search so as to avoid the additional query parsing overhead.\n *\n * A query object is yielded to the supplied function which should be used to\n * express the query to be run against the index.\n *\n * Note that although this function takes a callback parameter it is _not_ an\n * asynchronous operation, the callback is just yielded a query object to be\n * customized.\n *\n * @param {lunr.Index~queryBuilder} fn - A function that is used to build the query.\n * @returns {lunr.Index~Result[]}\n */\nlunr.Index.prototype.query = function (fn) {\n  // for each query clause\n  // * process terms\n  // * expand terms from token set\n  // * find matching documents and metadata\n  // * get document vectors\n  // * score documents\n\n  var query = new lunr.Query(this.fields),\n      matchingFields = Object.create(null),\n      queryVectors = Object.create(null),\n      termFieldCache = Object.create(null),\n      requiredMatches = Object.create(null),\n      prohibitedMatches = Object.create(null)\n\n  /*\n   * To support field level boosts a query vector is created per\n   * field. An empty vector is eagerly created to support negated\n   * queries.\n   */\n  for (var i = 0; i < this.fields.length; i++) {\n    queryVectors[this.fields[i]] = new lunr.Vector\n  }\n\n  fn.call(query, query)\n\n  for (var i = 0; i < query.clauses.length; i++) {\n    /*\n     * Unless the pipeline has been disabled for this term, which is\n     * the case for terms with wildcards, we need to pass the clause\n     * term through the search pipeline. A pipeline returns an array\n     * of processed terms. Pipeline functions may expand the passed\n     * term, which means we may end up performing multiple index lookups\n     * for a single query term.\n     */\n    var clause = query.clauses[i],\n        terms = null,\n        clauseMatches = lunr.Set.empty\n\n    if (clause.usePipeline) {\n      terms = this.pipeline.runString(clause.term, {\n        fields: clause.fields\n      })\n    } else {\n      terms = [clause.term]\n    }\n\n    for (var m = 0; m < terms.length; m++) {\n      var term = terms[m]\n\n      /*\n       * Each term returned from the pipeline needs to use the same query\n       * clause object, e.g. the same boost and or edit distance. The\n       * simplest way to do this is to re-use the clause object but mutate\n       * its term property.\n       */\n      clause.term = term\n\n      /*\n       * From the term in the clause we create a token set which will then\n       * be used to intersect the indexes token set to get a list of terms\n       * to lookup in the inverted index\n       */\n      var termTokenSet = lunr.TokenSet.fromClause(clause),\n          expandedTerms = this.tokenSet.intersect(termTokenSet).toArray()\n\n      /*\n       * If a term marked as required does not exist in the tokenSet it is\n       * impossible for the search to return any matches. We set all the field\n       * scoped required matches set to empty and stop examining any further\n       * clauses.\n       */\n      if (expandedTerms.length === 0 && clause.presence === lunr.Query.presence.REQUIRED) {\n        for (var k = 0; k < clause.fields.length; k++) {\n          var field = clause.fields[k]\n          requiredMatches[field] = lunr.Set.empty\n        }\n\n        break\n      }\n\n      for (var j = 0; j < expandedTerms.length; j++) {\n        /*\n         * For each term get the posting and termIndex, this is required for\n         * building the query vector.\n         */\n        var expandedTerm = expandedTerms[j],\n            posting = this.invertedIndex[expandedTerm],\n            termIndex = posting._index\n\n        for (var k = 0; k < clause.fields.length; k++) {\n          /*\n           * For each field that this query term is scoped by (by default\n           * all fields are in scope) we need to get all the document refs\n           * that have this term in that field.\n           *\n           * The posting is the entry in the invertedIndex for the matching\n           * term from above.\n           */\n          var field = clause.fields[k],\n              fieldPosting = posting[field],\n              matchingDocumentRefs = Object.keys(fieldPosting),\n              termField = expandedTerm + \"/\" + field,\n              matchingDocumentsSet = new lunr.Set(matchingDocumentRefs)\n\n          /*\n           * if the presence of this term is required ensure that the matching\n           * documents are added to the set of required matches for this clause.\n           *\n           */\n          if (clause.presence == lunr.Query.presence.REQUIRED) {\n            clauseMatches = clauseMatches.union(matchingDocumentsSet)\n\n            if (requiredMatches[field] === undefined) {\n              requiredMatches[field] = lunr.Set.complete\n            }\n          }\n\n          /*\n           * if the presence of this term is prohibited ensure that the matching\n           * documents are added to the set of prohibited matches for this field,\n           * creating that set if it does not yet exist.\n           */\n          if (clause.presence == lunr.Query.presence.PROHIBITED) {\n            if (prohibitedMatches[field] === undefined) {\n              prohibitedMatches[field] = lunr.Set.empty\n            }\n\n            prohibitedMatches[field] = prohibitedMatches[field].union(matchingDocumentsSet)\n\n            /*\n             * Prohibited matches should not be part of the query vector used for\n             * similarity scoring and no metadata should be extracted so we continue\n             * to the next field\n             */\n            continue\n          }\n\n          /*\n           * The query field vector is populated using the termIndex found for\n           * the term and a unit value with the appropriate boost applied.\n           * Using upsert because there could already be an entry in the vector\n           * for the term we are working with. In that case we just add the scores\n           * together.\n           */\n          queryVectors[field].upsert(termIndex, clause.boost, function (a, b) { return a + b })\n\n          /**\n           * If we've already seen this term, field combo then we've already collected\n           * the matching documents and metadata, no need to go through all that again\n           */\n          if (termFieldCache[termField]) {\n            continue\n          }\n\n          for (var l = 0; l < matchingDocumentRefs.length; l++) {\n            /*\n             * All metadata for this term/field/document triple\n             * are then extracted and collected into an instance\n             * of lunr.MatchData ready to be returned in the query\n             * results\n             */\n            var matchingDocumentRef = matchingDocumentRefs[l],\n                matchingFieldRef = new lunr.FieldRef (matchingDocumentRef, field),\n                metadata = fieldPosting[matchingDocumentRef],\n                fieldMatch\n\n            if ((fieldMatch = matchingFields[matchingFieldRef]) === undefined) {\n              matchingFields[matchingFieldRef] = new lunr.MatchData (expandedTerm, field, metadata)\n            } else {\n              fieldMatch.add(expandedTerm, field, metadata)\n            }\n\n          }\n\n          termFieldCache[termField] = true\n        }\n      }\n    }\n\n    /**\n     * If the presence was required we need to update the requiredMatches field sets.\n     * We do this after all fields for the term have collected their matches because\n     * the clause terms presence is required in _any_ of the fields not _all_ of the\n     * fields.\n     */\n    if (clause.presence === lunr.Query.presence.REQUIRED) {\n      for (var k = 0; k < clause.fields.length; k++) {\n        var field = clause.fields[k]\n        requiredMatches[field] = requiredMatches[field].intersect(clauseMatches)\n      }\n    }\n  }\n\n  /**\n   * Need to combine the field scoped required and prohibited\n   * matching documents into a global set of required and prohibited\n   * matches\n   */\n  var allRequiredMatches = lunr.Set.complete,\n      allProhibitedMatches = lunr.Set.empty\n\n  for (var i = 0; i < this.fields.length; i++) {\n    var field = this.fields[i]\n\n    if (requiredMatches[field]) {\n      allRequiredMatches = allRequiredMatches.intersect(requiredMatches[field])\n    }\n\n    if (prohibitedMatches[field]) {\n      allProhibitedMatches = allProhibitedMatches.union(prohibitedMatches[field])\n    }\n  }\n\n  var matchingFieldRefs = Object.keys(matchingFields),\n      results = [],\n      matches = Object.create(null)\n\n  /*\n   * If the query is negated (contains only prohibited terms)\n   * we need to get _all_ fieldRefs currently existing in the\n   * index. This is only done when we know that the query is\n   * entirely prohibited terms to avoid any cost of getting all\n   * fieldRefs unnecessarily.\n   *\n   * Additionally, blank MatchData must be created to correctly\n   * populate the results.\n   */\n  if (query.isNegated()) {\n    matchingFieldRefs = Object.keys(this.fieldVectors)\n\n    for (var i = 0; i < matchingFieldRefs.length; i++) {\n      var matchingFieldRef = matchingFieldRefs[i]\n      var fieldRef = lunr.FieldRef.fromString(matchingFieldRef)\n      matchingFields[matchingFieldRef] = new lunr.MatchData\n    }\n  }\n\n  for (var i = 0; i < matchingFieldRefs.length; i++) {\n    /*\n     * Currently we have document fields that match the query, but we\n     * need to return documents. The matchData and scores are combined\n     * from multiple fields belonging to the same document.\n     *\n     * Scores are calculated by field, using the query vectors created\n     * above, and combined into a final document score using addition.\n     */\n    var fieldRef = lunr.FieldRef.fromString(matchingFieldRefs[i]),\n        docRef = fieldRef.docRef\n\n    if (!allRequiredMatches.contains(docRef)) {\n      continue\n    }\n\n    if (allProhibitedMatches.contains(docRef)) {\n      continue\n    }\n\n    var fieldVector = this.fieldVectors[fieldRef],\n        score = queryVectors[fieldRef.fieldName].similarity(fieldVector),\n        docMatch\n\n    if ((docMatch = matches[docRef]) !== undefined) {\n      docMatch.score += score\n      docMatch.matchData.combine(matchingFields[fieldRef])\n    } else {\n      var match = {\n        ref: docRef,\n        score: score,\n        matchData: matchingFields[fieldRef]\n      }\n      matches[docRef] = match\n      results.push(match)\n    }\n  }\n\n  /*\n   * Sort the results objects by score, highest first.\n   */\n  return results.sort(function (a, b) {\n    return b.score - a.score\n  })\n}\n\n/**\n * Prepares the index for JSON serialization.\n *\n * The schema for this JSON blob will be described in a\n * separate JSON schema file.\n *\n * @returns {Object}\n */\nlunr.Index.prototype.toJSON = function () {\n  var invertedIndex = Object.keys(this.invertedIndex)\n    .sort()\n    .map(function (term) {\n      return [term, this.invertedIndex[term]]\n    }, this)\n\n  var fieldVectors = Object.keys(this.fieldVectors)\n    .map(function (ref) {\n      return [ref, this.fieldVectors[ref].toJSON()]\n    }, this)\n\n  return {\n    version: lunr.version,\n    fields: this.fields,\n    fieldVectors: fieldVectors,\n    invertedIndex: invertedIndex,\n    pipeline: this.pipeline.toJSON()\n  }\n}\n\n/**\n * Loads a previously serialized lunr.Index\n *\n * @param {Object} serializedIndex - A previously serialized lunr.Index\n * @returns {lunr.Index}\n */\nlunr.Index.load = function (serializedIndex) {\n  var attrs = {},\n      fieldVectors = {},\n      serializedVectors = serializedIndex.fieldVectors,\n      invertedIndex = Object.create(null),\n      serializedInvertedIndex = serializedIndex.invertedIndex,\n      tokenSetBuilder = new lunr.TokenSet.Builder,\n      pipeline = lunr.Pipeline.load(serializedIndex.pipeline)\n\n  if (serializedIndex.version != lunr.version) {\n    lunr.utils.warn(\"Version mismatch when loading serialised index. Current version of lunr '\" + lunr.version + \"' does not match serialized index '\" + serializedIndex.version + \"'\")\n  }\n\n  for (var i = 0; i < serializedVectors.length; i++) {\n    var tuple = serializedVectors[i],\n        ref = tuple[0],\n        elements = tuple[1]\n\n    fieldVectors[ref] = new lunr.Vector(elements)\n  }\n\n  for (var i = 0; i < serializedInvertedIndex.length; i++) {\n    var tuple = serializedInvertedIndex[i],\n        term = tuple[0],\n        posting = tuple[1]\n\n    tokenSetBuilder.insert(term)\n    invertedIndex[term] = posting\n  }\n\n  tokenSetBuilder.finish()\n\n  attrs.fields = serializedIndex.fields\n\n  attrs.fieldVectors = fieldVectors\n  attrs.invertedIndex = invertedIndex\n  attrs.tokenSet = tokenSetBuilder.root\n  attrs.pipeline = pipeline\n\n  return new lunr.Index(attrs)\n}\n/*!\n * lunr.Builder\n * Copyright (C) 2020 Oliver Nightingale\n */\n\n/**\n * lunr.Builder performs indexing on a set of documents and\n * returns instances of lunr.Index ready for querying.\n *\n * All configuration of the index is done via the builder, the\n * fields to index, the document reference, the text processing\n * pipeline and document scoring parameters are all set on the\n * builder before indexing.\n *\n * @constructor\n * @property {string} _ref - Internal reference to the document reference field.\n * @property {string[]} _fields - Internal reference to the document fields to index.\n * @property {object} invertedIndex - The inverted index maps terms to document fields.\n * @property {object} documentTermFrequencies - Keeps track of document term frequencies.\n * @property {object} documentLengths - Keeps track of the length of documents added to the index.\n * @property {lunr.tokenizer} tokenizer - Function for splitting strings into tokens for indexing.\n * @property {lunr.Pipeline} pipeline - The pipeline performs text processing on tokens before indexing.\n * @property {lunr.Pipeline} searchPipeline - A pipeline for processing search terms before querying the index.\n * @property {number} documentCount - Keeps track of the total number of documents indexed.\n * @property {number} _b - A parameter to control field length normalization, setting this to 0 disabled normalization, 1 fully normalizes field lengths, the default value is 0.75.\n * @property {number} _k1 - A parameter to control how quickly an increase in term frequency results in term frequency saturation, the default value is 1.2.\n * @property {number} termIndex - A counter incremented for each unique term, used to identify a terms position in the vector space.\n * @property {array} metadataWhitelist - A list of metadata keys that have been whitelisted for entry in the index.\n */\nlunr.Builder = function () {\n  this._ref = \"id\"\n  this._fields = Object.create(null)\n  this._documents = Object.create(null)\n  this.invertedIndex = Object.create(null)\n  this.fieldTermFrequencies = {}\n  this.fieldLengths = {}\n  this.tokenizer = lunr.tokenizer\n  this.pipeline = new lunr.Pipeline\n  this.searchPipeline = new lunr.Pipeline\n  this.documentCount = 0\n  this._b = 0.75\n  this._k1 = 1.2\n  this.termIndex = 0\n  this.metadataWhitelist = []\n}\n\n/**\n * Sets the document field used as the document reference. Every document must have this field.\n * The type of this field in the document should be a string, if it is not a string it will be\n * coerced into a string by calling toString.\n *\n * The default ref is 'id'.\n *\n * The ref should _not_ be changed during indexing, it should be set before any documents are\n * added to the index. Changing it during indexing can lead to inconsistent results.\n *\n * @param {string} ref - The name of the reference field in the document.\n */\nlunr.Builder.prototype.ref = function (ref) {\n  this._ref = ref\n}\n\n/**\n * A function that is used to extract a field from a document.\n *\n * Lunr expects a field to be at the top level of a document, if however the field\n * is deeply nested within a document an extractor function can be used to extract\n * the right field for indexing.\n *\n * @callback fieldExtractor\n * @param {object} doc - The document being added to the index.\n * @returns {?(string|object|object[])} obj - The object that will be indexed for this field.\n * @example <caption>Extracting a nested field</caption>\n * function (doc) { return doc.nested.field }\n */\n\n/**\n * Adds a field to the list of document fields that will be indexed. Every document being\n * indexed should have this field. Null values for this field in indexed documents will\n * not cause errors but will limit the chance of that document being retrieved by searches.\n *\n * All fields should be added before adding documents to the index. Adding fields after\n * a document has been indexed will have no effect on already indexed documents.\n *\n * Fields can be boosted at build time. This allows terms within that field to have more\n * importance when ranking search results. Use a field boost to specify that matches within\n * one field are more important than other fields.\n *\n * @param {string} fieldName - The name of a field to index in all documents.\n * @param {object} attributes - Optional attributes associated with this field.\n * @param {number} [attributes.boost=1] - Boost applied to all terms within this field.\n * @param {fieldExtractor} [attributes.extractor] - Function to extract a field from a document.\n * @throws {RangeError} fieldName cannot contain unsupported characters '/'\n */\nlunr.Builder.prototype.field = function (fieldName, attributes) {\n  if (/\\//.test(fieldName)) {\n    throw new RangeError (\"Field '\" + fieldName + \"' contains illegal character '/'\")\n  }\n\n  this._fields[fieldName] = attributes || {}\n}\n\n/**\n * A parameter to tune the amount of field length normalisation that is applied when\n * calculating relevance scores. A value of 0 will completely disable any normalisation\n * and a value of 1 will fully normalise field lengths. The default is 0.75. Values of b\n * will be clamped to the range 0 - 1.\n *\n * @param {number} number - The value to set for this tuning parameter.\n */\nlunr.Builder.prototype.b = function (number) {\n  if (number < 0) {\n    this._b = 0\n  } else if (number > 1) {\n    this._b = 1\n  } else {\n    this._b = number\n  }\n}\n\n/**\n * A parameter that controls the speed at which a rise in term frequency results in term\n * frequency saturation. The default value is 1.2. Setting this to a higher value will give\n * slower saturation levels, a lower value will result in quicker saturation.\n *\n * @param {number} number - The value to set for this tuning parameter.\n */\nlunr.Builder.prototype.k1 = function (number) {\n  this._k1 = number\n}\n\n/**\n * Adds a document to the index.\n *\n * Before adding fields to the index the index should have been fully setup, with the document\n * ref and all fields to index already having been specified.\n *\n * The document must have a field name as specified by the ref (by default this is 'id') and\n * it should have all fields defined for indexing, though null or undefined values will not\n * cause errors.\n *\n * Entire documents can be boosted at build time. Applying a boost to a document indicates that\n * this document should rank higher in search results than other documents.\n *\n * @param {object} doc - The document to add to the index.\n * @param {object} attributes - Optional attributes associated with this document.\n * @param {number} [attributes.boost=1] - Boost applied to all terms within this document.\n */\nlunr.Builder.prototype.add = function (doc, attributes) {\n  var docRef = doc[this._ref],\n      fields = Object.keys(this._fields)\n\n  this._documents[docRef] = attributes || {}\n  this.documentCount += 1\n\n  for (var i = 0; i < fields.length; i++) {\n    var fieldName = fields[i],\n        extractor = this._fields[fieldName].extractor,\n        field = extractor ? extractor(doc) : doc[fieldName],\n        tokens = this.tokenizer(field, {\n          fields: [fieldName]\n        }),\n        terms = this.pipeline.run(tokens),\n        fieldRef = new lunr.FieldRef (docRef, fieldName),\n        fieldTerms = Object.create(null)\n\n    this.fieldTermFrequencies[fieldRef] = fieldTerms\n    this.fieldLengths[fieldRef] = 0\n\n    // store the length of this field for this document\n    this.fieldLengths[fieldRef] += terms.length\n\n    // calculate term frequencies for this field\n    for (var j = 0; j < terms.length; j++) {\n      var term = terms[j]\n\n      if (fieldTerms[term] == undefined) {\n        fieldTerms[term] = 0\n      }\n\n      fieldTerms[term] += 1\n\n      // add to inverted index\n      // create an initial posting if one doesn't exist\n      if (this.invertedIndex[term] == undefined) {\n        var posting = Object.create(null)\n        posting[\"_index\"] = this.termIndex\n        this.termIndex += 1\n\n        for (var k = 0; k < fields.length; k++) {\n          posting[fields[k]] = Object.create(null)\n        }\n\n        this.invertedIndex[term] = posting\n      }\n\n      // add an entry for this term/fieldName/docRef to the invertedIndex\n      if (this.invertedIndex[term][fieldName][docRef] == undefined) {\n        this.invertedIndex[term][fieldName][docRef] = Object.create(null)\n      }\n\n      // store all whitelisted metadata about this token in the\n      // inverted index\n      for (var l = 0; l < this.metadataWhitelist.length; l++) {\n        var metadataKey = this.metadataWhitelist[l],\n            metadata = term.metadata[metadataKey]\n\n        if (this.invertedIndex[term][fieldName][docRef][metadataKey] == undefined) {\n          this.invertedIndex[term][fieldName][docRef][metadataKey] = []\n        }\n\n        this.invertedIndex[term][fieldName][docRef][metadataKey].push(metadata)\n      }\n    }\n\n  }\n}\n\n/**\n * Calculates the average document length for this index\n *\n * @private\n */\nlunr.Builder.prototype.calculateAverageFieldLengths = function () {\n\n  var fieldRefs = Object.keys(this.fieldLengths),\n      numberOfFields = fieldRefs.length,\n      accumulator = {},\n      documentsWithField = {}\n\n  for (var i = 0; i < numberOfFields; i++) {\n    var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]),\n        field = fieldRef.fieldName\n\n    documentsWithField[field] || (documentsWithField[field] = 0)\n    documentsWithField[field] += 1\n\n    accumulator[field] || (accumulator[field] = 0)\n    accumulator[field] += this.fieldLengths[fieldRef]\n  }\n\n  var fields = Object.keys(this._fields)\n\n  for (var i = 0; i < fields.length; i++) {\n    var fieldName = fields[i]\n    accumulator[fieldName] = accumulator[fieldName] / documentsWithField[fieldName]\n  }\n\n  this.averageFieldLength = accumulator\n}\n\n/**\n * Builds a vector space model of every document using lunr.Vector\n *\n * @private\n */\nlunr.Builder.prototype.createFieldVectors = function () {\n  var fieldVectors = {},\n      fieldRefs = Object.keys(this.fieldTermFrequencies),\n      fieldRefsLength = fieldRefs.length,\n      termIdfCache = Object.create(null)\n\n  for (var i = 0; i < fieldRefsLength; i++) {\n    var fieldRef = lunr.FieldRef.fromString(fieldRefs[i]),\n        fieldName = fieldRef.fieldName,\n        fieldLength = this.fieldLengths[fieldRef],\n        fieldVector = new lunr.Vector,\n        termFrequencies = this.fieldTermFrequencies[fieldRef],\n        terms = Object.keys(termFrequencies),\n        termsLength = terms.length\n\n\n    var fieldBoost = this._fields[fieldName].boost || 1,\n        docBoost = this._documents[fieldRef.docRef].boost || 1\n\n    for (var j = 0; j < termsLength; j++) {\n      var term = terms[j],\n          tf = termFrequencies[term],\n          termIndex = this.invertedIndex[term]._index,\n          idf, score, scoreWithPrecision\n\n      if (termIdfCache[term] === undefined) {\n        idf = lunr.idf(this.invertedIndex[term], this.documentCount)\n        termIdfCache[term] = idf\n      } else {\n        idf = termIdfCache[term]\n      }\n\n      score = idf * ((this._k1 + 1) * tf) / (this._k1 * (1 - this._b + this._b * (fieldLength / this.averageFieldLength[fieldName])) + tf)\n      score *= fieldBoost\n      score *= docBoost\n      scoreWithPrecision = Math.round(score * 1000) / 1000\n      // Converts 1.23456789 to 1.234.\n      // Reducing the precision so that the vectors take up less\n      // space when serialised. Doing it now so that they behave\n      // the same before and after serialisation. Also, this is\n      // the fastest approach to reducing a number's precision in\n      // JavaScript.\n\n      fieldVector.insert(termIndex, scoreWithPrecision)\n    }\n\n    fieldVectors[fieldRef] = fieldVector\n  }\n\n  this.fieldVectors = fieldVectors\n}\n\n/**\n * Creates a token set of all tokens in the index using lunr.TokenSet\n *\n * @private\n */\nlunr.Builder.prototype.createTokenSet = function () {\n  this.tokenSet = lunr.TokenSet.fromArray(\n    Object.keys(this.invertedIndex).sort()\n  )\n}\n\n/**\n * Builds the index, creating an instance of lunr.Index.\n *\n * This completes the indexing process and should only be called\n * once all documents have been added to the index.\n *\n * @returns {lunr.Index}\n */\nlunr.Builder.prototype.build = function () {\n  this.calculateAverageFieldLengths()\n  this.createFieldVectors()\n  this.createTokenSet()\n\n  return new lunr.Index({\n    invertedIndex: this.invertedIndex,\n    fieldVectors: this.fieldVectors,\n    tokenSet: this.tokenSet,\n    fields: Object.keys(this._fields),\n    pipeline: this.searchPipeline\n  })\n}\n\n/**\n * Applies a plugin to the index builder.\n *\n * A plugin is a function that is called with the index builder as its context.\n * Plugins can be used to customise or extend the behaviour of the index\n * in some way. A plugin is just a function, that encapsulated the custom\n * behaviour that should be applied when building the index.\n *\n * The plugin function will be called with the index builder as its argument, additional\n * arguments can also be passed when calling use. The function will be called\n * with the index builder as its context.\n *\n * @param {Function} plugin The plugin to apply.\n */\nlunr.Builder.prototype.use = function (fn) {\n  var args = Array.prototype.slice.call(arguments, 1)\n  args.unshift(this)\n  fn.apply(this, args)\n}\n/**\n * Contains and collects metadata about a matching document.\n * A single instance of lunr.MatchData is returned as part of every\n * lunr.Index~Result.\n *\n * @constructor\n * @param {string} term - The term this match data is associated with\n * @param {string} field - The field in which the term was found\n * @param {object} metadata - The metadata recorded about this term in this field\n * @property {object} metadata - A cloned collection of metadata associated with this document.\n * @see {@link lunr.Index~Result}\n */\nlunr.MatchData = function (term, field, metadata) {\n  var clonedMetadata = Object.create(null),\n      metadataKeys = Object.keys(metadata || {})\n\n  // Cloning the metadata to prevent the original\n  // being mutated during match data combination.\n  // Metadata is kept in an array within the inverted\n  // index so cloning the data can be done with\n  // Array#slice\n  for (var i = 0; i < metadataKeys.length; i++) {\n    var key = metadataKeys[i]\n    clonedMetadata[key] = metadata[key].slice()\n  }\n\n  this.metadata = Object.create(null)\n\n  if (term !== undefined) {\n    this.metadata[term] = Object.create(null)\n    this.metadata[term][field] = clonedMetadata\n  }\n}\n\n/**\n * An instance of lunr.MatchData will be created for every term that matches a\n * document. However only one instance is required in a lunr.Index~Result. This\n * method combines metadata from another instance of lunr.MatchData with this\n * objects metadata.\n *\n * @param {lunr.MatchData} otherMatchData - Another instance of match data to merge with this one.\n * @see {@link lunr.Index~Result}\n */\nlunr.MatchData.prototype.combine = function (otherMatchData) {\n  var terms = Object.keys(otherMatchData.metadata)\n\n  for (var i = 0; i < terms.length; i++) {\n    var term = terms[i],\n        fields = Object.keys(otherMatchData.metadata[term])\n\n    if (this.metadata[term] == undefined) {\n      this.metadata[term] = Object.create(null)\n    }\n\n    for (var j = 0; j < fields.length; j++) {\n      var field = fields[j],\n          keys = Object.keys(otherMatchData.metadata[term][field])\n\n      if (this.metadata[term][field] == undefined) {\n        this.metadata[term][field] = Object.create(null)\n      }\n\n      for (var k = 0; k < keys.length; k++) {\n        var key = keys[k]\n\n        if (this.metadata[term][field][key] == undefined) {\n          this.metadata[term][field][key] = otherMatchData.metadata[term][field][key]\n        } else {\n          this.metadata[term][field][key] = this.metadata[term][field][key].concat(otherMatchData.metadata[term][field][key])\n        }\n\n      }\n    }\n  }\n}\n\n/**\n * Add metadata for a term/field pair to this instance of match data.\n *\n * @param {string} term - The term this match data is associated with\n * @param {string} field - The field in which the term was found\n * @param {object} metadata - The metadata recorded about this term in this field\n */\nlunr.MatchData.prototype.add = function (term, field, metadata) {\n  if (!(term in this.metadata)) {\n    this.metadata[term] = Object.create(null)\n    this.metadata[term][field] = metadata\n    return\n  }\n\n  if (!(field in this.metadata[term])) {\n    this.metadata[term][field] = metadata\n    return\n  }\n\n  var metadataKeys = Object.keys(metadata)\n\n  for (var i = 0; i < metadataKeys.length; i++) {\n    var key = metadataKeys[i]\n\n    if (key in this.metadata[term][field]) {\n      this.metadata[term][field][key] = this.metadata[term][field][key].concat(metadata[key])\n    } else {\n      this.metadata[term][field][key] = metadata[key]\n    }\n  }\n}\n/**\n * A lunr.Query provides a programmatic way of defining queries to be performed\n * against a {@link lunr.Index}.\n *\n * Prefer constructing a lunr.Query using the {@link lunr.Index#query} method\n * so the query object is pre-initialized with the right index fields.\n *\n * @constructor\n * @property {lunr.Query~Clause[]} clauses - An array of query clauses.\n * @property {string[]} allFields - An array of all available fields in a lunr.Index.\n */\nlunr.Query = function (allFields) {\n  this.clauses = []\n  this.allFields = allFields\n}\n\n/**\n * Constants for indicating what kind of automatic wildcard insertion will be used when constructing a query clause.\n *\n * This allows wildcards to be added to the beginning and end of a term without having to manually do any string\n * concatenation.\n *\n * The wildcard constants can be bitwise combined to select both leading and trailing wildcards.\n *\n * @constant\n * @default\n * @property {number} wildcard.NONE - The term will have no wildcards inserted, this is the default behaviour\n * @property {number} wildcard.LEADING - Prepend the term with a wildcard, unless a leading wildcard already exists\n * @property {number} wildcard.TRAILING - Append a wildcard to the term, unless a trailing wildcard already exists\n * @see lunr.Query~Clause\n * @see lunr.Query#clause\n * @see lunr.Query#term\n * @example <caption>query term with trailing wildcard</caption>\n * query.term('foo', { wildcard: lunr.Query.wildcard.TRAILING })\n * @example <caption>query term with leading and trailing wildcard</caption>\n * query.term('foo', {\n *   wildcard: lunr.Query.wildcard.LEADING | lunr.Query.wildcard.TRAILING\n * })\n */\n\nlunr.Query.wildcard = new String (\"*\")\nlunr.Query.wildcard.NONE = 0\nlunr.Query.wildcard.LEADING = 1\nlunr.Query.wildcard.TRAILING = 2\n\n/**\n * Constants for indicating what kind of presence a term must have in matching documents.\n *\n * @constant\n * @enum {number}\n * @see lunr.Query~Clause\n * @see lunr.Query#clause\n * @see lunr.Query#term\n * @example <caption>query term with required presence</caption>\n * query.term('foo', { presence: lunr.Query.presence.REQUIRED })\n */\nlunr.Query.presence = {\n  /**\n   * Term's presence in a document is optional, this is the default value.\n   */\n  OPTIONAL: 1,\n\n  /**\n   * Term's presence in a document is required, documents that do not contain\n   * this term will not be returned.\n   */\n  REQUIRED: 2,\n\n  /**\n   * Term's presence in a document is prohibited, documents that do contain\n   * this term will not be returned.\n   */\n  PROHIBITED: 3\n}\n\n/**\n * A single clause in a {@link lunr.Query} contains a term and details on how to\n * match that term against a {@link lunr.Index}.\n *\n * @typedef {Object} lunr.Query~Clause\n * @property {string[]} fields - The fields in an index this clause should be matched against.\n * @property {number} [boost=1] - Any boost that should be applied when matching this clause.\n * @property {number} [editDistance] - Whether the term should have fuzzy matching applied, and how fuzzy the match should be.\n * @property {boolean} [usePipeline] - Whether the term should be passed through the search pipeline.\n * @property {number} [wildcard=lunr.Query.wildcard.NONE] - Whether the term should have wildcards appended or prepended.\n * @property {number} [presence=lunr.Query.presence.OPTIONAL] - The terms presence in any matching documents.\n */\n\n/**\n * Adds a {@link lunr.Query~Clause} to this query.\n *\n * Unless the clause contains the fields to be matched all fields will be matched. In addition\n * a default boost of 1 is applied to the clause.\n *\n * @param {lunr.Query~Clause} clause - The clause to add to this query.\n * @see lunr.Query~Clause\n * @returns {lunr.Query}\n */\nlunr.Query.prototype.clause = function (clause) {\n  if (!('fields' in clause)) {\n    clause.fields = this.allFields\n  }\n\n  if (!('boost' in clause)) {\n    clause.boost = 1\n  }\n\n  if (!('usePipeline' in clause)) {\n    clause.usePipeline = true\n  }\n\n  if (!('wildcard' in clause)) {\n    clause.wildcard = lunr.Query.wildcard.NONE\n  }\n\n  if ((clause.wildcard & lunr.Query.wildcard.LEADING) && (clause.term.charAt(0) != lunr.Query.wildcard)) {\n    clause.term = \"*\" + clause.term\n  }\n\n  if ((clause.wildcard & lunr.Query.wildcard.TRAILING) && (clause.term.slice(-1) != lunr.Query.wildcard)) {\n    clause.term = \"\" + clause.term + \"*\"\n  }\n\n  if (!('presence' in clause)) {\n    clause.presence = lunr.Query.presence.OPTIONAL\n  }\n\n  this.clauses.push(clause)\n\n  return this\n}\n\n/**\n * A negated query is one in which every clause has a presence of\n * prohibited. These queries require some special processing to return\n * the expected results.\n *\n * @returns boolean\n */\nlunr.Query.prototype.isNegated = function () {\n  for (var i = 0; i < this.clauses.length; i++) {\n    if (this.clauses[i].presence != lunr.Query.presence.PROHIBITED) {\n      return false\n    }\n  }\n\n  return true\n}\n\n/**\n * Adds a term to the current query, under the covers this will create a {@link lunr.Query~Clause}\n * to the list of clauses that make up this query.\n *\n * The term is used as is, i.e. no tokenization will be performed by this method. Instead conversion\n * to a token or token-like string should be done before calling this method.\n *\n * The term will be converted to a string by calling `toString`. Multiple terms can be passed as an\n * array, each term in the array will share the same options.\n *\n * @param {object|object[]} term - The term(s) to add to the query.\n * @param {object} [options] - Any additional properties to add to the query clause.\n * @returns {lunr.Query}\n * @see lunr.Query#clause\n * @see lunr.Query~Clause\n * @example <caption>adding a single term to a query</caption>\n * query.term(\"foo\")\n * @example <caption>adding a single term to a query and specifying search fields, term boost and automatic trailing wildcard</caption>\n * query.term(\"foo\", {\n *   fields: [\"title\"],\n *   boost: 10,\n *   wildcard: lunr.Query.wildcard.TRAILING\n * })\n * @example <caption>using lunr.tokenizer to convert a string to tokens before using them as terms</caption>\n * query.term(lunr.tokenizer(\"foo bar\"))\n */\nlunr.Query.prototype.term = function (term, options) {\n  if (Array.isArray(term)) {\n    term.forEach(function (t) { this.term(t, lunr.utils.clone(options)) }, this)\n    return this\n  }\n\n  var clause = options || {}\n  clause.term = term.toString()\n\n  this.clause(clause)\n\n  return this\n}\nlunr.QueryParseError = function (message, start, end) {\n  this.name = \"QueryParseError\"\n  this.message = message\n  this.start = start\n  this.end = end\n}\n\nlunr.QueryParseError.prototype = new Error\nlunr.QueryLexer = function (str) {\n  this.lexemes = []\n  this.str = str\n  this.length = str.length\n  this.pos = 0\n  this.start = 0\n  this.escapeCharPositions = []\n}\n\nlunr.QueryLexer.prototype.run = function () {\n  var state = lunr.QueryLexer.lexText\n\n  while (state) {\n    state = state(this)\n  }\n}\n\nlunr.QueryLexer.prototype.sliceString = function () {\n  var subSlices = [],\n      sliceStart = this.start,\n      sliceEnd = this.pos\n\n  for (var i = 0; i < this.escapeCharPositions.length; i++) {\n    sliceEnd = this.escapeCharPositions[i]\n    subSlices.push(this.str.slice(sliceStart, sliceEnd))\n    sliceStart = sliceEnd + 1\n  }\n\n  subSlices.push(this.str.slice(sliceStart, this.pos))\n  this.escapeCharPositions.length = 0\n\n  return subSlices.join('')\n}\n\nlunr.QueryLexer.prototype.emit = function (type) {\n  this.lexemes.push({\n    type: type,\n    str: this.sliceString(),\n    start: this.start,\n    end: this.pos\n  })\n\n  this.start = this.pos\n}\n\nlunr.QueryLexer.prototype.escapeCharacter = function () {\n  this.escapeCharPositions.push(this.pos - 1)\n  this.pos += 1\n}\n\nlunr.QueryLexer.prototype.next = function () {\n  if (this.pos >= this.length) {\n    return lunr.QueryLexer.EOS\n  }\n\n  var char = this.str.charAt(this.pos)\n  this.pos += 1\n  return char\n}\n\nlunr.QueryLexer.prototype.width = function () {\n  return this.pos - this.start\n}\n\nlunr.QueryLexer.prototype.ignore = function () {\n  if (this.start == this.pos) {\n    this.pos += 1\n  }\n\n  this.start = this.pos\n}\n\nlunr.QueryLexer.prototype.backup = function () {\n  this.pos -= 1\n}\n\nlunr.QueryLexer.prototype.acceptDigitRun = function () {\n  var char, charCode\n\n  do {\n    char = this.next()\n    charCode = char.charCodeAt(0)\n  } while (charCode > 47 && charCode < 58)\n\n  if (char != lunr.QueryLexer.EOS) {\n    this.backup()\n  }\n}\n\nlunr.QueryLexer.prototype.more = function () {\n  return this.pos < this.length\n}\n\nlunr.QueryLexer.EOS = 'EOS'\nlunr.QueryLexer.FIELD = 'FIELD'\nlunr.QueryLexer.TERM = 'TERM'\nlunr.QueryLexer.EDIT_DISTANCE = 'EDIT_DISTANCE'\nlunr.QueryLexer.BOOST = 'BOOST'\nlunr.QueryLexer.PRESENCE = 'PRESENCE'\n\nlunr.QueryLexer.lexField = function (lexer) {\n  lexer.backup()\n  lexer.emit(lunr.QueryLexer.FIELD)\n  lexer.ignore()\n  return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexTerm = function (lexer) {\n  if (lexer.width() > 1) {\n    lexer.backup()\n    lexer.emit(lunr.QueryLexer.TERM)\n  }\n\n  lexer.ignore()\n\n  if (lexer.more()) {\n    return lunr.QueryLexer.lexText\n  }\n}\n\nlunr.QueryLexer.lexEditDistance = function (lexer) {\n  lexer.ignore()\n  lexer.acceptDigitRun()\n  lexer.emit(lunr.QueryLexer.EDIT_DISTANCE)\n  return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexBoost = function (lexer) {\n  lexer.ignore()\n  lexer.acceptDigitRun()\n  lexer.emit(lunr.QueryLexer.BOOST)\n  return lunr.QueryLexer.lexText\n}\n\nlunr.QueryLexer.lexEOS = function (lexer) {\n  if (lexer.width() > 0) {\n    lexer.emit(lunr.QueryLexer.TERM)\n  }\n}\n\n// This matches the separator used when tokenising fields\n// within a document. These should match otherwise it is\n// not possible to search for some tokens within a document.\n//\n// It is possible for the user to change the separator on the\n// tokenizer so it _might_ clash with any other of the special\n// characters already used within the search string, e.g. :.\n//\n// This means that it is possible to change the separator in\n// such a way that makes some words unsearchable using a search\n// string.\nlunr.QueryLexer.termSeparator = lunr.tokenizer.separator\n\nlunr.QueryLexer.lexText = function (lexer) {\n  while (true) {\n    var char = lexer.next()\n\n    if (char == lunr.QueryLexer.EOS) {\n      return lunr.QueryLexer.lexEOS\n    }\n\n    // Escape character is '\\'\n    if (char.charCodeAt(0) == 92) {\n      lexer.escapeCharacter()\n      continue\n    }\n\n    if (char == \":\") {\n      return lunr.QueryLexer.lexField\n    }\n\n    if (char == \"~\") {\n      lexer.backup()\n      if (lexer.width() > 0) {\n        lexer.emit(lunr.QueryLexer.TERM)\n      }\n      return lunr.QueryLexer.lexEditDistance\n    }\n\n    if (char == \"^\") {\n      lexer.backup()\n      if (lexer.width() > 0) {\n        lexer.emit(lunr.QueryLexer.TERM)\n      }\n      return lunr.QueryLexer.lexBoost\n    }\n\n    // \"+\" indicates term presence is required\n    // checking for length to ensure that only\n    // leading \"+\" are considered\n    if (char == \"+\" && lexer.width() === 1) {\n      lexer.emit(lunr.QueryLexer.PRESENCE)\n      return lunr.QueryLexer.lexText\n    }\n\n    // \"-\" indicates term presence is prohibited\n    // checking for length to ensure that only\n    // leading \"-\" are considered\n    if (char == \"-\" && lexer.width() === 1) {\n      lexer.emit(lunr.QueryLexer.PRESENCE)\n      return lunr.QueryLexer.lexText\n    }\n\n    if (char.match(lunr.QueryLexer.termSeparator)) {\n      return lunr.QueryLexer.lexTerm\n    }\n  }\n}\n\nlunr.QueryParser = function (str, query) {\n  this.lexer = new lunr.QueryLexer (str)\n  this.query = query\n  this.currentClause = {}\n  this.lexemeIdx = 0\n}\n\nlunr.QueryParser.prototype.parse = function () {\n  this.lexer.run()\n  this.lexemes = this.lexer.lexemes\n\n  var state = lunr.QueryParser.parseClause\n\n  while (state) {\n    state = state(this)\n  }\n\n  return this.query\n}\n\nlunr.QueryParser.prototype.peekLexeme = function () {\n  return this.lexemes[this.lexemeIdx]\n}\n\nlunr.QueryParser.prototype.consumeLexeme = function () {\n  var lexeme = this.peekLexeme()\n  this.lexemeIdx += 1\n  return lexeme\n}\n\nlunr.QueryParser.prototype.nextClause = function () {\n  var completedClause = this.currentClause\n  this.query.clause(completedClause)\n  this.currentClause = {}\n}\n\nlunr.QueryParser.parseClause = function (parser) {\n  var lexeme = parser.peekLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  switch (lexeme.type) {\n    case lunr.QueryLexer.PRESENCE:\n      return lunr.QueryParser.parsePresence\n    case lunr.QueryLexer.FIELD:\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.TERM:\n      return lunr.QueryParser.parseTerm\n    default:\n      var errorMessage = \"expected either a field or a term, found \" + lexeme.type\n\n      if (lexeme.str.length >= 1) {\n        errorMessage += \" with value '\" + lexeme.str + \"'\"\n      }\n\n      throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n}\n\nlunr.QueryParser.parsePresence = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  switch (lexeme.str) {\n    case \"-\":\n      parser.currentClause.presence = lunr.Query.presence.PROHIBITED\n      break\n    case \"+\":\n      parser.currentClause.presence = lunr.Query.presence.REQUIRED\n      break\n    default:\n      var errorMessage = \"unrecognised presence operator'\" + lexeme.str + \"'\"\n      throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    var errorMessage = \"expecting term or field, found nothing\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.FIELD:\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.TERM:\n      return lunr.QueryParser.parseTerm\n    default:\n      var errorMessage = \"expecting term or field, found '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseField = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  if (parser.query.allFields.indexOf(lexeme.str) == -1) {\n    var possibleFields = parser.query.allFields.map(function (f) { return \"'\" + f + \"'\" }).join(', '),\n        errorMessage = \"unrecognised field '\" + lexeme.str + \"', possible fields: \" + possibleFields\n\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  parser.currentClause.fields = [lexeme.str]\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    var errorMessage = \"expecting term, found nothing\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      return lunr.QueryParser.parseTerm\n    default:\n      var errorMessage = \"expecting term, found '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseTerm = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  parser.currentClause.term = lexeme.str.toLowerCase()\n\n  if (lexeme.str.indexOf(\"*\") != -1) {\n    parser.currentClause.usePipeline = false\n  }\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    parser.nextClause()\n    return\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      parser.nextClause()\n      return lunr.QueryParser.parseTerm\n    case lunr.QueryLexer.FIELD:\n      parser.nextClause()\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.EDIT_DISTANCE:\n      return lunr.QueryParser.parseEditDistance\n    case lunr.QueryLexer.BOOST:\n      return lunr.QueryParser.parseBoost\n    case lunr.QueryLexer.PRESENCE:\n      parser.nextClause()\n      return lunr.QueryParser.parsePresence\n    default:\n      var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseEditDistance = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  var editDistance = parseInt(lexeme.str, 10)\n\n  if (isNaN(editDistance)) {\n    var errorMessage = \"edit distance must be numeric\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  parser.currentClause.editDistance = editDistance\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    parser.nextClause()\n    return\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      parser.nextClause()\n      return lunr.QueryParser.parseTerm\n    case lunr.QueryLexer.FIELD:\n      parser.nextClause()\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.EDIT_DISTANCE:\n      return lunr.QueryParser.parseEditDistance\n    case lunr.QueryLexer.BOOST:\n      return lunr.QueryParser.parseBoost\n    case lunr.QueryLexer.PRESENCE:\n      parser.nextClause()\n      return lunr.QueryParser.parsePresence\n    default:\n      var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\nlunr.QueryParser.parseBoost = function (parser) {\n  var lexeme = parser.consumeLexeme()\n\n  if (lexeme == undefined) {\n    return\n  }\n\n  var boost = parseInt(lexeme.str, 10)\n\n  if (isNaN(boost)) {\n    var errorMessage = \"boost must be numeric\"\n    throw new lunr.QueryParseError (errorMessage, lexeme.start, lexeme.end)\n  }\n\n  parser.currentClause.boost = boost\n\n  var nextLexeme = parser.peekLexeme()\n\n  if (nextLexeme == undefined) {\n    parser.nextClause()\n    return\n  }\n\n  switch (nextLexeme.type) {\n    case lunr.QueryLexer.TERM:\n      parser.nextClause()\n      return lunr.QueryParser.parseTerm\n    case lunr.QueryLexer.FIELD:\n      parser.nextClause()\n      return lunr.QueryParser.parseField\n    case lunr.QueryLexer.EDIT_DISTANCE:\n      return lunr.QueryParser.parseEditDistance\n    case lunr.QueryLexer.BOOST:\n      return lunr.QueryParser.parseBoost\n    case lunr.QueryLexer.PRESENCE:\n      parser.nextClause()\n      return lunr.QueryParser.parsePresence\n    default:\n      var errorMessage = \"Unexpected lexeme type '\" + nextLexeme.type + \"'\"\n      throw new lunr.QueryParseError (errorMessage, nextLexeme.start, nextLexeme.end)\n  }\n}\n\n  /**\n   * export the module via AMD, CommonJS or as a browser global\n   * Export code from https://github.com/umdjs/umd/blob/master/returnExports.js\n   */\n  ;(function (root, factory) {\n    if (typeof define === 'function' && define.amd) {\n      // AMD. Register as an anonymous module.\n      define(factory)\n    } else if (typeof exports === 'object') {\n      /**\n       * Node. Does not work with strict CommonJS, but\n       * only CommonJS-like enviroments that support module.exports,\n       * like Node.\n       */\n      module.exports = factory()\n    } else {\n      // Browser globals (root is window)\n      root.lunr = factory()\n    }\n  }(this, function () {\n    /**\n     * Just return a value to define the module export.\n     * This example returns an object, but the module\n     * can return a function as the exported value.\n     */\n    return lunr\n  }))\n})();\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A RTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport lunr from \"lunr\"\n\nimport { getElement } from \"~/browser/element/_\"\nimport \"~/polyfills\"\n\nimport { Search } from \"../../_\"\nimport { SearchConfig } from \"../../config\"\nimport {\n  SearchMessage,\n  SearchMessageType\n} from \"../message\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Add support for `iframe-worker` shim\n *\n * While `importScripts` is synchronous when executed inside of a web worker,\n * it's not possible to provide a synchronous shim implementation. The cool\n * thing is that awaiting a non-Promise will convert it into a Promise, so\n * extending the type definition to return a `Promise` shouldn't break anything.\n *\n * @see https://bit.ly/2PjDnXi - GitHub comment\n *\n * @param urls - Scripts to load\n *\n * @returns Promise resolving with no result\n */\ndeclare global {\n  function importScripts(...urls: string[]): Promise<void> | void\n}\n\n/* ----------------------------------------------------------------------------\n * Data\n * ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nlet index: Search\n\n/* ----------------------------------------------------------------------------\n * Helper functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Fetch (= import) multi-language support through `lunr-languages`\n *\n * This function automatically imports the stemmers necessary to process the\n * languages which are defined as part of the search configuration.\n *\n * If the worker runs inside of an `iframe` (when using `iframe-worker` as\n * a shim), the base URL for the stemmers to be loaded must be determined by\n * searching for the first `script` element with a `src` attribute, which will\n * contain the contents of this script.\n *\n * @param config - Search configuration\n *\n * @returns Promise resolving with no result\n */\nasync function setupSearchLanguages(\n  config: SearchConfig\n): Promise<void> {\n  let base = \"../lunr\"\n\n  /* Detect `iframe-worker` and fix base URL */\n  if (typeof parent !== \"undefined\" && \"IFrameWorker\" in parent) {\n    const worker = getElement<HTMLScriptElement>(\"script[src]\")!\n    const [path] = worker.src.split(\"/worker\")\n\n    /* Prefix base with path */\n    base = base.replace(\"..\", path)\n  }\n\n  /* Add scripts for languages */\n  const scripts = []\n  for (const lang of config.lang) {\n    switch (lang) {\n\n      /* Add segmenter for Japanese */\n      case \"ja\":\n        scripts.push(`${base}/tinyseg.js`)\n        break\n\n      /* Add segmenter for Hindi and Thai */\n      case \"hi\":\n      case \"th\":\n        scripts.push(`${base}/wordcut.js`)\n        break\n    }\n\n    /* Add language support */\n    if (lang !== \"en\")\n      scripts.push(`${base}/min/lunr.${lang}.min.js`)\n  }\n\n  /* Add multi-language support */\n  if (config.lang.length > 1)\n    scripts.push(`${base}/min/lunr.multi.min.js`)\n\n  /* Load scripts synchronously */\n  if (scripts.length)\n    await importScripts(\n      `${base}/min/lunr.stemmer.support.min.js`,\n      ...scripts\n    )\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Message handler\n *\n * @param message - Source message\n *\n * @returns Target message\n */\nexport async function handler(\n  message: SearchMessage\n): Promise<SearchMessage> {\n  switch (message.type) {\n\n    /* Search setup message */\n    case SearchMessageType.SETUP:\n      await setupSearchLanguages(message.data.config)\n      index = new Search(message.data)\n      return {\n        type: SearchMessageType.READY\n      }\n\n    /* Search query message */\n    case SearchMessageType.QUERY:\n      const query = message.data\n      try {\n        return {\n          type: SearchMessageType.RESULT,\n          data: index.search(query)\n        }\n\n      /* Return empty result in case of error */\n      } catch (err) {\n        console.warn(`Invalid query: ${query} \u2013 see https://bit.ly/2s3ChXG`)\n        console.warn(err)\n        return {\n          type: SearchMessageType.RESULT,\n          data: { items: [] }\n        }\n      }\n\n    /* All other messages */\n    default:\n      throw new TypeError(\"Invalid message type\")\n  }\n}\n\n/* ----------------------------------------------------------------------------\n * Worker\n * ------------------------------------------------------------------------- */\n\n/* Expose Lunr.js in global scope, or stemmers won't work */\nself.lunr = lunr\n\n/* Handle messages */\naddEventListener(\"message\", async ev => {\n  postMessage(await handler(ev.data))\n})\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Retrieve all elements matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Elements\n */\nexport function getElements<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T][]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T[]\n\nexport function getElements<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T[] {\n  return Array.from(node.querySelectorAll<T>(selector))\n}\n\n/**\n * Retrieve an element matching a query selector or throw a reference error\n *\n * Note that this function assumes that the element is present. If unsure if an\n * element is existent, use the `getOptionalElement` function instead.\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element\n */\nexport function getElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T]\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T\n\nexport function getElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T {\n  const el = getOptionalElement<T>(selector, node)\n  if (typeof el === \"undefined\")\n    throw new ReferenceError(\n      `Missing element: expected \"${selector}\" to be present`\n    )\n\n  /* Return element */\n  return el\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Retrieve an optional element matching the query selector\n *\n * @template T - Element type\n *\n * @param selector - Query selector\n * @param node - Node of reference\n *\n * @returns Element or nothing\n */\nexport function getOptionalElement<T extends keyof HTMLElementTagNameMap>(\n  selector: T, node?: ParentNode\n): HTMLElementTagNameMap[T] | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node?: ParentNode\n): T | undefined\n\nexport function getOptionalElement<T extends HTMLElement>(\n  selector: string, node: ParentNode = document\n): T | undefined {\n  return node.querySelector<T>(selector) || undefined\n}\n\n/**\n * Retrieve the currently active element\n *\n * @returns Element or nothing\n */\nexport function getActiveElement(): HTMLElement | undefined {\n  return document.activeElement instanceof HTMLElement\n    ? document.activeElement || undefined\n    : undefined\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Polyfills\n * ------------------------------------------------------------------------- */\n\n/* Polyfill `Object.entries` */\nif (!Object.entries)\n  Object.entries = function (obj: object) {\n    const data: [string, string][] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push([key, obj[key]])\n\n    /* Return entries */\n    return data\n  }\n\n/* Polyfill `Object.values` */\nif (!Object.values)\n  Object.values = function (obj: object) {\n    const data: string[] = []\n    for (const key of Object.keys(obj))\n      // @ts-expect-error - ignore property access warning\n      data.push(obj[key])\n\n    /* Return values */\n    return data\n  }\n\n/* ------------------------------------------------------------------------- */\n\n/* Polyfills for `Element` */\nif (typeof Element !== \"undefined\") {\n\n  /* Polyfill `Element.scrollTo` */\n  if (!Element.prototype.scrollTo)\n    Element.prototype.scrollTo = function (\n      x?: ScrollToOptions | number, y?: number\n    ): void {\n      if (typeof x === \"object\") {\n        this.scrollLeft = x.left!\n        this.scrollTop = x.top!\n      } else {\n        this.scrollLeft = x!\n        this.scrollTop = y!\n      }\n    }\n\n  /* Polyfill `Element.replaceWith` */\n  if (!Element.prototype.replaceWith)\n    Element.prototype.replaceWith = function (\n      ...nodes: Array<string | Node>\n    ): void {\n      const parent = this.parentNode\n      if (parent) {\n        if (nodes.length === 0)\n          parent.removeChild(this)\n\n        /* Replace children and create text nodes */\n        for (let i = nodes.length - 1; i >= 0; i--) {\n          let node = nodes[i]\n          if (typeof node === \"string\")\n            node = document.createTextNode(node)\n          else if (node.parentNode)\n            node.parentNode.removeChild(node)\n\n          /* Replace child or insert before previous sibling */\n          if (!i)\n            parent.replaceChild(node, this)\n          else\n            parent.insertBefore(this.previousSibling!, node)\n        }\n      }\n    }\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search configuration\n */\nexport interface SearchConfig {\n  lang: string[]                       /* Search languages */\n  separator: string                    /* Search separator */\n  pipeline: SearchPipelineFn[]         /* Search pipeline */\n}\n\n/**\n * Search document\n */\nexport interface SearchDocument {\n  location: string                     /* Document location */\n  title: string                        /* Document title */\n  text: string                         /* Document text */\n  tags?: string[]                      /* Document tags */\n  boost?: number                       /* Document boost */\n  parent?: SearchDocument              /* Document parent */\n}\n\n/**\n * Search options\n */\nexport interface SearchOptions {\n  suggest: boolean                     /* Search suggestions */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nexport interface SearchIndex {\n  config: SearchConfig                 /* Search configuration */\n  docs: SearchDocument[]               /* Search documents */\n  options: SearchOptions               /* Search options */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search pipeline function\n */\ntype SearchPipelineFn =\n  | \"trimmer\"                          /* Trimmer */\n  | \"stopWordFilter\"                   /* Stop word filter */\n  | \"stemmer\"                          /* Stemmer */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create a search document map\n *\n * This function creates a mapping of URLs (including anchors) to the actual\n * articles and sections. It relies on the invariant that the search index is\n * ordered with the main article appearing before all sections with anchors.\n * If this is not the case, the logic music be changed.\n *\n * @param docs - Search documents\n *\n * @returns Search document map\n */\nexport function setupSearchDocumentMap(\n  docs: SearchDocument[]\n): Map<string, SearchDocument> {\n  const map = new Map<string, SearchDocument>()\n  for (const doc of docs) {\n    const [path] = doc.location.split(\"#\")\n\n    /* Add document article */\n    const article = map.get(path)\n    if (typeof article === \"undefined\") {\n      map.set(path, doc)\n\n      /* Add document section */\n    } else {\n      map.set(doc.location, doc)\n      doc.parent = article\n    }\n  }\n\n  /* Return search document map */\n  return map\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Visitor function\n *\n * @param start - Start offset\n * @param end - End offset\n */\ntype VisitorFn = (\n  start: number, end: number\n) => void\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Split a string using the given separator\n *\n * @param input - Input value\n * @param separator - Separator\n * @param fn - Visitor function\n */\nexport function split(\n  input: string, separator: RegExp, fn: VisitorFn\n): void {\n  separator = new RegExp(separator, \"g\")\n\n  /* Split string using separator */\n  let match: RegExpExecArray | null\n  let index = 0\n  do {\n    match = separator.exec(input)\n\n    /* Emit non-empty range */\n    const until = match?.index ?? input.length\n    if (index < until)\n      fn(index, until)\n\n    /* Update last index */\n    if (match) {\n      const [term] = match\n      index = match.index + term.length\n\n      /* Support zero-length lookaheads */\n      if (term.length === 0)\n        separator.lastIndex = match.index + 1\n    }\n  } while (match)\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Extraction type\n *\n * This type defines the possible values that are encoded into the first two\n * bits of a section that is part of the blocks of a tokenization table. There\n * are three types of interest: HTML opening and closing tags, as well as the\n * actual text content we need to extract for indexing.\n */\nexport const enum Extract {\n  TAG_OPEN  = 0,                       /* HTML opening tag */\n  TEXT      = 1,                       /* Text content */\n  TAG_CLOSE = 2                        /* HTML closing tag */\n}\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Visitor function\n *\n * @param block - Block index\n * @param type - Extraction type\n * @param start - Start offset\n * @param end - End offset\n */\ntype VisitorFn = (\n  block: number, type: Extract, start: number, end: number\n) => void\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Split a string into markup and text sections\n *\n * This function scans a string and divides it up into sections of markup and\n * text. For each section, it invokes the given visitor function with the block\n * index, extraction type, as well as start and end offsets. Using a visitor\n * function (= streaming data) is ideal for minimizing pressure on the GC.\n *\n * @param input - Input value\n * @param fn - Visitor function\n */\nexport function extract(\n  input: string, fn: VisitorFn\n): void {\n\n  let block = 0                        /* Current block */\n  let start = 0                        /* Current start offset */\n  let end = 0                          /* Current end offset */\n\n  /* Split string into sections */\n  for (let stack = 0; end < input.length; end++) {\n\n    /* Opening tag after non-empty section */\n    if (input.charAt(end) === \"<\" && end > start) {\n      fn(block, Extract.TEXT, start, start = end)\n\n    /* Closing tag */\n    } else if (input.charAt(end) === \">\") {\n      if (input.charAt(start + 1) === \"/\") {\n        if (--stack === 0)\n          fn(block++, Extract.TAG_CLOSE, start, end + 1)\n\n      /* Tag is not self-closing */\n      } else if (input.charAt(end - 1) !== \"/\") {\n        if (stack++ === 0)\n          fn(block, Extract.TAG_OPEN, start, end + 1)\n      }\n\n      /* New section */\n      start = end + 1\n    }\n  }\n\n  /* Add trailing section */\n  if (end > start)\n    fn(block, Extract.TEXT, start, end)\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Position table\n */\nexport type PositionTable = number[][]\n\n/**\n * Position\n */\nexport type Position = number\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Highlight all occurrences in a string\n *\n * This function receives a field's value (e.g. like `title` or `text`), it's\n * position table that was generated during indexing, and the positions found\n * when executing the query. It then highlights all occurrences, and returns\n * their concatenation. In case of multiple blocks, two are returned.\n *\n * @param input - Input value\n * @param table - Table for indexing\n * @param positions - Occurrences\n * @param full - Full results\n *\n * @returns Highlighted string value\n */\nexport function highlight(\n  input: string, table: PositionTable, positions: Position[], full = false\n): string {\n  return highlightAll([input], table, positions, full).pop()!\n}\n\n/**\n * Highlight all occurrences in a set of strings\n *\n * @param inputs - Input values\n * @param table - Table for indexing\n * @param positions - Occurrences\n * @param full - Full results\n *\n * @returns Highlighted string values\n */\nexport function highlightAll(\n  inputs: string[], table: PositionTable, positions: Position[], full = false\n): string[] {\n\n  /* Map blocks to input values */\n  const mapping = [0]\n  for (let t = 1; t < table.length; t++) {\n    const prev = table[t - 1]\n    const next = table[t]\n\n    /* Check if table points to new block */\n    const p = prev[prev.length - 1] >>> 2 & 0x3FF\n    const q = next[0]               >>> 12\n\n    /* Add block to mapping */\n    mapping.push(+(p > q) + mapping[mapping.length - 1])\n  }\n\n  /* Highlight strings one after another */\n  return inputs.map((input, i) => {\n    let cursor = 0\n\n    /* Map occurrences to blocks */\n    const blocks = new Map<number, number[]>()\n    for (const p of positions.sort((a, b) => a - b)) {\n      const index = p & 0xFFFFF\n      const block = p >>> 20\n      if (mapping[block] !== i)\n        continue\n\n      /* Ensure presence of block group */\n      let group = blocks.get(block)\n      if (typeof group === \"undefined\")\n        blocks.set(block, group = [])\n\n      /* Add index to group */\n      group.push(index)\n    }\n\n    /* Just return string, if no occurrences */\n    if (blocks.size === 0)\n      return input\n\n    /* Compute slices */\n    const slices: string[] = []\n    for (const [block, indexes] of blocks) {\n      const t = table[block]\n\n      /* Extract positions and length */\n      const start  = t[0]            >>> 12\n      const end    = t[t.length - 1] >>> 12\n      const length = t[t.length - 1] >>> 2 & 0x3FF\n\n      /* Add prefix, if full results are desired */\n      if (full && start > cursor)\n        slices.push(input.slice(cursor, start))\n\n      /* Extract and highlight slice */\n      let slice = input.slice(start, end + length)\n      for (const j of indexes.sort((a, b) => b - a)) {\n\n        /* Retrieve offset and length of match */\n        const p = (t[j] >>> 12) - start\n        const q = (t[j] >>> 2 & 0x3FF) + p\n\n        /* Wrap occurrence */\n        slice = [\n          slice.slice(0, p),\n          \"<mark>\",\n          slice.slice(p, q),\n          \"</mark>\",\n          slice.slice(q)\n        ].join(\"\")\n      }\n\n      /* Update cursor */\n      cursor = end + length\n\n      /* Append slice and abort if we have two */\n      if (slices.push(slice) === 2)\n        break\n    }\n\n    /* Add suffix, if full results are desired */\n    if (full && cursor < input.length)\n      slices.push(input.slice(cursor))\n\n    /* Return highlighted slices */\n    return slices.join(\"\")\n  })\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { split } from \"../_\"\nimport {\n  Extract,\n  extract\n} from \"../extract\"\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Split a string or set of strings into tokens\n *\n * This tokenizer supersedes the default tokenizer that is provided by Lunr.js,\n * as it is aware of HTML tags and allows for multi-character splitting.\n *\n * It takes the given inputs, splits each of them into markup and text sections,\n * tokenizes and segments (if necessary) each of them, and then indexes them in\n * a table by using a compact bit representation. Bitwise techniques are used\n * to write and read from the table during indexing and querying.\n *\n * @see https://bit.ly/3W3Xw4J - Search: better, faster, smaller\n *\n * @param input - Input value(s)\n *\n * @returns Tokens\n */\nexport function tokenize(\n  input?: string | string[]\n): lunr.Token[] {\n  const tokens: lunr.Token[] = []\n  if (typeof input === \"undefined\")\n    return tokens\n\n  /* Tokenize strings one after another */\n  const inputs = Array.isArray(input) ? input : [input]\n  for (let i = 0; i < inputs.length; i++) {\n    const table = lunr.tokenizer.table\n    const total = table.length\n\n    /* Split string into sections and tokenize content blocks */\n    extract(inputs[i], (block, type, start, end) => {\n      table[block += total] ||= []\n      switch (type) {\n\n        /* Handle markup */\n        case Extract.TAG_OPEN:\n        case Extract.TAG_CLOSE:\n          table[block].push(\n            start       << 12 |\n            end - start <<  2 |\n            type\n          )\n          break\n\n        /* Handle text content */\n        case Extract.TEXT:\n          const section = inputs[i].slice(start, end)\n          split(section, lunr.tokenizer.separator, (index, until) => {\n\n            /**\n             * Apply segmenter after tokenization. Note that the segmenter will\n             * also split words at word boundaries, which is not what we want,\n             * so we need to check if we can somehow mitigate this behavior.\n             */\n            if (typeof lunr.segmenter !== \"undefined\") {\n              const subsection = section.slice(index, until)\n              if (/^[MHIK]$/.test(lunr.segmenter.ctype_(subsection))) {\n                const segments = lunr.segmenter.segment(subsection)\n                for (let s = 0, l = 0; s < segments.length; s++) {\n\n                  /* Add block to section */\n                  table[block] ||= []\n                  table[block].push(\n                    start + index + l  << 12 |\n                    segments[s].length <<  2 |\n                    type\n                  )\n\n                  /* Add token with position */\n                  tokens.push(new lunr.Token(\n                    segments[s].toLowerCase(), {\n                      position: block << 20 | table[block].length - 1\n                    }\n                  ))\n\n                  /* Keep track of length */\n                  l += segments[s].length\n                }\n                return\n              }\n            }\n\n            /* Add block to section */\n            table[block].push(\n              start + index << 12 |\n              until - index <<  2 |\n              type\n            )\n\n            /* Add token with position */\n            tokens.push(new lunr.Token(\n              section.slice(index, until).toLowerCase(), {\n                position: block << 20 | table[block].length - 1\n              }\n            ))\n          })\n      }\n    })\n  }\n\n  /* Return tokens */\n  return tokens\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Helper types\n * ------------------------------------------------------------------------- */\n\n/**\n * Visitor function\n *\n * @param value - String value\n *\n * @returns String term(s)\n */\ntype VisitorFn = (\n  value: string\n) => string | string[]\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Default transformation function\n *\n * 1. Trim excess whitespace from left and right.\n *\n * 2. Search for parts in quotation marks and prepend a `+` modifier to denote\n *    that the resulting document must contain all parts, converting the query\n *    to an `AND` query (as opposed to the default `OR` behavior). While users\n *    may expect parts enclosed in quotation marks to map to span queries, i.e.\n *    for which order is important, Lunr.js doesn't support them, so the best\n *    we can do is to convert the parts to an `AND` query.\n *\n * 3. Replace control characters which are not located at the beginning of the\n *    query or preceded by white space, or are not followed by a non-whitespace\n *    character or are at the end of the query string. Furthermore, filter\n *    unmatched quotation marks.\n *\n * 4. Split the query string at whitespace, then pass each part to the visitor\n *    function for tokenization, and append a wildcard to every resulting term\n *    that is not explicitly marked with a `+`, `-`, `~` or `^` modifier, since\n *    it ensures consistent and stable ranking when multiple terms are entered.\n *    Also, if a fuzzy or boost modifier are given, but no numeric value has\n *    been entered, default to 1 to not induce a query error.\n *\n * @param query - Query value\n * @param fn - Visitor function\n *\n * @returns Transformed query value\n */\nexport function transform(\n  query: string, fn: VisitorFn = term => term\n): string {\n  return query\n\n    /* => 1 */\n    .trim()\n\n    /* => 2 */\n    .split(/\"([^\"]+)\"/g)\n      .map((parts, index) => index & 1\n        ? parts.replace(/^\\b|^(?![^\\x00-\\x7F]|$)|\\s+/g, \" +\")\n        : parts\n      )\n      .join(\"\")\n\n    /* => 3 */\n    .replace(/\"|(?:^|\\s+)[*+\\-:^~]+(?=\\s+|$)/g, \"\")\n\n    /* => 4 */\n    .split(/\\s+/g)\n      .reduce((prev, term) => {\n        const next = fn(term)\n        return [...prev, ...Array.isArray(next) ? next : [next]]\n      }, [] as string[])\n      .map(term => /([~^]$)/.test(term) ? `${term}1` : term)\n      .map(term => /(^[+-]|[~^]\\d+$)/.test(term) ? term : `${term}*`)\n      .join(\" \")\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport { split } from \"../../internal\"\nimport { transform } from \"../transform\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search query clause\n */\nexport interface SearchQueryClause {\n  presence: lunr.Query.presence        /* Clause presence */\n  term: string                         /* Clause term */\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Search query terms\n */\nexport type SearchQueryTerms = Record<string, boolean>\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Transform search query\n *\n * This function lexes the given search query and applies the transformation\n * function to each term, preserving markup like `+` and `-` modifiers.\n *\n * @param query - Search query\n *\n * @returns Search query\n */\nexport function transformSearchQuery(\n  query: string\n): string {\n\n  /* Split query terms with tokenizer */\n  return transform(query, part => {\n    const terms: string[] = []\n\n    /* Initialize lexer and analyze part */\n    const lexer = new lunr.QueryLexer(part)\n    lexer.run()\n\n    /* Extract and tokenize term from lexeme */\n    for (const { type, str: term, start, end } of lexer.lexemes)\n      switch (type) {\n\n        /* Hack: remove colon - see https://bit.ly/3wD3T3I */\n        case \"FIELD\":\n          if (![\"title\", \"text\", \"tags\"].includes(term))\n            part = [\n              part.slice(0, end),\n              \" \",\n              part.slice(end + 1)\n            ].join(\"\")\n          break\n\n        /* Tokenize term */\n        case \"TERM\":\n          split(term, lunr.tokenizer.separator, (...range) => {\n            terms.push([\n              part.slice(0, start),\n              term.slice(...range),\n              part.slice(end)\n            ].join(\"\"))\n          })\n      }\n\n    /* Return terms */\n    return terms\n  })\n}\n\n/* ------------------------------------------------------------------------- */\n\n/**\n * Parse a search query for analysis\n *\n * Lunr.js itself has a bug where it doesn't detect or remove wildcards for\n * query clauses, so we must do this here.\n *\n * @see https://bit.ly/3DpTGtz - GitHub issue\n *\n * @param value - Query value\n *\n * @returns Search query clauses\n */\nexport function parseSearchQuery(\n  value: string\n): SearchQueryClause[] {\n  const query  = new lunr.Query([\"title\", \"text\", \"tags\"])\n  const parser = new lunr.QueryParser(value, query)\n\n  /* Parse Search query */\n  parser.parse()\n  for (const clause of query.clauses) {\n    clause.usePipeline = true\n\n    /* Handle leading wildcard */\n    if (clause.term.startsWith(\"*\")) {\n      clause.wildcard = lunr.Query.wildcard.LEADING\n      clause.term = clause.term.slice(1)\n    }\n\n    /* Handle trailing wildcard */\n    if (clause.term.endsWith(\"*\")) {\n      clause.wildcard = lunr.Query.wildcard.TRAILING\n      clause.term = clause.term.slice(0, -1)\n    }\n  }\n\n  /* Return query clauses */\n  return query.clauses\n}\n\n/**\n * Analyze the search query clauses in regard to the search terms found\n *\n * @param query - Search query clauses\n * @param terms - Search terms\n *\n * @returns Search query terms\n */\nexport function getSearchQueryTerms(\n  query: SearchQueryClause[], terms: string[]\n): SearchQueryTerms {\n  const clauses = new Set<SearchQueryClause>(query)\n\n  /* Match query clauses against terms */\n  const result: SearchQueryTerms = {}\n  for (let t = 0; t < terms.length; t++)\n    for (const clause of clauses)\n      if (terms[t].startsWith(clause.term)) {\n        result[clause.term] = true\n        clauses.delete(clause)\n      }\n\n  /* Annotate unmatched non-stopword query clauses */\n  for (const clause of clauses)\n    if (lunr.stopWordFilter?.(clause.term))\n      result[clause.term] = false\n\n  /* Return query terms */\n  return result\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Segment a search query using the inverted index\n *\n * This function implements a clever approach to text segmentation for Asian\n * languages, as it used the information already available in the search index.\n * The idea is to greedily segment the search query based on the tokens that are\n * already part of the index, as described in the linked issue.\n *\n * @see https://bit.ly/3lwjrk7 - GitHub issue\n *\n * @param query - Query value\n * @param index - Inverted index\n *\n * @returns Segmented query value\n */\nexport function segment(\n  query: string, index: object\n): Iterable<string> {\n  const segments = new Set<string>()\n\n  /* Segment search query */\n  const wordcuts = new Uint16Array(query.length)\n  for (let i = 0; i < query.length; i++)\n    for (let j = i + 1; j < query.length; j++) {\n      const value = query.slice(i, j)\n      if (value in index)\n        wordcuts[i] = j - i\n    }\n\n  /* Compute longest matches with minimum overlap */\n  const stack = [0]\n  for (let s = stack.length; s > 0;) {\n    const p = stack[--s]\n    for (let q = 1; q < wordcuts[p]; q++)\n      if (wordcuts[p + q] > wordcuts[p] - q) {\n        segments.add(query.slice(p, p + q))\n        stack[s++] = p + q\n      }\n\n    /* Continue at end of query string */\n    const q = p + wordcuts[p]\n    if (wordcuts[q] && q < query.length - 1)\n      stack[s++] = q\n\n    /* Add current segment */\n    segments.add(query.slice(p, q))\n  }\n\n  // @todo fix this case in the code block above, this is a hotfix\n  if (segments.has(\"\"))\n    return new Set([query])\n\n  /* Return segmented query value */\n  return segments\n}\n", "/*\n * Copyright (c) 2016-2023 Martin Donath <martin.donath@squidfunk.com>\n *\n * Permission is hereby granted, free of charge, to any person obtaining a copy\n * of this software and associated documentation files (the \"Software\"), to\n * deal in the Software without restriction, including without limitation the\n * rights to use, copy, modify, merge, publish, distribute, sublicense, and/or\n * sell copies of the Software, and to permit persons to whom the Software is\n * furnished to do so, subject to the following conditions:\n *\n * The above copyright notice and this permission notice shall be included in\n * all copies or substantial portions of the Software.\n *\n * THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n * FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE\n * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING\n * FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS\n * IN THE SOFTWARE.\n */\n\nimport {\n  SearchDocument,\n  SearchIndex,\n  SearchOptions,\n  setupSearchDocumentMap\n} from \"../config\"\nimport {\n  Position,\n  PositionTable,\n  highlight,\n  highlightAll,\n  tokenize\n} from \"../internal\"\nimport {\n  SearchQueryTerms,\n  getSearchQueryTerms,\n  parseSearchQuery,\n  segment,\n  transformSearchQuery\n} from \"../query\"\n\n/* ----------------------------------------------------------------------------\n * Types\n * ------------------------------------------------------------------------- */\n\n/**\n * Search item\n */\nexport interface SearchItem\n  extends SearchDocument\n{\n  score: number                        /* Score (relevance) */\n  terms: SearchQueryTerms              /* Search query terms */\n}\n\n/**\n * Search result\n */\nexport interface SearchResult {\n  items: SearchItem[][]                /* Search items */\n  suggest?: string[]                   /* Search suggestions */\n}\n\n/* ----------------------------------------------------------------------------\n * Functions\n * ------------------------------------------------------------------------- */\n\n/**\n * Create field extractor factory\n *\n * @param table - Position table map\n *\n * @returns Extractor factory\n */\nfunction extractor(table: Map<string, PositionTable>) {\n  return (name: keyof SearchDocument) => {\n    return (doc: SearchDocument) => {\n      if (typeof doc[name] === \"undefined\")\n        return undefined\n\n      /* Compute identifier and initialize table */\n      const id = [doc.location, name].join(\":\")\n      table.set(id, lunr.tokenizer.table = [])\n\n      /* Return field value */\n      return doc[name]\n    }\n  }\n}\n\n/**\n * Compute the difference of two lists of strings\n *\n * @param a - 1st list of strings\n * @param b - 2nd list of strings\n *\n * @returns Difference\n */\nfunction difference(a: string[], b: string[]): string[] {\n  const [x, y] = [new Set(a), new Set(b)]\n  return [\n    ...new Set([...x].filter(value => !y.has(value)))\n  ]\n}\n\n/* ----------------------------------------------------------------------------\n * Class\n * ------------------------------------------------------------------------- */\n\n/**\n * Search index\n */\nexport class Search {\n\n  /**\n   * Search document map\n   */\n  protected map: Map<string, SearchDocument>\n\n  /**\n   * Search options\n   */\n  protected options: SearchOptions\n\n  /**\n   * The underlying Lunr.js search index\n   */\n  protected index: lunr.Index\n\n  /**\n   * Internal position table map\n   */\n  protected table: Map<string, PositionTable>\n\n  /**\n   * Create the search integration\n   *\n   * @param data - Search index\n   */\n  public constructor({ config, docs, options }: SearchIndex) {\n    const field = extractor(this.table = new Map())\n\n    /* Set up document map and options */\n    this.map = setupSearchDocumentMap(docs)\n    this.options = options\n\n    /* Set up document index */\n    this.index = lunr(function () {\n      this.metadataWhitelist = [\"position\"]\n      this.b(0)\n\n      /* Set up (multi-)language support */\n      if (config.lang.length === 1 && config.lang[0] !== \"en\") {\n        // @ts-expect-error - namespace indexing not supported\n        this.use(lunr[config.lang[0]])\n      } else if (config.lang.length > 1) {\n        this.use(lunr.multiLanguage(...config.lang))\n      }\n\n      /* Set up custom tokenizer (must be after language setup) */\n      this.tokenizer = tokenize as typeof lunr.tokenizer\n      lunr.tokenizer.separator = new RegExp(config.separator)\n\n      /* Set up custom segmenter, if loaded */\n      lunr.segmenter = \"TinySegmenter\" in lunr\n        ? new lunr.TinySegmenter()\n        : undefined\n\n      /* Compute functions to be removed from the pipeline */\n      const fns = difference([\n        \"trimmer\", \"stopWordFilter\", \"stemmer\"\n      ], config.pipeline)\n\n      /* Remove functions from the pipeline for registered languages */\n      for (const lang of config.lang.map(language => (\n        // @ts-expect-error - namespace indexing not supported\n        language === \"en\" ? lunr : lunr[language]\n      )))\n        for (const fn of fns) {\n          this.pipeline.remove(lang[fn])\n          this.searchPipeline.remove(lang[fn])\n        }\n\n      /* Set up index reference */\n      this.ref(\"location\")\n\n      /* Set up index fields */\n      this.field(\"title\", { boost: 1e3, extractor: field(\"title\") })\n      this.field(\"text\",  { boost: 1e0, extractor: field(\"text\") })\n      this.field(\"tags\",  { boost: 1e6, extractor: field(\"tags\") })\n\n      /* Add documents to index */\n      for (const doc of docs)\n        this.add(doc, { boost: doc.boost })\n    })\n  }\n\n  /**\n   * Search for matching documents\n   *\n   * @param query - Search query\n   *\n   * @returns Search result\n   */\n  public search(query: string): SearchResult {\n\n    // Experimental Chinese segmentation\n    query = query.replace(/\\p{sc=Han}+/gu, value => {\n      return [...segment(value, this.index.invertedIndex)]\n        .join(\"* \")\n    })\n\n    // @todo: move segmenter (above) into transformSearchQuery\n    query = transformSearchQuery(query)\n    if (!query)\n      return { items: [] }\n\n    /* Parse query to extract clauses for analysis */\n    const clauses = parseSearchQuery(query)\n      .filter(clause => (\n        clause.presence !== lunr.Query.presence.PROHIBITED\n      ))\n\n    /* Perform search and post-process results */\n    const groups = this.index.search(query)\n\n      /* Apply post-query boosts based on title and search query terms */\n      .reduce<SearchItem[]>((item, { ref, score, matchData }) => {\n        let doc = this.map.get(ref)\n        if (typeof doc !== \"undefined\") {\n\n          /* Shallow copy document */\n          doc = { ...doc }\n          if (doc.tags)\n            doc.tags = [...doc.tags]\n\n          /* Compute and analyze search query terms */\n          const terms = getSearchQueryTerms(\n            clauses,\n            Object.keys(matchData.metadata)\n          )\n\n          /* Highlight matches in fields */\n          for (const field of this.index.fields) {\n            if (typeof doc[field] === \"undefined\")\n              continue\n\n            /* Collect positions from matches */\n            const positions: Position[] = []\n            for (const match of Object.values(matchData.metadata))\n              if (typeof match[field] !== \"undefined\")\n                positions.push(...match[field].position)\n\n            /* Skip highlighting, if no positions were collected */\n            if (!positions.length)\n              continue\n\n            /* Load table and determine highlighting method */\n            const table = this.table.get([doc.location, field].join(\":\"))!\n            const fn = Array.isArray(doc[field])\n              ? highlightAll\n              : highlight\n\n            // @ts-expect-error - stop moaning, TypeScript!\n            doc[field] = fn(doc[field], table, positions, field !== \"text\")\n          }\n\n          /* Highlight title and text and apply post-query boosts */\n          const boost = +!doc.parent +\n            Object.values(terms)\n              .filter(t => t).length /\n            Object.keys(terms).length\n\n          /* Append item */\n          item.push({\n            ...doc,\n            score: score * (1 + boost ** 2),\n            terms\n          })\n        }\n        return item\n      }, [])\n\n      /* Sort search results again after applying boosts */\n      .sort((a, b) => b.score - a.score)\n\n      /* Group search results by article */\n      .reduce((items, result) => {\n        const doc = this.map.get(result.location)\n        if (typeof doc !== \"undefined\") {\n          const ref = doc.parent\n            ? doc.parent.location\n            : doc.location\n          items.set(ref, [...items.get(ref) || [], result])\n        }\n        return items\n      }, new Map<string, SearchItem[]>())\n\n    /* Ensure that every item set has an article */\n    for (const [ref, items] of groups)\n      if (!items.find(item => item.location === ref)) {\n        const doc = this.map.get(ref)!\n        items.push({ ...doc, score: 0, terms: {} })\n      }\n\n    /* Generate search suggestions, if desired */\n    let suggest: string[] | undefined\n    if (this.options.suggest) {\n      const titles = this.index.query(builder => {\n        for (const clause of clauses)\n          builder.term(clause.term, {\n            fields: [\"title\"],\n            presence: lunr.Query.presence.REQUIRED,\n            wildcard: lunr.Query.wildcard.TRAILING\n          })\n      })\n\n      /* Retrieve suggestions for best match */\n      suggest = titles.length\n        ? Object.keys(titles[0].matchData.metadata)\n        : []\n    }\n\n    /* Return search result */\n    return {\n      items: [...groups.values()],\n      ...typeof suggest !== \"undefined\" && { suggest }\n    }\n  }\n}\n"],
+  "mappings": 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diff --git a/assets/stylesheets/main.45e1311d.min.css b/assets/stylesheets/main.45e1311d.min.css
new file mode 100644
index 000000000..550b8b828
--- /dev/null
+++ b/assets/stylesheets/main.45e1311d.min.css
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diff --git a/assets/stylesheets/main.72749a73.min.css b/assets/stylesheets/main.72749a73.min.css
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diff --git a/assets/stylesheets/palette.06af60db.min.css b/assets/stylesheets/palette.06af60db.min.css
new file mode 100644
index 000000000..a640d38ab
--- /dev/null
+++ b/assets/stylesheets/palette.06af60db.min.css
@@ -0,0 +1 @@
+@media screen{[data-md-color-scheme=slate]{--md-default-fg-color:hsla(var(--md-hue),15%,90%,0.82);--md-default-fg-color--light:hsla(var(--md-hue),15%,90%,0.56);--md-default-fg-color--lighter:hsla(var(--md-hue),15%,90%,0.32);--md-default-fg-color--lightest:hsla(var(--md-hue),15%,90%,0.12);--md-default-bg-color:hsla(var(--md-hue),15%,14%,1);--md-default-bg-color--light:hsla(var(--md-hue),15%,14%,0.54);--md-default-bg-color--lighter:hsla(var(--md-hue),15%,14%,0.26);--md-default-bg-color--lightest:hsla(var(--md-hue),15%,14%,0.07);--md-code-fg-color:hsla(var(--md-hue),18%,86%,0.82);--md-code-bg-color:hsla(var(--md-hue),15%,18%,1);--md-code-hl-color:#2977ff;--md-code-hl-color--light:#2977ff1a;--md-code-hl-number-color:#e6695b;--md-code-hl-special-color:#f06090;--md-code-hl-function-color:#c973d9;--md-code-hl-constant-color:#9383e2;--md-code-hl-keyword-color:#6791e0;--md-code-hl-string-color:#2fb170;--md-code-hl-name-color:var(--md-code-fg-color);--md-code-hl-operator-color:var(--md-default-fg-color--light);--md-code-hl-punctuation-color:var(--md-default-fg-color--light);--md-code-hl-comment-color:var(--md-default-fg-color--light);--md-code-hl-generic-color:var(--md-default-fg-color--light);--md-code-hl-variable-color:var(--md-default-fg-color--light);--md-typeset-color:var(--md-default-fg-color);--md-typeset-a-color:var(--md-primary-fg-color);--md-typeset-kbd-color:hsla(var(--md-hue),15%,90%,0.12);--md-typeset-kbd-accent-color:hsla(var(--md-hue),15%,90%,0.2);--md-typeset-kbd-border-color:hsla(var(--md-hue),15%,14%,1);--md-typeset-mark-color:#4287ff4d;--md-typeset-table-color:hsla(var(--md-hue),15%,95%,0.12);--md-typeset-table-color--light:hsla(var(--md-hue),15%,95%,0.035);--md-admonition-fg-color:var(--md-default-fg-color);--md-admonition-bg-color:var(--md-default-bg-color);--md-footer-bg-color:hsla(var(--md-hue),15%,10%,0.87);--md-footer-bg-color--dark:hsla(var(--md-hue),15%,8%,1);--md-shadow-z1:0 0.2rem 0.5rem #0000000d,0 0 0.05rem #0000001a;--md-shadow-z2:0 0.2rem 0.5rem #00000040,0 0 0.05rem #00000040;--md-shadow-z3:0 0.2rem 0.5rem #0006,0 0 0.05rem #00000059;color-scheme:dark}[data-md-color-scheme=slate] img[src$="#gh-light-mode-only"],[data-md-color-scheme=slate] 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new file mode 100644
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index 577d471ec..000000000
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+  
     
     
       
@@ -3254,12 +3397,13 @@
       
     
     
+    
+    
     <li class="md-nav__item md-nav__item--nested">
       
         
         
         
-        
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@@ -3321,21 +3465,27 @@
       
         <li class="md-nav__item">
   <a href="#1" class="md-nav__link">
-    1. &nbsp; 内容微调
+    <span class="md-ellipsis">
+      1. &nbsp; 内容微调
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#2" class="md-nav__link">
-    2. &nbsp; 内容创作
+    <span class="md-ellipsis">
+      2. &nbsp; 内容创作
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#3-docker" class="md-nav__link">
-    3. &nbsp; Docker 部署
+    <span class="md-ellipsis">
+      3. &nbsp; Docker 部署
+    </span>
   </a>
   
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
       </main>
       
@@ -3557,7 +3708,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../installation/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 16.1 &amp;nbsp; 编程环境安装" rel="prev">
+          <a href="../installation/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 16.1 &amp;nbsp; 编程环境安装">
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@@ -3574,7 +3725,7 @@ aria-label="页脚"
         
         
           
-          <a href="../terminology/" class="md-footer__link md-footer__link--next" aria-label="下一页: 16.3 &amp;nbsp; 术语表" rel="next">
+          <a href="../terminology/" class="md-footer__link md-footer__link--next" aria-label="下一页: 16.3 &amp;nbsp; 术语表">
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@@ -3613,7 +3764,7 @@ aria-label="页脚"
       
     
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diff --git a/chapter_appendix/index.html b/chapter_appendix/index.html
index be322e9de..d2c9320ac 100644
--- a/chapter_appendix/index.html
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@@ -3260,7 +3398,9 @@
       
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+    <span class="md-ellipsis">
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+  
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-          <a href="installation/" class="md-footer__link md-footer__link--next" aria-label="下一页: 16.1 &amp;nbsp; 编程环境安装" rel="next">
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diff --git a/chapter_appendix/installation/index.html b/chapter_appendix/installation/index.html
index 49cb9f6b9..7ec7fc6fb 100644
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-    2. &nbsp; C/C++ 环境
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+      2. &nbsp; C/C++ 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#3-java" class="md-nav__link">
-    3. &nbsp; Java 环境
+    <span class="md-ellipsis">
+      3. &nbsp; Java 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#4-c" class="md-nav__link">
-    4. &nbsp; C# 环境
+    <span class="md-ellipsis">
+      4. &nbsp; C# 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#5-go" class="md-nav__link">
-    5. &nbsp; Go 环境
+    <span class="md-ellipsis">
+      5. &nbsp; Go 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#6-swift" class="md-nav__link">
-    6. &nbsp; Swift 环境
+    <span class="md-ellipsis">
+      6. &nbsp; Swift 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#7-javascript" class="md-nav__link">
-    7. &nbsp; JavaScript 环境
+    <span class="md-ellipsis">
+      7. &nbsp; JavaScript 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#8-dart" class="md-nav__link">
-    8. &nbsp; Dart 环境
+    <span class="md-ellipsis">
+      8. &nbsp; Dart 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#9-rust" class="md-nav__link">
-    9. &nbsp; Rust 环境
+    <span class="md-ellipsis">
+      9. &nbsp; Rust 环境
+    </span>
   </a>
   
 </li>
@@ -3252,6 +3408,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../contribution/" class="md-nav__link">
         
@@ -3272,6 +3429,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../terminology/" class="md-nav__link">
         
@@ -3307,6 +3465,7 @@
   
   
   
+  
     
     
       
@@ -3316,12 +3475,13 @@
       
     
     
+    
+    
     <li class="md-nav__item md-nav__item--nested">
       
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
@@ -3383,14 +3543,18 @@
       
         <li class="md-nav__item">
   <a href="#1611-ide" class="md-nav__link">
-    16.1.1 &nbsp; 安装 IDE
+    <span class="md-ellipsis">
+      16.1.1 &nbsp; 安装 IDE
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1612" class="md-nav__link">
-    16.1.2 &nbsp; 安装语言环境
+    <span class="md-ellipsis">
+      16.1.2 &nbsp; 安装语言环境
+    </span>
   </a>
   
     <nav class="md-nav" aria-label="16.1.2   安装语言环境">
@@ -3398,63 +3562,81 @@
         
           <li class="md-nav__item">
   <a href="#1-python" class="md-nav__link">
-    1. &nbsp; Python 环境
+    <span class="md-ellipsis">
+      1. &nbsp; Python 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#2-cc" class="md-nav__link">
-    2. &nbsp; C/C++ 环境
+    <span class="md-ellipsis">
+      2. &nbsp; C/C++ 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#3-java" class="md-nav__link">
-    3. &nbsp; Java 环境
+    <span class="md-ellipsis">
+      3. &nbsp; Java 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#4-c" class="md-nav__link">
-    4. &nbsp; C# 环境
+    <span class="md-ellipsis">
+      4. &nbsp; C# 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#5-go" class="md-nav__link">
-    5. &nbsp; Go 环境
+    <span class="md-ellipsis">
+      5. &nbsp; Go 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#6-swift" class="md-nav__link">
-    6. &nbsp; Swift 环境
+    <span class="md-ellipsis">
+      6. &nbsp; Swift 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#7-javascript" class="md-nav__link">
-    7. &nbsp; JavaScript 环境
+    <span class="md-ellipsis">
+      7. &nbsp; JavaScript 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#8-dart" class="md-nav__link">
-    8. &nbsp; Dart 环境
+    <span class="md-ellipsis">
+      8. &nbsp; Dart 环境
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#9-rust" class="md-nav__link">
-    9. &nbsp; Rust 环境
+    <span class="md-ellipsis">
+      9. &nbsp; Rust 环境
+    </span>
   </a>
   
 </li>
@@ -3491,7 +3673,7 @@
       class="md-content__button md-icon"
     >
       
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+      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M441 58.9 453.1 71c9.4 9.4 9.4 24.6 0 33.9L424 134.1 377.9 88 407 58.9c9.4-9.4 24.6-9.4 33.9 0zM209.8 256.2 344 121.9l46.1 46.1-134.3 134.2c-2.9 2.9-6.5 5-10.4 6.1L186.9 325l16.7-58.5c1.1-3.9 3.2-7.5 6.1-10.4zM373.1 25 175.8 222.2c-8.7 8.7-15 19.4-18.3 31.1l-28.6 100c-2.4 8.4-.1 17.4 6.1 23.6s15.2 8.5 23.6 6.1l100-28.6c11.8-3.4 22.5-9.7 31.1-18.3L487 138.9c28.1-28.1 28.1-73.7 0-101.8L474.9 25c-28.1-28.1-73.7-28.1-101.8 0zM88 64c-48.6 0-88 39.4-88 88v272c0 48.6 39.4 88 88 88h272c48.6 0 88-39.4 88-88V312c0-13.3-10.7-24-24-24s-24 10.7-24 24v112c0 22.1-17.9 40-40 40H88c-22.1 0-40-17.9-40-40V152c0-22.1 17.9-40 40-40h112c13.3 0 24-10.7 24-24s-10.7-24-24-24H88z"/></svg>
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@@ -3690,13 +3872,14 @@ aria-label="页脚"
           
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
         </div>
         
           <button type="button" class="md-top md-icon" data-md-component="top" hidden>
-            
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
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@@ -3707,7 +3890,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 16 章 &amp;nbsp; 附录" rel="prev">
+          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 16 章 &amp;nbsp; 附录">
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@@ -3724,7 +3907,7 @@ aria-label="页脚"
         
         
           
-          <a href="../contribution/" class="md-footer__link md-footer__link--next" aria-label="下一页: 16.2 &amp;nbsp; 一起参与创作" rel="next">
+          <a href="../contribution/" class="md-footer__link md-footer__link--next" aria-label="下一页: 16.2 &amp;nbsp; 一起参与创作">
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@@ -3774,7 +3957,7 @@ aria-label="页脚"
       
     
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@@ -3785,7 +3968,7 @@ aria-label="页脚"
       
     
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diff --git a/chapter_array_and_linkedlist/index.html b/chapter_array_and_linkedlist/index.html
index 7bd1c6514..54194c7cc 100644
--- a/chapter_array_and_linkedlist/index.html
+++ b/chapter_array_and_linkedlist/index.html
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       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
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+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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diff --git a/chapter_array_and_linkedlist/list/index.html b/chapter_array_and_linkedlist/list/index.html
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@@ -3191,7 +3339,8 @@
 
             </a>
             
-              <label class="md-nav__link " for="__nav_17">
+              
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
@@ -3211,6 +3360,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../../chapter_appendix/installation/" class="md-nav__link">
         
@@ -3231,6 +3381,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../../chapter_appendix/contribution/" class="md-nav__link">
         
@@ -3251,6 +3402,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../../chapter_appendix/terminology/" class="md-nav__link">
         
@@ -3286,6 +3438,7 @@
   
   
   
+  
     
     
       
@@ -3295,12 +3448,13 @@
       
     
     
+    
+    
     <li class="md-nav__item md-nav__item--nested">
       
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
@@ -3362,7 +3516,9 @@
       
         <li class="md-nav__item">
   <a href="#431" class="md-nav__link">
-    4.3.1 &nbsp; 列表常用操作
+    <span class="md-ellipsis">
+      4.3.1 &nbsp; 列表常用操作
+    </span>
   </a>
   
     <nav class="md-nav" aria-label="4.3.1   列表常用操作">
@@ -3370,42 +3526,54 @@
         
           <li class="md-nav__item">
   <a href="#1" class="md-nav__link">
-    1. &nbsp; 初始化列表
+    <span class="md-ellipsis">
+      1. &nbsp; 初始化列表
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#2" class="md-nav__link">
-    2. &nbsp; 访问元素
+    <span class="md-ellipsis">
+      2. &nbsp; 访问元素
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#3" class="md-nav__link">
-    3. &nbsp; 插入与删除元素
+    <span class="md-ellipsis">
+      3. &nbsp; 插入与删除元素
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#4" class="md-nav__link">
-    4. &nbsp; 遍历列表
+    <span class="md-ellipsis">
+      4. &nbsp; 遍历列表
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#5" class="md-nav__link">
-    5. &nbsp; 拼接列表
+    <span class="md-ellipsis">
+      5. &nbsp; 拼接列表
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#6" class="md-nav__link">
-    6. &nbsp; 排序列表
+    <span class="md-ellipsis">
+      6. &nbsp; 排序列表
+    </span>
   </a>
   
 </li>
@@ -3417,7 +3585,9 @@
       
         <li class="md-nav__item">
   <a href="#432" class="md-nav__link">
-    4.3.2 &nbsp; 列表实现
+    <span class="md-ellipsis">
+      4.3.2 &nbsp; 列表实现
+    </span>
   </a>
   
 </li>
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+      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M441 58.9 453.1 71c9.4 9.4 9.4 24.6 0 33.9L424 134.1 377.9 88 407 58.9c9.4-9.4 24.6-9.4 33.9 0zM209.8 256.2 344 121.9l46.1 46.1-134.3 134.2c-2.9 2.9-6.5 5-10.4 6.1L186.9 325l16.7-58.5c1.1-3.9 3.2-7.5 6.1-10.4zM373.1 25 175.8 222.2c-8.7 8.7-15 19.4-18.3 31.1l-28.6 100c-2.4 8.4-.1 17.4 6.1 23.6s15.2 8.5 23.6 6.1l100-28.6c11.8-3.4 22.5-9.7 31.1-18.3L487 138.9c28.1-28.1 28.1-73.7 0-101.8L474.9 25c-28.1-28.1-73.7-28.1-101.8 0zM88 64c-48.6 0-88 39.4-88 88v272c0 48.6 39.4 88 88 88h272c48.6 0 88-39.4 88-88V312c0-13.3-10.7-24-24-24s-24 10.7-24 24v112c0 22.1-17.9 40-40 40H88c-22.1 0-40-17.9-40-40V152c0-22.1 17.9-40 40-40h112c13.3 0 24-10.7 24-24s-10.7-24-24-24H88z"/></svg>
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
         </div>
         
           <button type="button" class="md-top md-icon" data-md-component="top" hidden>
-            
-            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
       </main>
       
@@ -5562,7 +5733,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../linked_list/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 4.2 &amp;nbsp; 链表" rel="prev">
+          <a href="../linked_list/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 4.2 &amp;nbsp; 链表">
             <div class="md-footer__button md-icon">
               
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@@ -5579,7 +5750,7 @@ aria-label="页脚"
         
         
           
-          <a href="../ram_and_cache/" class="md-footer__link md-footer__link--next" aria-label="下一页: 4.4 &amp;nbsp; 内存与缓存 *" rel="next">
+          <a href="../ram_and_cache/" class="md-footer__link md-footer__link--next" aria-label="下一页: 4.4 &amp;nbsp; 内存与缓存 *">
             <div class="md-footer__title">
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@@ -5618,7 +5789,7 @@ aria-label="页脚"
       
     
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@@ -5640,7 +5811,7 @@ aria-label="页脚"
       
     
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+      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg>
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-    13.1.4 &nbsp; 常用术语
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-    13.1.6 &nbsp; 回溯典型例题
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@@ -5177,13 +5339,14 @@ aria-label="页脚"
           
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+  回到页面顶部
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@@ -5194,7 +5357,7 @@ aria-label="页脚"
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@@ -5211,7 +5374,7 @@ aria-label="页脚"
         
         
           
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@@ -5261,7 +5424,7 @@ aria-label="页脚"
       
     
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-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_backtracking/index.html b/chapter_backtracking/index.html
index 7c8c5d2fe..899634d12 100644
--- a/chapter_backtracking/index.html
+++ b/chapter_backtracking/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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diff --git a/chapter_backtracking/n_queens_problem/index.html b/chapter_backtracking/n_queens_problem/index.html
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+    </span>
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-    2. &nbsp; 列与对角线剪枝
+    <span class="md-ellipsis">
+      2. &nbsp; 列与对角线剪枝
+    </span>
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 </li>
       
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-    3. &nbsp; 代码实现
+    <span class="md-ellipsis">
+      3. &nbsp; 代码实现
+    </span>
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@@ -233,7 +233,7 @@
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@@ -278,7 +278,7 @@
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index 5f0c40d9b..83f08b9ec 100644
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-    3.1.2 &nbsp; 物理结构：连续与分散
+    <span class="md-ellipsis">
+      3.1.2 &nbsp; 物理结构：连续与分散
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diff --git a/chapter_data_structure/index.html b/chapter_data_structure/index.html
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diff --git a/chapter_data_structure/summary/index.html b/chapter_data_structure/summary/index.html
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@@ -3328,28 +3474,36 @@
       
         <li class="md-nav__item">
   <a href="#1" class="md-nav__link">
-    1. &nbsp; 判断是否为分治问题
+    <span class="md-ellipsis">
+      1. &nbsp; 判断是否为分治问题
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#2" class="md-nav__link">
-    2. &nbsp; 如何划分子树
+    <span class="md-ellipsis">
+      2. &nbsp; 如何划分子树
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#3" class="md-nav__link">
-    3. &nbsp; 基于变量描述子树区间
+    <span class="md-ellipsis">
+      3. &nbsp; 基于变量描述子树区间
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#4" class="md-nav__link">
-    4. &nbsp; 代码实现
+    <span class="md-ellipsis">
+      4. &nbsp; 代码实现
+    </span>
   </a>
   
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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@@ -4005,7 +4160,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
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+          <a href="../binary_search_recur/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 12.2 &amp;nbsp; 分治搜索策略">
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@@ -4022,7 +4177,7 @@ aria-label="页脚"
         
         
           
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diff --git a/chapter_divide_and_conquer/divide_and_conquer/index.html b/chapter_divide_and_conquer/divide_and_conquer/index.html
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-    12.1.1 &nbsp; 如何判断分治问题
+    <span class="md-ellipsis">
+      12.1.1 &nbsp; 如何判断分治问题
+    </span>
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-    12.1.2 &nbsp; 通过分治提升效率
+    <span class="md-ellipsis">
+      12.1.2 &nbsp; 通过分治提升效率
+    </span>
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-    1. &nbsp; 操作数量优化
+    <span class="md-ellipsis">
+      1. &nbsp; 操作数量优化
+    </span>
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-    2. &nbsp; 并行计算优化
+    <span class="md-ellipsis">
+      2. &nbsp; 并行计算优化
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-    12.1.3 &nbsp; 分治常见应用
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diff --git a/chapter_divide_and_conquer/index.html b/chapter_divide_and_conquer/index.html
index 246aac4fc..a1c1ab5d8 100644
--- a/chapter_divide_and_conquer/index.html
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-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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@@ -278,7 +278,7 @@
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-    14.3.1 &nbsp; 问题判断
+    <span class="md-ellipsis">
+      14.3.1 &nbsp; 问题判断
+    </span>
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-    14.3.2 &nbsp; 问题求解步骤
+    <span class="md-ellipsis">
+      14.3.2 &nbsp; 问题求解步骤
+    </span>
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-    1. &nbsp; 方法一：暴力搜索
+    <span class="md-ellipsis">
+      1. &nbsp; 方法一：暴力搜索
+    </span>
   </a>
   
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-    2. &nbsp; 方法二：记忆化搜索
+    <span class="md-ellipsis">
+      2. &nbsp; 方法二：记忆化搜索
+    </span>
   </a>
   
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   <a href="#3" class="md-nav__link">
-    3. &nbsp; 方法三：动态规划
+    <span class="md-ellipsis">
+      3. &nbsp; 方法三：动态规划
+    </span>
   </a>
   
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+    <span class="md-ellipsis">
+      4. &nbsp; 空间优化
+    </span>
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   <a href="#1431" class="md-nav__link">
-    14.3.1 &nbsp; 问题判断
+    <span class="md-ellipsis">
+      14.3.1 &nbsp; 问题判断
+    </span>
   </a>
   
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   <a href="#1432" class="md-nav__link">
-    14.3.2 &nbsp; 问题求解步骤
+    <span class="md-ellipsis">
+      14.3.2 &nbsp; 问题求解步骤
+    </span>
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           <li class="md-nav__item">
   <a href="#1" class="md-nav__link">
-    1. &nbsp; 方法一：暴力搜索
+    <span class="md-ellipsis">
+      1. &nbsp; 方法一：暴力搜索
+    </span>
   </a>
   
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           <li class="md-nav__item">
   <a href="#2" class="md-nav__link">
-    2. &nbsp; 方法二：记忆化搜索
+    <span class="md-ellipsis">
+      2. &nbsp; 方法二：记忆化搜索
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#3" class="md-nav__link">
-    3. &nbsp; 方法三：动态规划
+    <span class="md-ellipsis">
+      3. &nbsp; 方法三：动态规划
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#4" class="md-nav__link">
-    4. &nbsp; 空间优化
+    <span class="md-ellipsis">
+      4. &nbsp; 空间优化
+    </span>
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         <li class="md-nav__item">
   <a href="#2" class="md-nav__link">
-    2. &nbsp; 代码实现
+    <span class="md-ellipsis">
+      2. &nbsp; 代码实现
+    </span>
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 </li>
       
         <li class="md-nav__item">
   <a href="#3" class="md-nav__link">
-    3. &nbsp; 空间优化
+    <span class="md-ellipsis">
+      3. &nbsp; 空间优化
+    </span>
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-            回到页面顶部
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+  
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@@ -4358,7 +4509,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
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+          <a href="../unbounded_knapsack_problem/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 14.5 &amp;nbsp; 完全背包问题">
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@@ -4375,7 +4526,7 @@ aria-label="页脚"
         
         
           
-          <a href="../summary/" class="md-footer__link md-footer__link--next" aria-label="下一页: 14.7 &amp;nbsp; 小结" rel="next">
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diff --git a/chapter_dynamic_programming/index.html b/chapter_dynamic_programming/index.html
index cdb8a3c3a..86116e579 100644
--- a/chapter_dynamic_programming/index.html
+++ b/chapter_dynamic_programming/index.html
@@ -22,7 +22,7 @@
       
       
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-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
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         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_18" >
         
           
@@ -3328,28 +3474,36 @@
       
         <li class="md-nav__item">
   <a href="#1411" class="md-nav__link">
-    14.1.1 &nbsp; 方法一：暴力搜索
+    <span class="md-ellipsis">
+      14.1.1 &nbsp; 方法一：暴力搜索
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1412" class="md-nav__link">
-    14.1.2 &nbsp; 方法二：记忆化搜索
+    <span class="md-ellipsis">
+      14.1.2 &nbsp; 方法二：记忆化搜索
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1413" class="md-nav__link">
-    14.1.3 &nbsp; 方法三：动态规划
+    <span class="md-ellipsis">
+      14.1.3 &nbsp; 方法三：动态规划
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1414" class="md-nav__link">
-    14.1.4 &nbsp; 空间优化
+    <span class="md-ellipsis">
+      14.1.4 &nbsp; 空间优化
+    </span>
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+  回到页面顶部
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@@ -4849,7 +5004,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 14 章 &amp;nbsp; 动态规划" rel="prev">
+          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 14 章 &amp;nbsp; 动态规划">
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-          <a href="../dp_problem_features/" class="md-footer__link md-footer__link--next" aria-label="下一页: 14.2 &amp;nbsp; DP 问题特性" rel="next">
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       <a href="../../chapter_greedy/summary/" class="md-nav__link">
         
@@ -3119,6 +3257,7 @@
   
   
   
+  
     
     
       
@@ -3132,6 +3271,8 @@
       
         
       
+    
+    
     
     
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@@ -3139,7 +3280,6 @@
         
         
         
-        
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@@ -3157,7 +3297,8 @@
 
             </a>
             
-              <label class="md-nav__link " for="__nav_17">
+              
+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
                 <span class="md-nav__icon md-icon"></span>
               </label>
             
@@ -3177,6 +3318,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../../chapter_appendix/installation/" class="md-nav__link">
         
@@ -3197,6 +3339,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../../chapter_appendix/contribution/" class="md-nav__link">
         
@@ -3217,6 +3360,7 @@
   
   
   
+  
     <li class="md-nav__item">
       <a href="../../chapter_appendix/terminology/" class="md-nav__link">
         
@@ -3252,6 +3396,7 @@
   
   
   
+  
     
     
       
@@ -3261,12 +3406,13 @@
       
     
     
+    
+    
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@@ -3328,28 +3474,36 @@
       
         <li class="md-nav__item">
   <a href="#1" class="md-nav__link">
-    1. &nbsp; 方法一：暴力搜索
+    <span class="md-ellipsis">
+      1. &nbsp; 方法一：暴力搜索
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#2" class="md-nav__link">
-    2. &nbsp; 方法二：记忆化搜索
+    <span class="md-ellipsis">
+      2. &nbsp; 方法二：记忆化搜索
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#3" class="md-nav__link">
-    3. &nbsp; 方法三：动态规划
+    <span class="md-ellipsis">
+      3. &nbsp; 方法三：动态规划
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#4" class="md-nav__link">
-    4. &nbsp; 空间优化
+    <span class="md-ellipsis">
+      4. &nbsp; 空间优化
+    </span>
   </a>
   
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
       </main>
       
@@ -4753,7 +4908,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../dp_solution_pipeline/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 14.3 &amp;nbsp; DP 解题思路" rel="prev">
+          <a href="../dp_solution_pipeline/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 14.3 &amp;nbsp; DP 解题思路">
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@@ -4770,7 +4925,7 @@ aria-label="页脚"
         
         
           
-          <a href="../unbounded_knapsack_problem/" class="md-footer__link md-footer__link--next" aria-label="下一页: 14.5 &amp;nbsp; 完全背包问题" rel="next">
+          <a href="../unbounded_knapsack_problem/" class="md-footer__link md-footer__link--next" aria-label="下一页: 14.5 &amp;nbsp; 完全背包问题">
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-    14.5.1 &nbsp; 完全背包
+    <span class="md-ellipsis">
+      14.5.1 &nbsp; 完全背包
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-    1. &nbsp; 动态规划思路
+    <span class="md-ellipsis">
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+      2. &nbsp; 代码实现
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-    3. &nbsp; 空间优化
+    <span class="md-ellipsis">
+      3. &nbsp; 空间优化
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-    14.5.2 &nbsp; 零钱兑换问题
+    <span class="md-ellipsis">
+      14.5.2 &nbsp; 零钱兑换问题
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-    1. &nbsp; 动态规划思路
+    <span class="md-ellipsis">
+      1. &nbsp; 动态规划思路
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-    1. &nbsp; 动态规划思路
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-    3. &nbsp; 空间优化
+    <span class="md-ellipsis">
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+    <span class="md-ellipsis">
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-    1. &nbsp; 动态规划思路
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-    3. &nbsp; 空间优化
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-    14.5.3 &nbsp; 零钱兑换问题 II
+    <span class="md-ellipsis">
+      14.5.3 &nbsp; 零钱兑换问题 II
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-    1. &nbsp; 动态规划思路
+    <span class="md-ellipsis">
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diff --git a/chapter_graph/graph/index.html b/chapter_graph/graph/index.html
index 7cf51ee78..3664c6e12 100644
--- a/chapter_graph/graph/index.html
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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+  回到页面顶部
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@@ -3660,7 +3819,7 @@ aria-label="页脚"
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@@ -3677,7 +3836,7 @@ aria-label="页脚"
         
         
           
-          <a href="../graph_operations/" class="md-footer__link md-footer__link--next" aria-label="下一页: 9.2 &amp;nbsp; 图基础操作" rel="next">
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-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_graph/graph_operations/index.html b/chapter_graph/graph_operations/index.html
index da50f1861..1f9f1c466 100644
--- a/chapter_graph/graph_operations/index.html
+++ b/chapter_graph/graph_operations/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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-    9.2.1 &nbsp; 基于邻接矩阵的实现
+    <span class="md-ellipsis">
+      9.2.1 &nbsp; 基于邻接矩阵的实现
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#922" class="md-nav__link">
-    9.2.2 &nbsp; 基于邻接表的实现
+    <span class="md-ellipsis">
+      9.2.2 &nbsp; 基于邻接表的实现
+    </span>
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 </li>
       
         <li class="md-nav__item">
   <a href="#923" class="md-nav__link">
-    9.2.3 &nbsp; 效率对比
+    <span class="md-ellipsis">
+      9.2.3 &nbsp; 效率对比
+    </span>
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diff --git a/chapter_graph/graph_traversal/index.html b/chapter_graph/graph_traversal/index.html
index 2ea7989d7..ee735b6e8 100644
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-    9.3.1 &nbsp; 广度优先遍历
+    <span class="md-ellipsis">
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+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_graph/index.html b/chapter_graph/index.html
index 03fe4906a..4caed94c7 100644
--- a/chapter_graph/index.html
+++ b/chapter_graph/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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@@ -142,9 +139,9 @@
     
     
     
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
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@@ -152,9 +149,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_2">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
+      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -293,6 +293,7 @@
   
   
   
+  
     
     
       
@@ -306,6 +307,8 @@
       
         
       
+    
+    
     
     
     <li class="md-nav__item md-nav__item--nested">
@@ -313,7 +316,6 @@
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
         
           
@@ -331,7 +333,8 @@
 
             </a>
             
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@@ -293,6 +293,7 @@
   
   
   
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-    15.1.1 &nbsp; 贪心的优点与局限性
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-    15.1.3 &nbsp; 贪心解题步骤
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-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_greedy/index.html b/chapter_greedy/index.html
index 23eb7f683..300c792a8 100644
--- a/chapter_greedy/index.html
+++ b/chapter_greedy/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
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+    
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@@ -233,7 +233,7 @@
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index f2b1d834f..13ff7b9c6 100644
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+  
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@@ -4381,7 +4536,7 @@ aria-label="页脚"
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@@ -4398,7 +4553,7 @@ aria-label="页脚"
         
         
           
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diff --git a/chapter_hashing/hash_collision/index.html b/chapter_hashing/hash_collision/index.html
index cf3ed4e6d..469b5c41f 100644
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       <a href="../../chapter_appendix/installation/" class="md-nav__link">
         
@@ -3217,6 +3363,7 @@
   
   
   
+  
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       <a href="../../chapter_appendix/contribution/" class="md-nav__link">
         
@@ -3237,6 +3384,7 @@
   
   
   
+  
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       <a href="../../chapter_appendix/terminology/" class="md-nav__link">
         
@@ -3272,6 +3420,7 @@
   
   
   
+  
     
     
       
@@ -3281,12 +3430,13 @@
       
     
     
+    
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-        
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@@ -3348,14 +3498,18 @@
       
         <li class="md-nav__item">
   <a href="#621" class="md-nav__link">
-    6.2.1 &nbsp; 链式地址
+    <span class="md-ellipsis">
+      6.2.1 &nbsp; 链式地址
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#622" class="md-nav__link">
-    6.2.2 &nbsp; 开放寻址
+    <span class="md-ellipsis">
+      6.2.2 &nbsp; 开放寻址
+    </span>
   </a>
   
     <nav class="md-nav" aria-label="6.2.2   开放寻址">
@@ -3363,21 +3517,27 @@
         
           <li class="md-nav__item">
   <a href="#1" class="md-nav__link">
-    1. &nbsp; 线性探测
+    <span class="md-ellipsis">
+      1. &nbsp; 线性探测
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#2" class="md-nav__link">
-    2. &nbsp; 平方探测
+    <span class="md-ellipsis">
+      2. &nbsp; 平方探测
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#3" class="md-nav__link">
-    3. &nbsp; 多次哈希
+    <span class="md-ellipsis">
+      3. &nbsp; 多次哈希
+    </span>
   </a>
   
 </li>
@@ -3389,7 +3549,9 @@
       
         <li class="md-nav__item">
   <a href="#623" class="md-nav__link">
-    6.2.3 &nbsp; 编程语言的选择
+    <span class="md-ellipsis">
+      6.2.3 &nbsp; 编程语言的选择
+    </span>
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
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@@ -6384,7 +6547,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../hash_map/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 6.1 &amp;nbsp; 哈希表" rel="prev">
+          <a href="../hash_map/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 6.1 &amp;nbsp; 哈希表">
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@@ -6401,7 +6564,7 @@ aria-label="页脚"
         
         
           
-          <a href="../hash_algorithm/" class="md-footer__link md-footer__link--next" aria-label="下一页: 6.3 &amp;nbsp; 哈希算法" rel="next">
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+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_hashing/index.html b/chapter_hashing/index.html
index 4bb2f4b01..448469c83 100644
--- a/chapter_hashing/index.html
+++ b/chapter_hashing/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
+      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
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index 85b3d8ab0..ccb960868 100644
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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-            回到页面顶部
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+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
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@@ -3727,7 +3878,7 @@ aria-label="页脚"
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-          <a href="../heap/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 8.1 &amp;nbsp; 堆" rel="prev">
+          <a href="../heap/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 8.1 &amp;nbsp; 堆">
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@@ -3744,7 +3895,7 @@ aria-label="页脚"
         
         
           
-          <a href="../top_k/" class="md-footer__link md-footer__link--next" aria-label="下一页: 8.3 &amp;nbsp; Top-K 问题" rel="next">
+          <a href="../top_k/" class="md-footer__link md-footer__link--next" aria-label="下一页: 8.3 &amp;nbsp; Top-K 问题">
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diff --git a/chapter_heap/heap/index.html b/chapter_heap/heap/index.html
index 3d0238111..a17f0391c 100644
--- a/chapter_heap/heap/index.html
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-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
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-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
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+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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+    </span>
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+    </span>
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           <li class="md-nav__item">
   <a href="#4" class="md-nav__link">
-    4. &nbsp; 堆顶元素出堆
+    <span class="md-ellipsis">
+      4. &nbsp; 堆顶元素出堆
+    </span>
   </a>
   
 </li>
@@ -3403,7 +3567,9 @@
       
         <li class="md-nav__item">
   <a href="#813" class="md-nav__link">
-    8.1.3 &nbsp; 堆常见应用
+    <span class="md-ellipsis">
+      8.1.3 &nbsp; 堆常见应用
+    </span>
   </a>
   
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
         </div>
         
           <button type="button" class="md-top md-icon" data-md-component="top" hidden>
-            
-            <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
       </main>
       
@@ -5100,7 +5267,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 8 章 &amp;nbsp; 堆" rel="prev">
+          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 8 章 &amp;nbsp; 堆">
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@@ -5117,7 +5284,7 @@ aria-label="页脚"
         
         
           
-          <a href="../build_heap/" class="md-footer__link md-footer__link--next" aria-label="下一页: 8.2 &amp;nbsp; 建堆操作" rel="next">
+          <a href="../build_heap/" class="md-footer__link md-footer__link--next" aria-label="下一页: 8.2 &amp;nbsp; 建堆操作">
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@@ -5156,7 +5323,7 @@ aria-label="页脚"
       
     
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@@ -5167,7 +5334,7 @@ aria-label="页脚"
       
     
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@@ -5178,7 +5345,7 @@ aria-label="页脚"
       
     
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-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_heap/index.html b/chapter_heap/index.html
index f0ae33e82..53bb9a9cb 100644
--- a/chapter_heap/index.html
+++ b/chapter_heap/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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-            回到页面顶部
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+  
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@@ -3463,7 +3604,7 @@ aria-label="页脚"
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-          <a href="../chapter_tree/summary/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 7.6 &amp;nbsp; 小结" rel="prev">
+          <a href="../chapter_tree/summary/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 7.6 &amp;nbsp; 小结">
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-          <a href="heap/" class="md-footer__link md-footer__link--next" aria-label="下一页: 8.1 &amp;nbsp; 堆" rel="next">
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diff --git a/chapter_heap/summary/index.html b/chapter_heap/summary/index.html
index 6f966d264..aa31c49bc 100644
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-            回到页面顶部
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+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
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@@ -3524,7 +3671,7 @@ aria-label="页脚"
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-          <a href="../top_k/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 8.3 &amp;nbsp; Top-K 问题" rel="prev">
+          <a href="../top_k/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 8.3 &amp;nbsp; Top-K 问题">
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@@ -3541,7 +3688,7 @@ aria-label="页脚"
         
         
           
-          <a href="../../chapter_graph/" class="md-footer__link md-footer__link--next" aria-label="下一页: 第 9 章 &amp;nbsp; 图" rel="next">
+          <a href="../../chapter_graph/" class="md-footer__link md-footer__link--next" aria-label="下一页: 第 9 章 &amp;nbsp; 图">
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diff --git a/chapter_heap/top_k/index.html b/chapter_heap/top_k/index.html
index b1acf3815..65a4225d2 100644
--- a/chapter_heap/top_k/index.html
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+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
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-    8.3.1 &nbsp; 方法一：遍历选择
+    <span class="md-ellipsis">
+      8.3.1 &nbsp; 方法一：遍历选择
+    </span>
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         <li class="md-nav__item">
   <a href="#832" class="md-nav__link">
-    8.3.2 &nbsp; 方法二：排序
+    <span class="md-ellipsis">
+      8.3.2 &nbsp; 方法二：排序
+    </span>
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   <a href="#833" class="md-nav__link">
-    8.3.3 &nbsp; 方法三：堆
+    <span class="md-ellipsis">
+      8.3.3 &nbsp; 方法三：堆
+    </span>
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index 3d181b21c..646044371 100644
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@@ -3517,7 +3656,7 @@ aria-label="页脚"
         
         
           
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+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_introduction/index.html b/chapter_introduction/index.html
index 03e096e1d..0eb01d794 100644
--- a/chapter_introduction/index.html
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+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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diff --git a/chapter_introduction/summary/index.html b/chapter_introduction/summary/index.html
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-    1.2.1 &nbsp; 算法定义
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@@ -3210,6 +3351,7 @@
   
   
   
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-    0.1.1 &nbsp; 读者对象
+    <span class="md-ellipsis">
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+    </span>
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-    0.1.3 &nbsp; 致谢
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diff --git a/chapter_preface/index.html b/chapter_preface/index.html
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@@ -3259,6 +3405,7 @@
   
   
   
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+    
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@@ -3335,35 +3483,45 @@
       
         <li class="md-nav__item">
   <a href="#021" class="md-nav__link">
-    0.2.1 &nbsp; 行文风格约定
+    <span class="md-ellipsis">
+      0.2.1 &nbsp; 行文风格约定
+    </span>
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 </li>
       
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   <a href="#022" class="md-nav__link">
-    0.2.2 &nbsp; 在动画图解中高效学习
+    <span class="md-ellipsis">
+      0.2.2 &nbsp; 在动画图解中高效学习
+    </span>
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 </li>
       
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-    0.2.3 &nbsp; 在代码实践中加深理解
+    <span class="md-ellipsis">
+      0.2.3 &nbsp; 在代码实践中加深理解
+    </span>
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-    0.2.4 &nbsp; 在提问讨论中共同成长
+    <span class="md-ellipsis">
+      0.2.4 &nbsp; 在提问讨论中共同成长
+    </span>
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-    0.2.5 &nbsp; 算法学习路线
+    <span class="md-ellipsis">
+      0.2.5 &nbsp; 算法学习路线
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+          <a href="../about_the_book/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 0.1 &amp;nbsp; 关于本书">
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-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_reference/index.html b/chapter_reference/index.html
index 21693be26..c49f2b3b2 100644
--- a/chapter_reference/index.html
+++ b/chapter_reference/index.html
@@ -20,7 +20,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -28,10 +28,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -83,9 +83,6 @@
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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@@ -150,9 +147,9 @@
     
     
     
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -161,6 +158,9 @@
       
     
     
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+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -231,7 +231,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -276,7 +276,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -291,6 +291,7 @@
   
   
   
+  
     
     
       
@@ -304,6 +305,8 @@
       
         
       
+    
+    
     
     
     <li class="md-nav__item md-nav__item--nested">
@@ -311,7 +314,6 @@
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
         
           
@@ -329,7 +331,8 @@
 
             </a>
             
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diff --git a/chapter_searching/binary_search/index.html b/chapter_searching/binary_search/index.html
index 54f939075..4dd9fadaf 100644
--- a/chapter_searching/binary_search/index.html
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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@@ -233,7 +233,7 @@
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   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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   <div class="md-source__repository">
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@@ -293,6 +293,7 @@
   
   
   
+  
     
     
       
@@ -306,6 +307,8 @@
       
         
       
+    
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diff --git a/chapter_searching/binary_search_edge/index.html b/chapter_searching/binary_search_edge/index.html
index 26447a7a6..8daf09fad 100644
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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@@ -233,7 +233,7 @@
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   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
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@@ -293,6 +293,7 @@
   
   
   
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@@ -306,6 +307,8 @@
       
         
       
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+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
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diff --git a/chapter_searching/binary_search_insertion/index.html b/chapter_searching/binary_search_insertion/index.html
index f3317bef6..290adba57 100644
--- a/chapter_searching/binary_search_insertion/index.html
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-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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@@ -163,6 +160,9 @@
       
     
     
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@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
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@@ -4106,7 +4253,7 @@ aria-label="页脚"
       
     
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@@ -4120,10 +4267,11 @@ aria-label="页脚"
       <div class="md-dialog__inner md-typeset"></div>
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+    
+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_searching/index.html b/chapter_searching/index.html
index 635c27297..b2d65aa01 100644
--- a/chapter_searching/index.html
+++ b/chapter_searching/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
     <body dir="ltr" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo">
   
     
-    
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-    
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     <input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
     <label class="md-overlay" for="__drawer"></label>
@@ -142,9 +139,9 @@
     
     
     
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
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@@ -152,9 +149,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_2">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
+      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -233,7 +233,7 @@
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
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@@ -3643,7 +3794,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
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+          <a href="../replace_linear_by_hashing/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 10.4 &amp;nbsp; 哈希优化策略">
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@@ -3660,7 +3811,7 @@ aria-label="页脚"
         
         
           
-          <a href="../summary/" class="md-footer__link md-footer__link--next" aria-label="下一页: 10.6 &amp;nbsp; 小结" rel="next">
+          <a href="../summary/" class="md-footer__link md-footer__link--next" aria-label="下一页: 10.6 &amp;nbsp; 小结">
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@@ -3699,7 +3850,7 @@ aria-label="页脚"
       
     
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+    
+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_searching/summary/index.html b/chapter_searching/summary/index.html
index 5b7a36f42..5dfbbd5d5 100644
--- a/chapter_searching/summary/index.html
+++ b/chapter_searching/summary/index.html
@@ -22,7 +22,7 @@
       
       
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-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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@@ -142,9 +139,9 @@
     
     
     
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
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diff --git a/chapter_sorting/bubble_sort/index.html b/chapter_sorting/bubble_sort/index.html
index 4badb6c76..bdd334d3c 100644
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         <li class="md-nav__item">
   <a href="#1132" class="md-nav__link">
-    11.3.2 &nbsp; 效率优化
+    <span class="md-ellipsis">
+      11.3.2 &nbsp; 效率优化
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1133" class="md-nav__link">
-    11.3.3 &nbsp; 算法特性
+    <span class="md-ellipsis">
+      11.3.3 &nbsp; 算法特性
+    </span>
   </a>
   
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
       </main>
       
@@ -4038,7 +4189,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../selection_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.2 &amp;nbsp; 选择排序" rel="prev">
+          <a href="../selection_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.2 &amp;nbsp; 选择排序">
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@@ -4055,7 +4206,7 @@ aria-label="页脚"
         
         
           
-          <a href="../insertion_sort/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.4 &amp;nbsp; 插入排序" rel="next">
+          <a href="../insertion_sort/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.4 &amp;nbsp; 插入排序">
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@@ -4105,7 +4256,7 @@ aria-label="页脚"
       
     
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-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_sorting/bucket_sort/index.html b/chapter_sorting/bucket_sort/index.html
index 477e1664d..a4e625654 100644
--- a/chapter_sorting/bucket_sort/index.html
+++ b/chapter_sorting/bucket_sort/index.html
@@ -22,7 +22,7 @@
       
       
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-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
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@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../../assets/stylesheets/palette.a5377069.min.css">
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-    11.8.1 &nbsp; 算法流程
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+              <label class="md-nav__link " for="__nav_17" id="__nav_17_label" tabindex="0">
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               </label>
             
@@ -3177,6 +3318,7 @@
   
   
   
+  
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       <a href="../../chapter_appendix/installation/" class="md-nav__link">
         
@@ -3197,6 +3339,7 @@
   
   
   
+  
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       <a href="../../chapter_appendix/contribution/" class="md-nav__link">
         
@@ -3217,6 +3360,7 @@
   
   
   
+  
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@@ -3252,6 +3396,7 @@
   
   
   
+  
     
     
       
@@ -3261,12 +3406,13 @@
       
     
     
+    
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@@ -3328,28 +3474,36 @@
       
         <li class="md-nav__item">
   <a href="#1191" class="md-nav__link">
-    11.9.1 &nbsp; 简单实现
+    <span class="md-ellipsis">
+      11.9.1 &nbsp; 简单实现
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1192" class="md-nav__link">
-    11.9.2 &nbsp; 完整实现
+    <span class="md-ellipsis">
+      11.9.2 &nbsp; 完整实现
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1193" class="md-nav__link">
-    11.9.3 &nbsp; 算法特性
+    <span class="md-ellipsis">
+      11.9.3 &nbsp; 算法特性
+    </span>
   </a>
   
 </li>
       
         <li class="md-nav__item">
   <a href="#1194" class="md-nav__link">
-    11.9.4 &nbsp; 局限性
+    <span class="md-ellipsis">
+      11.9.4 &nbsp; 局限性
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-            回到页面顶部
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+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
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-          <a href="../bucket_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.8 &amp;nbsp; 桶排序" rel="prev">
+          <a href="../bucket_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.8 &amp;nbsp; 桶排序">
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+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_sorting/index.html b/chapter_sorting/index.html
index 46b436ecc..b5760a50b 100644
--- a/chapter_sorting/index.html
+++ b/chapter_sorting/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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     <label class="md-overlay" for="__drawer"></label>
@@ -142,9 +139,9 @@
     
     
     
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
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@@ -152,9 +149,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_2">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
+      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -293,6 +293,7 @@
   
   
   
+  
     
     
       
@@ -306,6 +307,8 @@
       
         
       
+    
+    
     
     
     <li class="md-nav__item md-nav__item--nested">
@@ -313,7 +316,6 @@
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
         
           
@@ -331,7 +333,8 @@
 
             </a>
             
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index ec5dcbf7a..ac4fd8a84 100644
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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     krahets/hello-algo
@@ -278,7 +278,7 @@
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@@ -293,6 +293,7 @@
   
   
   
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index 2bf6b2236..b6f43c317 100644
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@@ -233,7 +233,7 @@
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@@ -278,7 +278,7 @@
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diff --git a/chapter_sorting/quick_sort/index.html b/chapter_sorting/quick_sort/index.html
index 10f07e3a2..7163e260b 100644
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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-    11.5.3 &nbsp; 快速排序为什么快
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-    11.5.4 &nbsp; 基准数优化
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+      11.5.4 &nbsp; 基准数优化
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-    11.5.5 &nbsp; 尾递归优化
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-            回到页面顶部
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+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
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@@ -4759,7 +4918,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../insertion_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.4 &amp;nbsp; 插入排序" rel="prev">
+          <a href="../insertion_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.4 &amp;nbsp; 插入排序">
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@@ -4776,7 +4935,7 @@ aria-label="页脚"
         
         
           
-          <a href="../merge_sort/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.6 &amp;nbsp; 归并排序" rel="next">
+          <a href="../merge_sort/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.6 &amp;nbsp; 归并排序">
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+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_sorting/radix_sort/index.html b/chapter_sorting/radix_sort/index.html
index ff9539aaf..17a05278e 100644
--- a/chapter_sorting/radix_sort/index.html
+++ b/chapter_sorting/radix_sort/index.html
@@ -22,7 +22,7 @@
       
       
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-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
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-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
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+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
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@@ -4172,7 +4319,7 @@ aria-label="页脚"
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-          <a href="../counting_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.9 &amp;nbsp; 计数排序" rel="prev">
+          <a href="../counting_sort/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 11.9 &amp;nbsp; 计数排序">
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@@ -4189,7 +4336,7 @@ aria-label="页脚"
         
         
           
-          <a href="../summary/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.11 &amp;nbsp; 小结" rel="next">
+          <a href="../summary/" class="md-footer__link md-footer__link--next" aria-label="下一页: 11.11 &amp;nbsp; 小结">
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+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_sorting/selection_sort/index.html b/chapter_sorting/selection_sort/index.html
index 20ce05b03..138e705e5 100644
--- a/chapter_sorting/selection_sort/index.html
+++ b/chapter_sorting/selection_sort/index.html
@@ -22,7 +22,7 @@
       
       
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-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
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+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
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+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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+      11.1.2 &nbsp; 理想排序算法
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+  
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-          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 11 章 &amp;nbsp; 排序" rel="prev">
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diff --git a/chapter_sorting/summary/index.html b/chapter_sorting/summary/index.html
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@@ -3568,7 +3715,7 @@ aria-label="页脚"
         
         
           
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diff --git a/chapter_stack_and_queue/deque/index.html b/chapter_stack_and_queue/deque/index.html
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         <li class="md-nav__item">
   <a href="#531" class="md-nav__link">
-    5.3.1 &nbsp; 双向队列常用操作
+    <span class="md-ellipsis">
+      5.3.1 &nbsp; 双向队列常用操作
+    </span>
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         <li class="md-nav__item">
   <a href="#532" class="md-nav__link">
-    5.3.2 &nbsp; 双向队列实现 *
+    <span class="md-ellipsis">
+      5.3.2 &nbsp; 双向队列实现 *
+    </span>
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@@ -3356,14 +3508,18 @@
         
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   <a href="#1" class="md-nav__link">
-    1. &nbsp; 基于双向链表的实现
+    <span class="md-ellipsis">
+      1. &nbsp; 基于双向链表的实现
+    </span>
   </a>
   
 </li>
         
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-    2. &nbsp; 基于数组的实现
+    <span class="md-ellipsis">
+      2. &nbsp; 基于数组的实现
+    </span>
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@@ -3375,7 +3531,9 @@
       
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   <a href="#533" class="md-nav__link">
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-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_stack_and_queue/index.html b/chapter_stack_and_queue/index.html
index bc008868a..04c889429 100644
--- a/chapter_stack_and_queue/index.html
+++ b/chapter_stack_and_queue/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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@@ -142,9 +139,9 @@
     
     
     
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
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@@ -152,9 +149,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_2">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
+      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -293,6 +293,7 @@
   
   
   
+  
     
     
       
@@ -306,6 +307,8 @@
       
         
       
+    
+    
     
     
     <li class="md-nav__item md-nav__item--nested">
@@ -313,7 +316,6 @@
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
         
           
@@ -331,7 +333,8 @@
 
             </a>
             
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diff --git a/chapter_stack_and_queue/queue/index.html b/chapter_stack_and_queue/queue/index.html
index ce9108319..f75f5f5fc 100644
--- a/chapter_stack_and_queue/queue/index.html
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -233,7 +233,7 @@
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   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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@@ -293,6 +293,7 @@
   
   
   
+  
     
     
       
@@ -306,6 +307,8 @@
       
         
       
+    
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@@ -313,7 +316,6 @@
         
         
         
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+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
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diff --git a/chapter_stack_and_queue/stack/index.html b/chapter_stack_and_queue/stack/index.html
index 1f68cde16..a750764cf 100644
--- a/chapter_stack_and_queue/stack/index.html
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-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
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   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
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diff --git a/chapter_stack_and_queue/summary/index.html b/chapter_stack_and_queue/summary/index.html
index 28b32e317..a31d749a1 100644
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-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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+<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
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-            
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-            回到页面顶部
-          </button>
+  
+  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M13 20h-2V8l-5.5 5.5-1.42-1.42L12 4.16l7.92 7.92-1.42 1.42L13 8v12Z"/></svg>
+  回到页面顶部
+</button>
         
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@@ -4637,7 +4788,7 @@ aria-label="页脚"
       <nav class="md-footer__inner md-grid" aria-label="页脚" >
         
           
-          <a href="../binary_tree_traversal/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 7.2 &amp;nbsp; 二叉树遍历" rel="prev">
+          <a href="../binary_tree_traversal/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 7.2 &amp;nbsp; 二叉树遍历">
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@@ -4654,7 +4805,7 @@ aria-label="页脚"
         
         
           
-          <a href="../binary_search_tree/" class="md-footer__link md-footer__link--next" aria-label="下一页: 7.4 &amp;nbsp; 二叉搜索树" rel="next">
+          <a href="../binary_search_tree/" class="md-footer__link md-footer__link--next" aria-label="下一页: 7.4 &amp;nbsp; 二叉搜索树">
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@@ -4693,7 +4844,7 @@ aria-label="页脚"
       
     
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+      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"/></svg>
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+    
+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_tree/avl_tree/index.html b/chapter_tree/avl_tree/index.html
index 76af534dd..29fd8992a 100644
--- a/chapter_tree/avl_tree/index.html
+++ b/chapter_tree/avl_tree/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
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+          <a href="../binary_search_tree/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 7.4 &amp;nbsp; 二叉搜索树">
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@@ -6083,10 +6278,11 @@ aria-label="页脚"
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+    
+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
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+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
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diff --git a/chapter_tree/binary_search_tree/index.html b/chapter_tree/binary_search_tree/index.html
index 09f139236..a79c45bd9 100644
--- a/chapter_tree/binary_search_tree/index.html
+++ b/chapter_tree/binary_search_tree/index.html
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-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
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-      <link rel="stylesheet" href="../../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../../assets/stylesheets/palette.06af60db.min.css">
       
       
 
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
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+    
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   <div class="md-select">
     
@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
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diff --git a/chapter_tree/binary_tree/index.html b/chapter_tree/binary_tree/index.html
index 6b9593873..0e5df4d0e 100644
--- a/chapter_tree/binary_tree/index.html
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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-    3. &nbsp; 完满二叉树
+    <span class="md-ellipsis">
+      3. &nbsp; 完满二叉树
+    </span>
   </a>
   
 </li>
         
           <li class="md-nav__item">
   <a href="#4" class="md-nav__link">
-    4. &nbsp; 平衡二叉树
+    <span class="md-ellipsis">
+      4. &nbsp; 平衡二叉树
+    </span>
   </a>
   
 </li>
@@ -3457,7 +3633,9 @@
       
         <li class="md-nav__item">
   <a href="#714" class="md-nav__link">
-    7.1.4 &nbsp; 二叉树的退化
+    <span class="md-ellipsis">
+      7.1.4 &nbsp; 二叉树的退化
+    </span>
   </a>
   
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+  回到页面顶部
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@@ -4197,7 +4376,7 @@ aria-label="页脚"
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-          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 7 章 &amp;nbsp; 树" rel="prev">
+          <a href="../" class="md-footer__link md-footer__link--prev" aria-label="上一页: 第 7 章 &amp;nbsp; 树">
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@@ -4214,7 +4393,7 @@ aria-label="页脚"
         
         
           
-          <a href="../binary_tree_traversal/" class="md-footer__link md-footer__link--next" aria-label="下一页: 7.2 &amp;nbsp; 二叉树遍历" rel="next">
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diff --git a/chapter_tree/binary_tree_traversal/index.html b/chapter_tree/binary_tree_traversal/index.html
index c42516750..bea6d26f0 100644
--- a/chapter_tree/binary_tree_traversal/index.html
+++ b/chapter_tree/binary_tree_traversal/index.html
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-    7.2.1 &nbsp; 层序遍历
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-    1. &nbsp; 代码实现
+    <span class="md-ellipsis">
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+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/chapter_tree/index.html b/chapter_tree/index.html
index f5e5f43f0..50027bc58 100644
--- a/chapter_tree/index.html
+++ b/chapter_tree/index.html
@@ -22,7 +22,7 @@
       
       
       <link rel="icon" href="../assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -30,10 +30,10 @@
       
     
     
-      <link rel="stylesheet" href="../assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="../assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="../assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -85,9 +85,6 @@
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     <label class="md-overlay" for="__drawer"></label>
@@ -142,9 +139,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
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@@ -152,9 +149,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_2">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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@@ -163,6 +160,9 @@
       
     
     
+      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -233,7 +233,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -278,7 +278,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -293,6 +293,7 @@
   
   
   
+  
     
     
       
@@ -306,6 +307,8 @@
       
         
       
+    
+    
     
     
     <li class="md-nav__item md-nav__item--nested">
@@ -313,7 +316,6 @@
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
         
           
@@ -331,7 +333,8 @@
 
             </a>
             
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-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
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@@ -293,6 +293,7 @@
   
   
   
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@@ -306,6 +307,8 @@
       
         
       
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+    
+    <script id="__config" type="application/json">{"base": "../..", "features": ["content.action.edit", "content.code.annotate", "content.code.copy", "content.tabs.link", "content.tooltips", "navigation.indexes", "navigation.instant", "navigation.top", "navigation.footer", "navigation.tracking", "search.highlight", "search.share", "search.suggest", "toc.follow"], "search": "../../assets/javascripts/workers/search.f886a092.min.js", "translations": {"clipboard.copied": "\u5df2\u590d\u5236", "clipboard.copy": "\u590d\u5236", "search.result.more.one": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.more.other": "\u5728\u8be5\u9875\u4e0a\u8fd8\u6709 # \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.none": "\u6ca1\u6709\u627e\u5230\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.one": "\u627e\u5230 1 \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.other": "# \u4e2a\u7b26\u5408\u6761\u4ef6\u7684\u7ed3\u679c", "search.result.placeholder": "\u952e\u5165\u4ee5\u5f00\u59cb\u641c\u7d22", "search.result.term.missing": "\u7f3a\u5c11", "select.version": "\u9009\u62e9\u5f53\u524d\u7248\u672c"}}</script>
     
     
-      <script src="../../assets/javascripts/bundle.4e0fa4ba.min.js"></script>
+      <script src="../../assets/javascripts/bundle.d7c377c4.min.js"></script>
       
         <script src="../../javascripts/mathjax.js"></script>
       
diff --git a/index.html b/index.html
index 59bbd432f..e5f537bad 100644
--- a/index.html
+++ b/index.html
@@ -18,7 +18,7 @@
       
       
       <link rel="icon" href="assets/images/favicon.png">
-      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.4.1">
+      <meta name="generator" content="mkdocs-1.5.3, mkdocs-material-9.5.0">
     
     
       
@@ -26,10 +26,10 @@
       
     
     
-      <link rel="stylesheet" href="assets/stylesheets/main.72749a73.min.css">
+      <link rel="stylesheet" href="assets/stylesheets/main.45e1311d.min.css">
       
         
-        <link rel="stylesheet" href="assets/stylesheets/palette.a5377069.min.css">
+        <link rel="stylesheet" href="assets/stylesheets/palette.06af60db.min.css">
       
       
 
@@ -70,9 +70,6 @@
     <body dir="ltr" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo">
   
     
-    
-      <script>var palette=__md_get("__palette");if(palette&&"object"==typeof palette.color)for(var key of Object.keys(palette.color))document.body.setAttribute("data-md-color-"+key,palette.color[key])</script>
-    
     <input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
     <input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
     <label class="md-overlay" for="__drawer"></label>
@@ -122,9 +119,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_1">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="white" data-md-color-accent="indigo"  aria-label="Switch to dark mode"  type="radio" name="__palette" id="__palette_0">
     
-      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_2" hidden>
+      <label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
       </label>
     
@@ -132,9 +129,9 @@
     
     
     
-    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_2">
+    <input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="grey" data-md-color-accent="indigo"  aria-label="Switch to light mode"  type="radio" name="__palette" id="__palette_1">
     
-      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_1" hidden>
+      <label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
         <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M7.5 2c-1.79 1.15-3 3.18-3 5.5s1.21 4.35 3.03 5.5C4.46 13 2 10.54 2 7.5A5.5 5.5 0 0 1 7.5 2m11.57 1.5 1.43 1.43L4.93 20.5 3.5 19.07 19.07 3.5m-6.18 2.43L11.41 5 9.97 6l.42-1.7L9 3.24l1.75-.12.58-1.65L12 3.1l1.73.03-1.35 1.13.51 1.67m-3.3 3.61-1.16-.73-1.12.78.34-1.32-1.09-.83 1.36-.09.45-1.29.51 1.27 1.36.03-1.05.87.4 1.31M19 13.5a5.5 5.5 0 0 1-5.5 5.5c-1.22 0-2.35-.4-3.26-1.07l7.69-7.69c.67.91 1.07 2.04 1.07 3.26m-4.4 6.58 2.77-1.15-.24 3.35-2.53-2.2m4.33-2.7 1.15-2.77 2.2 2.54-3.35.23m1.15-4.96-1.14-2.78 3.34.24-2.2 2.54M9.63 18.93l2.77 1.15-2.53 2.19-.24-3.34Z"/></svg>
       </label>
     
@@ -143,6 +140,9 @@
       
     
     
+      <script>var media,input,key,value,palette=__md_get("__palette");if(palette&&palette.color){"(prefers-color-scheme)"===palette.color.media&&(media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']"),palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent"));for([key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
+    
+    
       <div class="md-header__option">
   <div class="md-select">
     
@@ -213,7 +213,7 @@
         <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -260,7 +260,7 @@
       <a href="https://github.com/krahets/hello-algo" title="前往仓库" class="md-source" data-md-component="source">
   <div class="md-source__icon md-icon">
     
-    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.4.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg>
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   </div>
   <div class="md-source__repository">
     krahets/hello-algo
@@ -275,6 +275,7 @@
   
   
   
+  
     
     
       
@@ -288,6 +289,8 @@
       
         
       
+    
+    
     
     
     <li class="md-nav__item md-nav__item--nested">
@@ -295,7 +298,6 @@
         
         
         
-        
         <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_1" >
         
           
@@ -313,7 +315,8 @@
 
             </a>
             
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index 0bb4b160e..b8c4edab7 100644
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+++ b/search/search_index.json
@@ -1 +1 @@
-{"config":{"lang":["en"],"separator":"[\\s\\u200b\\u3000\\-\u3001\u3002\uff0c\uff0e\uff1f\uff01\uff1b]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Home","text":"\u300a Hello \u7b97\u6cd5 \u300b <p>\u52a8\u753b\u56fe\u89e3\u3001\u4e00\u952e\u8fd0\u884c\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6559\u7a0b</p> <p> </p> <p>        \u5f00\u59cb\u9605\u8bfb           \u4e0b\u8f7d\u4ee3\u7801           \u4e0b\u8f7d PDF    </p> <p> </p>  \u63a8\u8350\u8bed  <p>Quote</p> <p>\u201c\u4e00\u672c\u901a\u4fd7\u6613\u61c2\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\uff0c\u5f15\u5bfc\u8bfb\u8005\u624b\u8111\u5e76\u7528\u5730\u5b66\u4e60\uff0c\u5f3a\u70c8\u63a8\u8350\u7b97\u6cd5\u521d\u5b66\u8005\u9605\u8bfb\u3002\u201d</p> <p>\u2014\u2014 \u9093\u4fca\u8f89\uff0c\u6e05\u534e\u5927\u5b66\u8ba1\u7b97\u673a\u7cfb\u6559\u6388</p> <p>Quote</p> <p>\u201c\u5982\u679c\u6211\u5f53\u5e74\u5b66\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u65f6\u5019\u6709\u300aHello \u7b97\u6cd5\u300b\uff0c\u5b66\u8d77\u6765\u5e94\u8be5\u4f1a\u7b80\u5355 10 \u500d\uff01\u201d</p> <p>\u2014\u2014 \u674e\u6c90\uff0c\u4e9a\u9a6c\u900a\u8d44\u6df1\u9996\u5e2d\u79d1\u5b66\u5bb6</p> \u52a8\u753b\u56fe\u89e3 <p>\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1</p> <p>\"A picture is worth a thousand words.\"</p> <p>\u201c\u4e00\u56fe\u80dc\u5343\u8a00\u201d</p> \u4e00\u952e\u8fd0\u884c <p>\u5341\u4f59\u79cd\u7f16\u7a0b\u8bed\u8a00\u4ee3\u7801\u53ef\u76f4\u63a5\u8fd0\u884c</p> <p>\"Talk is cheap. Show me the code.\"</p> <p>\u201c\u5c11\u5439\u725b\uff0c\u770b\u4ee3\u7801\u201d</p> \u4e92\u52a9\u5b66\u4e60 <p>\u6b22\u8fce\u8ba8\u8bba\u4e0e\u63d0\u95ee\u8bfb\u8005\u95f4\u643a\u624b\u5171\u8fdb</p> <p>\"\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\"</p> <p>\u201c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u201d</p>  \u5e8f  <p>\u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u201c\u5251\u6307 Offer\u201d\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u8bb8\u591a\u540c\u5b66\u7684\u559c\u7231\u548c\u652f\u6301\u3002\u5728\u4e0e\u8bfb\u8005\u4ea4\u6d41\u671f\u95f4\uff0c\u6211\u6700\u5e38\u6536\u5230\u7684\u4e00\u4e2a\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u7b97\u6cd5\u201d\u3002\u9010\u6e10\u5730\uff0c\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86\u6d53\u539a\u7684\u5174\u8da3\u3002</p> <p>\u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u4f3c\u4e4e\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u6cd5\uff0c\u7b80\u5355\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u540c\u5b66\u80fd\u591f\u987a\u5229\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u540c\u5b66\u5f88\u53ef\u80fd\u88ab\u70b8\u7684\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u505a\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u540c\u5b66\u6765\u8bf4\uff0c\u6bd5\u4e1a\u5b63\u3001\u6295\u9012\u7b80\u5386\u3001\u51c6\u5907\u7b14\u8bd5\u9762\u8bd5\u5df2\u7ecf\u6d88\u8017\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u5543\u539a\u91cd\u7684\u4e66\u5f80\u5f80\u53d8\u6210\u4e86\u4e00\u9879\u8270\u5de8\u7684\u6311\u6218\u3002</p> <p>\u5982\u679c\u4f60\u4e5f\u9762\u4e34\u7c7b\u4f3c\u7684\u56f0\u6270\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u627e\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u5373\u4f7f\u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u4e5f\u81f3\u5c11\u662f\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer \uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u3001\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5237\u9898\u548c\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u6784\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002</p> <p>\u6211\u6df1\u6df1\u8d5e\u540c\u8d39\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u4ece\u8fd9\u4e2a\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u672c\u4e66\u5e76\u975e\u5b8c\u5168\u201c\u514d\u8d39\u201d\u3002\u4e3a\u4e86\u4e0d\u8f9c\u8d1f\u4f60\u4e3a\u672c\u4e66\u6240\u4ed8\u51fa\u7684\u5b9d\u8d35\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u4f1a\u5c3d\u6211\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u6765\u5b8c\u6210\u8fd9\u672c\u4e66\u7684\u521b\u4f5c\u3002</p>  \u4f5c\u8005\u7b80\u4ecb  <p>\u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u53d1\u8868\u7684\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 30 \u4e07\u672c\u3002</p>  \u81f4\u8c22  <p>\u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u4f17\u591a\u8d21\u732e\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65ad\u5b8c\u5584\u3002\u611f\u8c22\u6bcf\u4e00\u4f4d\u6295\u5165\u65f6\u95f4\u4e0e\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\u6392\u5217\uff09\uff1a</p> <p> </p> <p>\u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 Gonglja\u3001gvenusleo\u3001hpstory\u3001justin\u2010tse\u3001krahets\u3001night-cruise\u3001nuomi1\u3001Reanon \u548c sjinzh \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\uff09\u3002\u611f\u8c22\u4ed6\u4eec\u4ed8\u51fa\u7684\u65f6\u95f4\u4e0e\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u4eec\u786e\u4fdd\u4e86\u5404\u8bed\u8a00\u4ee3\u7801\u7684\u89c4\u8303\u4e0e\u7edf\u4e00\u3002</p> <sub>Gonglja</sub><sub>C, C++</sub> <sub>gvenusleo</sub><sub>Dart</sub> <sub>hpstory</sub><sub>C#</sub> <sub>justin-tse</sub><sub>JS, TS</sub> <sub>krahets</sub><sub>Java, Python</sub> <sub>night-cruise</sub><sub>Rust</sub> <sub>nuomi1</sub><sub>Swift</sub> <sub>Reanon</sub><sub>Go, C</sub> <sub>sjinzh</sub><sub>Rust, Zig</sub>"},{"location":"chapter_appendix/","title":"\u7b2c 16 \u7ae0 \u00a0 \u9644\u5f55","text":""},{"location":"chapter_appendix/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5</li> <li>16.2 \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c</li> <li>16.3 \u00a0 \u672f\u8bed\u8868</li> </ul>"},{"location":"chapter_appendix/contribution/","title":"16.2 \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c","text":"<p>\u7531\u4e8e\u7b14\u8005\u80fd\u529b\u6709\u9650\uff0c\u4e66\u4e2d\u96be\u514d\u5b58\u5728\u4e00\u4e9b\u9057\u6f0f\u548c\u9519\u8bef\uff0c\u8bf7\u60a8\u8c05\u89e3\u3002\u5982\u679c\u60a8\u53d1\u73b0\u4e86\u7b14\u8bef\u3001\u94fe\u63a5\u5931\u6548\u3001\u5185\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u4e49\u3001\u89e3\u91ca\u4e0d\u6e05\u6670\u6216\u884c\u6587\u7ed3\u6784\u4e0d\u5408\u7406\u7b49\u95ee\u9898\uff0c\u8bf7\u534f\u52a9\u6211\u4eec\u8fdb\u884c\u4fee\u6b63\uff0c\u4ee5\u7ed9\u8bfb\u8005\u63d0\u4f9b\u66f4\u4f18\u8d28\u7684\u5b66\u4e60\u8d44\u6e90\u3002</p> <p>\u6240\u6709\u64b0\u7a3f\u4eba\u7684 GitHub ID \u5c06\u5728\u672c\u4e66\u4ed3\u5e93\u3001\u7f51\u9875\u7248\u548c PDF \u7248\u7684\u4e3b\u9875\u4e0a\u8fdb\u884c\u5c55\u793a\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\u3002</p> <p>\u5f00\u6e90\u7684\u9b45\u529b</p> <p>\u7eb8\u8d28\u56fe\u4e66\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u8f83\u4e45\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002</p> <p>\u800c\u5728\u672c\u5f00\u6e90\u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002</p>"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"<p>\u5982\u56fe 16-3 \u6240\u793a\uff0c\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7f16\u8f91\u56fe\u6807\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\u3002</p> <ol> <li>\u70b9\u51fb\u201c\u7f16\u8f91\u56fe\u6807\u201d\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\u7684\u63d0\u793a\uff0c\u8bf7\u540c\u610f\u8be5\u64cd\u4f5c\u3002</li> <li>\u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u68c0\u67e5\u5185\u5bb9\u7684\u6b63\u786e\u6027\uff0c\u5e76\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\u3002</li> <li>\u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\u3002\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol> <p></p> <p> \u56fe 16-3 \u00a0 \u9875\u9762\u7f16\u8f91\u6309\u952e </p> <p>\u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5c3d\u5feb\u91cd\u65b0\u7ed8\u5236\u5e76\u66ff\u6362\u56fe\u7247\u3002</p>"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"<p>\u5982\u679c\u60a8\u6709\u5174\u8da3\u53c2\u4e0e\u6b64\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u5c06\u4ee3\u7801\u7ffb\u8bd1\u6210\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u6269\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd\u4ee5\u4e0b Pull Request \u5de5\u4f5c\u6d41\u7a0b\u3002</p> <ol> <li>\u767b\u5f55 GitHub \uff0c\u5c06\u672c\u4e66\u7684\u4ee3\u7801\u4ed3\u5e93 Fork \u5230\u4e2a\u4eba\u8d26\u53f7\u4e0b\u3002</li> <li>\u8fdb\u5165\u60a8\u7684 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528 <code>git clone</code> \u547d\u4ee4\u5c06\u4ed3\u5e93\u514b\u9686\u81f3\u672c\u5730\u3002</li> <li>\u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u8fdb\u884c\u5b8c\u6574\u6d4b\u8bd5\uff0c\u9a8c\u8bc1\u4ee3\u7801\u7684\u6b63\u786e\u6027\u3002</li> <li>\u5c06\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u540e Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\u3002</li> <li>\u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol>"},{"location":"chapter_appendix/contribution/#3-docker","title":"3. \u00a0 Docker \u90e8\u7f72","text":"<p>\u5728 <code>hello-algo</code> \u6839\u76ee\u5f55\u4e0b\uff0c\u6267\u884c\u4ee5\u4e0b Docker \u811a\u672c\uff0c\u5373\u53ef\u5728 <code>http://localhost:8000</code> \u8bbf\u95ee\u672c\u9879\u76ee\uff1a</p> <pre><code>docker-compose up -d\n</code></pre> <p>\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\uff1a</p> <pre><code>docker-compose down\n</code></pre>"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88c5 IDE","text":"<p>\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u3001\u8f7b\u91cf\u7684 VSCode \u4f5c\u4e3a\u672c\u5730\u96c6\u6210\u5f00\u53d1\u73af\u5883\uff08IDE\uff09\u3002\u8bbf\u95ee VSCode \u5b98\u7f51\uff0c\u6839\u636e\u64cd\u4f5c\u7cfb\u7edf\u9009\u62e9\u76f8\u5e94\u7248\u672c\u7684 VSCode \u8fdb\u884c\u4e0b\u8f7d\u548c\u5b89\u88c5\u3002</p> <p></p> <p> \u56fe 16-1 \u00a0 \u4ece\u5b98\u7f51\u4e0b\u8f7d VSCode </p> <p>VSCode \u62e5\u6709\u5f3a\u5927\u7684\u6269\u5c55\u5305\u751f\u6001\u7cfb\u7edf\uff0c\u652f\u6301\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u7684\u8fd0\u884c\u548c\u8c03\u8bd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5b89\u88c5\u201cPython Extension Pack\u201d\u6269\u5c55\u5305\u4e4b\u540e\uff0c\u5373\u53ef\u8fdb\u884c Python \u4ee3\u7801\u8c03\u8bd5\u3002\u5b89\u88c5\u6b65\u9aa4\u5982\u56fe 16-2 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 16-2 \u00a0 \u5b89\u88c5 VSCode \u6269\u5c55\u5305 </p>"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88c5\u8bed\u8a00\u73af\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>python</code> \uff0c\u5b89\u88c5 Python Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165 <code>pip install black</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#2-cc","title":"2. \u00a0 C/C++ \u73af\u5883","text":"<ol> <li>Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e26 Clang \uff0c\u65e0\u987b\u5b89\u88c5\u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>c++</code> \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22 <code>Clang_format_fallback Style</code> \u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code> \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#3-java","title":"3. \u00a0 Java \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 &gt; JDK 9\uff09\u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>java</code> \uff0c\u5b89\u88c5 Extension Pack for Java \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#4-c","title":"4. \u00a0 C# \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 8.0 \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>C# Dev Kit</code> \uff0c\u5b89\u88c5 C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002</li> <li>\u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88c5\u6559\u7a0b\uff09\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#5-go","title":"5. \u00a0 Go \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>go</code> \uff0c\u5b89\u88c5 Go \u3002</li> <li>\u6309\u5feb\u6377\u952e <code>Ctrl + Shift + P</code> \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 <code>Go: Install/Update Tools</code> \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#6-swift","title":"6. \u00a0 Swift \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>swift</code> \uff0c\u5b89\u88c5 Swift for Visual Studio Code \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#7-javascript","title":"7. \u00a0 JavaScript \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>javascript</code> \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>Prettier</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#8-dart","title":"8. \u00a0 Dart \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Dart \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>dart</code> \uff0c\u5b89\u88c5 Dart \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#9-rust","title":"9. \u00a0 Rust \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>rust</code> \uff0c\u5b89\u88c5 rust-analyzer \u3002</li> </ol>"},{"location":"chapter_appendix/terminology/","title":"16.3 \u00a0 \u672f\u8bed\u8868","text":"<p>\u8868 16-1 \u5217\u51fa\u4e86\u4e66\u4e2d\u51fa\u73b0\u7684\u91cd\u8981\u672f\u8bed\u3002\u5efa\u8bae\u4f60\u540c\u65f6\u8bb0\u4f4f\u5b83\u4eec\u7684\u4e2d\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u9605\u8bfb\u82f1\u6587\u6587\u732e\u3002</p> <p> \u8868 16-1 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8bcd </p> \u4e2d\u6587 English \u4e2d\u6587 English \u7b97\u6cd5 algorithm \u5c42\u5e8f\u904d\u5386 level-order traversal \u6570\u636e\u7ed3\u6784 data structure \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis \u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal \u65f6\u95f4\u590d\u6742\u5ea6 time complexity \u4e8c\u53c9\u641c\u7d22\u6811 binary search tree \u7a7a\u95f4\u590d\u6742\u5ea6 space complexity \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree \u8fed\u4ee3 iteration \u5e73\u8861\u56e0\u5b50 balance factor \u9012\u5f52 recursion \u5806 heap \u5c3e\u9012\u5f52 tail recursion \u5927\u9876\u5806 max heap \u9012\u5f52\u6811 recursion tree \u5c0f\u9876\u5806 min heap \u5927 big- \u4f18\u5148\u961f\u5217 priority queue \u8bb0\u53f7 notation \u6e10\u8fd1\u4e0a\u754c asymptotic upper bound \u5806\u5316 heapify \u539f\u7801 sign\u2013magnitude \u56fe graph \u53cd\u7801 1\u2019s complement \u9876\u70b9 vertex \u8865\u7801 2\u2019s complement \u65e0\u5411\u56fe undirected graph \u6570\u7ec4 array \u6709\u5411\u56fe directed graph \u7d22\u5f15 index \u8fde\u901a\u56fe connected graph \u94fe\u8868 linked list \u975e\u8fde\u901a\u56fe disconnected graph \u94fe\u8868\u8282\u70b9 linked list node, list node \u6709\u6743\u56fe weighted graph \u5217\u8868 list \u90bb\u63a5 adjacency \u52a8\u6001\u6570\u7ec4 dynamic array \u8def\u5f84 path \u786c\u76d8 hard disk \u5165\u5ea6 in-degree \u5185\u5b58 random-access memory (RAM) \u51fa\u5ea6 out-degree \u7f13\u5b58 cache memory \u90bb\u63a5\u77e9\u9635 adjacency matrix \u7f13\u5b58\u672a\u547d\u4e2d cache miss \u90bb\u63a5\u8868 adjacency list \u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search \u6808 stack \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search \u961f\u5217 queue \u4e8c\u5206\u67e5\u627e binary search \u53cc\u5411\u961f\u5217 double-ended queue \u641c\u7d22\u7b97\u6cd5 searching algorithm \u54c8\u5e0c\u8868 hash table \u6392\u5e8f\u7b97\u6cd5 sorting algorithm \u6876 bucket \u9009\u62e9\u6392\u5e8f selection sort \u54c8\u5e0c\u51fd\u6570 hash function \u5192\u6ce1\u6392\u5e8f bubble sort \u54c8\u5e0c\u51b2\u7a81 hash collision \u63d2\u5165\u6392\u5e8f insertion sort \u8d1f\u8f7d\u56e0\u5b50 load factor \u5feb\u901f\u6392\u5e8f quick sort \u94fe\u5f0f\u5730\u5740 separate chaining \u5f52\u5e76\u6392\u5e8f merge sort \u5f00\u653e\u5bfb\u5740 open addressing \u5806\u6392\u5e8f heap sort \u7ebf\u6027\u63a2\u6d4b linear probing \u6876\u6392\u5e8f bucket sort \u61d2\u5220\u9664 lazy deletion \u8ba1\u6570\u6392\u5e8f counting sort \u4e8c\u53c9\u6811 binary tree \u57fa\u6570\u6392\u5e8f radix sort \u6811\u8282\u70b9 tree node \u5206\u6cbb divide and conquer \u5de6\u5b50\u8282\u70b9 left-child node \u6c49\u8bfa\u5854\u95ee\u9898 hanota problem \u53f3\u5b50\u8282\u70b9 right-child node \u56de\u6eaf\u7b97\u6cd5 backtracking algorithm \u7236\u8282\u70b9 parent node \u7ea6\u675f constraint \u5de6\u5b50\u6811 left subtree \u89e3 solution \u53f3\u5b50\u6811 right subtree \u72b6\u6001 state \u6839\u8282\u70b9 root node \u526a\u679d pruning \u53f6\u8282\u70b9 leaf node \u5168\u6392\u5217\u95ee\u9898 permutations problem \u8fb9 edge \u5b50\u96c6\u548c\u95ee\u9898 subset-sum problem \u5c42 level N \u7687\u540e\u95ee\u9898 N-queens problem \u5ea6 degree \u52a8\u6001\u89c4\u5212 dynamic programming \u9ad8\u5ea6 height \u521d\u59cb\u72b6\u6001 initial state \u6df1\u5ea6 depth \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b state-trasition equation \u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree \u80cc\u5305\u95ee\u9898 knapsack problem \u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 edit distance problem \u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree \u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm \u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree AVL \u6811 AVL tree \u7ea2\u9ed1\u6811 red-black tree"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5b9e\u7684\u7816\u5899\u3002</p> <p>\u6570\u7ec4\u7684\u7816\u5757\u6574\u9f50\u6392\u5217\uff0c\u9010\u4e2a\u7d27\u8d34\u3002\u94fe\u8868\u7684\u7816\u5757\u5206\u6563\u5404\u5904\uff0c\u8fde\u63a5\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u4e8e\u7816\u7f1d\u4e4b\u95f4\u3002</p>"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>4.1 \u00a0 \u6570\u7ec4</li> <li>4.2 \u00a0 \u94fe\u8868</li> <li>4.3 \u00a0 \u5217\u8868</li> <li>4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *</li> <li>4.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 \u6570\u7ec4","text":"<p>\u300c\u6570\u7ec4 array\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u8be5\u5143\u7d20\u7684\u300c\u7d22\u5f15 index\u300d\u3002\u56fe 4-1 \u5c55\u793a\u4e86\u6570\u7ec4\u7684\u4e3b\u8981\u672f\u8bed\u548c\u6982\u5ff5\u3002</p> <p></p> <p> \u56fe 4-1 \u00a0 \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p>"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u6570\u7ec4","text":"<p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9700\u6c42\u9009\u7528\u6570\u7ec4\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4f1a\u5c06\u6570\u7ec4\u5143\u7d20\u521d\u59cb\u5316\u4e3a \\(0\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code># \u521d\u59cb\u5316\u6570\u7ec4\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]  \n</code></pre> array.cpp<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\uff08\u9700\u8981\u624b\u52a8\u91ca\u653e\u7a7a\u95f4\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: Vec&lt;i32&gt; = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.zig<pre><code>// \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u56fe 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u8be5\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 4-2 \u00a0 \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 </p> <p>\u89c2\u5bdf\u56fe 4-2 \uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002\u4f46\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u4e3a \\(0\\) \u662f\u5408\u7406\u7684\u3002</p> <p>\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n\"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n# \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandom_index = random.randint(0, len(nums) - 1)\n# \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandom_num = nums[random_index]\nreturn random_num\n</code></pre> array.cpp<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n</code></pre> array.java<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\nRandom random = new();\n// \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = random.Next(nums.Length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n</code></pre> array.go<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nrandomIndex := rand.Intn(len(nums))\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nrandomNum = nums[randomIndex]\nreturn\n}\n</code></pre> array.swift<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n// \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nlet randomIndex = nums.indices.randomElement()!\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nlet randomNum = nums[randomIndex]\nreturn randomNum\n}\n</code></pre> array.js<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n</code></pre> array.ts<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nconst random_index = Math.floor(Math.random() * nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nconst random_num = nums[random_index];\nreturn random_num;\n}\n</code></pre> array.dart<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n// \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = Random().nextInt(nums.length);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n// \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nlet random_index = rand::thread_rng().gen_range(0..nums.len());\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nlet random_num = nums[random_index];\nrandom_num\n}\n</code></pre> array.c<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n// \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\nint randomIndex = rand() % size;\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nint randomNum = nums[randomIndex];\nreturn randomNum;\n}\n</code></pre> array.zig<pre><code>// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n// \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\nvar randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n// \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\nvar randomNum = nums[randomIndex];\nreturn randomNum;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u5b58\u653e\u4efb\u4f55\u6570\u636e\u3002\u5982\u56fe 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u5219\u9700\u8981\u5c06\u8be5\u5143\u7d20\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u4e4b\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 4-3 \u00a0 \u6570\u7ec4\u63d2\u5165\u5143\u7d20\u793a\u4f8b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5c3e\u90e8\u5143\u7d20\u201c\u4e22\u5931\u201d\u3002\u6211\u4eec\u5c06\u8fd9\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\u7559\u5728\u201c\u5217\u8868\u201d\u7ae0\u8282\u4e2d\u8ba8\u8bba\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n\"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n# \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in range(len(nums) - 1, index, -1):\nnums[i] = nums[i - 1]\n# \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i &gt; index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.length - 1; i &gt; index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = nums.Length - 1; i &gt; index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i := len(nums) - 1; i &gt; index; i-- {\nnums[i] = nums[i-1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in nums.indices.dropFirst(index).reversed() {\nnums[i] = nums[i - 1]\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i &gt; index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let i = nums.length - 1; i &gt; index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (var i = nums.length - 1; i &gt; index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\nnums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut Vec&lt;i32&gt;, num: i32, index: usize) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor i in (index + 1..nums.len()).rev() {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int i = size - 1; i &gt; index; i--) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar i = nums.len - 1;\nwhile (i &gt; index) : (i -= 1) {\nnums[i] = nums[i - 1];\n}\n// \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\nnums[index] = num;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u5220\u9664\u5143\u7d20","text":"<p>\u540c\u7406\uff0c\u5982\u56fe 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002</p> <p></p> <p> \u56fe 4-4 \u00a0 \u6570\u7ec4\u5220\u9664\u5143\u7d20\u793a\u4f8b </p> <p>\u8bf7\u6ce8\u610f\uff0c\u5220\u9664\u5143\u7d20\u5b8c\u6210\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u987b\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def remove(nums: list[int], index: int):\n\"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n# \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in range(index, len(nums) - 1):\nnums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i &lt; size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.java<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i &lt; nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.cs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i &lt; nums.Length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.go<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i := index; i &lt; len(nums)-1; i++ {\nnums[i] = nums[i+1]\n}\n}\n</code></pre> array.swift<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in nums.indices.dropFirst(index).dropLast() {\nnums[i] = nums[i + 1]\n}\n}\n</code></pre> array.js<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i &lt; nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.ts<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let i = index; i &lt; nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.dart<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (var i = index; i &lt; nums.length - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.rs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut Vec&lt;i32&gt;, index: usize) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor i in index..nums.len() - 1 {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.c<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int i = index; i &lt; size - 1; i++) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> array.zig<pre><code>// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n// \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar i = index;\nwhile (i &lt; nums.len - 1) : (i += 1) {\nnums[i] = nums[i + 1];\n}\n}\n</code></pre> <p>\u603b\u7684\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002</li> <li>\u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002</li> <li>\u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u65e0\u610f\u4e49\u201d\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5","title":"5. \u00a0 \u904d\u5386\u6570\u7ec4","text":"<p>\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e2\u53ef\u4ee5\u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def traverse(nums: list[int]):\n\"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\ncount = 0\n# \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i in range(len(nums)):\ncount += nums[i]\n# \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor num in nums:\ncount += num\n# \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\nfor i, num in enumerate(nums):\ncount += nums[i]\ncount += num\n</code></pre> array.cpp<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; size; i++) {\ncount += nums[i];\n}\n}\n</code></pre> array.java<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; nums.length; i++) {\ncount += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor (int num : nums) {\ncount += num;\n}\n}\n</code></pre> array.cs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; nums.Length; i++) {\ncount += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nforeach (int num in nums) {\ncount += num;\n}\n}\n</code></pre> array.go<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\ncount := 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i := 0; i &lt; len(nums); i++ {\ncount += nums[i]\n}\ncount = 0\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor _, num := range nums {\ncount += num\n}\n// \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\nfor i, num := range nums {\ncount += nums[i]\ncount += num\n}\n}\n</code></pre> array.swift<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\nvar count = 0\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i in nums.indices {\ncount += nums[i]\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor num in nums {\ncount += num\n}\n}\n</code></pre> array.js<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i &lt; nums.length; i++) {\ncount += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor (const num of nums) {\ncount += num;\n}\n}\n</code></pre> array.ts<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\nlet count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i &lt; nums.length; i++) {\ncount += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor (const num of nums) {\ncount += num;\n}\n}\n</code></pre> array.dart<pre><code>/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (var i = 0; i &lt; nums.length; i++) {\ncount += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor (int _num in nums) {\ncount += _num;\n}\n// \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\nnums.forEach((_num) {\ncount += _num;\n});\n}\n</code></pre> array.rs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &amp;[i32]) {\nlet mut _count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor i in 0..nums.len() {\n_count += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor num in nums {\n_count += num;\n}\n}\n</code></pre> array.c<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\nint count = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; size; i++) {\ncount += nums[i];\n}\n}\n</code></pre> array.zig<pre><code>// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\nvar count: i32 = 0;\n// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\nvar i: i32 = 0;\nwhile (i &lt; nums.len) : (i += 1) {\ncount += nums[i];\n}\ncount = 0;\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\nfor (nums) |num| {\ncount += num;\n}\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u627e\u5143\u7d20","text":"<p>\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\u9700\u8981\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u5224\u65ad\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5219\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002</p> <p>\u56e0\u4e3a\u6570\u7ec4\u662f\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u627e\u64cd\u4f5c\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u67e5\u627e\u201d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n\"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\nfor i in range(len(nums)):\nif nums[i] == target:\nreturn i\nreturn -1\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\nfor (int i = 0; i &lt; size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\nfor (int i = 0; i &lt; nums.length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\nfor (int i = 0; i &lt; nums.Length; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\nindex = -1\nfor i := 0; i &lt; len(nums); i++ {\nif nums[i] == target {\nindex = i\nbreak\n}\n}\nreturn\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\nfor i in nums.indices {\nif nums[i] == target {\nreturn i\n}\n}\nreturn -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\nfor (let i = 0; i &lt; nums.length; i++) {\nif (nums[i] === target) return i;\n}\nreturn -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\nfor (let i = 0; i &lt; nums.length; i++) {\nif (nums[i] === target) {\nreturn i;\n}\n}\nreturn -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\nfor (var i = 0; i &lt; nums.length; i++) {\nif (nums[i] == target) return i;\n}\nreturn -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\nfor i in 0..nums.len() {\nif nums[i] == target {\nreturn Some(i);\n}\n}\nNone\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\nfor (int i = 0; i &lt; size; i++) {\nif (nums[i] == target)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\nfor (nums, 0..) |num, i| {\nif (num == target) return @intCast(i);\n}\nreturn -1;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u6269\u5bb9\u6570\u7ec4","text":"<p>\u5728\u590d\u6742\u7684\u7cfb\u7edf\u73af\u5883\u4e2d\uff0c\u7a0b\u5e8f\u96be\u4ee5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u4ece\u800c\u65e0\u6cd5\u5b89\u5168\u5730\u6269\u5c55\u6570\u7ec4\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u4e0d\u53ef\u53d8\u7684\u3002</p> <p>\u5982\u679c\u6211\u4eec\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u590d\u5236\u5230\u65b0\u6570\u7ec4\u3002\u8fd9\u662f\u4e00\u4e2a \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\u975e\u5e38\u8017\u65f6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n\"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n# \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres = [0] * (len(nums) + enlarge)\n# \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in range(len(nums)):\nres[i] = nums[i]\n# \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n</code></pre> array.cpp<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint *res = new int[size + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i &lt; size; i++) {\nres[i] = nums[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] nums;\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre> array.java<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i &lt; nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre> array.cs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint[] res = new int[nums.Length + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i &lt; nums.Length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre> array.go<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nres := make([]int, len(nums)+enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i, num := range nums {\nres[i] = num\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n</code></pre> array.swift<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = Array(repeating: 0, count: nums.count + enlarge)\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor i in nums.indices {\nres[i] = nums[i]\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res\n}\n</code></pre> array.js<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i &lt; nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre> array.ts<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nconst res = new Array(nums.length + enlarge).fill(0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (let i = 0; i &lt; nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre> array.dart<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nList&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (var i = 0; i &lt; nums.length; i++) {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre> array.rs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec&lt;i32&gt;, enlarge: usize) -&gt; Vec&lt;i32&gt; {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nlet mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\nfor i in 0..nums.len() {\nres[i] = nums[i];\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nres\n}\n</code></pre> array.c<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nint *res = (int *)malloc(sizeof(int) * (size + enlarge));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i &lt; size; i++) {\nres[i] = nums[i];\n}\n// \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\nfor (int i = size; i &lt; size + enlarge; i++) {\nres[i] = 0;\n}\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre> array.zig<pre><code>// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\nvar res = try mem_allocator.alloc(i32, nums.len + enlarge);\n@memset(res, 0);\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(i32, res, nums);\n// \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\nreturn res;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u6570\u7ec4\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u4e14\u5143\u7d20\u7c7b\u578b\u76f8\u540c\u3002\u8fd9\u79cd\u505a\u6cd5\u5305\u542b\u4e30\u5bcc\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u7cfb\u7edf\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u4fe1\u606f\u6765\u4f18\u5316\u6570\u636e\u7ed3\u6784\u7684\u64cd\u4f5c\u6548\u7387\u3002</p> <ul> <li>\u7a7a\u95f4\u6548\u7387\u9ad8\uff1a\u6570\u7ec4\u4e3a\u6570\u636e\u5206\u914d\u4e86\u8fde\u7eed\u7684\u5185\u5b58\u5757\uff0c\u65e0\u987b\u989d\u5916\u7684\u7ed3\u6784\u5f00\u9500\u3002</li> <li>\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff1a\u6570\u7ec4\u5141\u8bb8\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4efb\u4f55\u5143\u7d20\u3002</li> <li>\u7f13\u5b58\u5c40\u90e8\u6027\uff1a\u5f53\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002</li> </ul> <p>\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u662f\u4e00\u628a\u53cc\u5203\u5251\uff0c\u5176\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u63d2\u5165\u4e0e\u5220\u9664\u6548\u7387\u4f4e\uff1a\u5f53\u6570\u7ec4\u4e2d\u5143\u7d20\u8f83\u591a\u65f6\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20\u3002</li> <li>\u957f\u5ea6\u4e0d\u53ef\u53d8\uff1a\u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u6269\u5bb9\u6570\u7ec4\u9700\u8981\u5c06\u6240\u6709\u6570\u636e\u590d\u5236\u5230\u65b0\u6570\u7ec4\uff0c\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u7a7a\u95f4\u6d6a\u8d39\uff1a\u5982\u679c\u6570\u7ec4\u5206\u914d\u7684\u5927\u5c0f\u8d85\u8fc7\u5b9e\u9645\u6240\u9700\uff0c\u90a3\u4e48\u591a\u4f59\u7684\u7a7a\u95f4\u5c31\u88ab\u6d6a\u8d39\u4e86\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3 \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"<p>\u6570\u7ec4\u662f\u4e00\u79cd\u57fa\u7840\u4e14\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\uff0c\u65e2\u9891\u7e41\u5e94\u7528\u5728\u5404\u7c7b\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u4e8e\u5b9e\u73b0\u5404\u79cd\u590d\u6742\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1a\u5982\u679c\u6211\u4eec\u60f3\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u968f\u673a\u62bd\u6837\u3002</li> <li>\u6392\u5e8f\u548c\u641c\u7d22\uff1a\u6570\u7ec4\u662f\u6392\u5e8f\u548c\u641c\u7d22\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u90fd\u4e3b\u8981\u5728\u6570\u7ec4\u4e0a\u8fdb\u884c\u3002</li> <li>\u67e5\u627e\u8868\uff1a\u5f53\u9700\u8981\u5feb\u901f\u67e5\u627e\u4e00\u4e2a\u5143\u7d20\u6216\u5176\u5bf9\u5e94\u5173\u7cfb\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u4f5c\u4e3a\u67e5\u627e\u8868\u3002\u5047\u5982\u6211\u4eec\u60f3\u5b9e\u73b0\u5b57\u7b26\u5230 ASCII \u7801\u7684\u6620\u5c04\uff0c\u5219\u53ef\u4ee5\u5c06\u5b57\u7b26\u7684 ASCII \u7801\u503c\u4f5c\u4e3a\u7d22\u5f15\uff0c\u5bf9\u5e94\u7684\u5143\u7d20\u5b58\u653e\u5728\u6570\u7ec4\u4e2d\u7684\u5bf9\u5e94\u4f4d\u7f6e\u3002</li> <li>\u673a\u5668\u5b66\u4e60\uff1a\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff1a\u6570\u7ec4\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002\u4f8b\u5982\uff0c\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 \u94fe\u8868","text":"<p>\u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5728\u4e00\u4e2a\u590d\u6742\u7684\u7cfb\u7edf\u8fd0\u884c\u73af\u5883\u4e0b\uff0c\u7a7a\u95f2\u7684\u5185\u5b58\u7a7a\u95f4\u53ef\u80fd\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6570\u7ec4\u975e\u5e38\u5927\u65f6\uff0c\u5185\u5b58\u53ef\u80fd\u65e0\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u6b64\u65f6\u94fe\u8868\u7684\u7075\u6d3b\u6027\u4f18\u52bf\u5c31\u4f53\u73b0\u51fa\u6765\u4e86\u3002</p> <p>\u300c\u94fe\u8868 linked list\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u901a\u8fc7\u201c\u5f15\u7528\u201d\u76f8\u8fde\u63a5\u3002\u5f15\u7528\u8bb0\u5f55\u4e86\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u901a\u8fc7\u5b83\u53ef\u4ee5\u4ece\u5f53\u524d\u8282\u70b9\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</p> <p>\u94fe\u8868\u7684\u8bbe\u8ba1\u4f7f\u5f97\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\uff0c\u5b83\u4eec\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p></p> <p> \u56fe 4-5 \u00a0 \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p> <p>\u89c2\u5bdf\u56fe 4-5 \uff0c\u94fe\u8868\u7684\u7ec4\u6210\u5355\u4f4d\u662f\u300c\u8282\u70b9 node\u300d\u5bf9\u8c61\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u5305\u542b\u4e24\u9879\u6570\u636e\uff1a\u8282\u70b9\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u201c\u5f15\u7528\u201d\u3002</p> <ul> <li>\u94fe\u8868\u7684\u9996\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5934\u8282\u70b9\u201d\uff0c\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5c3e\u8282\u70b9\u201d\u3002</li> <li>\u5c3e\u8282\u70b9\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u522b\u88ab\u8bb0\u4e3a \\(\\text{null}\\)\u3001\\(\\text{nullptr}\\) \u548c \\(\\text{None}\\) \u3002</li> <li>\u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u9488\u7684\u8bed\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u7684\u201c\u5f15\u7528\u201d\u5e94\u88ab\u66ff\u6362\u4e3a\u201c\u6307\u9488\u201d\u3002</li> </ul> <p>\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u94fe\u8868\u8282\u70b9 <code>ListNode</code> \u9664\u4e86\u5305\u542b\u503c\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u4e00\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class ListNode:\n\"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val               # \u8282\u70b9\u503c\nself.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u8282\u70b9\u503c\nListNode *next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\nListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  //\u6784\u9020\u51fd\u6570\nint val = x;         // \u8282\u70b9\u503c\nListNode? next;      // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\nVal  int       // \u8282\u70b9\u503c\nNext *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\nreturn &amp;ListNode{\nVal:  val,\nNext: nil,\n}\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nconstructor(val, next) {\nthis.val = (val === undefined ? 0 : val);       // \u8282\u70b9\u503c\nthis.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nconstructor(val?: number, next?: ListNode | null) {\nthis.val = val === undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\nListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\n#[derive(Debug)]\nstruct ListNode {\nval: i32, // \u8282\u70b9\u503c\nnext: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\nint val;               // \u8282\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\nListNode *node;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode-&gt;val = val;\nnode-&gt;next = NULL;\nreturn node;\n}\n</code></pre> <pre><code>// \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u8282\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\n}\n};\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u94fe\u8868","text":"<p>\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u5173\u7cfb\u3002\u521d\u59cb\u5316\u5b8c\u6210\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u901a\u8fc7\u5f15\u7528\u6307\u5411 <code>next</code> \u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code># \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.zig<pre><code>// \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> <p>\u6570\u7ec4\u6574\u4f53\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u6bd4\u5982\u6570\u7ec4 <code>nums</code> \u5305\u542b\u5143\u7d20 <code>nums[0]</code> \u548c <code>nums[1]</code> \u7b49\uff0c\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u72ec\u7acb\u7684\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\u7684\u3002\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u6bd4\u5982\u4ee5\u4e0a\u4ee3\u7801\u4e2d\u7684\u94fe\u8868\u53ef\u8bb0\u4f5c\u94fe\u8868 <code>n0</code> \u3002</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u8282\u70b9\u975e\u5e38\u5bb9\u6613\u3002\u5982\u56fe 4-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u8282\u70b9 <code>n0</code> \u548c <code>n1</code> \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9 <code>P</code> \uff0c\u5219\u53ea\u9700\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> <p>\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u6570\u7ec4\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u6548\u7387\u8f83\u4f4e\u3002</p> <p></p> <p> \u56fe 4-6 \u00a0 \u94fe\u8868\u63d2\u5165\u8282\u70b9\u793a\u4f8b </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n\"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\nn1 = n0.next\nP.next = n1\nn0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\nListNode *n1 = n0-&gt;next;\nP-&gt;next = n1;\nn0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\nn1 := n0.Next\nP.Next = n1\nn0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\nlet n1 = n0.next\nP.next = n1\nn0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\nconst n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\nListNode? n1 = n0.next;\nP.next = n1;\nn0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\nlet n1 =  n0.borrow_mut().next.take();\nP.borrow_mut().next = n1;\nn0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\nListNode *n1 = n0-&gt;next;\nP-&gt;next = n1;\nn0-&gt;next = P;\n}\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\nvar n1 = n0.?.next;\nP.?.next = n1;\nn0.?.next = P;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5982\u56fe 4-7 \u6240\u793a\uff0c\u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\u8282\u70b9 <code>P</code> \u4ecd\u7136\u6307\u5411 <code>n1</code> \uff0c\u4f46\u5b9e\u9645\u4e0a\u904d\u5386\u6b64\u94fe\u8868\u5df2\u7ecf\u65e0\u6cd5\u8bbf\u95ee\u5230 <code>P</code> \uff0c\u8fd9\u610f\u5473\u7740 <code>P</code> \u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u8be5\u94fe\u8868\u4e86\u3002</p> <p></p> <p> \u56fe 4-7 \u00a0 \u94fe\u8868\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def remove(n0: ListNode):\n\"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\nif not n0.next:\nreturn\n# n0 -&gt; P -&gt; n1\nP = n0.next\nn1 = P.next\nn0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\nif (n0-&gt;next == nullptr)\nreturn;\n// n0 -&gt; P -&gt; n1\nListNode *P = n0-&gt;next;\nListNode *n1 = P-&gt;next;\nn0-&gt;next = n1;\n// \u91ca\u653e\u5185\u5b58\ndelete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -&gt; P -&gt; n1\nListNode P = n0.next;\nListNode n1 = P.next;\nn0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\nif (n0.next == null)\nreturn;\n// n0 -&gt; P -&gt; n1\nListNode P = n0.next;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\nif n0.Next == nil {\nreturn\n}\n// n0 -&gt; P -&gt; n1\nP := n0.Next\nn1 := P.Next\nn0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\nif n0.next == nil {\nreturn\n}\n// n0 -&gt; P -&gt; n1\nlet P = n0.next\nlet n1 = P?.next\nn0.next = n1\nP?.next = nil\n}\n</code></pre> linked_list.js<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\nif (!n0.next) return;\n// n0 -&gt; P -&gt; n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\nif (!n0.next) {\nreturn;\n}\n// n0 -&gt; P -&gt; n1\nconst P = n0.next;\nconst n1 = P.next;\nn0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\nif (n0.next == null) return;\n// n0 -&gt; P -&gt; n1\nListNode P = n0.next!;\nListNode? n1 = P.next;\nn0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\nif n0.borrow().next.is_none() {return};\n// n0 -&gt; P -&gt; n1\nlet P = n0.borrow_mut().next.take();\nif let Some(node) = P {\nlet n1 = node.borrow_mut().next.take();\nn0.borrow_mut().next = n1;\n}\n}\n</code></pre> linked_list.c<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\nif (!n0-&gt;next)\nreturn;\n// n0 -&gt; P -&gt; n1\nListNode *P = n0-&gt;next;\nListNode *n1 = P-&gt;next;\nn0-&gt;next = n1;\n// \u91ca\u653e\u5185\u5b58\nfree(P);\n}\n</code></pre> linked_list.zig<pre><code>// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\nif (n0.?.next == null) return;\n// n0 -&gt; P -&gt; n1\nvar P = n0.?.next;\nvar n1 = P.?.next;\nn0.?.next = n1;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8bbf\u95ee\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u8bbf\u95ee\u8282\u70b9\u7684\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u4e00\u8282\u6240\u8ff0\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u7a0b\u5e8f\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8bbf\u95ee\u94fe\u8868\u7684\u7b2c \\(i\\) \u4e2a\u8282\u70b9\u9700\u8981\u5faa\u73af \\(i - 1\\) \u8f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n\"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\nfor _ in range(index):\nif not head:\nreturn None\nhead = head.next\nreturn head\n</code></pre> linked_list.cpp<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\nfor (int i = 0; i &lt; index; i++) {\nif (head == nullptr)\nreturn nullptr;\nhead = head-&gt;next;\n}\nreturn head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\nfor (int i = 0; i &lt; index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\nfor (int i = 0; i &lt; index; i++) {\nif (head == null)\nreturn null;\nhead = head.next;\n}\nreturn head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\nfor i := 0; i &lt; index; i++ {\nif head == nil {\nreturn nil\n}\nhead = head.Next\n}\nreturn head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\nvar head: ListNode? = head\nfor _ in 0 ..&lt; index {\nif head == nil {\nreturn nil\n}\nhead = head?.next\n}\nreturn head\n}\n</code></pre> linked_list.js<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\nfor (let i = 0; i &lt; index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\nfor (let i = 0; i &lt; index; i++) {\nif (!head) {\nreturn null;\n}\nhead = head.next;\n}\nreturn head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\nfor (var i = 0; i &lt; index; i++) {\nif (head == null) return null;\nhead = head.next;\n}\nreturn head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\nif index &lt;= 0 {return head};\nif let Some(node) = &amp;head.borrow_mut().next {\nreturn access(node.clone(), index - 1);\n}\nreturn head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\nwhile (head &amp;&amp; head-&gt;next &amp;&amp; index) {\nhead = head-&gt;next;\nindex--;\n}\nreturn head;\n}\n</code></pre> linked_list.zig<pre><code>// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\nvar head = node;\nvar i: i32 = 0;\nwhile (i &lt; index) : (i += 1) {\nhead = head.?.next;\nif (head == null) return null;\n}\nreturn head;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u5176\u4e2d\u503c\u4e3a <code>target</code> \u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8be5\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u8fc7\u7a0b\u4e5f\u5c5e\u4e8e\u7ebf\u6027\u67e5\u627e\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n\"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\nindex = 0\nwhile head:\nif head.val == target:\nreturn index\nhead = head.next\nindex += 1\nreturn -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\nint index = 0;\nwhile (head != nullptr) {\nif (head-&gt;val == target)\nreturn index;\nhead = head-&gt;next;\nindex++;\n}\nreturn -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target)\nreturn index;\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\nindex := 0\nfor head != nil {\nif head.Val == target {\nreturn index\n}\nhead = head.Next\nindex++\n}\nreturn -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\nvar head: ListNode? = head\nvar index = 0\nwhile head != nil {\nif head?.val == target {\nreturn index\n}\nhead = head?.next\nindex += 1\n}\nreturn -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\nlet index = 0;\nwhile (head !== null) {\nif (head.val === target) {\nreturn index;\n}\nhead = head.next;\nindex += 1;\n}\nreturn -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\nint index = 0;\nwhile (head != null) {\nif (head.val == target) {\nreturn index;\n}\nhead = head.next;\nindex++;\n}\nreturn -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\nif head.borrow().val == target {return index};\nif let Some(node) = &amp;head.borrow_mut().next {\nreturn find(node.clone(), target, index + 1);\n}\nreturn -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\nint index = 0;\nwhile (head) {\nif (head-&gt;val == target)\nreturn index;\nhead = head-&gt;next;\nindex++;\n}\nreturn -1;\n}\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\nvar head = node;\nvar index: i32 = 0;\nwhile (head != null) {\nif (head.?.val == target) return index;\nhead = head.?.next;\nindex += 1;\n}\nreturn -1;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u6570\u7ec4 vs. \u94fe\u8868","text":"<p>\u8868 4-1 \u603b\u7ed3\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u7684\u5404\u9879\u7279\u70b9\u5e76\u5bf9\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u4e8e\u5b83\u4eec\u91c7\u7528\u4e24\u79cd\u76f8\u53cd\u7684\u5b58\u50a8\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u79cd\u6027\u8d28\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73b0\u5bf9\u7acb\u7684\u7279\u70b9\u3002</p> <p> \u8868 4-1 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u5206\u6563\u5185\u5b58\u7a7a\u95f4 \u5bb9\u91cf\u6269\u5c55 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u53ef\u7075\u6d3b\u6269\u5c55 \u5185\u5b58\u6548\u7387 \u5143\u7d20\u5360\u7528\u5185\u5b58\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8d39\u7a7a\u95f4 \u5143\u7d20\u5360\u7528\u5185\u5b58\u591a \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"<p>\u5982\u56fe 4-8 \u6240\u793a\uff0c\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u4e09\u79cd\u3002</p> <ul> <li>\u5355\u5411\u94fe\u8868\uff1a\u5373\u524d\u9762\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a \\(\\text{None}\\) \u3002</li> <li>\u73af\u5f62\u94fe\u8868\uff1a\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002</li> <li>\u53cc\u5411\u94fe\u8868\uff1a\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u5f15\u7528\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u4e2a\u8282\u70b9\uff09\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class ListNode:\n\"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                # \u8282\u70b9\u503c\nself.next: ListNode | None = None  # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\nself.prev: ListNode | None = None  # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\nint val;         // \u8282\u70b9\u503c\nListNode *next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\nListNode *prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\nListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\nListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  // \u6784\u9020\u51fd\u6570\nint val = x;    // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\nVal  int             // \u8282\u70b9\u503c\nNext *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\nPrev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\nreturn &amp;DoublyListNode{\nVal:  val,\nNext: nil,\nPrev: nil,\n}\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\nvar prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\ninit(x: Int) { // \u6784\u9020\u51fd\u6570\nval = x\n}\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nconstructor(val, next, prev) {\nthis.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nval: number;\nnext: ListNode | null;\nprev: ListNode | null;\nconstructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\nthis.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\nthis.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\nthis.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\nint val;        // \u8282\u70b9\u503c\nListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\nListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\nListNode(this.val, [this.next, this.prev]);  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\u578b */\n#[derive(Debug)]\nstruct ListNode {\nval: i32, // \u8282\u70b9\u503c\nnext: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\nprev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n/* \u6784\u9020\u51fd\u6570 */\nimpl ListNode {\nfn new(val: i32) -&gt; Self {\nListNode {\nval,\nnext: None,\nprev: None,\n}\n}\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\nint val;               // \u8282\u70b9\u503c\nstruct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\nstruct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\nListNode *node;\nnode = (ListNode *) malloc(sizeof(ListNode));\nnode-&gt;val = val;\nnode-&gt;next = NULL;\nnode-&gt;prev = NULL;\nreturn node;\n}\n</code></pre> <pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = 0, // \u8282\u70b9\u503c\nnext: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\nprev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n</code></pre> <p></p> <p> \u56fe 4-8 \u00a0 \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u94fe\u8868\u5178\u578b\u5e94\u7528","text":"<p>\u5355\u5411\u94fe\u8868\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u548c\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u6808\u4e0e\u961f\u5217\uff1a\u5f53\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u90fd\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\u65f6\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u540e\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u6808\uff1b\u5f53\u63d2\u5165\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\uff0c\u5220\u9664\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u53e6\u4e00\u7aef\u8fdb\u884c\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u5148\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u961f\u5217\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u94fe\u5f0f\u5730\u5740\u662f\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u7684\u4e3b\u6d41\u65b9\u6848\u4e4b\u4e00\uff0c\u5728\u8be5\u65b9\u6848\u4e2d\uff0c\u6240\u6709\u51b2\u7a81\u7684\u5143\u7d20\u90fd\u4f1a\u88ab\u653e\u5230\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002</li> <li>\u56fe\uff1a\u90bb\u63a5\u8868\u662f\u8868\u793a\u56fe\u7684\u4e00\u79cd\u5e38\u7528\u65b9\u5f0f\uff0c\u5176\u4e2d\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u4e0e\u4e00\u4e2a\u94fe\u8868\u76f8\u5173\u8054\uff0c\u94fe\u8868\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u4ee3\u8868\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u5176\u4ed6\u9876\u70b9\u3002</li> </ul> <p>\u53cc\u5411\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5feb\u901f\u67e5\u627e\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u5143\u7d20\u7684\u573a\u666f\u3002</p> <ul> <li>\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\uff1a\u6bd4\u5982\u5728\u7ea2\u9ed1\u6811\u3001B \u6811\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u8bbf\u95ee\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5728\u8282\u70b9\u4e2d\u4fdd\u5b58\u4e00\u4e2a\u6307\u5411\u7236\u8282\u70b9\u7684\u5f15\u7528\u6765\u5b9e\u73b0\uff0c\u7c7b\u4f3c\u4e8e\u53cc\u5411\u94fe\u8868\u3002</li> <li>\u6d4f\u89c8\u5668\u5386\u53f2\uff1a\u5728\u7f51\u9875\u6d4f\u89c8\u5668\u4e2d\uff0c\u5f53\u7528\u6237\u70b9\u51fb\u524d\u8fdb\u6216\u540e\u9000\u6309\u94ae\u65f6\uff0c\u6d4f\u89c8\u5668\u9700\u8981\u77e5\u9053\u7528\u6237\u8bbf\u95ee\u8fc7\u7684\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u7f51\u9875\u3002\u53cc\u5411\u94fe\u8868\u7684\u7279\u6027\u4f7f\u5f97\u8fd9\u79cd\u64cd\u4f5c\u53d8\u5f97\u7b80\u5355\u3002</li> <li>LRU \u7b97\u6cd5\uff1a\u5728\u7f13\u5b58\u6dd8\u6c70\uff08LRU\uff09\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u3002\u8fd9\u65f6\u5019\u4f7f\u7528\u53cc\u5411\u94fe\u8868\u5c31\u975e\u5e38\u5408\u9002\u3002</li> </ul> <p>\u73af\u5f62\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5468\u671f\u6027\u64cd\u4f5c\u7684\u573a\u666f\uff0c\u6bd4\u5982\u64cd\u4f5c\u7cfb\u7edf\u7684\u8d44\u6e90\u8c03\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\uff1a\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\u662f\u4e00\u79cd\u5e38\u89c1\u7684 CPU \u8c03\u5ea6\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5bf9\u4e00\u7ec4\u8fdb\u7a0b\u8fdb\u884c\u5faa\u73af\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u88ab\u8d4b\u4e88\u4e00\u4e2a\u65f6\u95f4\u7247\uff0c\u5f53\u65f6\u95f4\u7247\u7528\u5b8c\u65f6\uff0cCPU \u5c06\u5207\u6362\u5230\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u3002\u8fd9\u79cd\u5faa\u73af\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u73af\u5f62\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u6570\u636e\u7f13\u51b2\u533a\uff1a\u5728\u67d0\u4e9b\u6570\u636e\u7f13\u51b2\u533a\u7684\u5b9e\u73b0\u4e2d\uff0c\u4e5f\u53ef\u80fd\u4f1a\u4f7f\u7528\u73af\u5f62\u94fe\u8868\u3002\u6bd4\u5982\u5728\u97f3\u9891\u3001\u89c6\u9891\u64ad\u653e\u5668\u4e2d\uff0c\u6570\u636e\u6d41\u53ef\u80fd\u4f1a\u88ab\u5206\u6210\u591a\u4e2a\u7f13\u51b2\u5757\u5e76\u653e\u5165\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\uff0c\u4ee5\u4fbf\u5b9e\u73b0\u65e0\u7f1d\u64ad\u653e\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 \u5217\u8868","text":"<p>\u300c\u5217\u8868 list\u300d\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8bbf\u95ee\u3001\u4fee\u6539\u3001\u6dfb\u52a0\u3001\u5220\u9664\u548c\u904d\u5386\u7b49\u64cd\u4f5c\uff0c\u65e0\u987b\u4f7f\u7528\u8005\u8003\u8651\u5bb9\u91cf\u9650\u5236\u7684\u95ee\u9898\u3002\u5217\u8868\u53ef\u4ee5\u57fa\u4e8e\u94fe\u8868\u6216\u6570\u7ec4\u5b9e\u73b0\u3002</p> <ul> <li>\u94fe\u8868\u5929\u7136\u53ef\u4ee5\u88ab\u770b\u4f5c\u4e00\u4e2a\u5217\u8868\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\u64cd\u4f5c\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u52a8\u6001\u6269\u5bb9\u3002</li> <li>\u6570\u7ec4\u4e5f\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\uff0c\u4f46\u7531\u4e8e\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u53ea\u80fd\u88ab\u770b\u4f5c\u4e00\u4e2a\u5177\u6709\u957f\u5ea6\u9650\u5236\u7684\u5217\u8868\u3002</li> </ul> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u5217\u8868\u65f6\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6027\u8d28\u4f1a\u5bfc\u81f4\u5217\u8868\u7684\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u65e0\u6cd5\u4e8b\u5148\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u4ece\u800c\u96be\u4ee5\u9009\u62e9\u5408\u9002\u7684\u5217\u8868\u957f\u5ea6\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u5219\u5f88\u53ef\u80fd\u65e0\u6cd5\u6ee1\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002</p> <p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u300c\u52a8\u6001\u6570\u7ec4 dynamic array\u300d\u6765\u5b9e\u73b0\u5217\u8868\u3002\u5b83\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u5404\u9879\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u52a8\u6001\u6269\u5bb9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u5217\u8868\u90fd\u662f\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684 <code>list</code> \u3001Java \u4e2d\u7684 <code>ArrayList</code> \u3001C++ \u4e2d\u7684 <code>vector</code> \u548c C# \u4e2d\u7684 <code>List</code> \u7b49\u3002\u5728\u63a5\u4e0b\u6765\u7684\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u5c06\u628a\u201c\u5217\u8868\u201d\u548c\u201c\u52a8\u6001\u6570\u7ec4\u201d\u89c6\u4e3a\u7b49\u540c\u7684\u6982\u5ff5\u3002</p>"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u5217\u8868","text":"<p>\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u8fd9\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u65e0\u521d\u59cb\u503c\nvector&lt;int&gt; nums1;\n// \u6709\u521d\u59cb\u503c\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u521d\u59cb\u5316\u5217\u8868\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u5217\u8868\u672c\u8d28\u4e0a\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u8bbf\u95ee\u5143\u7d20\nnum: int = nums[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cpp<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.java<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list_test.go<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nnum := nums[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.swift<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num = nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.js<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.ts<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.dart<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.rs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num: i32 = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;             // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u8bbf\u95ee\u5143\u7d20\nvar num = nums.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0  \n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20","text":"<p>\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u6e05\u7a7a\u5217\u8868\nnums.clear()\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n# \u5220\u9664\u5143\u7d20\nnums.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cpp<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.java<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.Clear();\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);\n/* \u5220\u9664\u5143\u7d20 */\nnums.RemoveAt(3);\n</code></pre> list_test.go<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums = nil\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.swift<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.removeAll()\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.js<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.ts<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.dart<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.rs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);    // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u6e05\u7a7a\u5217\u8868\nnums.clearRetainingCapacity();\n// \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n// \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n// \u5220\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u904d\u5386\u5217\u8868","text":"<p>\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(nums)):\ncount += nums[i]\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\nfor num in nums:\ncount += num\n</code></pre> list.cpp<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\ncount += nums[i];\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\ncount += num;\n}\n</code></pre> list.java<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\ncount += nums.get(i);\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (int num : nums) {\ncount += num;\n}\n</code></pre> list.cs<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\ncount += nums[i];\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\ncount += num;\n}\n</code></pre> list_test.go<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\ncount += nums[i]\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\ncount += num\n}\n</code></pre> list.swift<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor i in nums.indices {\ncount += nums[i]\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor num in nums {\ncount += num\n}\n</code></pre> list.js<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\ncount += nums[i];\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\ncount += num;\n}\n</code></pre> list.ts<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\ncount += nums[i];\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\ncount += num;\n}\n</code></pre> list.dart<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\ncount += nums[i];\n}\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\ncount += num;\n}\n</code></pre> list.rs<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nlet mut _count = 0;\nfor i in 0..nums.len() {\n_count += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\n_count = 0;\nfor num in &amp;nums {\n_count += num;\n}\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\ncount += nums[i];\n}\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\ncount += num;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u5217\u8868","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868 <code>nums1</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cpp<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list_test.go<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.swift<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.js<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.ts<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.dart<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.rs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u5217\u8868","text":"<p>\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u548c\u201c\u53cc\u6307\u9488\u201d\u7b97\u6cd5\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u6392\u5e8f\u5217\u8868\nnums.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cpp<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort(nums.begin(), nums.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.java<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(nums);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list_test.go<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(nums)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.swift<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.js<pre><code>/* \u6392\u5e8f\u5217\u8868 */  nums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.ts<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.dart<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u5217\u8868\u5b9e\u73b0","text":"<p>\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u4e86\u5217\u8868\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u4eec\u7684\u5b9e\u73b0\u6bd4\u8f83\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002</p> <p>\u4e3a\u4e86\u52a0\u6df1\u5bf9\u5217\u8868\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5c1d\u8bd5\u5b9e\u73b0\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u4e2a\u91cd\u70b9\u8bbe\u8ba1\u3002</p> <ul> <li>\u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002</li> <li>\u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf <code>size</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002</li> <li>\u6269\u5bb9\u673a\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u65f6\u5217\u8868\u5bb9\u91cf\u5df2\u6ee1\uff0c\u5219\u9700\u8981\u8fdb\u884c\u6269\u5bb9\u3002\u5148\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u518d\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_list.py<pre><code>class MyList:\n\"\"\"\u5217\u8868\u7c7b\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\nself._arr: list[int] = [0] * self._capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nself._size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nself._extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\nreturn self._size\ndef capacity(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\nreturn self._capacity\ndef get(self, index: int) -&gt; int:\n\"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n# \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index &lt; 0 or index &gt;= self._size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nreturn self._arr[index]\ndef set(self, num: int, index: int):\n\"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\nif index &lt; 0 or index &gt;= self._size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nself._arr[index] = num\ndef add(self, num: int):\n\"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.size() == self.capacity():\nself.extend_capacity()\nself._arr[self._size] = num\nself._size += 1\ndef insert(self, num: int, index: int):\n\"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\nif index &lt; 0 or index &gt;= self._size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n# \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self._size == self.capacity():\nself.extend_capacity()\n# \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in range(self._size - 1, index - 1, -1):\nself._arr[j + 1] = self._arr[j]\nself._arr[index] = num\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself._size += 1\ndef remove(self, index: int) -&gt; int:\n\"\"\"\u5220\u9664\u5143\u7d20\"\"\"\nif index &lt; 0 or index &gt;= self._size:\nraise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\nnum = self._arr[index]\n# \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in range(index, self._size - 1):\nself._arr[j] = self._arr[j + 1]\n# \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself._size -= 1\n# \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\ndef extend_capacity(self):\n\"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n# \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 __extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nself._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n# \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself._capacity = len(self._arr)\ndef to_array(self) -&gt; list[int]:\n\"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\nreturn self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\nprivate:\nint *arr;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\nint arrSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\narr = new int[arrCapacity];\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~MyList() {\ndelete[] arr;\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() {\nreturn arrSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() {\nreturn arrCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index &lt; 0 || index &gt;= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int num) {\nif (index &lt; 0 || index &gt;= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\narr[index] = num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\narr[size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\narrSize++;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int num) {\nif (index &lt; 0 || index &gt;= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size() == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size() - 1; j &gt;= index; j--) {\narr[j + 1] = arr[j];\n}\narr[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\narrSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index &lt; 0 || index &gt;= size())\nthrow out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = arr[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j &lt; size() - 1; j++) {\narr[j] = arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\narrSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\nint newCapacity = capacity() * extendRatio;\nint *tmp = arr;\narr = new int[newCapacity];\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nfor (int i = 0; i &lt; size(); i++) {\narr[i] = tmp[i];\n}\n// \u91ca\u653e\u5185\u5b58\ndelete[] tmp;\narrCapacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\nvector&lt;int&gt; toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector&lt;int&gt; vec(size());\nfor (int i = 0; i &lt; size(); i++) {\nvec[i] = arr[i];\n}\nreturn vec;\n}\n};\n</code></pre> my_list.java<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\nprivate int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int capacity = 10; // \u5217\u8868\u5bb9\u91cf\nprivate int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\narr = new int[capacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\npublic int size() {\nreturn size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int capacity() {\nreturn capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index &lt; 0 || index &gt;= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void set(int index, int num) {\nif (index &lt; 0 || index &gt;= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\narr[index] = num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\narr[size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void insert(int index, int num) {\nif (index &lt; 0 || index &gt;= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size == capacity())\nextendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = size - 1; j &gt;= index; j--) {\narr[j + 1] = arr[j];\n}\narr[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int remove(int index) {\nif (index &lt; 0 || index &gt;= size)\nthrow new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = arr[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j &lt; size - 1; j++) {\narr[j] = arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nsize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\narr = Arrays.copyOf(arr, capacity() * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\ncapacity = arr.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] toArray() {\nint size = size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] arr = new int[size];\nfor (int i = 0; i &lt; size; i++) {\narr[i] = get(i);\n}\nreturn arr;\n}\n}\n</code></pre> my_list.cs<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\nprivate int[] arr;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate int arrCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\nprivate int arrSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate readonly int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\npublic MyList() {\narr = new int[arrCapacity];\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic int Size() {\nreturn arrSize;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic int Capacity() {\nreturn arrCapacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic int Get(int index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index &lt; 0 || index &gt;= arrSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic void Set(int index, int num) {\nif (index &lt; 0 || index &gt;= arrSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\narr[index] = num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic void Add(int num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (arrSize == arrCapacity)\nExtendCapacity();\narr[arrSize] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\narrSize++;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic void Insert(int index, int num) {\nif (index &lt; 0 || index &gt;= arrSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (arrSize == arrCapacity)\nExtendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (int j = arrSize - 1; j &gt;= index; j--) {\narr[j + 1] = arr[j];\n}\narr[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\narrSize++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic int Remove(int index) {\nif (index &lt; 0 || index &gt;= arrSize)\nthrow new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\nint num = arr[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (int j = index; j &lt; arrSize - 1; j++) {\narr[j] = arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\narrSize--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic void ExtendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nArray.Resize(ref arr, arrCapacity * extendRatio);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\narrCapacity = arr.Length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic int[] ToArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] arr = new int[arrSize];\nfor (int i = 0; i &lt; arrSize; i++) {\narr[i] = Get(i);\n}\nreturn arr;\n}\n}\n</code></pre> my_list.go<pre><code>/* \u5217\u8868\u7c7b */\ntype myList struct {\narrCapacity int\narr         []int\narrSize     int\nextendRatio int\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\nreturn &amp;myList{\narrCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\narr:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\narrSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio: 2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\nreturn l.arrSize\n}\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\nreturn l.arrCapacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index &lt; 0 || index &gt;= l.arrSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn l.arr[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\nif index &lt; 0 || index &gt;= l.arrSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nl.arr[index] = num\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.arrSize == l.arrCapacity {\nl.extendCapacity()\n}\nl.arr[l.arrSize] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.arrSize++\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\nif index &lt; 0 || index &gt;= l.arrSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif l.arrSize == l.arrCapacity {\nl.extendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j := l.arrSize - 1; j &gt;= index; j-- {\nl.arr[j+1] = l.arr[j]\n}\nl.arr[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.arrSize++\n}\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\nif index &lt; 0 || index &gt;= l.arrSize {\npanic(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nnum := l.arr[index]\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j := index; j &lt; l.arrSize-1; j++ {\nl.arr[j] = l.arr[j+1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nl.arrSize--\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nl.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nl.arrCapacity = len(l.arr)\n}\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nreturn l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\nprivate var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\nprivate var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\narr = Array(repeating: 0, count: _capacity)\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nfunc size() -&gt; Int {\n_size\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc capacity() -&gt; Int {\n_capacity\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc get(index: Int) -&gt; Int {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\nif index &lt; 0 || index &gt;= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nreturn arr[index]\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc set(index: Int, num: Int) {\nif index &lt; 0 || index &gt;= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\narr[index] = num\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc add(num: Int) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\narr[_size] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc insert(index: Int, num: Int) {\nif index &lt; 0 || index &gt;= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif _size == _capacity {\nextendCapacity()\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in sequence(first: _size - 1, next: { $0 &gt;= index + 1 ? $0 - 1 : nil }) {\narr[j + 1] = arr[j]\n}\narr[index] = num\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size += 1\n}\n/* \u5220\u9664\u5143\u7d20 */\n@discardableResult\nfunc remove(index: Int) -&gt; Int {\nif index &lt; 0 || index &gt;= _size {\nfatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n}\nlet num = arr[index]\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in index ..&lt; (_size - 1) {\narr[j] = arr[j + 1]\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size -= 1\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num\n}\n/* \u5217\u8868\u6269\u5bb9 */\nfunc extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\narr = arr + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = arr.count\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nfunc toArray() -&gt; [Int] {\nvar arr = Array(repeating: 0, count: _size)\nfor i in 0 ..&lt; _size {\narr[i] = get(index: i)\n}\nreturn arr\n}\n}\n</code></pre> my_list.js<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n#arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n#capacity = 10; // \u5217\u8868\u5bb9\u91cf\n#size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n#extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#arr = new Array(this.#capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nsize() {\nreturn this.#size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\ncapacity() {\nreturn this.#capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nget(index) {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.#arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nset(index, num) {\nif (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.#arr[index] = num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nadd(num) {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.#arr[this.#size] = num;\nthis.#size++;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\ninsert(index, num) {\nif (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this.#size === this.#capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this.#size - 1; j &gt;= index; j--) {\nthis.#arr[j + 1] = this.#arr[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#arr[index] = num;\nthis.#size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nremove(index) {\nif (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.#arr[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j &lt; this.#size - 1; j++) {\nthis.#arr[j] = this.#arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.#size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nextendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.#arr = this.#arr.concat(\nnew Array(this.capacity() * (this.#extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis.#capacity = this.#arr.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\ntoArray() {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(size);\nfor (let i = 0; i &lt; size; i++) {\narr[i] = this.get(i);\n}\nreturn arr;\n}\n}\n</code></pre> my_list.ts<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\nprivate arr: Array&lt;number&gt;; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nprivate _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\nprivate _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nprivate extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.arr = new Array(this._capacity);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npublic size(): number {\nreturn this._size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npublic capacity(): number {\nreturn this._capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npublic get(index: number): number {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nreturn this.arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npublic set(index: number, num: number): void {\nif (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nthis.arr[index] = num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npublic add(num: number): void {\n// \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\nif (this._size === this._capacity) this.extendCapacity();\n// \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\nthis.arr[this._size] = num;\nthis._size++;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npublic insert(index: number, num: number): void {\nif (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (this._size === this._capacity) {\nthis.extendCapacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (let j = this._size - 1; j &gt;= index; j--) {\nthis.arr[j + 1] = this.arr[j];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis.arr[index] = num;\nthis._size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\npublic remove(index: number): number {\nif (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\nlet num = this.arr[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (let j = index; j &lt; this._size - 1; j++) {\nthis.arr[j] = this.arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nthis._size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npublic extendCapacity(): void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nthis.arr = this.arr.concat(\nnew Array(this.capacity() * (this.extendRatio - 1))\n);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nthis._capacity = this.arr.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npublic toArray(): number[] {\nlet size = this.size();\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(size);\nfor (let i = 0; i &lt; size; i++) {\narr[i] = this.get(i);\n}\nreturn arr;\n}\n}\n</code></pre> my_list.dart<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\nlate List&lt;int&gt; _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint _capacity = 10; // \u5217\u8868\u5bb9\u91cf\nint _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nint _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n/* \u6784\u9020\u65b9\u6cd5 */\nMyList() {\n_arr = List.filled(_capacity, 0);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\nint size() =&gt; _size;\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity() =&gt; _capacity;\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(int index) {\nif (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\nreturn _arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(int index, int _num) {\nif (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n_arr[index] = _num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(int _num) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (_size == _capacity) extendCapacity();\n_arr[_size] = _num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size++;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(int index, int _num) {\nif (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (_size == _capacity) extendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor (var j = _size - 1; j &gt;= index; j--) {\n_arr[j + 1] = _arr[j];\n}\n_arr[index] = _num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\nint remove(int index) {\nif (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\nint _num = _arr[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor (var j = index; j &lt; _size - 1; j++) {\n_arr[j] = _arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n_size--;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn _num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity() {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\nfinal _newNums = List.filled(_capacity * _extendRatio, 0);\n// \u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nList.copyRange(_newNums, 0, _arr);\n// \u66f4\u65b0 _arr \u7684\u5f15\u7528\n_arr = _newNums;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n_capacity = _arr.length;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\nList&lt;int&gt; toArray() {\nList&lt;int&gt; arr = [];\nfor (var i = 0; i &lt; _size; i++) {\narr.add(get(i));\n}\nreturn arr;\n}\n}\n</code></pre> my_list.rs<pre><code>/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\narr: Vec&lt;i32&gt;,       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\ncapacity: usize,      // \u5217\u8868\u5bb9\u91cf\nsize: usize,          // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextend_ratio: usize,  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n#[allow(unused,unused_comparisons)]\nimpl MyList {\n/* \u6784\u9020\u65b9\u6cd5 */\npub fn new(capacity: usize) -&gt; Self {\nlet mut vec = Vec::new(); vec.resize(capacity, 0);\nSelf {\narr: vec,\ncapacity,\nsize: 0,\nextend_ratio: 2,\n}\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\npub fn size(&amp;self) -&gt; usize {\nreturn self.size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\npub fn capacity(&amp;self) -&gt; usize {\nreturn self.capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\npub fn get(&amp;self, index: usize) -&gt; i32 {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif index &gt;= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\nreturn self.arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\npub fn set(&amp;mut self, index: usize, num: i32) {\nif index &gt;= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\nself.arr[index] = num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\npub fn add(&amp;mut self, num: i32) {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.size == self.capacity() {\nself.extend_capacity();\n}\nself.arr[self.size] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.size += 1;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\npub fn insert(&amp;mut self, index: usize, num: i32) {\nif index &gt;= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif self.size == self.capacity() {\nself.extend_capacity();\n}\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfor j in (index..self.size).rev() {\nself.arr[j + 1] = self.arr[j];\n}\nself.arr[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.size += 1;\n}\n/* \u5220\u9664\u5143\u7d20 */\npub fn remove(&amp;mut self, index: usize) -&gt; i32 {\nif index &gt;= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\nlet num = self.arr[index];\n// \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nfor j in (index..self.size - 1) {\nself.arr[j] = self.arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.size -= 1;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\npub fn extend_capacity(&amp;mut self) {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nlet new_capacity = self.capacity * self.extend_ratio;\nself.arr.resize(new_capacity, 0);\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.capacity = new_capacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\npub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nlet mut arr = Vec::new();\nfor i in 0..self.size {\narr.push(self.get(i));\n}\narr\n}\n}\n</code></pre> my_list.c<pre><code>/* \u5217\u8868\u7c7b */\ntypedef struct {\nint *arr;        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\nint capacity;    // \u5217\u8868\u5bb9\u91cf\nint size;        // \u5217\u8868\u5927\u5c0f\nint extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\nMyList *nums = malloc(sizeof(MyList));\nnums-&gt;capacity = 10;\nnums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\nnums-&gt;size = 0;\nnums-&gt;extendRatio = 2;\nreturn nums;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\nfree(nums-&gt;arr);\nfree(nums);\n}\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\nreturn nums-&gt;size;\n}\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\nreturn nums-&gt;capacity;\n}\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\nassert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\nreturn nums-&gt;arr[index];\n}\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\nassert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\nnums-&gt;arr[index] = num;\n}\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\nif (size(nums) == capacity(nums)) {\nextendCapacity(nums); // \u6269\u5bb9\n}\nnums-&gt;arr[size(nums)] = num;\nnums-&gt;size++;\n}\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\nassert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (size(nums) == capacity(nums)) {\nextendCapacity(nums); // \u6269\u5bb9\n}\nfor (int i = size(nums); i &gt; index; --i) {\nnums-&gt;arr[i] = nums-&gt;arr[i - 1];\n}\nnums-&gt;arr[index] = num;\nnums-&gt;size++;\n}\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\nassert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\nint num = nums-&gt;arr[index];\nfor (int i = index; i &lt; size(nums) - 1; i++) {\nnums-&gt;arr[i] = nums-&gt;arr[i + 1];\n}\nnums-&gt;size--;\nreturn num;\n}\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n// \u5148\u5206\u914d\u7a7a\u95f4\nint newCapacity = capacity(nums) * nums-&gt;extendRatio;\nint *extend = (int *)malloc(sizeof(int) * newCapacity);\nint *temp = nums-&gt;arr;\n// \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\nfor (int i = 0; i &lt; size(nums); i++)\nextend[i] = nums-&gt;arr[i];\n// \u91ca\u653e\u65e7\u6570\u636e\nfree(temp);\n// \u66f4\u65b0\u65b0\u6570\u636e\nnums-&gt;arr = extend;\nnums-&gt;capacity = newCapacity;\n}\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\nreturn nums-&gt;arr;\n}\n</code></pre> my_list.zig<pre><code>// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\nreturn struct {\nconst Self = @This();\narr: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\narrCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\nnumSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\nextendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n@memset(self.arr, @as(T, 0));\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\npub fn size(self: *Self) usize {\nreturn self.numSize;\n}\n// \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.arrCapacity;\n}\n// \u8bbf\u95ee\u5143\u7d20\npub fn get(self: *Self, index: usize) T {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nreturn self.arr[index];\n}  // \u66f4\u65b0\u5143\u7d20\npub fn set(self: *Self, index: usize, num: T) void {\n// \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\nif (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nself.arr[index] = num;\n}  // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\npub fn add(self: *Self, num: T) !void {\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\nself.arr[self.size()] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize += 1;\n}  // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\npub fn insert(self: *Self, index: usize, num: T) !void {\nif (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n// \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\nif (self.size() == self.capacity()) try self.extendCapacity();\n// \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nvar j = self.size() - 1;\nwhile (j &gt;= index) : (j -= 1) {\nself.arr[j + 1] = self.arr[j];\n}\nself.arr[index] = num;\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize += 1;\n}\n// \u5220\u9664\u5143\u7d20\npub fn remove(self: *Self, index: usize) T {\nif (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\nvar num = self.arr[index];\n// \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\nvar j = index;\nwhile (j &lt; self.size() - 1) : (j += 1) {\nself.arr[j] = self.arr[j + 1];\n}\n// \u66f4\u65b0\u5143\u7d20\u6570\u91cf\nself.numSize -= 1;\n// \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\nreturn num;\n}\n// \u5217\u8868\u6269\u5bb9\npub fn extendCapacity(self: *Self) !void {\n// \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\nvar newCapacity = self.capacity() * self.extendRatio;\nvar extend = try self.mem_allocator.alloc(T, newCapacity);\n@memset(extend, @as(T, 0));\n// \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\nstd.mem.copy(T, extend, self.arr);\nself.arr = extend;\n// \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\nself.arrCapacity = newCapacity;\n}\n// \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar arr = try self.mem_allocator.alloc(T, self.size());\n@memset(arr, @as(T, 0));\nfor (arr, 0..) |*num, i| {\nnum.* = self.get(i);\n}\nreturn arr;\n}\n};\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *","text":"<p>\u5728\u672c\u7ae0\u7684\u524d\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u8fd9\u4e24\u79cd\u57fa\u7840\u4e14\u91cd\u8981\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u4eec\u5206\u522b\u4ee3\u8868\u4e86\u201c\u8fde\u7eed\u5b58\u50a8\u201d\u548c\u201c\u5206\u6563\u5b58\u50a8\u201d\u8fd9\u4e24\u79cd\u4e0d\u540c\u7684\u7269\u7406\u7ed3\u6784\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u7269\u7406\u7ed3\u6784\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u51b3\u5b9a\u4e86\u7a0b\u5e8f\u5bf9\u5185\u5b58\u548c\u7f13\u5b58\u7684\u4f7f\u7528\u6548\u7387\uff0c\u8fdb\u800c\u5f71\u54cd\u7b97\u6cd5\u7a0b\u5e8f\u7684\u6574\u4f53\u6027\u80fd\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u8bbe\u5907","text":"<p>\u8ba1\u7b97\u673a\u4e2d\u5305\u62ec\u4e09\u79cd\u4e0d\u540c\u7c7b\u578b\u7684\u5b58\u50a8\u8bbe\u5907\uff1a\u300c\u786c\u76d8 hard disk\u300d\u3001\u300c\u5185\u5b58 random-access memory, RAM\u300d\u3001\u300c\u7f13\u5b58 cache memory\u300d\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u4eec\u5728\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6027\u80fd\u7279\u70b9\u3002</p> <p> \u8868 4-2 \u00a0 \u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u8bbe\u5907 </p> \u786c\u76d8 \u5185\u5b58 \u7f13\u5b58 \u7528\u9014 \u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5305\u62ec\u64cd\u4f5c\u7cfb\u7edf\u3001\u7a0b\u5e8f\u3001\u6587\u4ef6\u7b49 \u4e34\u65f6\u5b58\u50a8\u5f53\u524d\u8fd0\u884c\u7684\u7a0b\u5e8f\u548c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e \u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u51cf\u5c11 CPU \u8bbf\u95ee\u5185\u5b58\u7684\u6b21\u6570 \u6613\u5931\u6027 \u65ad\u7535\u540e\u6570\u636e\u4e0d\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u5bb9\u91cf \u8f83\u5927\uff0cTB \u7ea7\u522b \u8f83\u5c0f\uff0cGB \u7ea7\u522b \u975e\u5e38\u5c0f\uff0cMB \u7ea7\u522b \u901f\u5ea6 \u8f83\u6162\uff0c\u51e0\u767e\u5230\u51e0\u5343 MB/s \u8f83\u5feb\uff0c\u51e0\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u51e0\u5341\u5230\u51e0\u767e GB/s \u4ef7\u683c \u8f83\u4fbf\u5b9c\uff0c\u51e0\u6bdb\u5230\u51e0\u5143 / GB \u8f83\u8d35\uff0c\u51e0\u5341\u5230\u51e0\u767e\u5143 / GB \u975e\u5e38\u8d35\uff0c\u968f CPU \u6253\u5305\u8ba1\u4ef7 <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf\u60f3\u8c61\u4e3a\u56fe 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7ed3\u6784\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9876\u7aef\u7684\u5b58\u50a8\u8bbe\u5907\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u8fd9\u79cd\u591a\u5c42\u7ea7\u7684\u8bbe\u8ba1\u5e76\u975e\u5076\u7136\uff0c\u800c\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u5de5\u7a0b\u5e08\u4eec\u7ecf\u8fc7\u6df1\u601d\u719f\u8651\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u786c\u76d8\u96be\u4ee5\u88ab\u5185\u5b58\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u5185\u5b58\u4e2d\u7684\u6570\u636e\u5728\u65ad\u7535\u540e\u4f1a\u4e22\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9002\u5408\u957f\u671f\u5b58\u50a8\u6570\u636e\uff1b\u5176\u6b21\uff0c\u5185\u5b58\u7684\u6210\u672c\u5927\u7ea6\u662f\u786c\u76d8\u7684\u51e0\u5341\u500d\uff0c\u8fd9\u4f7f\u5f97\u5b83\u96be\u4ee5\u5728\u6d88\u8d39\u8005\u5e02\u573a\u666e\u53ca\u3002</li> <li>\u7f13\u5b58\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96be\u4ee5\u517c\u5f97\u3002\u968f\u7740 L1\u3001L2\u3001L3 \u7f13\u5b58\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u4f1a\u53d8\u5927\uff0c\u4e0e CPU \u6838\u5fc3\u4e4b\u95f4\u7684\u7269\u7406\u8ddd\u79bb\u4f1a\u53d8\u8fdc\uff0c\u4ece\u800c\u5bfc\u81f4\u6570\u636e\u4f20\u8f93\u65f6\u95f4\u589e\u52a0\uff0c\u5143\u7d20\u8bbf\u95ee\u5ef6\u8fdf\u53d8\u9ad8\u3002\u5728\u5f53\u524d\u6280\u672f\u4e0b\uff0c\u591a\u5c42\u7ea7\u7684\u7f13\u5b58\u7ed3\u6784\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u95f4\u7684\u6700\u4f73\u5e73\u8861\u70b9\u3002</li> </ul> <p></p> <p> \u56fe 4-9 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf </p> <p>Note</p> <p>\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u5c42\u6b21\u7ed3\u6784\u4f53\u73b0\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u95f4\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u79cd\u6743\u8861\u666e\u904d\u5b58\u5728\u4e8e\u6240\u6709\u5de5\u4e1a\u9886\u57df\uff0c\u5b83\u8981\u6c42\u6211\u4eec\u5728\u4e0d\u540c\u7684\u4f18\u52bf\u548c\u9650\u5236\u4e4b\u95f4\u627e\u5230\u6700\u4f73\u7684\u5e73\u8861\u70b9\u3002</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u786c\u76d8\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u5927\u91cf\u6570\u636e\uff0c\u5185\u5b58\u7528\u4e8e\u4e34\u65f6\u5b58\u50a8\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\uff0c\u800c\u7f13\u5b58\u5219\u7528\u4e8e\u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5e8f\u8fd0\u884c\u6548\u7387\u3002\u8fd9\u4e09\u8005\u5171\u540c\u534f\u4f5c\uff0c\u786e\u4fdd\u8ba1\u7b97\u673a\u7cfb\u7edf\u7684\u9ad8\u6548\u8fd0\u884c\u3002</p> <p>\u5982\u56fe 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4f1a\u4ece\u786c\u76d8\u4e2d\u88ab\u8bfb\u53d6\u5230\u5185\u5b58\u4e2d\uff0c\u4f9b\u7ed9 CPU \u8ba1\u7b97\u4f7f\u7528\u3002\u7f13\u5b58\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u901a\u8fc7\u667a\u80fd\u5730\u4ece\u5185\u5b58\u52a0\u8f7d\u6570\u636e\uff0c\u7ed9 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u6570\u636e\u8bfb\u53d6\uff0c\u4ece\u800c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\uff0c\u51cf\u5c11\u5bf9\u8f83\u6162\u7684\u5185\u5b58\u7684\u4f9d\u8d56\u3002</p> <p></p> <p> \u56fe 4-10 \u00a0 \u786c\u76d8\u3001\u5185\u5b58\u548c\u7f13\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u6d41\u901a </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u5185\u5b58\u6548\u7387","text":"<p>\u5728\u5185\u5b58\u7a7a\u95f4\u5229\u7528\u65b9\u9762\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5177\u6709\u5404\u81ea\u7684\u4f18\u52bf\u548c\u5c40\u9650\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u5185\u5b58\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u5757\u5185\u5b58\u4e0d\u80fd\u88ab\u591a\u4e2a\u7a0b\u5e8f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u4eec\u5e0c\u671b\u6570\u636e\u7ed3\u6784\u80fd\u591f\u5c3d\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u95f4\u3002\u6570\u7ec4\u7684\u5143\u7d20\u7d27\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u989d\u5916\u7684\u7a7a\u95f4\u6765\u5b58\u50a8\u94fe\u8868\u8282\u70b9\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u56e0\u6b64\u7a7a\u95f4\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u6570\u7ec4\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u591f\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u7684\u6d6a\u8d39\uff0c\u6570\u7ec4\u6269\u5bb9\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u94fe\u8868\u4ee5\u201c\u8282\u70b9\u201d\u4e3a\u5355\u4f4d\u8fdb\u884c\u52a8\u6001\u5185\u5b58\u5206\u914d\u548c\u56de\u6536\uff0c\u8fd9\u79cd\u65b9\u5f0f\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u7075\u6d3b\u6027\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u968f\u7740\u53cd\u590d\u7533\u8bf7\u4e0e\u91ca\u653e\u5185\u5b58\uff0c\u7a7a\u95f2\u5185\u5b58\u7684\u788e\u7247\u5316\u7a0b\u5ea6\u4f1a\u8d8a\u6765\u8d8a\u9ad8\uff0c\u4ece\u800c\u5bfc\u81f4\u5185\u5b58\u7684\u5229\u7528\u6548\u7387\u964d\u4f4e\u3002\u6570\u7ec4\u7531\u4e8e\u5176\u8fde\u7eed\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u76f8\u5bf9\u4e0d\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002\u76f8\u53cd\uff0c\u94fe\u8868\u7684\u5143\u7d20\u662f\u5206\u6563\u5b58\u50a8\u7684\uff0c\u5728\u9891\u7e41\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u4e2d\uff0c\u66f4\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u7f13\u5b58\u6548\u7387","text":"<p>\u7f13\u5b58\u867d\u7136\u5728\u7a7a\u95f4\u5bb9\u91cf\u4e0a\u8fdc\u5c0f\u4e8e\u5185\u5b58\uff0c\u4f46\u5b83\u6bd4\u5185\u5b58\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5e8f\u6267\u884c\u901f\u5ea6\u4e0a\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u4e8e\u7f13\u5b58\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u5b83\u53ea\u80fd\u5b58\u50a8\u4e00\u5c0f\u90e8\u5206\u9891\u7e41\u8bbf\u95ee\u7684\u6570\u636e\u3002\u56e0\u6b64\uff0c\u5f53 CPU \u5c1d\u8bd5\u8bbf\u95ee\u7684\u6570\u636e\u4e0d\u5728\u7f13\u5b58\u4e2d\u65f6\uff0c\u5c31\u4f1a\u53d1\u751f\u300c\u7f13\u5b58\u672a\u547d\u4e2d cache miss\u300d\uff0c\u6b64\u65f6 CPU \u4e0d\u5f97\u4e0d\u4ece\u901f\u5ea6\u8f83\u6162\u7684\u5185\u5b58\u4e2d\u52a0\u8f7d\u6240\u9700\u6570\u636e\u3002</p> <p>\u663e\u7136\uff0c\u201c\u7f13\u5b58\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8bfb\u5199\u6570\u636e\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5e8f\u6027\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u4eec\u5c06 CPU \u4ece\u7f13\u5b58\u4e2d\u6210\u529f\u83b7\u53d6\u6570\u636e\u7684\u6bd4\u4f8b\u79f0\u4e3a\u300c\u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate\u300d\uff0c\u8fd9\u4e2a\u6307\u6807\u901a\u5e38\u7528\u6765\u8861\u91cf\u7f13\u5b58\u6548\u7387\u3002</p> <p>\u4e3a\u4e86\u5c3d\u53ef\u80fd\u8fbe\u5230\u66f4\u9ad8\u6548\u7387\uff0c\u7f13\u5b58\u4f1a\u91c7\u53d6\u4ee5\u4e0b\u6570\u636e\u52a0\u8f7d\u673a\u5236\u3002</p> <ul> <li>\u7f13\u5b58\u884c\uff1a\u7f13\u5b58\u4e0d\u662f\u5355\u4e2a\u5b57\u8282\u5730\u5b58\u50a8\u4e0e\u52a0\u8f7d\u6570\u636e\uff0c\u800c\u662f\u4ee5\u7f13\u5b58\u884c\u4e3a\u5355\u4f4d\u3002\u76f8\u6bd4\u4e8e\u5355\u4e2a\u5b57\u8282\u7684\u4f20\u8f93\uff0c\u7f13\u5b58\u884c\u7684\u4f20\u8f93\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u5904\u7406\u5668\u4f1a\u5c1d\u8bd5\u9884\u6d4b\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\uff08\u4f8b\u5982\u987a\u5e8f\u8bbf\u95ee\u3001\u56fa\u5b9a\u6b65\u957f\u8df3\u8dc3\u8bbf\u95ee\u7b49\uff09\uff0c\u5e76\u6839\u636e\u7279\u5b9a\u6a21\u5f0f\u5c06\u6570\u636e\u52a0\u8f7d\u81f3\u7f13\u5b58\u4e4b\u4e2d\uff0c\u4ece\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u9644\u8fd1\u7684\u6570\u636e\u53ef\u80fd\u4e5f\u4f1a\u8fd1\u671f\u88ab\u8bbf\u95ee\u3002\u56e0\u6b64\uff0c\u7f13\u5b58\u5728\u52a0\u8f7d\u67d0\u4e00\u6570\u636e\u65f6\uff0c\u4e5f\u4f1a\u5c06\u5176\u9644\u8fd1\u7684\u6570\u636e\u52a0\u8f7d\u8fdb\u6765\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> <li>\u65f6\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\u3002\u7f13\u5b58\u5229\u7528\u8fd9\u4e00\u539f\u7406\uff0c\u901a\u8fc7\u4fdd\u7559\u6700\u8fd1\u8bbf\u95ee\u8fc7\u7684\u6570\u636e\u6765\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u6548\u7387\u4e5f\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u4f53\u73b0\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\uff1a\u94fe\u8868\u5143\u7d20\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u7a7a\u95f4\u66f4\u591a\uff0c\u5bfc\u81f4\u7f13\u5b58\u4e2d\u5bb9\u7eb3\u7684\u6709\u6548\u6570\u636e\u91cf\u66f4\u5c11\u3002</li> <li>\u7f13\u5b58\u884c\uff1a\u94fe\u8868\u6570\u636e\u5206\u6563\u5728\u5185\u5b58\u5404\u5904\uff0c\u800c\u7f13\u5b58\u662f\u201c\u6309\u884c\u52a0\u8f7d\u201d\u7684\uff0c\u56e0\u6b64\u52a0\u8f7d\u5230\u7684\u65e0\u6548\u6570\u636e\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u6570\u7ec4\u6bd4\u94fe\u8868\u7684\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9884\u6d4b\u6027\u201d\uff0c\u5373\u7cfb\u7edf\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c06\u88ab\u52a0\u8f7d\u7684\u6570\u636e\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u6570\u7ec4\u88ab\u5b58\u50a8\u5728\u96c6\u4e2d\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u88ab\u52a0\u8f7d\u6570\u636e\u7684\u9644\u8fd1\u6570\u636e\u66f4\u6709\u53ef\u80fd\u5373\u5c06\u88ab\u8bbf\u95ee\u3002</li> </ul> <p>\u603b\u4f53\u800c\u8a00\uff0c\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u4f18\u4e8e\u94fe\u8868\u3002\u8fd9\u4f7f\u5f97\u5728\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u5f80\u5f80\u66f4\u53d7\u6b22\u8fce\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u7f13\u5b58\u6548\u7387\u5e76\u4e0d\u610f\u5473\u7740\u6570\u7ec4\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u4f18\u4e8e\u94fe\u8868\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\u9009\u62e9\u54ea\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5b9e\u73b0\u201c\u6808\u201d\u6570\u636e\u7ed3\u6784\uff08\u4e0b\u4e00\u7ae0\u4f1a\u8be6\u7ec6\u4ecb\u7ecd\uff09\uff0c\u4f46\u5b83\u4eec\u9002\u7528\u4e8e\u4e0d\u540c\u573a\u666f\u3002</p> <ul> <li>\u5728\u505a\u7b97\u6cd5\u9898\u65f6\uff0c\u6211\u4eec\u4f1a\u503e\u5411\u4e8e\u9009\u62e9\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\uff0c\u56e0\u4e3a\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u968f\u673a\u8bbf\u95ee\u7684\u80fd\u529b\uff0c\u4ee3\u4ef7\u4ec5\u662f\u9700\u8981\u9884\u5148\u4e3a\u6570\u7ec4\u5206\u914d\u4e00\u5b9a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u5982\u679c\u6570\u636e\u91cf\u975e\u5e38\u5927\u3001\u52a8\u6001\u6027\u5f88\u9ad8\u3001\u6808\u7684\u9884\u671f\u5927\u5c0f\u96be\u4ee5\u4f30\u8ba1\uff0c\u90a3\u4e48\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u5c31\u66f4\u52a0\u5408\u9002\u3002\u94fe\u8868\u80fd\u591f\u5c06\u5927\u91cf\u6570\u636e\u5206\u6563\u5b58\u50a8\u4e8e\u5185\u5b58\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u5e76\u4e14\u907f\u514d\u4e86\u6570\u7ec4\u6269\u5bb9\u4ea7\u751f\u7684\u989d\u5916\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_array_and_linkedlist/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff1a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u7279\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002</li> <li>\u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002</li> <li>\u94fe\u8868\u901a\u8fc7\u66f4\u6539\u5f15\u7528\uff08\u6307\u9488\uff09\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002</li> <li>\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u73af\u5f62\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\uff0c\u5b83\u4eec\u5206\u522b\u5177\u6709\u5404\u81ea\u7684\u5e94\u7528\u573a\u666f\u3002</li> <li>\u5217\u8868\u662f\u4e00\u79cd\u652f\u6301\u589e\u5220\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\uff0c\u5176\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002</li> <li>\u5217\u8868\u7684\u51fa\u73b0\u5927\u5e45\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> <li>\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u6570\u7ec4\u63d0\u4f9b\u66f4\u9ad8\u7684\u5185\u5b58\u7a7a\u95f4\u6548\u7387\uff0c\u800c\u94fe\u8868\u5219\u5728\u5185\u5b58\u4f7f\u7528\u4e0a\u66f4\u52a0\u7075\u6d3b\u3002</li> <li>\u7f13\u5b58\u901a\u8fc7\u7f13\u5b58\u884c\u3001\u9884\u53d6\u673a\u5236\u4ee5\u53ca\u7a7a\u95f4\u548c\u65f6\u95f4\u5c40\u90e8\u6027\u7b49\u6570\u636e\u52a0\u8f7d\u673a\u5236\uff0c\u4e3a CPU \u63d0\u4f9b\u5feb\u901f\u6570\u636e\u8bbf\u95ee\uff0c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\u3002</li> <li>\u7531\u4e8e\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u94fe\u8868\u66f4\u9ad8\u6548\u3002\u5728\u9009\u62e9\u6570\u636e\u7ed3\u6784\u65f6\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u548c\u573a\u666f\u505a\u51fa\u6070\u5f53\u9009\u62e9\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f</p> <p>\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5806\u4e0a\u7684\u6570\u7ec4\u90fd\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u6570\u636e\u64cd\u4f5c\u6548\u7387\u662f\u57fa\u672c\u4e00\u81f4\u7684\u3002\u7136\u800c\uff0c\u6808\u548c\u5806\u5177\u6709\u5404\u81ea\u7684\u7279\u70b9\uff0c\u4ece\u800c\u5bfc\u81f4\u4ee5\u4e0b\u4e0d\u540c\u70b9\u3002</p> <ol> <li>\u5206\u914d\u548c\u91ca\u653e\u6548\u7387\uff1a\u6808\u662f\u4e00\u5757\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff1b\u800c\u5806\u5185\u5b58\u76f8\u5bf9\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u52a8\u6001\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u4e0a\u7684\u5206\u914d\u548c\u91ca\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u6808\u4e0a\u7684\u6162\u3002</li> <li>\u5927\u5c0f\u9650\u5236\uff1a\u6808\u5185\u5b58\u76f8\u5bf9\u8f83\u5c0f\uff0c\u5806\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\u3002\u56e0\u6b64\u5806\u66f4\u52a0\u9002\u5408\u5b58\u50a8\u5927\u578b\u6570\u7ec4\u3002</li> <li>\u7075\u6d3b\u6027\uff1a\u6808\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u9700\u8981\u5728\u7f16\u8bd1\u65f6\u786e\u5b9a\uff0c\u800c\u5806\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u786e\u5b9a\u3002</li> </ol> <p>\u4e3a\u4ec0\u4e48\u6570\u7ec4\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u540c\u7c7b\u578b\u5462\uff1f</p> <p>\u94fe\u8868\u7531\u8282\u70b9\u7ec4\u6210\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u5f15\u7528\uff08\u6307\u9488\uff09\u8fde\u63a5\uff0c\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 int\u3001double\u3001string\u3001object \u7b49\u3002</p> <p>\u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u540c\u65f6\u5305\u542b int \u548c long \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 bytes \u548c 8 bytes \uff0c\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd\u957f\u5ea6\u7684\u5143\u7d20\u3002</p> <pre><code># \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\n</code></pre> <p>\u5220\u9664\u8282\u70b9\u540e\uff0c\u662f\u5426\u9700\u8981\u628a <code>P.next</code> \u8bbe\u4e3a \\(\\text{None}\\) \u5462\uff1f</p> <p>\u4e0d\u4fee\u6539 <code>P.next</code> \u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u8282\u70b9\u904d\u5386\u5230\u5c3e\u8282\u70b9\u5df2\u7ecf\u4e0d\u4f1a\u9047\u5230 <code>P</code> \u4e86\u3002\u8fd9\u610f\u5473\u7740\u8282\u70b9 <code>P</code> \u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u8282\u70b9 <code>P</code> \u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8be5\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u3002</p> <p>\u4ece\u5783\u573e\u56de\u6536\u7684\u89d2\u5ea6\u770b\uff0c\u5bf9\u4e8e Java\u3001Python\u3001Go \u7b49\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\u7684\u8bed\u8a00\u6765\u8bf4\uff0c\u8282\u70b9 <code>P</code> \u662f\u5426\u88ab\u56de\u6536\u53d6\u51b3\u4e8e\u662f\u5426\u4ecd\u5b58\u5728\u6307\u5411\u5b83\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662f <code>P.next</code> \u7684\u503c\u3002\u5728 C \u548c C++ \u7b49\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u624b\u52a8\u91ca\u653e\u8282\u70b9\u5185\u5b58\u3002</p> <p>\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f</p> <p>\u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u8282\u70b9\u3001\u5c3e\u8282\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002</p> <p>\u56fe\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u8282\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u8282\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f</p> <p>\u8be5\u793a\u610f\u56fe\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p> <ul> <li>\u4e0d\u540c\u7c7b\u578b\u7684\u8282\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 int\u3001long\u3001double \u548c\u5b9e\u4f8b\u5bf9\u8c61\u7b49\u3002</li> <li>\u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002</li> </ul> <p>\u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f</p> <p>\u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002</p> <p>\u201c\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f</p> <p>\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002</p> <p>\u5728 Python \u4e2d\u521d\u59cb\u5316 <code>n = [1, 2, 3]</code> \u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 <code>m = [2, 1, 3]</code> \u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddf <code>n</code> \u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u7684\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48 <code>m</code> \u8fd8\u662f\u6570\u7ec4\u5417\uff1f</p> <p>\u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9 <code>n = [n1, n2, n3, n4, n5]</code> \uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9 5 \u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002</p> <p>\u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p>C++ STL \u91cc\u9762\u7684 <code>std::list</code> \u5df2\u7ecf\u5b9e\u73b0\u4e86\u53cc\u5411\u94fe\u8868\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u7b97\u6cd5\u4e66\u4e0a\u4e0d\u600e\u4e48\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u4e3a\u6709\u4ec0\u4e48\u5c40\u9650\u6027\u5462\uff1f</p> <p>\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u5f80\u5f80\u66f4\u9752\u7750\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u65f6\u624d\u4f7f\u7528\u94fe\u8868\uff0c\u4e3b\u8981\u6709\u4e24\u4e2a\u539f\u56e0\u3002</p> <ul> <li>\u7a7a\u95f4\u5f00\u9500\uff1a\u7531\u4e8e\u6bcf\u4e2a\u5143\u7d20\u9700\u8981\u4e24\u4e2a\u989d\u5916\u7684\u6307\u9488\uff08\u4e00\u4e2a\u7528\u4e8e\u524d\u4e00\u4e2a\u5143\u7d20\uff0c\u4e00\u4e2a\u7528\u4e8e\u540e\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u6240\u4ee5 <code>std::list</code> \u901a\u5e38\u6bd4 <code>std::vector</code> \u66f4\u5360\u7528\u7a7a\u95f4\u3002</li> <li>\u7f13\u5b58\u4e0d\u53cb\u597d\uff1a\u7531\u4e8e\u6570\u636e\u4e0d\u662f\u8fde\u7eed\u5b58\u653e\u7684\uff0c\u56e0\u6b64 <code>std::list</code> \u5bf9\u7f13\u5b58\u7684\u5229\u7528\u7387\u8f83\u4f4e\u3002\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c<code>std::vector</code> \u7684\u6027\u80fd\u4f1a\u66f4\u597d\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u94fe\u8868\u7684\u60c5\u51b5\u4e3b\u8981\u662f\u4e8c\u53c9\u6811\u548c\u56fe\u3002\u6808\u548c\u961f\u5217\u5f80\u5f80\u4f1a\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684 <code>stack</code> \u548c <code>queue</code> \uff0c\u800c\u975e\u94fe\u8868\u3002</p> <p>\u521d\u59cb\u5316\u5217\u8868 <code>res = [0] * self.size()</code> \u64cd\u4f5c\uff0c\u4f1a\u5bfc\u81f4 <code>res</code> \u7684\u6bcf\u4e2a\u5143\u7d20\u5f15\u7528\u76f8\u540c\u7684\u5730\u5740\u5417\uff1f</p> <p>\u4e0d\u4f1a\u3002\u4f46\u4e8c\u7ef4\u6570\u7ec4\u4f1a\u6709\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f8b\u5982\u521d\u59cb\u5316\u4e8c\u7ef4\u5217\u8868 <code>res = [[0] * self.size()]</code> \uff0c\u5219\u591a\u6b21\u5f15\u7528\u4e86\u540c\u4e00\u4e2a\u5217\u8868 <code>[0]</code> \u3002</p> <p>\u5728\u5220\u9664\u8282\u70b9\u4e2d\uff0c\u9700\u8981\u65ad\u5f00\u8be5\u8282\u70b9\u4e0e\u5176\u540e\u7ee7\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u6307\u5411\u5417\uff1f</p> <p>\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\uff08\u505a\u9898\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65ad\u5f00\u6ca1\u6709\u5173\u7cfb\uff0c\u53ea\u8981\u4fdd\u8bc1\u7a0b\u5e8f\u7684\u903b\u8f91\u662f\u6b63\u786e\u7684\u5c31\u884c\u3002\u4ece\u6807\u51c6\u5e93\u7684\u89d2\u5ea6\u770b\uff0c\u65ad\u5f00\u66f4\u52a0\u5b89\u5168\u3001\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65ad\u5f00\uff0c\u5047\u8bbe\u88ab\u5220\u9664\u8282\u70b9\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u4e48\u5b83\u4e5f\u4f1a\u5f71\u54cd\u540e\u7ee7\u8282\u70b9\u7684\u5185\u5b58\u56de\u6536\u3002</p>"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"<p>Abstract</p> <p>\u6211\u4eec\u5982\u540c\u8ff7\u5bab\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u8fdb\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u4f1a\u9047\u5230\u56f0\u96be\u3002</p> <p>\u56de\u6eaf\u7684\u529b\u91cf\u8ba9\u6211\u4eec\u80fd\u591f\u91cd\u65b0\u5f00\u59cb\uff0c\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u6700\u7ec8\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002</p>"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5</li> <li>13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898</li> <li>13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898</li> <li>13.4 \u00a0 N \u7687\u540e\u95ee\u9898</li> <li>13.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/","title":"13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5","text":"<p>\u300c\u56de\u6eaf\u7b97\u6cd5 backtracking algorithm\u300d\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002</p> <p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5229\u7528\u524d\u5e8f\u904d\u5386\u6784\u9020\u4e00\u4e2a\u56de\u6eaf\u95ee\u9898\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p>\u4f8b\u9898\u4e00</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002</p> <p>\u5bf9\u4e8e\u6b64\u9898\uff0c\u6211\u4eec\u524d\u5e8f\u904d\u5386\u8fd9\u68f5\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u7ed3\u679c\u5217\u8868 <code>res</code> \u4e4b\u4e2d\u3002\u76f8\u5173\u8fc7\u7a0b\u5b9e\u73b0\u5982\u56fe 13-1 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\nif root is None:\nreturn\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(root)\npre_order(root.left)\npre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr) {\nreturn;\n}\nif (root-&gt;val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(root);\n}\npreOrder(root-&gt;left);\npreOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(root);\n}\npreOrder(root.left);\npreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid PreOrder(TreeNode? root) {\nif (root == null) {\nreturn;\n}\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(root);\n}\nPreOrder(root.left);\nPreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\nif root == nil {\nreturn\n}\nif (root.Val).(int) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, root)\n}\npreOrderI(root.Left, res)\npreOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(root)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\nif (root === null) {\nreturn;\n}\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push(root);\n}\npreOrder(root.left, res);\npreOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\nif (root === null) {\nreturn;\n}\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push(root);\n}\npreOrder(root.left, res);\npreOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\nif (root == null) {\nreturn;\n}\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(root);\n}\npreOrder(root.left, res);\npreOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\nif root.is_none() {\nreturn;\n}\nif let Some(node) = root {\nif node.borrow().val == 7 {\n// \u8bb0\u5f55\u89e3\nres.push(node.clone());\n}\npre_order(res, node.borrow().left.clone());\npre_order(res, node.borrow().right.clone());\n}\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\nif (root == NULL) {\nreturn;\n}\nif (root-&gt;val == 7) {\n// \u8bb0\u5f55\u89e3\nres[resSize++] = root;\n}\npreOrder(root-&gt;left);\npreOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> <p></p> <p> \u56fe 13-1 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"<p>\u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002</p> <p>\u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684 <code>return</code> \u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u4ec5\u4ec5\u5305\u62ec\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002</p> <p>\u4f8b\u9898\u4e8c</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002</p> <p>\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868 <code>path</code> \u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236 <code>path</code> \u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868 <code>res</code> \u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c<code>res</code> \u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\nif root is None:\nreturn\n# \u5c1d\u8bd5\npath.append(root)\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(list(path))\npre_order(root.left)\npre_order(root.right)\n# \u56de\u9000\npath.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push_back(root);\nif (root-&gt;val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(path);\n}\npreOrder(root-&gt;left);\npreOrder(root-&gt;right);\n// \u56de\u9000\npath.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\nif (root == null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(new ArrayList&lt;&gt;(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid PreOrder(TreeNode? root) {\nif (root == null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.Add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(new List&lt;TreeNode&gt;(path));\n}\nPreOrder(root.left);\nPreOrder(root.right);\n// \u56de\u9000\npath.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\nif root == nil {\nreturn\n}\n// \u5c1d\u8bd5\n*path = append(*path, root)\nif root.Val.(int) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, *path)\n}\npreOrderII(root.Left, res, path)\npreOrderII(root.Right, res, path)\n// \u56de\u9000\n*path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u5c1d\u8bd5\npath.append(root)\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(path)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n// \u56de\u9000\npath.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\nif (root === null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\nroot: TreeNode | null,\npath: TreeNode[],\nres: TreeNode[][]\n): void {\nif (root === null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(\nTreeNode? root,\nList&lt;TreeNode&gt; path,\nList&lt;List&lt;TreeNode&gt;&gt; res,\n) {\nif (root == null) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(List.from(path));\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfn pre_order(res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;, path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\nif root.is_none() {\nreturn;\n}\nif let Some(node) = root {\n// \u5c1d\u8bd5\npath.push(node.clone());\nif node.borrow().val == 7 {\n// \u8bb0\u5f55\u89e3\nres.push(path.clone());\n}\npre_order(res, path, node.borrow().left.clone());\npre_order(res, path, node.borrow().right.clone());\n// \u56de\u9000\npath.remove(path.len() -  1);\n}\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\nif (root == NULL) {\nreturn;\n}\n// \u5c1d\u8bd5\npath[pathSize++] = root;\nif (root-&gt;val == 7) {\n// \u8bb0\u5f55\u89e3\nfor (int i = 0; i &lt; pathSize; ++i) {\nres[resSize][i] = path[i];\n}\nresSize++;\n}\npreOrder(root-&gt;left);\npreOrder(root-&gt;right);\n// \u56de\u9000\npathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> <p>\u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ece <code>path</code> \u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002</p> <p>\u89c2\u5bdf\u56fe 13-2 \u6240\u793a\u7684\u8fc7\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u4e92\u4e3a\u9006\u5411\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 13-2 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"<p>\u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002</p> <p>\u4f8b\u9898\u4e09</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u5e76\u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002</p> <p>\u4e3a\u4e86\u6ee1\u8db3\u4ee5\u4e0a\u7ea6\u675f\u6761\u4ef6\uff0c\u6211\u4eec\u9700\u8981\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\uff0c\u5219\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n\"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n# \u526a\u679d\nif root is None or root.val == 3:\nreturn\n# \u5c1d\u8bd5\npath.append(root)\nif root.val == 7:\n# \u8bb0\u5f55\u89e3\nres.append(list(path))\npre_order(root.left)\npre_order(root.right)\n# \u56de\u9000\npath.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n// \u526a\u679d\nif (root == nullptr || root-&gt;val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push_back(root);\nif (root-&gt;val == 7) {\n// \u8bb0\u5f55\u89e3\nres.push_back(path);\n}\npreOrder(root-&gt;left);\npreOrder(root-&gt;right);\n// \u56de\u9000\npath.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n// \u526a\u679d\nif (root == null || root.val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(new ArrayList&lt;&gt;(path));\n}\npreOrder(root.left);\npreOrder(root.right);\n// \u56de\u9000\npath.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid PreOrder(TreeNode? root) {\n// \u526a\u679d\nif (root == null || root.val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.Add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.Add(new List&lt;TreeNode&gt;(path));\n}\nPreOrder(root.left);\nPreOrder(root.right);\n// \u56de\u9000\npath.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n// \u526a\u679d\nif root == nil || root.Val == 3 {\nreturn\n}\n// \u5c1d\u8bd5\n*path = append(*path, root)\nif root.Val.(int) == 7 {\n// \u8bb0\u5f55\u89e3\n*res = append(*res, *path)\n}\npreOrderIII(root.Left, res, path)\npreOrderIII(root.Right, res, path)\n// \u56de\u9000\n*path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n// \u526a\u679d\nguard let root = root, root.val != 3 else {\nreturn\n}\n// \u5c1d\u8bd5\npath.append(root)\nif root.val == 7 {\n// \u8bb0\u5f55\u89e3\nres.append(path)\n}\npreOrder(root: root.left)\npreOrder(root: root.right)\n// \u56de\u9000\npath.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n// \u526a\u679d\nif (root === null || root.val === 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\nroot: TreeNode | null,\npath: TreeNode[],\nres: TreeNode[][]\n): void {\n// \u526a\u679d\nif (root === null || root.val === 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.push(root);\nif (root.val === 7) {\n// \u8bb0\u5f55\u89e3\nres.push([...path]);\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(\nTreeNode? root,\nList&lt;TreeNode&gt; path,\nList&lt;List&lt;TreeNode&gt;&gt; res,\n) {\nif (root == null || root.val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath.add(root);\nif (root.val == 7) {\n// \u8bb0\u5f55\u89e3\nres.add(List.from(path));\n}\npreOrder(root.left, path, res);\npreOrder(root.right, path, res);\n// \u56de\u9000\npath.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfn pre_order(res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;, path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n// \u526a\u679d\nif root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\nreturn;\n}\nif let Some(node) = root {\n// \u5c1d\u8bd5\npath.push(node.clone());\nif node.borrow().val == 7 {\n// \u8bb0\u5f55\u89e3\nres.push(path.clone());\n}\npre_order(res, path, node.borrow().left.clone());\npre_order(res, path, node.borrow().right.clone());\n// \u56de\u9000\npath.remove(path.len() -  1);\n}\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n// \u526a\u679d\nif (root == NULL || root-&gt;val == 3) {\nreturn;\n}\n// \u5c1d\u8bd5\npath[pathSize++] = root;\nif (root-&gt;val == 7) {\n// \u8bb0\u5f55\u89e3\nfor (int i = 0; i &lt; pathSize; i++) {\nres[resSize][i] = path[i];\n}\nresSize++;\n}\npreOrder(root-&gt;left);\npreOrder(root-&gt;right);\n// \u56de\u9000\npathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> <p>\u201c\u526a\u679d\u201d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5982\u56fe 13-3 \u6240\u793a\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u641c\u7d22\u6548\u7387\u3002</p> <p></p> <p> \u56fe 13-3 \u00a0 \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u4ee3\u7801","text":"<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c1d\u8bd5\u5c06\u56de\u6eaf\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\uff0c\u63d0\u5347\u4ee3\u7801\u7684\u901a\u7528\u6027\u3002</p> <p>\u5728\u4ee5\u4e0b\u6846\u67b6\u4ee3\u7801\u4e2d\uff0c<code>state</code> \u8868\u793a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0c<code>choices</code> \u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n# \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif is_solution(state):\n# \u8bb0\u5f55\u89e3\nrecord_solution(state, res)\n# \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice):\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice)\nbacktrack(state, choices, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice)\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice : choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice : choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (IsSolution(state)) {\n// \u8bb0\u5f55\u89e3\nRecordSolution(state, res);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (Choice choice in choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (IsValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nMakeChoice(state, choice);\nBacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nUndoChoice(state, choice);\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif isSolution(state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res)\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor _, choice := range choices {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state, choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice)\nbacktrack(state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice)\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif isSolution(state: state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state: state, res: &amp;res)\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state: state, choice: choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state: &amp;state, choice: choice)\nbacktrack(state: &amp;state, choices: choices, res: &amp;res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state: &amp;state, choice: choice)\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let choice of choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let choice of choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Choice choice in choices) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif is_solution(state) {\n// \u8bb0\u5f55\u89e3\nrecord_solution(state, res);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice);\nbacktrack(state, choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice);\n}\n}\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n// \u5224\u65ad\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res, numRes);\n// \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; numChoices; i++) {\n// \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, &amp;choices[i])) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, &amp;choices[i]);\nbacktrack(state, choices, numChoices, res, numRes);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, &amp;choices[i]);\n}\n}\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u57fa\u4e8e\u6846\u67b6\u4ee3\u7801\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u72b6\u6001 <code>state</code> \u4e3a\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9 <code>choices</code> \u4e3a\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679c <code>res</code> \u662f\u8def\u5f84\u5217\u8868\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n\"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\nreturn state and state[-1].val == 7\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n\"\"\"\u8bb0\u5f55\u89e3\"\"\"\nres.append(list(state))\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n\"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\nreturn choice is not None and choice.val != 3\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n\"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\nstate.append(choice)\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n\"\"\"\u6062\u590d\u72b6\u6001\"\"\"\nstate.pop()\ndef backtrack(\nstate: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n# \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif is_solution(state):\n# \u8bb0\u5f55\u89e3\nrecord_solution(state, res)\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice):\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\nreturn !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\nres.push_back(state);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\nreturn choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\nstate.push_back(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\nstate.pop_back();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (TreeNode *choice : choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nvector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\nbacktrack(state, nextChoices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\nreturn !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\nres.add(new ArrayList&lt;&gt;(state));\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\nreturn choice != null &amp;&amp; choice.val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\nstate.add(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\nstate.remove(state.size() - 1);\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (TreeNode choice : choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, Arrays.asList(choice.left, choice.right), res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\nreturn state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\nres.Add(new List&lt;TreeNode&gt;(state));\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\nreturn choice != null &amp;&amp; choice.val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\nstate.Add(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\nstate.RemoveAt(state.Count - 1);\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (IsSolution(state)) {\n// \u8bb0\u5f55\u89e3\nRecordSolution(state, res);\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (TreeNode choice in choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (IsValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nMakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nBacktrack(state, [choice.left!, choice.right!], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nUndoChoice(state, choice);\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\nreturn len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n*res = append(*res, *state)\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\nreturn choice != nil &amp;&amp; choice.Val != 3\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n*state = append(*state, choice)\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n*state = (*state)[:len(*state)-1]\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif isSolution(state) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor _, choice := range *choices {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state, choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\ntemp := make([]*TreeNode, 0)\ntemp = append(temp, choice.Left, choice.Right)\nbacktrackIII(state, &amp;temp, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice)\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n!state.isEmpty &amp;&amp; state.last!.val == 7\n}\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\nres.append(state)\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\nchoice != nil &amp;&amp; choice!.val != 3\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\nstate.append(choice)\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\nstate.removeLast()\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif isSolution(state: state) {\nrecordSolution(state: state, res: &amp;res)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif isValid(state: state, choice: choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state: &amp;state, choice: choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state: &amp;state, choice: choice)\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\nreturn state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\nres.push([...state]);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\nreturn choice !== null &amp;&amp; choice.val !== 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\nstate.push(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\nstate.pop();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state);\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\nreturn state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\nres.push([...state]);\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\nreturn choice !== null &amp;&amp; choice.val !== 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\nstate.push(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\nstate.pop();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\nstate: TreeNode[],\nchoices: TreeNode[],\nres: TreeNode[][]\n): void {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice.left, choice.right], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state);\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List&lt;TreeNode&gt; state) {\nreturn state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\nres.add(List.from(state));\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\nreturn choice != null &amp;&amp; choice.val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\nstate.add(choice!);\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\nstate.removeLast();\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(\nList&lt;TreeNode&gt; state,\nList&lt;TreeNode?&gt; choices,\nList&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution(state)) {\n// \u8bb0\u5f55\u89e3\nrecordSolution(state, res);\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (TreeNode? choice in choices) {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(state, choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(state, choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, [choice!.left, choice.right], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice(state, choice);\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\nreturn !state.is_empty() &amp;&amp; state.get(state.len() - 1).unwrap().borrow().val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nfn record_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;) {\nres.push(state.clone());\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; bool {\nreturn choice.borrow().val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\nstate.push(choice);\n}\n/* \u6062\u590d\u72b6\u6001 */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\nstate.remove(state.len() - 1);\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfn backtrack(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choices: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif is_solution(state) {\n// \u8bb0\u5f55\u89e3\nrecord_solution(state, res);\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif is_valid(state, choice.clone()) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmake_choice(state, choice.clone());\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, &amp;mut vec![choice.borrow().left.clone().unwrap(), choice.borrow().right.clone().unwrap()], res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundo_choice(state, choice.clone());\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(void) {\nreturn pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(void) {\nfor (int i = 0; i &lt; pathSize; i++) {\nres[resSize][i] = path[i];\n}\nresSize++;\n}\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\nreturn choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(TreeNode *choice) {\npath[pathSize++] = choice;\n}\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(void) {\npathSize--;\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n// \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\nif (isSolution()) {\n// \u8bb0\u5f55\u89e3\nrecordSolution();\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; 2; i++) {\nTreeNode *choice = choices[i];\n// \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\nif (isValid(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nmakeChoice(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nTreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\nbacktrack(nextChoices);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nundoChoice();\n}\n}\n}\n</code></pre> preorder_traversal_iii_template.zig<pre><code>[class]{}-[func]{isSolution}\n[class]{}-[func]{recordSolution}\n[class]{}-[func]{isValid}\n[class]{}-[func]{makeChoice}\n[class]{}-[func]{undoChoice}\n[class]{}-[func]{backtrack}\n</code></pre> <p>\u6839\u636e\u9898\u610f\uff0c\u6211\u4eec\u5728\u627e\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u540e\u5e94\u8be5\u7ee7\u7eed\u641c\u7d22\uff0c\u56e0\u6b64\u9700\u8981\u5c06\u8bb0\u5f55\u89e3\u4e4b\u540e\u7684 <code>return</code> \u8bed\u53e5\u5220\u9664\u3002\u56fe 13-4 \u5bf9\u6bd4\u4e86\u4fdd\u7559\u6216\u5220\u9664 <code>return</code> \u8bed\u53e5\u7684\u641c\u7d22\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-4 \u00a0 \u4fdd\u7559\u4e0e\u5220\u9664 return \u7684\u641c\u7d22\u8fc7\u7a0b\u5bf9\u6bd4 </p> <p>\u76f8\u6bd4\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u4ee3\u7801\u5b9e\u73b0\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u4ee3\u7801\u5b9e\u73b0\u867d\u7136\u663e\u5f97\u5570\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u56de\u6eaf\u95ee\u9898\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u53ea\u9700\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49 <code>state</code> \u548c <code>choices</code> \uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u5373\u53ef\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u672f\u8bed","text":"<p>\u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002</p> <p> \u8868 13-1 \u00a0 \u5e38\u89c1\u7684\u56de\u6eaf\u7b97\u6cd5\u672f\u8bed </p> \u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u4e2a\u6216\u591a\u4e2a \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u6240\u6709\u8def\u5f84 \u7ea6\u675f\u6761\u4ef6\uff08constraint\uff09 \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u8282\u70b9 \\(3\\) \u72b6\u6001\uff08state\uff09 \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373 <code>path</code> \u8282\u70b9\u5217\u8868 \u5c1d\u8bd5\uff08attempt\uff09 \u5c1d\u8bd5\u662f\u6839\u636e\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u8282\u70b9\u3001\u7ed3\u675f\u8282\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22 <p>Tip</p> <p>\u95ee\u9898\u3001\u89e3\u3001\u72b6\u6001\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002</p> <p>\u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002</p> <ul> <li>\u65f6\u95f4\uff1a\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u904d\u5386\u72b6\u6001\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u6307\u6570\u9636\u6216\u9636\u4e58\u9636\u3002</li> <li>\u7a7a\u95f4\uff1a\u5728\u9012\u5f52\u8c03\u7528\u4e2d\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5f53\u6df1\u5ea6\u5f88\u5927\u65f6\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u5f88\u5927\u3002</li> </ul> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u4f18\u5316\u6548\u7387\uff0c\u5e38\u89c1\u7684\u6548\u7387\u4f18\u5316\u65b9\u6cd5\u6709\u4e24\u79cd\u3002</p> <ul> <li>\u526a\u679d\uff1a\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002</li> <li>\u542f\u53d1\u5f0f\u641c\u7d22\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/#1316","title":"13.1.6 \u00a0 \u56de\u6eaf\u5178\u578b\u4f8b\u9898","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002</p> <p>\u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>\u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u6839\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002</li> </ul> <p>\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002</p> <ul> <li>\\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002</li> <li>\u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002</li> <li>\u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002</li> </ul> <p>\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002</li> <li>\u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u4f18\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff0c\u4ee5\u8fbe\u5230\u66f4\u9ad8\u7684\u65f6\u95f4\u6548\u7387\u3002</li> <li>\u65c5\u884c\u5546\u662f\u4e00\u4e2a\u8457\u540d\u7684 NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\uff0c\u53ef\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_backtracking/n_queens_problem/","title":"13.4 \u00a0 N \u7687\u540e\u95ee\u9898","text":"<p>Question</p> <p>\u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u540c\u5904\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u6761\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002</p> <p>\u5982\u56fe 13-15 \u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9 <code>choices</code> \u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001 <code>state</code> \u3002</p> <p></p> <p> \u56fe 13-15 \u00a0 4 \u7687\u540e\u95ee\u9898\u7684\u89e3 </p> <p>\u56fe 13-16 \u5c55\u793a\u4e86\u672c\u9898\u7684\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u5bf9\u89d2\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf <code>\\</code> \u548c\u6b21\u5bf9\u89d2\u7ebf <code>/</code> \u4e24\u79cd\u3002</p> <p></p> <p> \u56fe 13-16 \u00a0 n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6 </p>"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"<p>\u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002</p> <p>\u56fe 13-17 \u6240\u793a\u4e3a \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u753b\u5e45\u9650\u5236\uff0c\u56fe 13-17 \u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u4e2a\u641c\u7d22\u5206\u652f\uff0c\u5e76\u4e14\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u8fdb\u884c\u4e86\u526a\u679d\u3002</p> <p></p> <p> \u56fe 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002</p>"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u4e0e\u5bf9\u89d2\u7ebf\u526a\u679d","text":"<p>\u4e3a\u4e86\u6ee1\u8db3\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4 <code>cols</code> \u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7 <code>cols</code> \u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u8fdb\u884c\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0 <code>cols</code> \u7684\u72b6\u6001\u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\((row, col)\\) \uff0c\u9009\u5b9a\u77e9\u9635\u4e2d\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u4e3a\u6052\u5b9a\u503c\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5219\u5b83\u4eec\u4e00\u5b9a\u5904\u5728\u540c\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002\u5229\u7528\u8be5\u89c4\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u56fe 13-18 \u6240\u793a\u7684\u6570\u7ec4 <code>diags1</code> \u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002</p> <p>\u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u540c\u6837\u4e5f\u53ef\u4ee5\u501f\u52a9\u6570\u7ec4 <code>diags2</code> \u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002</p> <p></p> <p> \u56fe 13-18 \u00a0 \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f </p>"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u8bf7\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635\u4e2d \\(row - col\\) \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u8303\u56f4\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7684\u6570\u91cf\u90fd\u4e3a \\(2n - 1\\) \uff0c\u5373\u6570\u7ec4 <code>diags1</code> \u548c <code>diags2</code> \u7684\u957f\u5ea6\u90fd\u4e3a \\(2n - 1\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig n_queens.py<pre><code>def backtrack(\nrow: int,\nn: int,\nstate: list[list[str]],\nres: list[list[list[str]]],\ncols: list[bool],\ndiags1: list[bool],\ndiags2: list[bool],\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e\"\"\"\n# \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n:\nres.append([list(row) for row in state])\nreturn\n# \u904d\u5386\u6240\u6709\u5217\nfor col in range(n):\n# \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 = row - col + n - 1\ndiag2 = row + col\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif not cols[col] and not diags1[diag1] and not diags2[diag2]:\n# \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\"\ncols[col] = diags1[diag1] = diags2[diag2] = True\n# \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2)\n# \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\"\ncols[col] = diags1[diag1] = diags2[diag2] = False\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n\"\"\"\u6c42\u89e3 N \u7687\u540e\"\"\"\n# \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nstate = [[\"#\" for _ in range(n)] for _ in range(n)]\ncols = [False] * n  # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\ndiags1 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\ndiags2 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nres = []\nbacktrack(0, n, state, res, cols, diags1, diags2)\nreturn res\n</code></pre> n_queens.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\nvector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col &lt; n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\";\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\";\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nvector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\nvector&lt;bool&gt; cols(n, false);           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nvector&lt;bool&gt; diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nvector&lt;bool&gt; diags2(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n</code></pre> n_queens.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\nboolean[] cols, boolean[] diags1, boolean[] diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nList&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\nfor (List&lt;String&gt; sRow : state) {\ncopyState.add(new ArrayList&lt;&gt;(sRow));\n}\nres.add(copyState);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col &lt; n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate.get(row).set(col, \"Q\");\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate.get(row).set(col, \"#\");\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nList&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\nfor (int i = 0; i &lt; n; i++) {\nList&lt;String&gt; row = new ArrayList&lt;&gt;();\nfor (int j = 0; j &lt; n; j++) {\nrow.add(\"#\");\n}\nstate.add(row);\n}\nboolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nboolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nboolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n</code></pre> n_queens.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\nbool[] cols, bool[] diags1, bool[] diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nList&lt;List&lt;string&gt;&gt; copyState = [];\nforeach (List&lt;string&gt; sRow in state) {\ncopyState.Add(new List&lt;string&gt;(sRow));\n}\nres.Add(copyState);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col &lt; n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\";\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nBacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\";\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nList&lt;List&lt;string&gt;&gt; state = [];\nfor (int i = 0; i &lt; n; i++) {\nList&lt;string&gt; row = [];\nfor (int j = 0; j &lt; n; j++) {\nrow.Add(\"#\");\n}\nstate.Add(row);\n}\nbool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nbool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nbool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\nBacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n</code></pre> n_queens.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nnewState := make([][]string, len(*state))\nfor i, _ := range newState {\nnewState[i] = make([]string, len((*state)[0]))\ncopy(newState[i], (*state)[i])\n}\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col := 0; col &lt; n; col++ {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 := row - col + n - 1\ndiag2 := row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n(*state)[row][col] = \"Q\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row+1, n, state, res, cols, diags1, diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n(*state)[row][col] = \"#\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n}\n}\n}\n/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nnewState := make([][]string, len(*state))\nfor i, _ := range newState {\nnewState[i] = make([]string, len((*state)[0]))\ncopy(newState[i], (*state)[i])\n}\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col := 0; col &lt; n; col++ {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\ndiag1 := row - col + n - 1\ndiag2 := row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n(*state)[row][col] = \"Q\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row+1, n, state, res, cols, diags1, diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n(*state)[row][col] = \"#\"\n(*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n}\n}\n}\nfunc nQueens(n int) [][][]string {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nstate := make([][]string, n)\nfor i := 0; i &lt; n; i++ {\nrow := make([]string, n)\nfor i := 0; i &lt; n; i++ {\nrow[i] = \"#\"\n}\nstate[i] = row\n}\n// \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\ncols := make([]bool, n)\ndiags1 := make([]bool, 2*n-1)\ndiags2 := make([]bool, 2*n-1)\nres := make([][][]string, 0)\nbacktrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\nreturn res\n}\n</code></pre> n_queens.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col in 0 ..&lt; n {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nlet diag1 = row - col + n - 1\nlet diag2 = row + col\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\"\ncols[col] = true\ndiags1[diag1] = true\ndiags2[diag2] = true\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\"\ncols[col] = false\ndiags1[diag1] = false\ndiags2[diag2] = false\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nvar state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\nvar cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nvar diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nvar diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nvar res: [[[String]]] = []\nbacktrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\nreturn res\n}\n</code></pre> n_queens.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row === n) {\nres.push(state.map((row) =&gt; row.slice()));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (let col = 0; col &lt; n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nconst diag1 = row - col + n - 1;\nconst diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = 'Q';\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = '#';\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nconst state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\nconst cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nconst diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nconst diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nconst res = [];\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n</code></pre> n_queens.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(\nrow: number,\nn: number,\nstate: string[][],\nres: string[][][],\ncols: boolean[],\ndiags1: boolean[],\ndiags2: boolean[]\n): void {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row === n) {\nres.push(state.map((row) =&gt; row.slice()));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (let col = 0; col &lt; n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nconst diag1 = row - col + n - 1;\nconst diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = 'Q';\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = '#';\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nconst state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\nconst cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nconst diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nconst diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nconst res: string[][][] = [];\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n</code></pre> n_queens.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(\nint row,\nint n,\nList&lt;List&lt;String&gt;&gt; state,\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\nList&lt;bool&gt; cols,\nList&lt;bool&gt; diags1,\nList&lt;bool&gt; diags2,\n) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nList&lt;List&lt;String&gt;&gt; copyState = [];\nfor (List&lt;String&gt; sRow in state) {\ncopyState.add(List.from(sRow));\n}\nres.add(copyState);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col &lt; n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = \"Q\";\ncols[col] = true;\ndiags1[diag1] = true;\ndiags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = \"#\";\ncols[col] = false;\ndiags1[diag1] = false;\ndiags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nList&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\nList&lt;bool&gt; cols = List.filled(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nList&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nList&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\nbacktrack(0, n, state, res, cols, diags1, diags2);\nreturn res;\n}\n</code></pre> n_queens.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfn backtrack(row: usize, n: usize, state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;, res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\ncols: &amp;mut [bool], diags1: &amp;mut [bool], diags2: &amp;mut [bool]) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif row == n {\nlet mut copy_state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\nfor s_row in state.clone() {\ncopy_state.push(s_row);\n}\nres.push(copy_state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor col in 0..n {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nlet diag1 = row + n - 1 - col;\nlet diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate.get_mut(row).unwrap()[col] = \"Q\".into();\n(cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate.get_mut(row).unwrap()[col] = \"#\".into();\n(cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nlet mut state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\nfor _ in 0..n {\nlet mut row: Vec&lt;String&gt; = Vec::new();\nfor _ in 0..n {\nrow.push(\"#\".into());\n}\nstate.push(row);\n}\nlet mut cols = vec![false; n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nlet mut diags1 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nlet mut diags2 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nlet mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\nbacktrack(0, n, &amp;mut state, &amp;mut res, &amp;mut cols, &amp;mut diags1, &amp;mut diags2);\nres\n}\n</code></pre> n_queens.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\nbool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n// \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (row == n) {\nres[*resSize] = (char **)malloc(sizeof(char *) * n);\nfor (int i = 0; i &lt; n; ++i) {\nres[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\nstrcpy(res[*resSize][i], state[i]);\n}\n(*resSize)++;\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u5217\nfor (int col = 0; col &lt; n; col++) {\n// \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\nint diag1 = row - col + n - 1;\nint diag2 = row + col;\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\nif (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n// \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\nstate[row][col] = 'Q';\ncols[col] = diags1[diag1] = diags2[diag2] = true;\n// \u653e\u7f6e\u4e0b\u4e00\u884c\nbacktrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n// \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\nstate[row][col] = '#';\ncols[col] = diags1[diag1] = diags2[diag2] = false;\n}\n}\n}\n/* \u6c42\u89e3 N \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\nchar state[MAX_SIZE][MAX_SIZE];\n// \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\nfor (int i = 0; i &lt; n; ++i) {\nfor (int j = 0; j &lt; n; ++j) {\nstate[i][j] = '#';\n}\nstate[i][n] = '\\0';\n}\nbool cols[MAX_SIZE] = {false};           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\nbool diags1[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nbool diags2[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\nchar ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n*returnSize = 0;\nbacktrack(0, n, state, res, returnSize, cols, diags1, diags2);\nreturn res;\n}\n</code></pre> n_queens.zig<pre><code>[class]{}-[func]{backtrack}\n[class]{}-[func]{nQueens}\n</code></pre> <p>\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u4e2a\u9009\u62e9\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\u6570\u7ec4 <code>state</code> \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0c\u6570\u7ec4 <code>cols</code>\u3001<code>diags1</code> \u548c <code>diags2</code> \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"<p>\u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u8868 13-2 \u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u636e\uff0c\u5305\u62ec\u8f93\u5165\u6570\u7ec4\u548c\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002</p> <p> \u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b </p> \u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u65e0\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \uff0c\u518d\u9009\u62e9 \\(3\\) \uff0c\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002</p> <p>\u4ece\u56de\u6eaf\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408 <code>choices</code> \u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001 <code>state</code> \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u8bf7\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64 <code>state</code> \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5e94\u8be5\u662f\u552f\u4e00\u7684\u3002</p> <p>\u5982\u56fe 13-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001 <code>state</code> \u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-5 \u00a0 \u5168\u6392\u5217\u7684\u9012\u5f52\u6811 </p>"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u590d\u9009\u62e9\u526a\u679d","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u6211\u4eec\u8003\u8651\u5f15\u5165\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4 <code>selected</code> \uff0c\u5176\u4e2d <code>selected[i]</code> \u8868\u793a <code>choices[i]</code> \u662f\u5426\u5df2\u88ab\u9009\u62e9\uff0c\u5e76\u57fa\u4e8e\u5b83\u5b9e\u73b0\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002</p> <ul> <li>\u5728\u505a\u51fa\u9009\u62e9 <code>choice[i]</code> \u540e\uff0c\u6211\u4eec\u5c31\u5c06 <code>selected[i]</code> \u8d4b\u503c\u4e3a \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9009\u62e9\u3002</li> <li>\u904d\u5386\u9009\u62e9\u5217\u8868 <code>choices</code> \u65f6\uff0c\u8df3\u8fc7\u6240\u6709\u5df2\u88ab\u9009\u62e9\u7684\u8282\u70b9\uff0c\u5373\u526a\u679d\u3002</li> </ul> <p>\u5982\u56fe 13-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002</p> <p></p> <p> \u56fe 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 13-6 \u53d1\u73b0\uff0c\u8be5\u526a\u679d\u64cd\u4f5c\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u51cf\u5c0f\u81f3 \\(O(n!)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u6574\u4f53\u4ee3\u7801\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u5b83\u4eec\u5c55\u5f00\u5728 <code>backtrack()</code> \u51fd\u6570\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_i.py<pre><code>def backtrack(\nstate: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n# \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(state) == len(choices):\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i, choice in enumerate(choices):\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif not selected[i]:\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = True\nstate.append(choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = False\nstate.pop()\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n\"\"\"\u5168\u6392\u5217 I\"\"\"\nres = []\nbacktrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\nreturn res\n</code></pre> permutations_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.size()) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choices.size(); i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push_back(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop_back();\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\nvector&lt;int&gt; state;\nvector&lt;bool&gt; selected(nums.size(), false);\nvector&lt;vector&lt;int&gt;&gt; res;\nbacktrack(state, nums, selected, res);\nreturn res;\n}\n</code></pre> permutations_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.length) {\nres.add(new ArrayList&lt;Integer&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.size() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\nList&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\nbacktrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\nreturn res;\n}\n</code></pre> permutations_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.Count == choices.Length) {\nres.Add(new List&lt;int&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choices.Length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.Add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nBacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.RemoveAt(state.Count - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\nList&lt;List&lt;int&gt;&gt; res = [];\nBacktrack([], nums, new bool[nums.Length], res);\nreturn res;\n}\n</code></pre> permutations_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(*state) == len(*choices) {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i := 0; i &lt; len(*choices); i++ {\nchoice := (*choices)[i]\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif !(*selected)[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n(*selected)[i] = true\n*state = append(*state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackI(state, choices, selected, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n(*selected)[i] = false\n*state = (*state)[:len(*state)-1]\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\nres := make([][]int, 0)\nstate := make([]int, 0)\nselected := make([]bool, len(nums))\nbacktrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\nreturn res\n}\n</code></pre> permutations_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.count == choices.count {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (i, choice) in choices.enumerated() {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif !selected[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true\nstate.append(choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false\nstate.removeLast()\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\nvar state: [Int] = []\nvar selected = Array(repeating: false, count: nums.count)\nvar res: [[Int]] = []\nbacktrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\nreturn res\n}\n</code></pre> permutations_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nchoices.forEach((choice, i) =&gt; {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\nconst res = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n</code></pre> permutations_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\nstate: number[],\nchoices: number[],\nselected: boolean[],\nres: number[][]\n): void {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nchoices.forEach((choice, i) =&gt; {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\nconst res: number[][] = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n</code></pre> permutations_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\nList&lt;int&gt; state,\nList&lt;int&gt; choices,\nList&lt;bool&gt; selected,\nList&lt;List&lt;int&gt;&gt; res,\n) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length == choices.length) {\nres.add(List.from(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.removeLast();\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\nList&lt;List&lt;int&gt;&gt; res = [];\nbacktrack([], nums, List.filled(nums.length, false), res);\nreturn res;\n}\n</code></pre> permutations_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.len() == choices.len() {\nres.push(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i in 0..choices.len() {\nlet choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif !selected[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state.clone(), choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.len() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\nlet mut res = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nbacktrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\nres\n}\n</code></pre> permutations_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (stateSize == choicesSize) {\nres[*resSize] = (int *)malloc(choicesSize * sizeof(int));\nfor (int i = 0; i &lt; choicesSize; i++) {\nres[*resSize][i] = state[i];\n}\n(*resSize)++;\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choicesSize; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\nif (!selected[i]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nselected[i] = true;\nstate[stateSize] = choice;\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\n}\n}\n}\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\nint *state = (int *)malloc(numsSize * sizeof(int));\nbool *selected = (bool *)malloc(numsSize * sizeof(bool));\nfor (int i = 0; i &lt; numsSize; i++) {\nselected[i] = false;\n}\nint **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n*returnSize = 0;\nbacktrack(state, 0, nums, numsSize, selected, res, returnSize);\nfree(state);\nfree(selected);\nreturn res;\n}\n</code></pre> permutations_i.zig<pre><code>[class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsI}\n</code></pre>"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u8651\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002</p> <p>\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u5143\u7d20 \\(1\\) \uff0c\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a \\(1\\) \u8bb0\u4e3a \\(\\hat{1}\\) \u3002</p> <p>\u5982\u56fe 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u590d\u7684\u3002</p> <p></p> <p> \u56fe 13-7 \u00a0 \u91cd\u590d\u6392\u5217 </p> <p>\u90a3\u4e48\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u6ca1\u6709\u5fc5\u8981\uff0c\u5e94\u5f53\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u89c2\u5bdf\u56fe 13-8 \uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u4e4b\u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u5e94\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u4fdd\u8bc1\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002</p> <p></p> <p> \u56fe 13-8 \u00a0 \u91cd\u590d\u6392\u5217\u526a\u679d </p>"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>duplicated</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_ii.py<pre><code>def backtrack(\nstate: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n# \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(state) == len(choices):\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nduplicated = set[int]()\nfor i, choice in enumerate(choices):\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif not selected[i] and choice not in duplicated:\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice)  # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = True\nstate.append(choice)\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = False\nstate.pop()\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n\"\"\"\u5168\u6392\u5217 II\"\"\"\nres = []\nbacktrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\nreturn res\n</code></pre> permutations_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.size()) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nunordered_set&lt;int&gt; duplicated;\nfor (int i = 0; i &lt; choices.size(); i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push_back(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop_back();\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\nvector&lt;int&gt; state;\nvector&lt;bool&gt; selected(nums.size(), false);\nvector&lt;vector&lt;int&gt;&gt; res;\nbacktrack(state, nums, selected, res);\nreturn res;\n}\n</code></pre> permutations_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.size() == choices.length) {\nres.add(new ArrayList&lt;Integer&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nSet&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\nfor (int i = 0; i &lt; choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.size() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\nList&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\nbacktrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\nreturn res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.Count == choices.Length) {\nres.Add(new List&lt;int&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nHashSet&lt;int&gt; duplicated = [];\nfor (int i = 0; i &lt; choices.Length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.Add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nBacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.RemoveAt(state.Count - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\nList&lt;List&lt;int&gt;&gt; res = [];\nBacktrack([], nums, new bool[nums.Length], res);\nreturn res;\n}\n</code></pre> permutations_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif len(*state) == len(*choices) {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nduplicated := make(map[int]struct{}, 0)\nfor i := 0; i &lt; len(*choices); i++ {\nchoice := (*choices)[i]\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n// \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nduplicated[choice] = struct{}{}\n(*selected)[i] = true\n*state = append(*state, choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackI(state, choices, selected, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n(*selected)[i] = false\n*state = (*state)[:len(*state)-1]\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\nres := make([][]int, 0)\nstate := make([]int, 0)\nselected := make([]bool, len(nums))\nbacktrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\nreturn res\n}\n</code></pre> permutations_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.count == choices.count {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nvar duplicated: Set&lt;Int&gt; = []\nfor (i, choice) in choices.enumerated() {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !selected[i], !duplicated.contains(choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true\nstate.append(choice)\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false\nstate.removeLast()\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\nvar state: [Int] = []\nvar selected = Array(repeating: false, count: nums.count)\nvar res: [[Int]] = []\nbacktrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\nreturn res\n}\n</code></pre> permutations_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nconst duplicated = new Set();\nchoices.forEach((choice, i) =&gt; {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\nconst res = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\nstate: number[],\nchoices: number[],\nselected: boolean[],\nres: number[][]\n): void {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length === choices.length) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nconst duplicated = new Set();\nchoices.forEach((choice, i) =&gt; {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.pop();\n}\n});\n}\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\nconst res: number[][] = [];\nbacktrack([], nums, Array(nums.length).fill(false), res);\nreturn res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\nList&lt;int&gt; state,\nList&lt;int&gt; choices,\nList&lt;bool&gt; selected,\nList&lt;List&lt;int&gt;&gt; res,\n) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (state.length == choices.length) {\nres.add(List.from(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nSet&lt;int&gt; duplicated = {};\nfor (int i = 0; i &lt; choices.length; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.add(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.removeLast();\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\nList&lt;List&lt;int&gt;&gt; res = [];\nbacktrack([], nums, List.filled(nums.length, false), res);\nreturn res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif state.len() == choices.len() {\nres.push(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nlet mut duplicated = HashSet::&lt;i32&gt;::new();\nfor i in 0..choices.len() {\nlet choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated.insert(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate.push(choice);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state.clone(), choices, selected, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\nstate.remove(state.len() - 1);\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\nlet mut res = Vec::new();\nbacktrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\nres\n}\n</code></pre> permutations_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n// \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\nif (stateSize == choicesSize) {\nres[*resSize] = (int *)malloc(choicesSize * sizeof(int));\nfor (int i = 0; i &lt; choicesSize; i++) {\nres[*resSize][i] = state[i];\n}\n(*resSize)++;\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nbool duplicated[MAX_SIZE] = {false};\nfor (int i = 0; i &lt; choicesSize; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\nif (!selected[i] &amp;&amp; !duplicated[choice]) {\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nduplicated[choice] = true; // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\nselected[i] = true;\nstate[stateSize] = choice;\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nselected[i] = false;\n}\n}\n}\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\nint *state = (int *)malloc(numsSize * sizeof(int));\nbool *selected = (bool *)malloc(numsSize * sizeof(bool));\nfor (int i = 0; i &lt; numsSize; i++) {\nselected[i] = false;\n}\nint **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n*returnSize = 0;\nbacktrack(state, 0, nums, numsSize, selected, res, returnSize);\nfree(state);\nfree(selected);\nreturn res;\n}\n</code></pre> permutations_ii.zig<pre><code>[class]{}-[func]{backtrack}\n[class]{}-[func]{permutationsII}\n</code></pre> <p>\u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\)  \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002</p> <p>\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002<code>selected</code> \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2a <code>duplicated</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u4e24\u79cd\u526a\u679d\u5bf9\u6bd4","text":"<p>\u8bf7\u6ce8\u610f\uff0c\u867d\u7136 <code>selected</code> \u548c <code>duplicated</code> \u90fd\u7528\u4e8e\u526a\u679d\uff0c\u4f46\u4e24\u8005\u7684\u76ee\u6807\u4e0d\u540c\u3002</p> <ul> <li>\u91cd\u590d\u9009\u62e9\u526a\u679d\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a <code>selected</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u9632\u6b62 <code>choices</code> \u4e2d\u7684\u4efb\u4e00\u5143\u7d20\u5728 <code>state</code> \u4e2d\u91cd\u590d\u51fa\u73b0\u3002</li> <li>\u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u6bcf\u4e2a\u8c03\u7528\u7684 <code>backtrack</code> \u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2a <code>duplicated</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u672c\u8f6e\u904d\u5386\uff08<code>for</code> \u5faa\u73af\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u7684\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> </ul> <p>\u56fe 13-9 \u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-9 \u00a0 \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4 </p>"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u8f93\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u91cd\u590d\u9009\u53d6\u3002</li> <li>\u5b50\u96c6\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#1","title":"1. \u00a0 \u53c2\u8003\u5168\u6392\u5217\u89e3\u6cd5","text":"<p>\u7c7b\u4f3c\u4e8e\u5168\u6392\u5217\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002</p> <p>\u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u9009\u53d6\uff0c\u56e0\u6b64\u65e0\u987b\u501f\u52a9 <code>selected</code> \u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_i_naive.py<pre><code>def backtrack(\nstate: list[int],\ntarget: int,\ntotal: int,\nchoices: list[int],\nres: list[list[int]],\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif total == target:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i in range(len(choices)):\n# \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total + choices[i] &gt; target:\ncontinue\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\ntotal = 0  # \u5b50\u96c6\u548c\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res)\nreturn res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (size_t i = 0; i &lt; choices.size(); i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] &gt; target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\nvector&lt;int&gt; state;       // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0;           // \u5b50\u96c6\u548c\nvector&lt;vector&lt;int&gt;&gt; res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.add(new ArrayList&lt;&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choices.length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] &gt; target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\nList&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0; // \u5b50\u96c6\u548c\nList&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.Add(new List&lt;int&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choices.Length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] &gt; target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nBacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\nList&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0; // \u5b50\u96c6\u548c\nList&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nBacktrack(state, target, total, nums, res);\nreturn res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == total {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i := 0; i &lt; len(*choices); i++ {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total+(*choices)[i] &gt; target {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\ntotal := 0              // \u5b50\u96c6\u548c\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\nreturn res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif total == target {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i in stride(from: 0, to: choices.count, by: 1) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total + choices[i] &gt; target {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.append(choices[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.removeLast()\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\nvar state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nlet total = 0 // \u5b50\u96c6\u548c\nvar res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\nreturn res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total === target) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let i = 0; i &lt; choices.length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] &gt; target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\nconst state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nconst total = 0; // \u5b50\u96c6\u548c\nconst res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\nstate: number[],\ntarget: number,\ntotal: number,\nchoices: number[],\nres: number[][]\n): void {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total === target) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (let i = 0; i &lt; choices.length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] &gt; target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\nconst state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nconst total = 0; // \u5b50\u96c6\u548c\nconst res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\nList&lt;int&gt; state,\nint target,\nint total,\nList&lt;int&gt; choices,\nList&lt;List&lt;int&gt;&gt; res,\n) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nres.add(List.from(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choices.length; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] &gt; target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.removeLast();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\nList&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nint total = 0; // \u5143\u7d20\u548c\nList&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, res);\nreturn res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec&lt;i32&gt;, target: i32, total: i32, choices: &amp;[i32], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif total == target {\nres.push(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor i in 0..choices.len() {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif total + choices[i] &gt; target {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state.clone(), target, total + choices[i], choices, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\nlet state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nlet total = 0; // \u5b50\u96c6\u548c\nlet mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, total, nums, &amp;mut res);\nres\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (total == target) {\nfor (int i = 0; i &lt; stateSize; i++) {\nres[resSize][i] = state[i];\n}\nresColSizes[resSize++] = stateSize;\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; choicesSize; i++) {\n// \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\nif (total + choices[i] &gt; target) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\nstate[stateSize++] = choices[i];\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(target, total + choices[i], choices, choicesSize);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstateSize--;\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\nresSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6570\u91cf\u4e3a0\nbacktrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.zig<pre><code>[class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumINaive}\n</code></pre> <p>\u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u533a\u5206\u9009\u62e9\u987a\u5e8f\u3002\u5982\u56fe 13-10 \u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5bf9\u5e94\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-10 \u00a0 \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d </p> <p>\u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u4e24\u65b9\u9762\u539f\u56e0\u3002</p> <ul> <li>\u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53 <code>target</code> \u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002</li> <li>\u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u975e\u5e38\u8017\u65f6\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#2","title":"2. \u00a0 \u91cd\u590d\u5b50\u96c6\u526a\u679d","text":"<p>\u6211\u4eec\u8003\u8651\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u56fe 13-11 \uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u51b5\u3002</p> <ol> <li>\u5f53\u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) \u548c \\(4\\) \u65f6\uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\dots]\\) \u3002</li> <li>\u4e4b\u540e\uff0c\u5f53\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \u65f6\uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c <code>1.</code> \u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p>\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c42\u7684\u9009\u62e9\u90fd\u662f\u4ece\u5de6\u5230\u53f3\u88ab\u9010\u4e2a\u5c1d\u8bd5\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002</p> <ol> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002</li> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002</li> <li>\u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u4e0e\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p></p> <p> \u56fe 13-11 \u00a0 \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p> <p>\u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u4f1a\u9020\u6210\u91cd\u590d\uff0c\u5e94\u5f53\u526a\u679d\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf <code>start</code> \uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u59cb\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\u3002\u8fd9\u6837\u505a\u5c31\u53ef\u4ee5\u8ba9\u9009\u62e9\u5e8f\u5217\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4ece\u800c\u4fdd\u8bc1\u5b50\u96c6\u552f\u4e00\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4ee5\u4e0b\u4e24\u9879\u4f18\u5316\u3002</p> <ul> <li>\u5728\u5f00\u542f\u641c\u7d22\u524d\uff0c\u5148\u5c06\u6570\u7ec4 <code>nums</code> \u6392\u5e8f\u3002\u5728\u904d\u5386\u6240\u6709\u9009\u62e9\u65f6\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7 <code>target</code> \u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 <code>target</code> \u3002</li> <li>\u7701\u53bb\u5143\u7d20\u548c\u53d8\u91cf <code>total</code> \uff0c\u901a\u8fc7\u5728 <code>target</code> \u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53 <code>target</code> \u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_i.py<pre><code>def backtrack(\nstate: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\n# \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i in range(start, len(choices)):\n# \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n# \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] &lt; 0:\nbreak\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart = 0  # \u904d\u5386\u8d77\u59cb\u70b9\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res)\nreturn res\n</code></pre> subset_sum_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i &lt; choices.size(); i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\nvector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\nvector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(new ArrayList&lt;&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\nList&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.Add(new List&lt;int&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i &lt; choices.Length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nBacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\nList&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArray.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nBacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i := start; i &lt; len(*choices); i++ {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target-(*choices)[i] &lt; 0 {\nbreak\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart := 0              // \u904d\u5386\u8d77\u59cb\u70b9\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\nreturn res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i in stride(from: start, to: choices.count, by: 1) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] &lt; 0 {\nbreak\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.removeLast()\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\nvar state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nlet nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nlet start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\nvar res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\nreturn res\n}\n</code></pre> subset_sum_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target === 0) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (let i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\nconst state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nconst start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nconst res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\nstate: number[],\ntarget: number,\nchoices: number[],\nstart: number,\nres: number[][]\n): void {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target === 0) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (let i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\nconst state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nconst start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nconst res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\nList&lt;int&gt; state,\nint target,\nList&lt;int&gt; choices,\nint start,\nList&lt;List&lt;int&gt;&gt; res,\n) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(List.from(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.removeLast();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\nList&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec&lt;i32&gt;, target: i32, choices: &amp;[i32], start: usize, res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nres.push(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor i in start..choices.len() {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] &lt; 0 {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state.clone(), target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\nlet state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nlet start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nlet mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, &amp;mut res);\nres\n}\n</code></pre> subset_sum_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nfor (int i = 0; i &lt; stateSize; ++i) {\nres[resSize][i] = state[i];\n}\nresColSizes[resSize++] = stateSize;\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\nfor (int i = start; i &lt; choicesSize; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate[stateSize] = choices[i];\nstateSize++;\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(target - choices[i], choices, choicesSize, i);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstateSize--;\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\nqsort(nums, numsSize, sizeof(int), cmp); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0;                           // \u904d\u5386\u8d77\u59cb\u70b9\nbacktrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.zig<pre><code>[class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumI}\n</code></pre> <p>\u56fe 13-12 \u6240\u793a\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u8651\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002</p> <p>\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5728\u56fe 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u8fd9\u610f\u5473\u7740\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002</p> <p>\u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6bcf\u4e2a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf <code>start</code> \u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u65e2\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u5143\u7d20\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_ii.py<pre><code>def backtrack(\nstate: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n\"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n# \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0:\nres.append(list(state))\nreturn\n# \u904d\u5386\u6240\u6709\u9009\u62e9\n# \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n# \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i in range(start, len(choices)):\n# \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n# \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] &lt; 0:\nbreak\n# \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i &gt; start and choices[i] == choices[i - 1]:\ncontinue\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n# \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res)\n# \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop()\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n\"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\nstate = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart = 0  # \u904d\u5386\u8d77\u59cb\u70b9\nres = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res)\nreturn res\n</code></pre> subset_sum_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.push_back(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i &lt; choices.size(); i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push_back(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop_back();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\nvector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\nvector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(new ArrayList&lt;&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.remove(state.size() - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\nList&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.Add(new List&lt;int&gt;(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i &lt; choices.Length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.Add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nBacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.RemoveAt(state.Count - 1);\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\nList&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nArray.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nBacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nnewState := append([]int{}, *state...)\n*res = append(*res, newState)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i := start; i &lt; len(*choices); i++ {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target-(*choices)[i] &lt; 0 {\nbreak\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n*state = append(*state, (*choices)[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n*state = (*state)[:len(*state)-1]\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\nstate := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nsort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nstart := 0              // \u904d\u5386\u8d77\u59cb\u70b9\nres := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\nreturn res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nres.append(state)\nreturn\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i in stride(from: start, to: choices.count, by: 1) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] &lt; 0 {\nbreak\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i &gt; start, choices[i] == choices[i - 1] {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.append(choices[i])\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.removeLast()\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\nvar state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nlet nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nlet start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\nvar res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\nreturn res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target === 0) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (let i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\nconst state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nconst start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nconst res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\nstate: number[],\ntarget: number,\nchoices: number[],\nstart: number,\nres: number[][]\n): void {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target === 0) {\nres.push([...state]);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (let i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\nconst state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nconst start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nconst res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\nList&lt;int&gt; state,\nint target,\nList&lt;int&gt; choices,\nint start,\nList&lt;List&lt;int&gt;&gt; res,\n) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nres.add(List.from(state));\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i &lt; choices.length; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif (target - choices[i] &lt; 0) {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.add(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state, target - choices[i], choices, i + 1, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.removeLast();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\nList&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nint start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nList&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, res);\nreturn res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(mut state: Vec&lt;i32&gt;, target: i32, choices: &amp;[i32], start: usize, res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif target == 0 {\nres.push(state);\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor i in start..choices.len() {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n// \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\nif target - choices[i] &lt; 0 {\nbreak;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate.push(choices[i]);\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(state.clone(), target - choices[i], choices, i, res);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstate.pop();\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\nlet state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\nnums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nlet start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\nlet mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\nbacktrack(state, target, nums, start, &amp;mut res);\nres\n}\n</code></pre> subset_sum_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n// \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\nif (target == 0) {\nfor (int i = 0; i &lt; stateSize; i++) {\nres[resSize][i] = state[i];\n}\nresColSizes[resSize++] = stateSize;\nreturn;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\n// \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n// \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\nfor (int i = start; i &lt; choicesSize; i++) {\n// \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\nif (target - choices[i] &lt; 0) {\ncontinue;\n}\n// \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\nif (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\nstate[stateSize] = choices[i];\nstateSize++;\n// \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\nbacktrack(target - choices[i], choices, choicesSize, i + 1);\n// \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\nstateSize--;\n}\n}\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n// \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\nqsort(nums, numsSize, sizeof(int), cmp);\n// \u5f00\u59cb\u56de\u6eaf\nbacktrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.zig<pre><code>[class]{}-[func]{backtrack}\n[class]{}-[func]{subsetSumII}\n</code></pre> <p>\u56fe 13-14 \u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002</p> <p></p> <p> \u56fe 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002</li> <li>\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002</li> <li>\u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> <li>\u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u59cb\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002</li> <li>\u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002</li> <li>\\(n\\) \u7687\u540e\u95ee\u9898\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</li> <li>\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002</li> </ul>"},{"location":"chapter_backtracking/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u600e\u4e48\u7406\u89e3\u56de\u6eaf\u548c\u9012\u5f52\u7684\u5173\u7cfb\uff1f</p> <p>\u603b\u7684\u6765\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u79cd\u201c\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u9012\u5f52\u66f4\u50cf\u662f\u4e00\u4e2a\u201c\u5de5\u5177\u201d\u3002</p> <ul> <li>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u9012\u5f52\u7684\u5e94\u7528\u573a\u666f\u4e4b\u4e00\uff0c\u662f\u9012\u5f52\u5728\u641c\u7d22\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002</li> <li>\u9012\u5f52\u7684\u7ed3\u6784\u4f53\u73b0\u4e86\u201c\u5b50\u95ee\u9898\u5206\u89e3\u201d\u7684\u89e3\u9898\u8303\u5f0f\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\uff08\u8bb0\u5fc6\u5316\u9012\u5f52\uff09\u7b49\u95ee\u9898\u3002</li> </ul>"},{"location":"chapter_computational_complexity/","title":"\u7b2c 2 \u7ae0 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>Abstract</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u72b9\u5982\u6d69\u701a\u7684\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u65f6\u7a7a\u5411\u5bfc\u3002</p> <p>\u5b83\u5e26\u9886\u6211\u4eec\u5728\u65f6\u95f4\u4e0e\u7a7a\u95f4\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5bfb\u627e\u66f4\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002</p>"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</li> <li>2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52</li> <li>2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6</li> <li>2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6</li> <li>2.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52","text":"<p>\u5728\u7b97\u6cd5\u4e2d\uff0c\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u662f\u5f88\u5e38\u89c1\u7684\uff0c\u5176\u4e0e\u590d\u6742\u5ea6\u5206\u6790\u606f\u606f\u76f8\u5173\u3002\u56e0\u6b64\uff0c\u5728\u5c55\u5f00\u4ecb\u7ecd\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5e8f\u4e2d\u5b9e\u73b0\u91cd\u590d\u6267\u884c\u4efb\u52a1\uff0c\u5373\u4e24\u79cd\u57fa\u672c\u7684\u7a0b\u5e8f\u63a7\u5236\u7ed3\u6784\uff1a\u8fed\u4ee3\u3001\u9012\u5f52\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"<p>\u300c\u8fed\u4ee3 iteration\u300d\u662f\u4e00\u79cd\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u7684\u63a7\u5236\u7ed3\u6784\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5e8f\u4f1a\u5728\u6ee1\u8db3\u4e00\u5b9a\u7684\u6761\u4ef6\u4e0b\u91cd\u590d\u6267\u884c\u67d0\u6bb5\u4ee3\u7801\uff0c\u76f4\u5230\u8fd9\u4e2a\u6761\u4ef6\u4e0d\u518d\u6ee1\u8db3\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u5faa\u73af","text":"<p><code>for</code> \u5faa\u73af\u662f\u6700\u5e38\u89c1\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9002\u5408\u5728\u9884\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6570\u65f6\u4f7f\u7528\u3002</p> <p>\u4ee5\u4e0b\u51fd\u6570\u57fa\u4e8e <code>for</code> \u5faa\u73af\u5b9e\u73b0\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7ed3\u679c\u4f7f\u7528\u53d8\u91cf <code>res</code> \u8bb0\u5f55\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2d <code>range(a, b)</code> \u5bf9\u5e94\u7684\u533a\u95f4\u662f\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\uff0c\u5bf9\u5e94\u7684\u904d\u5386\u8303\u56f4\u4e3a \\(a, a + 1, \\dots, b-1\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n\"\"\"for \u5faa\u73af\"\"\"\nres = 0\n# \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor i in range(1, n + 1):\nres += i\nreturn res\n</code></pre> iteration.cpp<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\nint res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; ++i) {\nres += i;\n}\nreturn res;\n}\n</code></pre> iteration.java<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\nint res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\nres += i;\n}\nreturn res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u5faa\u73af */\nint ForLoop(int n) {\nint res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\nres += i;\n}\nreturn res;\n}\n</code></pre> iteration.go<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n int) int {\nres := 0\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor i := 1; i &lt;= n; i++ {\nres += i\n}\nreturn res\n}\n</code></pre> iteration.swift<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n: Int) -&gt; Int {\nvar res = 0\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor i in 1 ... n {\nres += i\n}\nreturn res\n}\n</code></pre> iteration.js<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n) {\nlet res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (let i = 1; i &lt;= n; i++) {\nres += i;\n}\nreturn res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\nlet res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (let i = 1; i &lt;= n; i++) {\nres += i;\n}\nreturn res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\nint res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\nres += i;\n}\nreturn res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u5faa\u73af */\nfn for_loop(n: i32) -&gt; i32 {\nlet mut res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor i in 1..=n {\nres += i;\n}\nres\n} </code></pre> iteration.c<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\nint res = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\nres += i;\n}\nreturn res;\n}\n</code></pre> iteration.zig<pre><code>// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\nvar res: i32 = 0;\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor (1..n+1) |i| {\nres = res + @as(i32, @intCast(i));\n}\nreturn res;\n} </code></pre> <p>\u56fe 2-1 \u662f\u8be5\u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-1 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u6b64\u6c42\u548c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u6210\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u8fd9\u4e2a\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u76f8\u5173\u5185\u5bb9\u5c06\u4f1a\u5728\u4e0b\u4e00\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u5faa\u73af","text":"<p>\u4e0e <code>for</code> \u5faa\u73af\u7c7b\u4f3c\uff0c<code>while</code> \u5faa\u73af\u4e5f\u662f\u4e00\u79cd\u5b9e\u73b0\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u7a0b\u5e8f\u6bcf\u8f6e\u90fd\u4f1a\u5148\u68c0\u67e5\u6761\u4ef6\uff0c\u5982\u679c\u6761\u4ef6\u4e3a\u771f\uff0c\u5219\u7ee7\u7eed\u6267\u884c\uff0c\u5426\u5219\u5c31\u7ed3\u675f\u5faa\u73af\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u7528 <code>while</code> \u5faa\u73af\u6765\u5b9e\u73b0\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n\"\"\"while \u5faa\u73af\"\"\"\nres = 0\ni = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n# \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile i &lt;= n:\nres += i\ni += 1  # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\nreturn res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += i;\ni++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += i;\ni++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af */\nint WhileLoop(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += i;\ni += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\nres := 0\n// \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\ni := 1\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nfor i &lt;= n {\nres += i\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++\n}\nreturn res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -&gt; Int {\nvar res = 0\nvar i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile i &lt;= n {\nres += i\ni += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n) {\nlet res = 0;\nlet i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += i;\ni++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\nlet res = 0;\nlet i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += i;\ni++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += i;\ni++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af */\nfn while_loop(n: i32) -&gt; i32 {\nlet mut res = 0;\nlet mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile i &lt;= n {\nres += i;\ni += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nres\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += i;\ni++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n}\nreturn res;\n}\n</code></pre> iteration.zig<pre><code>// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\nvar res: i32 = 0;\nvar i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\nwhile (i &lt;= n) {\nres += @intCast(i);\ni += 1;\n}\nreturn res;\n}\n</code></pre> <p><code>while</code> \u5faa\u73af\u6bd4 <code>for</code> \u5faa\u73af\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u8bbe\u8ba1\u6761\u4ef6\u53d8\u91cf\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9aa4\u3002</p> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6761\u4ef6\u53d8\u91cf \\(i\\) \u6bcf\u8f6e\u8fdb\u884c\u4e24\u6b21\u66f4\u65b0\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528 <code>for</code> \u5faa\u73af\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n\"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\nres = 0\ni = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n# \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile i &lt;= n:\nres += i\n# \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni += 1\ni *= 2\nreturn res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile (i &lt;= n) {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++;\ni *= 2;\n}\nreturn res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile (i &lt;= n) {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++;\ni *= 2;\n}\nreturn res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 2, 4, 5...\nwhile (i &lt;= n) {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni += 1; i *= 2;\n}\nreturn res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\nres := 0\n// \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\ni := 1\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nfor i &lt;= n {\nres += i\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++\ni *= 2\n}\nreturn res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\nvar res = 0\nvar i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile i &lt;= n {\nres += i\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni += 1\ni *= 2\n}\nreturn res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\nlet res = 0;\nlet i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile (i &lt;= n) {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++;\ni *= 2;\n}\nreturn res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\nlet res = 0;\nlet i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile (i &lt;= n) {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++;\ni *= 2;\n}\nreturn res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile (i &lt;= n) {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++;\ni *= 2;\n}\nreturn res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\nlet mut res = 0;\nlet mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile i &lt;= n {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni += 1;\ni *= 2;\n}\nres\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\nint res = 0;\nint i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile (i &lt;= n) {\nres += i;\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni++;\ni *= 2;\n}\nreturn res;\n}\n</code></pre> iteration.zig<pre><code>//  while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\nvar res: i32 = 0;\nvar i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n// \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\nwhile (i &lt;= n) {\nres += @intCast(i);\n// \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\ni += 1;\ni *= 2;\n}\nreturn res;\n}\n</code></pre> <p>\u603b\u7684\u6765\u8bf4\uff0c<code>for</code> \u5faa\u73af\u7684\u4ee3\u7801\u66f4\u52a0\u7d27\u51d1\uff0c<code>while</code> \u5faa\u73af\u66f4\u52a0\u7075\u6d3b\uff0c\u4e24\u8005\u90fd\u53ef\u4ee5\u5b9e\u73b0\u8fed\u4ee3\u7ed3\u6784\u3002\u9009\u62e9\u4f7f\u7528\u54ea\u4e00\u4e2a\u5e94\u8be5\u6839\u636e\u7279\u5b9a\u95ee\u9898\u7684\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5d4c\u5957\u5faa\u73af","text":"<p>\u6211\u4eec\u53ef\u4ee5\u5728\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\u5185\u5d4c\u5957\u53e6\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\uff0c\u4e0b\u9762\u4ee5 <code>for</code> \u5faa\u73af\u4e3a\u4f8b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n\"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\nres = \"\"\n# \u5faa\u73af i = 1, 2, ..., n-1, n\nfor i in range(1, n + 1):\n# \u5faa\u73af j = 1, 2, ..., n-1, n\nfor j in range(1, n + 1):\nres += f\"({i}, {j}), \"\nreturn res\n</code></pre> iteration.cpp<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\nostringstream res;\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; ++i) {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (int j = 1; j &lt;= n; ++j) {\nres &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n}\n}\nreturn res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\nStringBuilder res = new StringBuilder();\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (int j = 1; j &lt;= n; j++) {\nres.append(\"(\" + i + \", \" + j + \"), \");\n}\n}\nreturn res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\nStringBuilder res = new();\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (int j = 1; j &lt;= n; j++) {\nres.Append($\"({i}, {j}), \");\n}\n}\nreturn res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\nres := \"\"\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor i := 1; i &lt;= n; i++ {\nfor j := 1; j &lt;= n; j++ {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nres += fmt.Sprintf(\"(%d, %d), \", i, j)\n}\n}\nreturn res\n}\n</code></pre> iteration.swift<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -&gt; String {\nvar res = \"\"\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor i in 1 ... n {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor j in 1 ... n {\nres.append(\"(\\(i), \\(j)), \")\n}\n}\nreturn res\n}\n</code></pre> iteration.js<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\nlet res = '';\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (let i = 1; i &lt;= n; i++) {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (let j = 1; j &lt;= n; j++) {\nres += `(${i}, ${j}), `;\n}\n}\nreturn res;\n}\n</code></pre> iteration.ts<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\nlet res = '';\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (let i = 1; i &lt;= n; i++) {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (let j = 1; j &lt;= n; j++) {\nres += `(${i}, ${j}), `;\n}\n}\nreturn res;\n}\n</code></pre> iteration.dart<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\nString res = \"\";\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (int j = 1; j &lt;= n; j++) {\nres += \"($i, $j), \";\n}\n}\nreturn res;\n}\n</code></pre> iteration.rs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -&gt; String {\nlet mut res = vec![];\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor i in 1..=n {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor j in 1..=n {\nres.push(format!(\"({}, {}), \", i, j));\n}\n}\nres.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n// n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\nint size = n * n * 26 + 1;\nchar *res = malloc(size * sizeof(char));\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (int i = 1; i &lt;= n; i++) {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (int j = 1; j &lt;= n; j++) {\nchar tmp[26];\nsnprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\nstrncat(res, tmp, size - strlen(res) - 1);\n}\n}\nreturn res;\n}\n</code></pre> iteration.zig<pre><code>// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\nvar res = std.ArrayList(u8).init(allocator);\ndefer res.deinit();\nvar buffer: [20]u8 = undefined;\n// \u5faa\u73af i = 1, 2, ..., n-1, n\nfor (1..n+1) |i| {\n// \u5faa\u73af j = 1, 2, ..., n-1, n\nfor (1..n+1) |j| {\nvar _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\ntry res.appendSlice(_str);\n}\n}\nreturn res.toOwnedSlice();\n}\n</code></pre> <p>\u56fe 2-2 \u662f\u8be5\u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-2 \u00a0 \u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u548c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u5173\u7cfb\u201d\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u6dfb\u52a0\u5d4c\u5957\u5faa\u73af\uff0c\u6bcf\u4e00\u6b21\u5d4c\u5957\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7ef4\u201d\uff0c\u5c06\u4f1a\u4f7f\u65f6\u95f4\u590d\u6742\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u5173\u7cfb\u201d\u201c\u56db\u6b21\u65b9\u5173\u7cfb\u201d\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u9012\u5f52","text":"<p>\u300c\u9012\u5f52 recursion\u300d\u662f\u4e00\u79cd\u7b97\u6cd5\u7b56\u7565\uff0c\u901a\u8fc7\u51fd\u6570\u8c03\u7528\u81ea\u8eab\u6765\u89e3\u51b3\u95ee\u9898\u3002\u5b83\u4e3b\u8981\u5305\u542b\u4e24\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9012\uff1a\u7a0b\u5e8f\u4e0d\u65ad\u6df1\u5165\u5730\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u4f20\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\uff0c\u76f4\u5230\u8fbe\u5230\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u3002</li> <li>\u5f52\uff1a\u89e6\u53d1\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u540e\uff0c\u7a0b\u5e8f\u4ece\u6700\u6df1\u5c42\u7684\u9012\u5f52\u51fd\u6570\u5f00\u59cb\u9010\u5c42\u8fd4\u56de\uff0c\u6c47\u805a\u6bcf\u4e00\u5c42\u7684\u7ed3\u679c\u3002</li> </ol> <p>\u800c\u4ece\u5b9e\u73b0\u7684\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u4ee3\u7801\u4e3b\u8981\u5305\u542b\u4e09\u4e2a\u8981\u7d20\u3002</p> <ol> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u7528\u4e8e\u51b3\u5b9a\u4ec0\u4e48\u65f6\u5019\u7531\u201c\u9012\u201d\u8f6c\u201c\u5f52\u201d\u3002</li> <li>\u9012\u5f52\u8c03\u7528\uff1a\u5bf9\u5e94\u201c\u9012\u201d\uff0c\u51fd\u6570\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u8f93\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\u3002</li> <li>\u8fd4\u56de\u7ed3\u679c\uff1a\u5bf9\u5e94\u201c\u5f52\u201d\uff0c\u5c06\u5f53\u524d\u9012\u5f52\u5c42\u7ea7\u7684\u7ed3\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c42\u3002</li> </ol> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u53ea\u9700\u8c03\u7528\u51fd\u6570 <code>recur(n)</code>  \uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8ba1\u7b97\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n\"\"\"\u9012\u5f52\"\"\"\n# \u7ec8\u6b62\u6761\u4ef6\nif n == 1:\nreturn 1\n# \u9012\uff1a\u9012\u5f52\u8c03\u7528\nres = recur(n - 1)\n# \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res\n</code></pre> recursion.cpp<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 1)\nreturn 1;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nint res = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 1)\nreturn 1;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nint res = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u9012\u5f52 */\nint Recur(int n) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 1)\nreturn 1;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nint res = Recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u9012\u5f52 */\nfunc recur(n int) int {\n// \u7ec8\u6b62\u6761\u4ef6\nif n == 1 {\nreturn 1\n}\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nres := recur(n - 1)\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u9012\u5f52 */\nfunc recur(n: Int) -&gt; Int {\n// \u7ec8\u6b62\u6761\u4ef6\nif n == 1 {\nreturn 1\n}\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nlet res = recur(n: n - 1)\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res\n}\n</code></pre> recursion.js<pre><code>/* \u9012\u5f52 */\nfunction recur(n) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n === 1) return 1;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nconst res = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u9012\u5f52 */\nfunction recur(n: number): number {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n === 1) return 1;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nconst res = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 1) return 1;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nint res = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u9012\u5f52 */\nfn recur(n: i32) -&gt; i32 {\n// \u7ec8\u6b62\u6761\u4ef6\nif n == 1 {\nreturn 1;\n}\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nlet res = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nn + res\n}\n</code></pre> recursion.c<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 1)\nreturn 1;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nint res = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> recursion.zig<pre><code>// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 1) {\nreturn 1;\n}\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nvar res: i32 = recur(n - 1);\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nreturn n + res;\n}\n</code></pre> <p>\u56fe 2-3 \u5c55\u793a\u4e86\u8be5\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 2-3 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u867d\u7136\u4ece\u8ba1\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u4e0e\u9012\u5f52\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u4f46\u5b83\u4eec\u4ee3\u8868\u4e86\u4e24\u79cd\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u51b3\u95ee\u9898\u7684\u8303\u5f0f\u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u4ece\u6700\u57fa\u7840\u7684\u6b65\u9aa4\u5f00\u59cb\uff0c\u7136\u540e\u4e0d\u65ad\u91cd\u590d\u6216\u7d2f\u52a0\u8fd9\u4e9b\u6b65\u9aa4\uff0c\u76f4\u5230\u4efb\u52a1\u5b8c\u6210\u3002</li> <li>\u9012\u5f52\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u8fd9\u4e9b\u5b50\u95ee\u9898\u548c\u539f\u95ee\u9898\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u6765\u5c06\u5b50\u95ee\u9898\u7ee7\u7eed\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u51b5\u65f6\u505c\u6b62\uff08\u57fa\u672c\u60c5\u51b5\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002</li> </ul> <p>\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u6570\u4e3a\u4f8b\uff0c\u8bbe\u95ee\u9898 \\(f(n) = 1 + 2 + \\dots + n\\) \u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u5728\u5faa\u73af\u4e2d\u6a21\u62df\u6c42\u548c\u8fc7\u7a0b\uff0c\u4ece \\(1\\) \u904d\u5386\u5230 \\(n\\) \uff0c\u6bcf\u8f6e\u6267\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002</li> <li>\u9012\u5f52\uff1a\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65ad\uff08\u9012\u5f52\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u51b5 \\(f(1) = 1\\) \u65f6\u7ec8\u6b62\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","title":"1. \u00a0 \u8c03\u7528\u6808","text":"<p>\u9012\u5f52\u51fd\u6570\u6bcf\u6b21\u8c03\u7528\u81ea\u8eab\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u4e3a\u65b0\u5f00\u542f\u7684\u51fd\u6570\u5206\u914d\u5185\u5b58\uff0c\u4ee5\u5b58\u50a8\u5c40\u90e8\u53d8\u91cf\u3001\u8c03\u7528\u5730\u5740\u548c\u5176\u4ed6\u4fe1\u606f\u7b49\u3002\u8fd9\u5c06\u5bfc\u81f4\u4e24\u65b9\u9762\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u90fd\u5b58\u50a8\u5728\u79f0\u4e3a\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u7684\u5185\u5b58\u533a\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u6570\u8fd4\u56de\u540e\u624d\u4f1a\u88ab\u91ca\u653e\u3002\u56e0\u6b64\uff0c\u9012\u5f52\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u9012\u5f52\u8c03\u7528\u51fd\u6570\u4f1a\u4ea7\u751f\u989d\u5916\u7684\u5f00\u9500\u3002\u56e0\u6b64\u9012\u5f52\u901a\u5e38\u6bd4\u5faa\u73af\u7684\u65f6\u95f4\u6548\u7387\u66f4\u4f4e\u3002</li> </ul> <p>\u5982\u56fe 2-4 \u6240\u793a\uff0c\u5728\u89e6\u53d1\u7ec8\u6b62\u6761\u4ef6\u524d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684\u9012\u5f52\u51fd\u6570\uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002</p> <p></p> <p> \u56fe 2-4 \u00a0 \u9012\u5f52\u8c03\u7528\u6df1\u5ea6 </p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u7f16\u7a0b\u8bed\u8a00\u5141\u8bb8\u7684\u9012\u5f52\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u8fc7\u6df1\u7684\u9012\u5f52\u53ef\u80fd\u5bfc\u81f4\u6808\u6ea2\u51fa\u9519\u8bef\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u9012\u5f52","text":"<p>\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u6570\u5728\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u6b65\u624d\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5219\u8be5\u51fd\u6570\u53ef\u4ee5\u88ab\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u4f18\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u95f4\u6548\u7387\u4e0a\u4e0e\u8fed\u4ee3\u76f8\u5f53\u3002\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u300c\u5c3e\u9012\u5f52 tail recursion\u300d\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u5f53\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u7684\u51fd\u6570\u540e\uff0c\u9700\u8981\u7ee7\u7eed\u6267\u884c\u4ee3\u7801\uff0c\u56e0\u6b64\u7cfb\u7edf\u9700\u8981\u4fdd\u5b58\u4e0a\u4e00\u5c42\u8c03\u7528\u7684\u4e0a\u4e0b\u6587\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u9012\u5f52\u8c03\u7528\u662f\u51fd\u6570\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u4e2a\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u540e\uff0c\u65e0\u987b\u7ee7\u7eed\u6267\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7edf\u65e0\u987b\u4fdd\u5b58\u4e0a\u4e00\u5c42\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u3002</li> </ul> <p>\u4ee5\u8ba1\u7b97 \\(1 + 2 + \\dots + n\\) \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7ed3\u679c\u53d8\u91cf <code>res</code> \u8bbe\u4e3a\u51fd\u6570\u53c2\u6570\uff0c\u4ece\u800c\u5b9e\u73b0\u5c3e\u9012\u5f52\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def tail_recur(n, res):\n\"\"\"\u5c3e\u9012\u5f52\"\"\"\n# \u7ec8\u6b62\u6761\u4ef6\nif n == 0:\nreturn res\n# \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 0)\nreturn res;\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 0)\nreturn res;\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 0)\nreturn res;\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n// \u7ec8\u6b62\u6761\u4ef6\nif n == 0 {\nreturn res\n}\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n// \u7ec8\u6b62\u6761\u4ef6\nif n == 0 {\nreturn res\n}\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n === 0) return res;\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n === 0) return res;\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 0) return res;\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n// \u7ec8\u6b62\u6761\u4ef6\nif n == 0 {\nreturn res;\n}\n// \u5c3e\u9012\u5f52\u8c03\u7528\ntail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 0)\nreturn res;\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.zig<pre><code>// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n// \u7ec8\u6b62\u6761\u4ef6\nif (n == 0) {\nreturn res;\n}\n// \u5c3e\u9012\u5f52\u8c03\u7528\nreturn tailRecur(n - 1, res + n);\n}\n</code></pre> <p>\u5c3e\u9012\u5f52\u7684\u6267\u884c\u8fc7\u7a0b\u5982\u56fe 2-5 \u6240\u793a\u3002\u5bf9\u6bd4\u666e\u901a\u9012\u5f52\u548c\u5c3e\u9012\u5f52\uff0c\u4e24\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u6267\u884c\u70b9\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u6bcf\u5c42\u8fd4\u56de\u540e\u90fd\u8981\u518d\u6267\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u53ea\u9700\u5c42\u5c42\u8fd4\u56de\u3002</li> </ul> <p></p> <p> \u56fe 2-5 \u00a0 \u5c3e\u9012\u5f52\u8fc7\u7a0b </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8bb8\u591a\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u5e76\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\u3002\u4f8b\u5982\uff0cPython \u9ed8\u8ba4\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u6570\u662f\u5c3e\u9012\u5f52\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u4f1a\u9047\u5230\u6808\u6ea2\u51fa\u95ee\u9898\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u9012\u5f52\u6811","text":"<p>\u5f53\u5904\u7406\u4e0e\u201c\u5206\u6cbb\u201d\u76f8\u5173\u7684\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u9012\u5f52\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c2\u3001\u4ee3\u7801\u66f4\u52a0\u6613\u8bfb\u3002\u4ee5\u201c\u6590\u6ce2\u90a3\u5951\u6570\u5217\u201d\u4e3a\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8be5\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u3002</p> <p>\u8bbe\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u4e3a \\(f(n)\\) \uff0c\u6613\u5f97\u4e24\u4e2a\u7ed3\u8bba\u3002</p> <ul> <li>\u6570\u5217\u7684\u524d\u4e24\u4e2a\u6570\u5b57\u4e3a \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002</li> <li>\u6570\u5217\u4e2d\u7684\u6bcf\u4e2a\u6570\u5b57\u662f\u524d\u4e24\u4e2a\u6570\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002</li> </ul> <p>\u6309\u7167\u9012\u63a8\u5173\u7cfb\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5c06\u524d\u4e24\u4e2a\u6570\u5b57\u4f5c\u4e3a\u7ec8\u6b62\u6761\u4ef6\uff0c\u4fbf\u53ef\u5199\u51fa\u9012\u5f52\u4ee3\u7801\u3002\u8c03\u7528 <code>fib(n)</code> \u5373\u53ef\u5f97\u5230\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n\"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n# \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif n == 1 or n == 2:\nreturn n - 1\n# \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nres = fib(n - 1) + fib(n - 2)\n# \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res\n</code></pre> recursion.cpp<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n == 1 || n == 2)\nreturn n - 1;\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nint res = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> recursion.java<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n == 1 || n == 2)\nreturn n - 1;\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nint res = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> recursion.cs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n == 1 || n == 2)\nreturn n - 1;\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nint res = Fib(n - 1) + Fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> recursion.go<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif n == 1 || n == 2 {\nreturn n - 1\n}\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nres := fib(n-1) + fib(n-2)\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res\n}\n</code></pre> recursion.swift<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -&gt; Int {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif n == 1 || n == 2 {\nreturn n - 1\n}\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nlet res = fib(n: n - 1) + fib(n: n - 2)\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res\n}\n</code></pre> recursion.js<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n === 1 || n === 2) return n - 1;\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nconst res = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> recursion.ts<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n === 1 || n === 2) return n - 1;\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nconst res = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> recursion.dart<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n == 1 || n == 2) return n - 1;\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nint res = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> recursion.rs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -&gt; i32 {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif n == 1 || n == 2 {\nreturn n - 1;\n}\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nlet res = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c\nres\n}\n</code></pre> recursion.c<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n == 1 || n == 2)\nreturn n - 1;\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nint res = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> recursion.zig<pre><code>// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n// \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\nif (n == 1 or n == 2) {\nreturn n - 1;\n}\n// \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\nvar res: i32 = fib(n - 1) + fib(n - 2);\n// \u8fd4\u56de\u7ed3\u679c f(n)\nreturn res;\n}\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u6211\u4eec\u5728\u51fd\u6570\u5185\u9012\u5f52\u8c03\u7528\u4e86\u4e24\u4e2a\u51fd\u6570\uff0c\u8fd9\u610f\u5473\u7740\u4ece\u4e00\u4e2a\u8c03\u7528\u4ea7\u751f\u4e86\u4e24\u4e2a\u8c03\u7528\u5206\u652f\u3002\u5982\u56fe 2-6 \u6240\u793a\uff0c\u8fd9\u6837\u4e0d\u65ad\u9012\u5f52\u8c03\u7528\u4e0b\u53bb\uff0c\u6700\u7ec8\u5c06\u4ea7\u751f\u4e00\u68f5\u5c42\u6570\u4e3a \\(n\\) \u7684\u300c\u9012\u5f52\u6811 recursion tree\u300d\u3002</p> <p></p> <p> \u56fe 2-6 \u00a0 \u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u9012\u5f52\u6811 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9012\u5f52\u4f53\u73b0\u4e86\u201c\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u5b50\u95ee\u9898\u201d\u7684\u601d\u7ef4\u8303\u5f0f\uff0c\u8fd9\u79cd\u5206\u6cbb\u7b56\u7565\u81f3\u5173\u91cd\u8981\u3002</p> <ul> <li>\u4ece\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u7d22\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u7b49\u8bb8\u591a\u91cd\u8981\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u5e94\u7528\u4e86\u8fd9\u79cd\u601d\u7ef4\u65b9\u5f0f\u3002</li> <li>\u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u5929\u7136\u9002\u5408\u5904\u7406\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u76f8\u5173\u95ee\u9898\uff0c\u56e0\u4e3a\u5b83\u4eec\u975e\u5e38\u9002\u5408\u7528\u5206\u6cbb\u601d\u60f3\u8fdb\u884c\u5206\u6790\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","title":"2.2.3 \u00a0 \u4e24\u8005\u5bf9\u6bd4","text":"<p>\u603b\u7ed3\u4ee5\u4e0a\u5185\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5b9e\u73b0\u3001\u6027\u80fd\u548c\u9002\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002</p> <p> \u8868 2-1 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52\u7279\u70b9\u5bf9\u6bd4 </p> \u8fed\u4ee3 \u9012\u5f52 \u5b9e\u73b0\u65b9\u5f0f \u5faa\u73af\u7ed3\u6784 \u51fd\u6570\u8c03\u7528\u81ea\u8eab \u65f6\u95f4\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f83\u9ad8\uff0c\u65e0\u51fd\u6570\u8c03\u7528\u5f00\u9500 \u6bcf\u6b21\u51fd\u6570\u8c03\u7528\u90fd\u4f1a\u4ea7\u751f\u5f00\u9500 \u5185\u5b58\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4 \u7d2f\u79ef\u51fd\u6570\u8c03\u7528\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u6808\u5e27\u7a7a\u95f4 \u9002\u7528\u95ee\u9898 \u9002\u7528\u4e8e\u7b80\u5355\u5faa\u73af\u4efb\u52a1\uff0c\u4ee3\u7801\u76f4\u89c2\u3001\u53ef\u8bfb\u6027\u597d \u9002\u7528\u4e8e\u5b50\u95ee\u9898\u5206\u89e3\uff0c\u5982\u6811\u3001\u56fe\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u4ee3\u7801\u7ed3\u6784\u7b80\u6d01\u3001\u6e05\u6670 <p>Tip</p> <p>\u5982\u679c\u611f\u89c9\u4ee5\u4e0b\u5185\u5bb9\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5728\u8bfb\u5b8c\u201c\u6808\u201d\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\u3002</p> <p>\u90a3\u4e48\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5177\u6709\u4ec0\u4e48\u5185\u5728\u8054\u7cfb\u5462\uff1f\u4ee5\u4e0a\u8ff0\u7684\u9012\u5f52\u51fd\u6570\u4e3a\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u9012\u5f52\u7684\u201c\u5f52\u201d\u9636\u6bb5\u8fdb\u884c\u3002\u8fd9\u610f\u5473\u7740\u6700\u521d\u88ab\u8c03\u7528\u7684\u51fd\u6570\u5b9e\u9645\u4e0a\u662f\u6700\u540e\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u8fd9\u79cd\u5de5\u4f5c\u673a\u5236\u4e0e\u6808\u7684\u201c\u5148\u5165\u540e\u51fa\u201d\u539f\u5219\u662f\u5f02\u66f2\u540c\u5de5\u7684\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u201c\u8c03\u7528\u6808\u201d\u548c\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u8fd9\u7c7b\u9012\u5f52\u672f\u8bed\u5df2\u7ecf\u6697\u793a\u4e86\u9012\u5f52\u4e0e\u6808\u4e4b\u95f4\u7684\u5bc6\u5207\u5173\u7cfb\u3002</p> <ol> <li>\u9012\uff1a\u5f53\u51fd\u6570\u88ab\u8c03\u7528\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5728\u201c\u8c03\u7528\u6808\u201d\u4e0a\u4e3a\u8be5\u51fd\u6570\u5206\u914d\u65b0\u7684\u6808\u5e27\uff0c\u7528\u4e8e\u5b58\u50a8\u51fd\u6570\u7684\u5c40\u90e8\u53d8\u91cf\u3001\u53c2\u6570\u3001\u8fd4\u56de\u5730\u5740\u7b49\u6570\u636e\u3002</li> <li>\u5f52\uff1a\u5f53\u51fd\u6570\u5b8c\u6210\u6267\u884c\u5e76\u8fd4\u56de\u65f6\uff0c\u5bf9\u5e94\u7684\u6808\u5e27\u4f1a\u4ece\u201c\u8c03\u7528\u6808\u201d\u4e0a\u88ab\u79fb\u9664\uff0c\u6062\u590d\u4e4b\u524d\u51fd\u6570\u7684\u6267\u884c\u73af\u5883\u3002</li> </ol> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u8c03\u7528\u6808\u7684\u884c\u4e3a\uff0c\u4ece\u800c\u5c06\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u5f62\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n\"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n# \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nstack = []\nres = 0\n# \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor i in range(n, 0, -1):\n# \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.append(i)\n# \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile stack:\n# \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.pop()\n# res = 1+2+3+...+n\nreturn res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nstack&lt;int&gt; stack;\nint res = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor (int i = n; i &gt; 0; i--) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.push(i);\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile (!stack.empty()) {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.top();\nstack.pop();\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\nint res = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor (int i = n; i &gt; 0; i--) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.push(i);\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile (!stack.isEmpty()) {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.pop();\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nStack&lt;int&gt; stack = new();\nint res = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor (int i = n; i &gt; 0; i--) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.Push(i);\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile (stack.Count &gt; 0) {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.Pop();\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nstack := list.New()\nres := 0\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor i := n; i &gt; 0; i-- {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.PushBack(i)\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nfor stack.Len() != 0 {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.Back().Value.(int)\nstack.Remove(stack.Back())\n}\n// res = 1+2+3+...+n\nreturn res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nvar stack: [Int] = []\nvar res = 0\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor i in stride(from: n, to: 0, by: -1) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.append(i)\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile !stack.isEmpty {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.removeLast()\n}\n// res = 1+2+3+...+n\nreturn res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nconst stack = [];\nlet res = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor (let i = 1; i &lt;= n; i++) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.push(i);\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile (stack.length) { // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.pop();\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \nconst stack: number[] = [];\nlet res: number = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor (let i = 1; i &lt;= n; i++) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.push(i);\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile (stack.length) { // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.pop();\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nList&lt;int&gt; stack = [];\nint res = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor (int i = n; i &gt; 0; i--) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.add(i);\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile (!stack.isEmpty) {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.removeLast();\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nlet mut stack = Vec::new();\nlet mut res = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor i in (1..=n).rev() {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack.push(i);\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile !stack.is_empty() {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack.pop().unwrap();\n}\n// res = 1+2+3+...+n\nres\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\nint stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\nint top = -1;    // \u6808\u9876\u7d22\u5f15\nint res = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nfor (int i = n; i &gt; 0; i--) {\n// \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\nstack[1 + top++] = i;\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nwhile (top &gt;= 0) {\n// \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\nres += stack[top--];\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n// \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\nvar stack: [n]i32 = undefined;\nvar res: i32 = 0;\n// \u9012\uff1a\u9012\u5f52\u8c03\u7528\nvar i: usize = n;\nwhile (i &gt; 0) {\nstack[i - 1] = @intCast(i);\ni -= 1;\n}\n// \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\nvar index: usize = n;\nwhile (index &gt; 0) {\nindex -= 1;\nres += stack[index];\n}\n// res = 1+2+3+...+n\nreturn res;\n}\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u5f53\u9012\u5f52\u88ab\u8f6c\u6362\u4e3a\u8fed\u4ee3\u540e\uff0c\u4ee3\u7801\u53d8\u5f97\u66f4\u52a0\u590d\u6742\u4e86\u3002\u5c3d\u7ba1\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\uff0c\u4f46\u4e5f\u4e0d\u4e00\u5b9a\u503c\u5f97\u8fd9\u6837\u505a\uff0c\u6709\u4ee5\u4e0b\u4e24\u70b9\u539f\u56e0\u3002</p> <ul> <li>\u8f6c\u5316\u540e\u7684\u4ee3\u7801\u53ef\u80fd\u66f4\u52a0\u96be\u4ee5\u7406\u89e3\uff0c\u53ef\u8bfb\u6027\u66f4\u5dee\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\u7684\u884c\u4e3a\u53ef\u80fd\u975e\u5e38\u56f0\u96be\u3002</li> </ul> <p>\u603b\u4e4b\uff0c\u9009\u62e9\u8fed\u4ee3\u8fd8\u662f\u9012\u5f52\u53d6\u51b3\u4e8e\u7279\u5b9a\u95ee\u9898\u7684\u6027\u8d28\u3002\u5728\u7f16\u7a0b\u5b9e\u8df5\u4e2d\uff0c\u6743\u8861\u4e24\u8005\u7684\u4f18\u52a3\u5e76\u6839\u636e\u60c5\u5883\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u662f\u81f3\u5173\u91cd\u8981\u7684\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"<p>\u5728\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002</p> <ol> <li>\u627e\u5230\u95ee\u9898\u89e3\u6cd5\uff1a\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002</li> <li>\u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\uff1a\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002</li> </ol> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u4e3a\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u4e24\u4e2a\u7ef4\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002</li> <li>\u7a7a\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002</li> </ul> <p>\u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u8fd9\u6837\u6211\u4eec\u624d\u80fd\u5c06\u5404\u79cd\u7b97\u6cd5\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8fdb\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002</p> <p>\u6548\u7387\u8bc4\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\uff1a\u5b9e\u9645\u6d4b\u8bd5\u3001\u7406\u8bba\u4f30\u7b97\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5b9e\u9645\u6d4b\u8bd5","text":"<p>\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 <code>A</code> \u548c\u7b97\u6cd5 <code>B</code> \uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u7684\u5c40\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u6bd4\u5982\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u7edf\u8ba1\u5e73\u5747\u6548\u7387\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u800c\u8fd9\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8bba\u4f30\u7b97","text":"<p>\u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis\u300d\uff0c\u7b80\u79f0\u300c\u590d\u6742\u5ea6\u5206\u6790\u300d\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u80fd\u591f\u4f53\u73b0\u7b97\u6cd5\u8fd0\u884c\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u7b97\u6cd5\u6267\u884c\u6240\u9700\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\u3002</p> <ul> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u201d\u5206\u522b\u5bf9\u5e94\u300c\u65f6\u95f4\u590d\u6742\u5ea6 time complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u3002</li> <li>\u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u7740\u590d\u6742\u5ea6\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u4f53\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002</li> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\u7684\u5177\u4f53\u503c\uff0c\u800c\u662f\u65f6\u95f4\u6216\u7a7a\u95f4\u589e\u957f\u7684\u201c\u5feb\u6162\u201d\u3002</li> </ul> <p>\u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u4f53\u73b0\u5728\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002</li> <li>\u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002</li> </ul> <p>Tip</p> <p>\u5982\u679c\u4f60\u4ecd\u5bf9\u590d\u6742\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u987b\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u8861\u91cf\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002</p> <p>\u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u6700\u5148\u4ecb\u7ecd\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u4ee5\u4fbf\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"<p>\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\uff0c\u53ea\u9700\u5c06\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u66ff\u6362\u4e3a\u201c\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u201d\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"<p>\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u8f93\u5165\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\u3002</li> <li>\u6682\u5b58\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\u3002</li> <li>\u8f93\u51fa\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\u3002</li> </ul> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u7edf\u8ba1\u8303\u56f4\u662f\u201c\u6682\u5b58\u7a7a\u95f4\u201d\u52a0\u4e0a\u201c\u8f93\u51fa\u7a7a\u95f4\u201d\u3002</p> <p>\u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u6682\u5b58\u6570\u636e\uff1a\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002</li> <li>\u6808\u5e27\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002</li> <li>\u6307\u4ee4\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002</li> </ul> <p>\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u7edf\u8ba1\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u8f93\u51fa\u6570\u636e\u4e09\u90e8\u5206\uff0c\u5982\u56fe 2-15 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 2-15 \u00a0 \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4 </p> <p>\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class Node:\n\"\"\"\u7c7b\"\"\"\ndef __init__(self, x: int):\nself.val: int = x              # \u8282\u70b9\u503c\nself.next: Node | None = None  # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\ndef function() -&gt; int:\n\"\"\"\u51fd\u6570\"\"\"\n# \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0\ndef algorithm(n) -&gt; int:  # \u8f93\u5165\u6570\u636e\nA = 0                 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\nb = 0                 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnode = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc = function()        # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn A + b + c      # \u8f93\u51fa\u6570\u636e\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\nstruct Node {\nint val;\nNode *next;\nNode(int x) : val(x), next(nullptr) {}\n};\n/* \u51fd\u6570 */\nint func() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(int x) { val = x; }\n}\n/* \u51fd\u6570 */\nint function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0;\n}\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nfinal int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node(int x) {\nint val = x;\nNode next;\n}\n/* \u51fd\u6570 */\nint Function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0;\n}\nint Algorithm(int n) {        // \u8f93\u5165\u6570\u636e\nconst int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = new(0);       // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = Function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\ntype node struct {\nval  int\nnext *node\n}\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\nreturn &amp;node{val: val}\n}\n/* \u51fd\u6570 */\nfunc function() int {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0\n}\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\nconst a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nb := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nnewNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nc := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\nvar val: Int\nvar next: Node?\ninit(x: Int) {\nval = x\n}\n}\n/* \u51fd\u6570 */\nfunc function() -&gt; Int {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0\n}\nfunc algorithm(n: Int) -&gt; Int { // \u8f93\u5165\u6570\u636e\nlet a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nvar b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nlet node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nlet c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\nval;\nnext;\nconstructor(val) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\nconst a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\nval: number;\nnext: Node | null;\nconstructor(val?: number) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n}\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\nconst a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nconst node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nconst c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\nint val;\nNode next;\nNode(this.val, [this.next]);\n}\n/* \u51fd\u6570 */\nint function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0;\n}\nint algorithm(int n) {  // \u8f93\u5165\u6570\u636e\nconst int a = 0;      // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;            // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nNode node = Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nint c = function();   // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;     // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\nval: i32,\nnext: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n/* \u521b\u5efa Node \u7ed3\u6784\u4f53 */\nimpl Node {\nfn new(val: i32) -&gt; Self {\nSelf { val: val, next: None }\n}\n}\n/* \u51fd\u6570 */\nfn function() -&gt; i32 {      // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0;\n}\nfn algorithm(n: i32) -&gt; i32 {       // \u8f93\u5165\u6570\u636e\nconst a: i32 = 0;               // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nlet mut b = 0;                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nlet node = Node::new(0);        // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\nlet c = function();             // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;               // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u51fd\u6570 */\nint func() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0;\n}\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\nconst int a = 0;   // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\nint b = 0;         // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\nint c = func();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\nreturn a + b + c;  // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002</p> <p>\u800c\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\u3002</p> <ol> <li>\u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\uff1a\u5f53 \\(n &lt; 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n &gt; 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 <code>nums</code> \u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u4ee5\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\uff1a\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 <code>nums</code> \u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def algorithm(n: int):\na = 0               # O(1)\nb = [0] * 10000     # O(1)\nif n &gt; 10:\nnums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 0;               // O(1)\nvector&lt;int&gt; b(10000);    // O(1)\nif (n &gt; 10)\nvector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n &gt; 10)\nint[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\nint a = 0;                   // O(1)\nint[] b = new int[10000];    // O(1)\nif (n &gt; 10) {\nint[] nums = new int[n]; // O(n)\n}\n}\n</code></pre> <pre><code>func algorithm(n int) {\na := 0                      // O(1)\nb := make([]int, 10000)     // O(1)\nvar nums []int\nif n &gt; 10 {\nnums := make([]int, n)  // O(n)\n}\nfmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\nlet a = 0 // O(1)\nlet b = Array(repeating: 0, count: 10000) // O(1)\nif n &gt; 10 {\nlet nums = Array(repeating: 0, count: n) // O(n)\n}\n}\n</code></pre> <pre><code>function algorithm(n) {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n &gt; 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\nconst a = 0;                   // O(1)\nconst b = new Array(10000);    // O(1)\nif (n &gt; 10) {\nconst nums = new Array(n); // O(n)\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 0;                            // O(1)\nList&lt;int&gt; b = List.filled(10000, 0);  // O(1)\nif (n &gt; 10) {\nList&lt;int&gt; nums = List.filled(n, 0); // O(n)\n}\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\nlet a = 0;                              // O(1)\nlet b = [0; 10000];                     // O(1)\nif n &gt; 10 {\nlet nums = vec![0; n as usize];     // O(n)\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 0;               // O(1)\nint b[10000];            // O(1)\nif (n &gt; 10)\nint nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def function() -&gt; int:\n# \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0\ndef loop(n: int):\n\"\"\"\u5faa\u73af O(1)\"\"\"\nfor _ in range(n):\nfunction()\ndef recur(n: int) -&gt; int:\n\"\"\"\u9012\u5f52 O(n)\"\"\"\nif n == 1: return\nreturn recur(n - 1)\n</code></pre> <pre><code>int func() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i &lt; n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i &lt; n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid Loop(int n) {\nfor (int i = 0; i &lt; n; i++) {\nFunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nint Recur(int n) {\nif (n == 1) return 1;\nreturn Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\nfor i := 0; i &lt; n; i++ {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\nif n == 1 {\nreturn\n}\nrecur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0\n}\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\nfor _ in 0 ..&lt; n {\nfunction()\n}\n}\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\nif n == 1 {\nreturn\n}\nrecur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\nfor (let i = 0; i &lt; n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\nfor (let i = 0; i &lt; n; i++) {\nconstFunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\nif (n === 1) return;\nreturn recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i &lt; n; i++) {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nfn loop(n: i32) {\nfor i in 0..n {\nfunction();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(n: i32) {\nif n == 1 {\nreturn;\n}\nrecur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\nfor (int i = 0; i &lt; n; i++) {\nfunc();\n}\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\nif (n == 1) return;\nreturn recur(n - 1);\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u51fd\u6570 <code>loop()</code> \u548c <code>recur()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u4f46\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u3002</p> <ul> <li>\u51fd\u6570 <code>loop()</code> \u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 <code>function()</code> \uff0c\u6bcf\u8f6e\u4e2d\u7684 <code>function()</code> \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002</li> <li>\u9012\u5f52\u51fd\u6570 <code>recur()</code> \u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>recur()</code> \uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243","title":"2.4.3 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u56fe 2-16 \u5c55\u793a\u4e86\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-16 \u00a0 \u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u56e0\u6b64\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def function() -&gt; int:\n\"\"\"\u51fd\u6570\"\"\"\n# \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0\ndef constant(n: int):\n\"\"\"\u5e38\u6570\u9636\"\"\"\n# \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\na = 0\nnums = [0] * 10000\nnode = ListNode(0)\n# \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nc = 0\n# \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in range(n):\nfunction()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u6570 */\nint func() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nvector&lt;int&gt; nums(10000);\nListNode node(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nfunc();\n}\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u6570 */\nint function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nfunction();\n}\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u6570 */\nint Function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nint a = 0;\nint b = 0;\nint[] nums = new int[10000];\nListNode node = new(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nFunction();\n}\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u6570 */\nfunc function() int {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\nreturn 0\n}\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0\nb := 0\nnums := make([]int, 10000)\nListNode := newNode(0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nvar c int\nfor i := 0; i &lt; n; i++ {\nc = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor i := 0; i &lt; n; i++ {\nfunction()\n}\nfmt.Println(a, b, nums, c, ListNode)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u6570 */\n@discardableResult\nfunc function() -&gt; Int {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0\n}\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nlet a = 0\nvar b = 0\nlet nums = Array(repeating: 0, count: 10000)\nlet node = ListNode(x: 0)\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..&lt; n {\nlet c = 0\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor _ in 0 ..&lt; n {\nfunction()\n}\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u6570 */\nfunction constFunc() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i &lt; n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i &lt; n; i++) {\nconstFunc();\n}\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u6570 */\nfunction constFunc(): number {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a = 0;\nconst b = 0;\nconst nums = new Array(10000);\nconst node = new ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i &lt; n; i++) {\nconst c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (let i = 0; i &lt; n; i++) {\nconstFunc();\n}\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u6570 */\nint function() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nfinal int a = 0;\nint b = 0;\nList&lt;int&gt; nums = List.filled(10000, 0);\nListNode node = ListNode(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (var i = 0; i &lt; n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (var i = 0; i &lt; n; i++) {\nfunction();\n}\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u6570 */\nfn function() -&gt;i32 {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst A: i32 = 0;\nlet b = 0;\nlet nums = vec![0; 10000];\nlet node = ListNode::new(0);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor i in 0..n {\nlet c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor i in 0..n {\nfunction();\n}\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u6570 */\nint func() {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst int a = 0;\nint b = 0;\nint nums[1000];\nListNode *node = newListNode(0);\nfree(node);\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nint c = 0;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\nfor (int i = 0; i &lt; n; i++) {\nfunc();\n}\n}\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u6570\nfn function() i32 {\n// \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\nreturn 0;\n}\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n// \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\nconst a: i32 = 0;\nvar b: i32 = 0;\nvar nums = [_]i32{0}**10000;\nvar node = inc.ListNode(i32){.val = 0};\nvar i: i32 = 0;\n// \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\nwhile (i &lt; n) : (i += 1) {\nvar c: i32 = 0;\n_ = c;\n}\n// \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\ni = 0;\nwhile (i &lt; n) : (i += 1) {\n_ = function();\n}\n_ = a;\n_ = b;\n_ = nums;\n_ = node;\n}\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def linear(n: int):\n\"\"\"\u7ebf\u6027\u9636\"\"\"\n# \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nnums = [0] * n\n# \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nhmap = dict[int, str]()\nfor i in range(n):\nhmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvector&lt;int&gt; nums(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvector&lt;ListNode&gt; nodes;\nfor (int i = 0; i &lt; n; i++) {\nnodes.push_back(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nunordered_map&lt;int, string&gt; map;\nfor (int i = 0; i &lt; n; i++) {\nmap[i] = to_string(i);\n}\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\nfor (int i = 0; i &lt; n; i++) {\nnodes.add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\nfor (int i = 0; i &lt; n; i++) {\nmap.put(i, String.valueOf(i));\n}\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint[] nums = new int[n];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList&lt;ListNode&gt; nodes = [];\nfor (int i = 0; i &lt; n; i++) {\nnodes.Add(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nDictionary&lt;int, string&gt; map = [];\nfor (int i = 0; i &lt; n; i++) {\nmap.Add(i, i.ToString());\n}\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n_ = make([]int, n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes []*node\nfor i := 0; i &lt; n; i++ {\nnodes = append(nodes, newNode(i))\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nm := make(map[int]string, n)\nfor i := 0; i &lt; n; i++ {\nm[i] = strconv.Itoa(i)\n}\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nlet nums = Array(repeating: 0, count: n)\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes = [];\nfor (let i = 0; i &lt; n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i &lt; n; i++) {\nmap.set(i, i.toString());\n}\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nconst nums = new Array(n);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst nodes: ListNode[] = [];\nfor (let i = 0; i &lt; n; i++) {\nnodes.push(new ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nconst map = new Map();\nfor (let i = 0; i &lt; n; i++) {\nmap.set(i, i.toString());\n}\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nList&lt;int&gt; nums = List.filled(n, 0);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nList&lt;ListNode&gt; nodes = [];\nfor (var i = 0; i &lt; n; i++) {\nnodes.add(ListNode(i));\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nMap&lt;int, String&gt; map = HashMap();\nfor (var i = 0; i &lt; n; i++) {\nmap.putIfAbsent(i, () =&gt; i.toString());\n}\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nlet mut nums = vec![0; n as usize];\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet mut nodes = Vec::new();\nfor i in 0..n {\nnodes.push(ListNode::new(i))\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nlet mut map = HashMap::new();\nfor i in 0..n {\nmap.insert(i, i.to_string());\n}\n}\n</code></pre> space_complexity.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\nint key;\nint val;\nUT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nint *nums = malloc(sizeof(int) * n);\nfree(nums);\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nListNode **nodes = malloc(sizeof(ListNode *) * n);\nfor (int i = 0; i &lt; n; i++) {\nnodes[i] = newListNode(i);\n}\n// \u5185\u5b58\u91ca\u653e\nfor (int i = 0; i &lt; n; i++) {\nfree(nodes[i]);\n}\nfree(nodes);\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nHashTable *h = NULL;\nfor (int i = 0; i &lt; n; i++) {\nHashTable *tmp = malloc(sizeof(HashTable));\ntmp-&gt;key = i;\ntmp-&gt;val = i;\nHASH_ADD_INT(h, key, tmp);\n}\n// \u5185\u5b58\u91ca\u653e\nHashTable *curr, *tmp;\nHASH_ITER(hh, h, curr, tmp) {\nHASH_DEL(h, curr);\nfree(curr);\n}\n}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n// \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\nvar nums = [_]i32{0}**n;\n// \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar nodes = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i &lt; n) : (i += 1) {\ntry nodes.append(i);\n}\n// \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\nvar map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\ndefer map.deinit();\nvar j: i32 = 0;\nwhile (j &lt; n) : (j += 1) {\nconst string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\ndefer std.heap.page_allocator.free(string);\ntry map.put(i, string);\n}\n_ = nums;\n}\n</code></pre> <p>\u5982\u56fe 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5373\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>linear_recur()</code> \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def linear_recur(n: int):\n\"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nprint(\"\u9012\u5f52 n =\", n)\nif n == 1:\nreturn\nlinear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\ncout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; endl;\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nSystem.out.println(\"\u9012\u5f52 n = \" + n);\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n);\nif (n == 1) return;\nLinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\nfmt.Println(\"\u9012\u5f52 n =\", n)\nif n == 1 {\nreturn\n}\nspaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\nprint(\"\u9012\u5f52 n = \\(n)\")\nif n == 1 {\nreturn\n}\nlinearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\nconsole.log(`\u9012\u5f52 n = ${n}`);\nif (n === 1) return;\nlinearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nprint('\u9012\u5f52 n = $n');\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\nprintln!(\"\u9012\u5f52 n = {}\", n);\nif n == 1 {return};\nlinear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\nprintf(\"\u9012\u5f52 n = %d\\r\\n\", n);\nif (n == 1)\nreturn;\nlinearRecur(n - 1);\n}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\nstd.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\nif (n == 1) return;\nlinearRecur(n - 1);\n}\n</code></pre> <p></p> <p> \u56fe 2-17 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def quadratic(n: int):\n\"\"\"\u5e73\u65b9\u9636\"\"\"\n# \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nnum_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvector&lt;vector&lt;int&gt;&gt; numMatrix;\nfor (int i = 0; i &lt; n; i++) {\nvector&lt;int&gt; tmp;\nfor (int j = 0; j &lt; n; j++) {\ntmp.push_back(0);\n}\nnumMatrix.push_back(tmp);\n}\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[][] numMatrix = new int[n][n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\nfor (int i = 0; i &lt; n; i++) {\nList&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\nfor (int j = 0; j &lt; n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nint[,] numMatrix = new int[n, n];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList&lt;List&lt;int&gt;&gt; numList = [];\nfor (int i = 0; i &lt; n; i++) {\nList&lt;int&gt; tmp = [];\nfor (int j = 0; j &lt; n; j++) {\ntmp.Add(0);\n}\nnumList.Add(tmp);\n}\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nnumMatrix := make([][]int, n)\nfor i := 0; i &lt; n; i++ {\nnumMatrix[i] = make([]int, n)\n}\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nlet numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() =&gt; Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i &lt; n; i++) {\nconst tmp = [];\nfor (let j = 0; j &lt; n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numMatrix = Array(n)\n.fill(null)\n.map(() =&gt; Array(n).fill(null));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nconst numList = [];\nfor (let i = 0; i &lt; n; i++) {\nconst tmp = [];\nfor (let j = 0; j &lt; n; j++) {\ntmp.push(0);\n}\nnumList.push(tmp);\n}\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nList&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nList&lt;List&lt;int&gt;&gt; numList = [];\nfor (var i = 0; i &lt; n; i++) {\nList&lt;int&gt; tmp = [];\nfor (int j = 0; j &lt; n; j++) {\ntmp.add(0);\n}\nnumList.add(tmp);\n}\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n// \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\nlet num_matrix = vec![vec![0; n as usize]; n as usize];\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nlet mut num_list = Vec::new();\nfor i in 0..n {\nlet mut tmp = Vec::new();\nfor j in 0..n {\ntmp.push(0);\n}\nnum_list.push(tmp);\n}\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nint **numMatrix = malloc(sizeof(int *) * n);\nfor (int i = 0; i &lt; n; i++) {\nint *tmp = malloc(sizeof(int) * n);\nfor (int j = 0; j &lt; n; j++) {\ntmp[j] = 0;\n}\nnumMatrix[i] = tmp;\n}\n// \u5185\u5b58\u91ca\u653e\nfor (int i = 0; i &lt; n; i++) {\nfree(numMatrix[i]);\n}\nfree(numMatrix);\n}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n// \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\nvar nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\ndefer nodes.deinit();\nvar i: i32 = 0;\nwhile (i &lt; n) : (i += 1) {\nvar tmp = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer tmp.deinit();\nvar j: i32 = 0;\nwhile (j &lt; n) : (j += 1) {\ntry tmp.append(0);\n}\ntry nodes.append(tmp);\n}\n}\n</code></pre> <p>\u5982\u56fe 2-18 \u6240\u793a\uff0c\u8be5\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5728\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(n / 2\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n\"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n &lt;= 0:\nreturn 0\n# \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nnums = [0] * n\nreturn quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n &lt;= 0)\nreturn 0;\nvector&lt;int&gt; nums(n);\ncout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u957f\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\nreturn quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n &lt;= 0)\nreturn 0;\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nint[] nums = new int[n];\nSystem.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\nreturn quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\nif (n &lt;= 0) return 0;\nint[] nums = new int[n];\nConsole.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\nreturn QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\nif n &lt;= 0 {\nreturn 0\n}\nnums := make([]int, n)\nfmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\nreturn spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\nif n &lt;= 0 {\nreturn 0\n}\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nlet nums = Array(repeating: 0, count: n)\nprint(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\nreturn quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\nif (n &lt;= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\nif (n &lt;= 0) return 0;\nconst nums = new Array(n);\nconsole.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\nreturn quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n &lt;= 0) return 0;\nList&lt;int&gt; nums = List.filled(n, 0);\nprint('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\nreturn quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\nif n &lt;= 0 {return 0};\n// \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\nlet nums = vec![0; n as usize];\nprintln!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\nreturn quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\nif (n &lt;= 0)\nreturn 0;\nint *nums = malloc(sizeof(int) * n);\nprintf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\nint res = quadraticRecur(n - 1);\nfree(nums);\nreturn res;\n}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\nif (n &lt;= 0) return 0;\nvar nums = [_]i32{0}**n;\nstd.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\nreturn quadraticRecur(n - 1);\n}\n</code></pre> <p></p> <p> \u56fe 2-18 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u89c2\u5bdf\u56fe 2-19 \uff0c\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u201c\u6ee1\u4e8c\u53c9\u6811\u201d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n\"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\nif n == 0:\nreturn None\nroot = TreeNode(0)\nroot.left = build_tree(n - 1)\nroot.right = build_tree(n - 1)\nreturn root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\nif (n == 0)\nreturn nullptr;\nTreeNode *root = new TreeNode(0);\nroot-&gt;left = buildTree(n - 1);\nroot-&gt;right = buildTree(n - 1);\nreturn root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\nif (n == 0)\nreturn null;\nTreeNode root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\nif (n == 0) return null;\nTreeNode root = new(0) {\nleft = BuildTree(n - 1),\nright = BuildTree(n - 1)\n};\nreturn root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\nif n == 0 {\nreturn nil\n}\nroot := newTreeNode(0)\nroot.left = buildTree(n - 1)\nroot.right = buildTree(n - 1)\nreturn root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\nif n == 0 {\nreturn nil\n}\nlet root = TreeNode(x: 0)\nroot.left = buildTree(n: n - 1)\nroot.right = buildTree(n: n - 1)\nreturn root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\nif (n === 0) return null;\nconst root = new TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\nif (n == 0) return null;\nTreeNode root = TreeNode(0);\nroot.left = buildTree(n - 1);\nroot.right = buildTree(n - 1);\nreturn root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\nif n == 0 {return None};\nlet root = TreeNode::new(0);\nroot.borrow_mut().left = build_tree(n - 1);\nroot.borrow_mut().right = build_tree(n - 1);\nreturn Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\nif (n == 0)\nreturn NULL;\nTreeNode *root = newTreeNode(0);\nroot-&gt;left = buildTree(n - 1);\nroot-&gt;right = buildTree(n - 1);\nreturn root;\n}\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\nif (n == 0) return null;\nconst root = try mem_allocator.create(inc.TreeNode(i32));\nroot.init(0);\nroot.left = try buildTree(mem_allocator, n - 1);\nroot.right = try buildTree(mem_allocator, n - 1);\nreturn root;\n}\n</code></pre> <p></p> <p> \u56fe 2-19 \u00a0 \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3002\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u518d\u4f8b\u5982\u5c06\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\uff0c\u8f93\u5165\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"<p>\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96be\u3002</p> <p>\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</p> <p>\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<p>\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</p> <ul> <li>\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e24\u4e2a\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002</li> <li>\u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002</li> </ul> <p>\u65f6\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u7684\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002</li> <li>\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002</li> <li>\u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</li> <li>\u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002</li> <li>\u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002</li> <li>\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002</li> </ul> <p>\u7a7a\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u4f5c\u7528\u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002</li> <li>\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u7eb3\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u6307\u4ee4\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u523b\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f</p> <p>\u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u901a\u5e38\u8ba4\u4e3a\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002</p> <p>\u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f</p> <p>\u300c\u51fd\u6570 function\u300d\u53ef\u4ee5\u88ab\u72ec\u7acb\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002\u300c\u65b9\u6cd5 method\u300d\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002</p> <p>\u4e0b\u9762\u4ee5\u51e0\u79cd\u5e38\u89c1\u7684\u7f16\u7a0b\u8bed\u8a00\u4e3a\u4f8b\u6765\u8bf4\u660e\u3002</p> <ul> <li>C \u8bed\u8a00\u662f\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\u8bed\u8a00\uff0c\u6ca1\u6709\u9762\u5411\u5bf9\u8c61\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u6570\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u521b\u5efa\u7ed3\u6784\u4f53\uff08struct\uff09\u6765\u6a21\u62df\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u4e0e\u7ed3\u6784\u4f53\u76f8\u5173\u8054\u7684\u51fd\u6570\u5c31\u76f8\u5f53\u4e8e\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002</li> <li>Java \u548c C# \u662f\u9762\u5411\u5bf9\u8c61\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u4ee3\u7801\u5757\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u4e3a\u67d0\u4e2a\u7c7b\u7684\u4e00\u90e8\u5206\u3002\u9759\u6001\u65b9\u6cd5\u7684\u884c\u4e3a\u7c7b\u4f3c\u4e8e\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u88ab\u7ed1\u5b9a\u5728\u7c7b\u4e0a\uff0c\u4e0d\u80fd\u8bbf\u95ee\u7279\u5b9a\u7684\u5b9e\u4f8b\u53d8\u91cf\u3002</li> <li>C++ \u548c Python \u65e2\u652f\u6301\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\uff08\u51fd\u6570\uff09\uff0c\u4e5f\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff08\u65b9\u6cd5\uff09\u3002</li> </ul> <p>\u56fe\u89e3\u201c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f</p> <p>\u4e0d\u662f\uff0c\u8be5\u56fe\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002</p> <p>\u5047\u8bbe\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u6bcf\u6761\u66f2\u7ebf\u7684\u503c\u4e0e\u51fd\u6570\u5bf9\u5e94\u4e0d\u4e0a\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6761\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u4e8e\u5c06\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\u3002\u4f46\u5bf9\u4e8e \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u65f6\u589e\u957f\u8d8b\u52bf\u5df2\u7ecf\u5360\u4e3b\u5bfc\u4e86\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u4eec\u60f3\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f</p> <ol> <li>\u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002</li> <li>\u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c <code>+</code> \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c <code>*</code> \u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5c <code>print()</code> \u9700\u8981 5 ns \u7b49\u3002</li> <li>\u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002</li> </ol> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int):\na = 2      # 1 ns\na = a + 1  # 1 ns\na = a * 2  # 10 ns\n# \u5faa\u73af n \u6b21\nfor _ in range(n):  # 1 ns\nprint(0)        # 5 ns\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\ncout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nSystem.out.println(0);     // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid Algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nConsole.WriteLine(0);      // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\na := 2     // 1 ns\na = a + 1  // 1 ns\na = a * 2  // 10 ns\n// \u5faa\u73af n \u6b21\nfor i := 0; i &lt; n; i++ {  // 1 ns\nfmt.Println(a)        // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\nvar a = 2 // 1 ns\na = a + 1 // 1 ns\na = a * 2 // 10 ns\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..&lt; n { // 1 ns\nprint(0) // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\nvar a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\nvar a: number = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nconsole.log(0); // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2; // 1 ns\na = a + 1; // 1 ns\na = a * 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprint(0); // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: i32) {\nlet mut a = 2;      // 1 ns\na = a + 1;          // 1 ns\na = a * 2;          // 10 ns\n// \u5faa\u73af n \u6b21\nfor _ in 0..n {     // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprintln!(\"{}\", 0);  // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\nint a = 2;  // 1 ns\na = a + 1;  // 1 ns\na = a * 2;  // 10 ns\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\nprintf(\"%d\", 0);            // 5 ns\n}\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: usize) void {\nvar a: i32 = 2; // 1 ns\na += 1; // 1 ns\na *= 2; // 10 ns\n// \u5faa\u73af n \u6b21\nfor (0..n) |_| { // 1 ns\nstd.debug.print(\"{}\\n\", .{0}); // 5 ns\n}\n}\n</code></pre> <p>\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e3a \\((6n + 12)\\) ns \uff1a</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>\u4f46\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u5c06\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002</p> <p>\u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int):\nprint(0)\n# \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int):\nfor _ in range(n):\nprint(0)\n# \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int):\nfor _ in range(1000000):\nprint(0)\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\ncout &lt;&lt; 0 &lt;&lt; endl;\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i &lt; n; i++) {\ncout &lt;&lt; 0 &lt;&lt; endl;\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i &lt; 1000000; i++) {\ncout &lt;&lt; 0 &lt;&lt; endl;\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nSystem.out.println(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i &lt; n; i++) {\nSystem.out.println(0);\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i &lt; 1000000; i++) {\nSystem.out.println(0);\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmA(int n) {\nConsole.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid AlgorithmB(int n) {\nfor (int i = 0; i &lt; n; i++) {\nConsole.WriteLine(0);\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmC(int n) {\nfor (int i = 0; i &lt; 1000000; i++) {\nConsole.WriteLine(0);\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\nfmt.Println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\nfor i := 0; i &lt; n; i++ {\nfmt.Println(0)\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\nfor i := 0; i &lt; 1000000; i++ {\nfmt.Println(0)\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\nprint(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\nfor _ in 0 ..&lt; n {\nprint(0)\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\nfor _ in 0 ..&lt; 1000000 {\nprint(0)\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\nfor (let i = 0; i &lt; n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\nfor (let i = 0; i &lt; 1000000; i++) {\nconsole.log(0);\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\nconsole.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\nfor (let i = 0; i &lt; n; i++) {\nconsole.log(0);\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\nfor (let i = 0; i &lt; 1000000; i++) {\nconsole.log(0);\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\nprint(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\nfor (int i = 0; i &lt; n; i++) {\nprint(0);\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\nfor (int i = 0; i &lt; 1000000; i++) {\nprint(0);\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: i32) {\nprintln!(\"{}\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) {\nfor _ in 0..n {\nprintln!(\"{}\", 0);\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) {\nfor _ in 0..1000000 {\nprintln!(\"{}\", 0);\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\nprintf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\nfor (int i = 0; i &lt; n; i++) {\nprintf(\"%d\", 0);\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\nfor (int i = 0; i &lt; 1000000; i++) {\nprintf(\"%d\", 0);\n}\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: usize) void {\n_ = n;\nstd.debug.print(\"{}\\n\", .{0});\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) void {\nfor (0..n) |_| {\nstd.debug.print(\"{}\\n\", .{0});\n}\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) void {\n_ = n;\nfor (0..1000000) |_| { std.debug.print(\"{}\\n\", .{0});\n}\n}\n</code></pre> <p>\u56fe 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u4e2a\u7b97\u6cd5\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u7b97\u6cd5 <code>A</code> \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>B</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>C</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u867d\u7136\u8fd0\u884c\u65f6\u95f4\u5f88\u957f\uff0c\u4f46\u5b83\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c <code>A</code> \u76f8\u540c\uff0c\u4ecd\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> </ul> <p></p> <p> \u56fe 2-7 \u00a0 \u7b97\u6cd5 A\u3001B \u548c C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf </p> <p>\u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u70b9\u5462\uff1f</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5 <code>B</code> \u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n &gt; 1\\) \u65f6\u6bd4\u7b97\u6cd5 <code>A</code> \u66f4\u6162\uff0c\u5728 \\(n &gt; 1000000\\) \u65f6\u6bd4\u7b97\u6cd5 <code>C</code> \u66f4\u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u201c\u7ebf\u6027\u9636\u201d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u8fd0\u884c\u65f6\u95f4\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u4e00\u6765\u4f30\u7b97\u96be\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>A</code> \u548c <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>B</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4 <code>C</code> \u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>B</code> \u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 <code>C</code> \u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232","title":"2.3.2 \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u8f93\u5165\u5927\u5c0f\u4e3a \\(n\\) \u7684\u51fd\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def algorithm(n: int):\na = 1      # +1\na = a + 1  # +1\na = a * 2  # +1\n# \u5faa\u73af n \u6b21\nfor i in range(n):  # +1\nprint(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\ncout &lt;&lt; 0 &lt;&lt; endl;    // +1\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nSystem.out.println(0);    // +1\n}\n}\n</code></pre> <pre><code>void Algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nConsole.WriteLine(0);   // +1\n}\n}\n</code></pre> <pre><code>func algorithm(n int) {\na := 1      // +1\na = a + 1   // +1\na = a * 2   // +1\n// \u5faa\u73af n \u6b21\nfor i := 0; i &lt; n; i++ {   // +1\nfmt.Println(a)         // +1\n}\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\nvar a = 1 // +1\na = a + 1 // +1\na = a * 2 // +1\n// \u5faa\u73af n \u6b21\nfor _ in 0 ..&lt; n { // +1\nprint(0) // +1\n}\n}\n</code></pre> <pre><code>function algorithm(n) {\nvar a = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\nvar a: number = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nconsole.log(0); // +1\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1; // +1\na = a + 1; // +1\na = a * 2; // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprint(0); // +1\n}\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\nlet mut a = 1;   // +1\na = a + 1;      // +1\na = a * 2;      // +1\n// \u5faa\u73af n \u6b21\nfor _ in 0..n { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprintln!(\"{}\", 0); // +1\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1;  // +1\na = a + 1;  // +1\na = a * 2;  // +1\n// \u5faa\u73af n \u6b21\nfor (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nprintf(\"%d\", 0);            // +1\n}\n}  </code></pre> <pre><code>fn algorithm(n: usize) void {\nvar a: i32 = 1; // +1\na += 1; // +1\na *= 2; // +1\n// \u5faa\u73af n \u6b21\nfor (0..n) |_| { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\nstd.debug.print(\"{}\\n\", .{0}); // +1\n}\n}\n</code></pre> <p>\u8bbe\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0a\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a\uff1a</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u5176\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002</p> <p>\u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 big-\\(O\\) notation\u300d\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c asymptotic upper bound\u300d\u3002</p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u786e\u7684\u6570\u5b66\u5b9a\u4e49\u3002</p> <p>\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c</p> <p>\u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n &gt; n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a \\(T(n) = O(f(n))\\) \u3002</p> <p>\u5982\u56fe 2-8 \u6240\u793a\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002</p> <p></p> <p> \u56fe 2-8 \u00a0 \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c </p>"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u987b\u62c5\u5fc3\u3002\u6211\u4eec\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65ad\u7684\u5b9e\u8df5\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u5176\u6570\u5b66\u610f\u4e49\u3002</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"<p>\u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\u3002</p> <ol> <li>\u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u4e0e \\(n\\) \u65e0\u5173\uff0c\u6240\u4ee5\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002</li> <li>\u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002</li> <li>\u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u7b2c <code>1.</code> \u70b9\u548c\u7b2c <code>2.</code> \u70b9\u7684\u6280\u5de7\u3002</li> </ol> <p>\u7ed9\u5b9a\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u6765\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def algorithm(n: int):\na = 1      # +0\uff08\u6280\u5de7 1\uff09\na = a + n  # +0\uff08\u6280\u5de7 1\uff09\n# +n\uff08\u6280\u5de7 2\uff09\nfor i in range(5 * n + 1):\nprint(0)\n# +n*n\uff08\u6280\u5de7 3\uff09\nfor i in range(2 * n):\nfor j in range(n + 1):\nprint(0)\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i &lt; 5 * n + 1; i++) {\ncout &lt;&lt; 0 &lt;&lt; endl;\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i &lt; 2 * n; i++) {\nfor (int j = 0; j &lt; n + 1; j++) {\ncout &lt;&lt; 0 &lt;&lt; endl;\n}\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i &lt; 5 * n + 1; i++) {\nSystem.out.println(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i &lt; 2 * n; i++) {\nfor (int j = 0; j &lt; n + 1; j++) {\nSystem.out.println(0);\n}\n}\n}\n</code></pre> <pre><code>void Algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i &lt; 5 * n + 1; i++) {\nConsole.WriteLine(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i &lt; 2 * n; i++) {\nfor (int j = 0; j &lt; n + 1; j++) {\nConsole.WriteLine(0);\n}\n}\n}\n</code></pre> <pre><code>func algorithm(n int) {\na := 1     // +0\uff08\u6280\u5de7 1\uff09\na = a + n  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor i := 0; i &lt; 5 * n + 1; i++ {\nfmt.Println(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor i := 0; i &lt; 2 * n; i++ {\nfor j := 0; j &lt; n + 1; j++ {\nfmt.Println(0)\n}\n}\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\nvar a = 1 // +0\uff08\u6280\u5de7 1\uff09\na = a + n // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor _ in 0 ..&lt; (5 * n + 1) {\nprint(0)\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor _ in 0 ..&lt; (2 * n) {\nfor _ in 0 ..&lt; (n + 1) {\nprint(0)\n}\n}\n}\n</code></pre> <pre><code>function algorithm(n) {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i &lt; 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i &lt; 2 * n; i++) {\nfor (let j = 0; j &lt; n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\nlet a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (let i = 0; i &lt; 5 * n + 1; i++) {\nconsole.log(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (let i = 0; i &lt; 2 * n; i++) {\nfor (let j = 0; j &lt; n + 1; j++) {\nconsole.log(0);\n}\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1; // +0\uff08\u6280\u5de7 1\uff09\na = a + n; // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i &lt; 5 * n + 1; i++) {\nprint(0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i &lt; 2 * n; i++) {\nfor (int j = 0; j &lt; n + 1; j++) {\nprint(0);\n}\n}\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\nlet mut a = 1;     // +0\uff08\u6280\u5de7 1\uff09\na = a + n;        // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor i in 0..(5 * n + 1) {\nprintln!(\"{}\", 0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor i in 0..(2 * n) {\nfor j in 0..(n + 1) {\nprintln!(\"{}\", 0);\n}\n}\n}\n</code></pre> <pre><code>void algorithm(int n) {\nint a = 1;  // +0\uff08\u6280\u5de7 1\uff09\na = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor (int i = 0; i &lt; 5 * n + 1; i++) {\nprintf(\"%d\", 0);\n}\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor (int i = 0; i &lt; 2 * n; i++) {\nfor (int j = 0; j &lt; n + 1; j++) {\nprintf(\"%d\", 0);\n}\n}\n}\n</code></pre> <pre><code>fn algorithm(n: usize) void {\nvar a: i32 = 1;     // +0\uff08\u6280\u5de7 1\uff09\na = a + @as(i32, @intCast(n));        // +0\uff08\u6280\u5de7 1\uff09\n// +n\uff08\u6280\u5de7 2\uff09\nfor(0..(5 * n + 1)) |_| {\nstd.debug.print(\"{}\\n\", .{0}); }\n// +n*n\uff08\u6280\u5de7 3\uff09\nfor(0..(2 * n)) |_| {\nfor(0..(n + 1)) |_| {\nstd.debug.print(\"{}\\n\", .{0}); }\n}\n}\n</code></pre> <p>\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\uff0c\u4e24\u8005\u63a8\u7b97\u51fa\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002</p> <p>\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002</p> <p> \u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6570\u91cf\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5982\u56fe 2-9 \u6240\u793a\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u7ebf\u6027\u5bf9\u6570\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} &lt; \\text{\u9636\u4e58\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-9 \u00a0 \u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002</p> <p>\u5728\u4ee5\u4e0b\u51fd\u6570\u4e2d\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf <code>size</code> \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n\"\"\"\u5e38\u6570\u9636\"\"\"\ncount = 0\nsize = 100000\nfor _ in range(size):\ncount += 1\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i &lt; size; i++)\ncount++;\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i &lt; size; i++)\ncount++;\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6570\u9636 */\nint Constant(int n) {\nint count = 0;\nint size = 100000;\nfor (int i = 0; i &lt; size; i++)\ncount++;\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\ncount := 0\nsize := 100000\nfor i := 0; i &lt; size; i++ {\ncount++\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -&gt; Int {\nvar count = 0\nlet size = 100_000\nfor _ in 0 ..&lt; size {\ncount += 1\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n) {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i &lt; size; i++) count++;\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\nlet count = 0;\nconst size = 100000;\nfor (let i = 0; i &lt; size; i++) count++;\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nfor (var i = 0; i &lt; size; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -&gt; i32 {\n_ = n;\nlet mut count = 0;\nlet size = 100_000;\nfor _ in 0..size {\ncount += 1;\n}\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\nint count = 0;\nint size = 100000;\nint i = 0;\nfor (int i = 0; i &lt; size; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n_ = n;\nvar count: i32 = 0;\nconst size: i32 = 100_000;\nvar i: i32 = 0;\nwhile(i&lt;size) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n</code></pre>"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n\"\"\"\u7ebf\u6027\u9636\"\"\"\ncount = 0\nfor _ in range(n):\ncount += 1\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i &lt; n; i++)\ncount++;\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i &lt; n; i++)\ncount++;\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\nint count = 0;\nfor (int i = 0; i &lt; n; i++)\ncount++;\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\ncount := 0\nfor i := 0; i &lt; n; i++ {\ncount++\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -&gt; Int {\nvar count = 0\nfor _ in 0 ..&lt; n {\ncount += 1\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\nlet count = 0;\nfor (let i = 0; i &lt; n; i++) count++;\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\nlet count = 0;\nfor (let i = 0; i &lt; n; i++) count++;\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (var i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -&gt; i32 {\nlet mut count = 0;\nfor _ in 0..n {\ncount += 1;\n}\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\nint count = 0;\nfor (int i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n</code></pre> <p>\u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n\"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\ncount = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor num in nums:\ncount += 1\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int num : nums) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nforeach (int num in nums) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor range nums {\ncount++\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor _ in nums {\ncount += 1\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i &lt; nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (let i = 0; i &lt; nums.length; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (var _num in nums) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\nlet mut count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor _ in nums {\ncount += 1;\n}\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (int i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\nvar count: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\nfor (nums) |_| {\ncount += 1;\n}\nreturn count;\n}\n</code></pre> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u4e2a\u793a\u4f8b\u4e2d\uff0c\u53d8\u91cf \\(n\\) \u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u4e3a\u6570\u636e\u5927\u5c0f\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n\"\"\"\u5e73\u65b9\u9636\"\"\"\ncount = 0\n# \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i in range(n):\nfor j in range(n):\ncount += 1\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\ncount := 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor i := 0; i &lt; n; i++ {\nfor j := 0; j &lt; n; j++ {\ncount++\n}\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -&gt; Int {\nvar count = 0\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor _ in 0 ..&lt; n {\nfor _ in 0 ..&lt; n {\ncount += 1\n}\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i &lt; n; i++) {\nfor (let j = 0; j &lt; n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\nlet count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (let i = 0; i &lt; n; i++) {\nfor (let j = 0; j &lt; n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -&gt; i32 {\nlet mut count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor _ in 0..n {\nfor _ in 0..n {\ncount += 1;\n}\n}\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\nint count = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; n; j++) {\ncount++;\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\nwhile (i &lt; n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j &lt; n) : (j += 1) {\ncount += 1;\n}\n}\nreturn count;\n}\n</code></pre> <p>\u56fe 2-10 \u5bf9\u6bd4\u4e86\u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u4e09\u79cd\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p></p> <p> \u56fe 2-10 \u00a0 \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4ee5\u5192\u6ce1\u6392\u5e8f\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n\"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\ncount = 0  # \u8ba1\u6570\u5668\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(len(nums) - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j in range(i):\nif nums[j] &gt; nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp: int = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\nint count = 0;  // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n(nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\ncount := 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i &gt; 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j &lt; i; j++ {\nif nums[j] &gt; nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\ntmp := nums[j]\nnums[j] = nums[j+1]\nnums[j+1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\nvar count = 0 // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in 0 ..&lt; i {\nif nums[j] &gt; nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\ncount += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (let j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\nlet count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (let j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (var i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (var j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\nlet mut count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in (1..nums.len()).rev() {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in 0..i {\nif nums[j] &gt; nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\nint count = 0; // \u8ba1\u6570\u5668\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = n - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\nvar count: i32 = 0;  // \u8ba1\u6570\u5668 \n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: i32 = @as(i32, @intCast(nums.len)) - 1;\nwhile (i &gt; 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j &lt; i) : (j += 1) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\ncount += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n}\n}\n}\nreturn count;\n}\n</code></pre>"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002</p> <p>\u56fe 2-11 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u7ec6\u80de\u5206\u88c2\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n\"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount = 0\nbase = 1\n# \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in range(n):\nfor _ in range(base):\ncount += 1\nbase *= 2\n# count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\nint count = 0, bas = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; bas; j++) {\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\ncount, base := 0, 1\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor i := 0; i &lt; n; i++ {\nfor j := 0; j &lt; base; j++ {\ncount++\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -&gt; Int {\nvar count = 0\nvar base = 1\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in 0 ..&lt; n {\nfor _ in 0 ..&lt; base {\ncount += 1\n}\nbase *= 2\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\nlet count = 0,\nbase = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i &lt; n; i++) {\nfor (let j = 0; j &lt; base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\nlet count = 0,\nbase = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (let i = 0; i &lt; n; i++) {\nfor (let j = 0; j &lt; base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0, base = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (var i = 0; i &lt; n; i++) {\nfor (var j = 0; j &lt; base; j++) {\ncount++;\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -&gt; i32 {\nlet mut count = 0;\nlet mut base = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor _ in 0..n {\nfor _ in 0..base {\ncount += 1\n}\nbase *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\nint count = 0;\nint bas = 1;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nfor (int i = 0; i &lt; n; i++) {\nfor (int j = 0; j &lt; bas; j++) {\ncount++;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\nvar count: i32 = 0;\nvar bas: i32 = 1;\nvar i: i32 = 0;\n// \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\nwhile (i &lt; n) : (i += 1) {\nvar j: i32 = 0;\nwhile (j &lt; bas) : (j += 1) {\ncount += 1;\n}\nbas *= 2;\n}\n// count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\nreturn count;\n}\n</code></pre> <p></p> <p> \u56fe 2-11 \u00a0 \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u5176\u9012\u5f52\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n\"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 1:\nreturn 1\nreturn exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\nif (n == 1) return 1;\nreturn ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\nif n == 1 {\nreturn 1\n}\nreturn expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\nif (n === 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\nif (n === 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\nif n == 1 {\nreturn 1;\n}\nexp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\nif (n == 1)\nreturn 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\nif (n == 1) return 1;\nreturn expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> <p>\u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7a77\u4e3e\u6cd5\uff08\u66b4\u529b\u641c\u7d22\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f83\u5e38\u89c1\u3002\u5bf9\u4e8e\u6570\u636e\u89c4\u6a21\u8f83\u5927\u7684\u95ee\u9898\uff0c\u6307\u6570\u9636\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u6216\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u6765\u89e3\u51b3\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u60c5\u51b5\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002</p> <p>\u56fe 2-12 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 n)\\) \uff0c\u7b80\u8bb0\u4e3a \\(O(\\log n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def logarithmic(n: float) -&gt; int:\n\"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\ncount = 0\nwhile n &gt; 1:\nn = n / 2\ncount += 1\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n &gt; 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n &gt; 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(float n) {\nint count = 0;\nwhile (n &gt; 1) {\nn /= 2;\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\ncount := 0\nfor n &gt; 1 {\nn = n / 2\ncount++\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -&gt; Int {\nvar count = 0\nvar n = n\nwhile n &gt; 1 {\nn = n / 2\ncount += 1\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\nlet count = 0;\nwhile (n &gt; 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\nlet count = 0;\nwhile (n &gt; 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\nint count = 0;\nwhile (n &gt; 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: f32) -&gt; i32 {\nlet mut count = 0;\nwhile n &gt; 1.0 {\nn = n / 2.0;\ncount += 1;\n}\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\nint count = 0;\nwhile (n &gt; 1) {\nn = n / 2;\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\nvar count: i32 = 0;\nvar n_var = n;\nwhile (n_var &gt; 1)\n{\nn_var = n_var / 2;\ncount +=1;\n}\nreturn count;\n}\n</code></pre> <p></p> <p> \u56fe 2-12 \u00a0 \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def log_recur(n: float) -&gt; int:\n\"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n &lt;= 1:\nreturn 0\nreturn log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n &lt;= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n &lt;= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(float n) {\nif (n &lt;= 1) return 0;\nreturn LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\nif n &lt;= 1 {\nreturn 0\n}\nreturn logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -&gt; Int {\nif n &lt;= 1 {\nreturn 0\n}\nreturn logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\nif (n &lt;= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\nif (n &lt;= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\nif (n &lt;= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: f32) -&gt; i32 {\nif n &lt;= 1.0 {\nreturn 0;\n}\nlog_recur(n / 2.0) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\nif (n &lt;= 1)\nreturn 0;\nreturn logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\nif (n &lt;= 1) return 0;\nreturn logRecur(n / 2) + 1;\n}\n</code></pre> <p>\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u7b97\u6cd5\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u957f\u7f13\u6162\uff0c\u662f\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\u7684\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\\(O(\\log n)\\) \u7684\u5e95\u6570\u662f\u591a\u5c11\uff1f</p> <p>\u51c6\u786e\u6765\u8bf4\uff0c\u201c\u4e00\u5206\u4e3a \\(m\\)\u201d\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u901a\u8fc7\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6570\u3001\u76f8\u7b49\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5e95\u6570 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u54cd\u590d\u6742\u5ea6\u7684\u524d\u63d0\u4e0b\u8f6c\u6362\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u7701\u7565\u5e95\u6570 \\(m\\) \uff0c\u5c06\u5bf9\u6570\u9636\u76f4\u63a5\u8bb0\u4e3a \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"<p>\u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def linear_log_recur(n: float) -&gt; int:\n\"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\nif n &lt;= 1:\nreturn 1\ncount: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\nfor _ in range(n):\ncount += 1\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n &lt;= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n &lt;= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(float n) {\nif (n &lt;= 1) return 1;\nint count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\nfor (int i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\nif n &lt;= 1 {\nreturn 1\n}\ncount := linearLogRecur(n/2) + linearLogRecur(n/2)\nfor i := 0.0; i &lt; n; i++ {\ncount++\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -&gt; Int {\nif n &lt;= 1 {\nreturn 1\n}\nvar count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\nfor _ in stride(from: 0, to: n, by: 1) {\ncount += 1\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\nif (n &lt;= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\nif (n &lt;= 1) return 1;\nlet count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (let i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\nif (n &lt;= 1) return 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (var i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: f32) -&gt; i32 {\nif n &lt;= 1.0 {\nreturn 1;\n}\nlet mut count = linear_log_recur(n / 2.0) + linear_log_recur(n / 2.0);\nfor _ in 0 ..n as i32 {\ncount += 1;\n}\nreturn count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\nif (n &lt;= 1)\nreturn 1;\nint count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nfor (int i = 0; i &lt; n; i++) {\ncount++;\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\nif (n &lt;= 1) return 1;\nvar count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\nvar i: f32 = 0;\nwhile (i &lt; n) : (i += 1) {\ncount += 1;\n}\nreturn count;\n}\n</code></pre> <p>\u56fe 2-13 \u5c55\u793a\u4e86\u7ebf\u6027\u5bf9\u6570\u9636\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u53c9\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u603b\u6570\u90fd\u4e3a \\(n\\) \uff0c\u6811\u5171\u6709 \\(\\log_2 n + 1\\) \u5c42\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p></p> <p> \u56fe 2-13 \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u9636\u4e58\u9636 \\(O(n!)\\)","text":"<p>\u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>\u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u5982\u56fe 2-14 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u505c\u6b62\u5206\u88c2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n\"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\nif n == 0:\nreturn 1\ncount = 0\n# \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in range(n):\ncount += factorial_recur(n - 1)\nreturn count\n</code></pre> time_complexity.cpp<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i &lt; n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i &lt; n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (int i = 0; i &lt; n; i++) {\ncount += FactorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\nif n == 0 {\nreturn 1\n}\ncount := 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor i := 0; i &lt; n; i++ {\ncount += factorialRecur(n - 1)\n}\nreturn count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\nif n == 0 {\nreturn 1\n}\nvar count = 0\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in 0 ..&lt; n {\ncount += factorialRecur(n: n - 1)\n}\nreturn count\n}\n</code></pre> time_complexity.js<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\nif (n === 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i &lt; n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\nif (n === 0) return 1;\nlet count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (let i = 0; i &lt; n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0) return 1;\nint count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor (var i = 0; i &lt; n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\nif n == 0 {\nreturn 1;\n}\nlet mut count = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nfor _ in 0..n {\ncount += factorial_recur(n - 1);\n}\ncount\n}\n</code></pre> time_complexity.c<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\nif (n == 0)\nreturn 1;\nint count = 0;\nfor (int i = 0; i &lt; n; i++) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\nif (n == 0) return 1;\nvar count: i32 = 0;\nvar i: i32 = 0;\n// \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\nwhile (i &lt; n) : (i += 1) {\ncount += factorialRecur(n - 1);\n}\nreturn count;\n}\n</code></pre> <p></p> <p> \u56fe 2-14 \u00a0 \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u5f53 \\(n \\geq 4\\) \u65f6\u6052\u6709 \\(n! &gt; 2^n\\) \uff0c\u6240\u4ee5\u9636\u4e58\u9636\u6bd4\u6307\u6570\u9636\u589e\u957f\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f83\u5927\u65f6\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u7b97\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d <code>nums</code> \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u6bcf\u4e2a\u6570\u5b57\u53ea\u51fa\u73b0\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff0c\u4efb\u52a1\u76ee\u6807\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u5f53 <code>nums = [?, ?, ..., 1]</code> \uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u5f53 <code>nums = [1, ?, ?, ...]</code> \uff0c\u5373\u5f53\u9996\u4e2a\u5143\u7d20\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \u3002</li> </ul> <p>\u201c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\u3002\u76f8\u5e94\u5730\uff0c\u201c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8bb0\u53f7\u8868\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n\"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n# \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\nnums = [i for i in range(1, n + 1)]\n# \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrandom.shuffle(nums)\nreturn nums\ndef find_one(nums: list[int]) -&gt; int:\n\"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\nfor i in range(len(nums)):\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n# \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1:\nreturn i\nreturn -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector&lt;int&gt; randomNumbers(int n) {\nvector&lt;int&gt; nums(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i &lt; n; i++) {\nnums[i] = i + 1;\n}\n// \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\nunsigned seed = chrono::system_clock::now().time_since_epoch().count();\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nshuffle(nums.begin(), nums.end(), default_random_engine(seed));\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\nfor (int i = 0; i &lt; nums.size(); i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\nInteger[] nums = new Integer[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i &lt; n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nCollections.shuffle(Arrays.asList(nums));\n// Integer[] -&gt; int[]\nint[] res = new int[n];\nfor (int i = 0; i &lt; n; i++) {\nres[i] = nums[i];\n}\nreturn res;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\nfor (int i = 0; i &lt; nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\nint[] nums = new int[n];\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i &lt; n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = 0; i &lt; nums.Length; i++) {\nint index = new Random().Next(i, nums.Length);\n(nums[i], nums[index]) = (nums[index], nums[i]);\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\nfor (int i = 0; i &lt; nums.Length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\nnums := make([]int, n)\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor i := 0; i &lt; n; i++ {\nnums[i] = i + 1\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nrand.Shuffle(len(nums), func(i, j int) {\nnums[i], nums[j] = nums[j], nums[i]\n})\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\nfor i := 0; i &lt; len(nums); i++ {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nvar nums = Array(1 ... n)\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle()\nreturn nums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\nfor i in nums.indices {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn i\n}\n}\nreturn -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i &lt; n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i &lt; n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\nfor (let i = 0; i &lt; nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\nconst nums = Array(n);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (let i = 0; i &lt; n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (let i = 0; i &lt; n; i++) {\nconst r = Math.floor(Math.random() * (i + 1));\nconst temp = nums[i];\nnums[i] = nums[r];\nnums[r] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\nfor (let i = 0; i &lt; nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] === 1) {\nreturn i;\n}\n}\nreturn -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList&lt;int&gt; randomNumbers(int n) {\nfinal nums = List.filled(n, 0);\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (var i = 0; i &lt; n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle();\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\nfor (var i = 0; i &lt; nums.length; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1) return i;\n}\nreturn -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nlet mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nnums.shuffle(&amp;mut thread_rng());\nnums\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\nfor i in 0..nums.len() {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif nums[i] == 1 {\nreturn Some(i);\n}\n}\nNone\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n// \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\nint *nums = (int *)malloc(n * sizeof(int));\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (int i = 0; i &lt; n; i++) {\nnums[i] = i + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nfor (int i = n - 1; i &gt; 0; i--) {\nint j = rand() % (i + 1);\nint temp = nums[i];\nnums[i] = nums[j];\nnums[j] = temp;\n}\nreturn nums;\n}\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\nfor (int i = 0; i &lt; n; i++) {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (nums[i] == 1)\nreturn i;\n}\nreturn -1;\n}\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\nvar nums: [n]i32 = undefined;\n// \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\nfor (&amp;nums, 0..) |*num, i| {\nnum.* = @as(i32, @intCast(i)) + 1;\n}\n// \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\nconst rand = std.crypto.random;\nrand.shuffle(i32, &amp;nums);\nreturn nums;\n}\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\nfor (nums, 0..) |num, i| {\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n// \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\nif (num == 1) return @intCast(i);\n}\nreturn -1;\n}\n</code></pre> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6211\u4eec\u5728\u5b9e\u9645\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u800c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u6548\u7387\u5b89\u5168\u503c\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002</p> <p>\u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u4e8e\u201c\u7279\u6b8a\u7684\u6570\u636e\u5206\u5e03\u201d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u5e76\u4e0d\u80fd\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002</p> <p>\u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5c31\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002</p> <p>\u4f46\u5bf9\u4e8e\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5f80\u5f80\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002</p> <p>\u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f</p> <p>\u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u8bb2\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002</p>"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u6570\u636e\u7ed3\u6784","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u5982\u540c\u4e00\u526f\u7a33\u56fa\u800c\u591a\u6837\u7684\u6846\u67b6\u3002</p> <p>\u5b83\u4e3a\u6570\u636e\u7684\u6709\u5e8f\u7ec4\u7ec7\u63d0\u4f9b\u4e86\u84dd\u56fe\uff0c\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u7840\u4e0a\u751f\u52a8\u8d77\u6765\u3002</p>"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b</li> <li>3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b</li> <li>3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *</li> <li>3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *</li> <li>3.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"<p>\u5f53\u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\u65f6\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \u3002</li> <li>\u6d6e\u70b9\u6570\u7c7b\u578b <code>float</code>\u3001<code>double</code> \uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\u3002</li> <li>\u5b57\u7b26\u7c7b\u578b <code>char</code> \uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002</li> <li>\u5e03\u5c14\u7c7b\u578b <code>bool</code> \uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\u3002</li> </ul> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u5373\u4e3a \\(1\\) \u6bd4\u7279\u3002\u5728\u7edd\u5927\u591a\u6570\u73b0\u4ee3\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bit\uff09\u7ec4\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u4e3a\u4f8b\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code> \u5360\u7528 \\(1\\) byte = \\(8\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u6570\u5b57\u3002</li> <li>\u6574\u6570\u7c7b\u578b <code>int</code> \u5360\u7528 \\(4\\) bytes = \\(32\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\u3002</li> </ul> <p>\u8868 3-1 \u5217\u4e3e\u4e86 Java \u4e2d\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u987b\u6b7b\u8bb0\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002</p> <p> \u8868 3-1 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u548c\u53d6\u503c\u8303\u56f4 </p> \u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570 <code>byte</code> 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 <code>float</code> 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 <code>char</code> 2 bytes \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 <code>bool</code> 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>\u8bf7\u6ce8\u610f\uff0c\u8868 3-1 \u9488\u5bf9\u7684\u662f Java \u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u60c5\u51b5\u3002\u6bcf\u79cd\u7f16\u7a0b\u8bed\u8a00\u6709\u5404\u81ea\u7684\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\uff0c\u5b83\u4eec\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5728 Python \u4e2d\uff0c\u6574\u6570\u7c7b\u578b <code>int</code> \u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\uff1b\u6d6e\u70b9\u6570 <code>float</code> \u662f\u53cc\u7cbe\u5ea6 64 \u4f4d\uff1b\u6ca1\u6709 <code>char</code> \u7c7b\u578b\uff0c\u5355\u4e2a\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32 <code>str</code> \u3002</li> <li>C \u548c C++ \u672a\u660e\u786e\u89c4\u5b9a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5927\u5c0f\uff0c\u800c\u56e0\u5b9e\u73b0\u548c\u5e73\u53f0\u5404\u5f02\u3002\u8868 3-1 \u9075\u5faa LP64 \u6570\u636e\u6a21\u578b\uff0c\u5176\u7528\u4e8e\u5305\u62ec Linux \u548c macOS \u5728\u5185\u7684 Unix 64 \u4f4d\u64cd\u4f5c\u7cfb\u7edf\u3002</li> <li>\u5b57\u7b26 <code>char</code> \u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u4e3a 1 \u5b57\u8282\uff0c\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\u53d6\u51b3\u4e8e\u7279\u5b9a\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u201c\u5b57\u7b26\u7f16\u7801\u201d\u7ae0\u8282\u3002</li> <li>\u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002</li> </ul> <p>\u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u8fd9\u53e5\u8bdd\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\u800c\u975e\u201c\u6570\u636e\u201d\u3002</p> <p>\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570 <code>int</code>\u3001\u5c0f\u6570 <code>float</code> \u6216\u662f\u5b57\u7b26 <code>char</code> \uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u7528\u76f8\u540c\u7684\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\uff09\u6765\u5b58\u50a8\u4e0e\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5305\u62ec <code>int</code>\u3001<code>float</code>\u3001<code>char</code>\u3001<code>bool</code> \u7b49\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet bools = Array(repeating: Bool(), count: 5)\n</code></pre> <pre><code>// JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0, 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"<p>\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26 <code>char</code> \u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"<p>\u300cASCII \u7801\u300d\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u79f0\u4e3a American Standard Code for Information Interchange\uff08\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5982\u56fe 3-6 \u6240\u793a\uff0cASCII \u7801\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002</p> <p></p> <p> \u56fe 3-6 \u00a0 ASCII \u7801 </p> <p>\u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u96c6\u300cEASCII\u300d\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002</p> <p>\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"<p>\u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6c49\u5b57\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86\u300cGB2312\u300d\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002</p> <p>\u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002\u300cGBK\u300d\u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\u6269\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7684\u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"<p>\u968f\u7740\u8ba1\u7b97\u673a\u6280\u672f\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u8ba1\u7b97\u673a\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002</p> <p>\u90a3\u4e2a\u65f6\u4ee3\u7684\u7814\u7a76\u4eba\u5458\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002</p> <p>\u300cUnicode\u300d\u7684\u4e2d\u6587\u540d\u79f0\u4e3a\u201c\u7edf\u4e00\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3 100 \u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002</p> <p>\u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360\u7528 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002</p> <p>Unicode \u662f\u4e00\u79cd\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f</p> <p>\u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u56fe 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002</p> <p></p> <p> \u56fe 3-7 \u00a0 Unicode \u7f16\u7801\u793a\u4f8b </p> <p>\u7136\u800c ASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u7684\u4e24\u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7f16\u7801","text":"<p>\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u5ea6\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700 1 \u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u5b57\u8282\u3002</p> <p>UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff0c\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002</li> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n &gt; 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002</li> </ul> <p>\u56fe 3-8 \u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u89c2\u5bdf\u53d1\u73b0\uff0c\u7531\u4e8e\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7edf\u53ef\u4ee5\u901a\u8fc7\u8bfb\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u4e2a\u6570\u6765\u89e3\u6790\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002</p> <p>\u4f46\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u5224\u65ad\u51fa\u5f02\u5e38\u3002</p> <p>\u4e4b\u6240\u4ee5\u5c06 \\(10\\) \u5f53\u4f5c\u6821\u9a8c\u7b26\uff0c\u662f\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\uff1a\u5047\u8bbe\u4e00\u4e2a\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \uff0c\u8bf4\u660e\u8be5\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(1\\) \uff0c\u5bf9\u5e94 ASCII \u7801\u3002\u800c ASCII \u7801\u7684\u6700\u9ad8\u4f4d\u5e94\u8be5\u662f \\(0\\) \uff0c\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</p> <p></p> <p> \u56fe 3-8 \u00a0 UTF-8 \u7f16\u7801\u793a\u4f8b </p> <p>\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ul> <li>UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002</li> <li>UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e ASCII \u5b57\u7b26\u5360\u6bd4\u8f83\u9ad8\u7684\u6587\u672c\u3002</li> </ul> <p>\u4ece\u5b58\u50a8\u7a7a\u95f4\u5360\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 2 \u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u5b57\u8282\u3002</p> <p>\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u4f18\u5148\u652f\u6301 UTF-8 \u3002</p>"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"<p>\u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7684\u7f16\u7801\u3002\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u8fd9\u79cd\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1aUTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002</li> <li>\u5b57\u7b26\u8ba1\u6570\uff1a\u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002</li> <li>\u5b57\u7b26\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u8bb8\u591a\u56e0\u7d20\u3002</p> <ul> <li>Java \u7684 <code>String</code> \u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002</li> <li>JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8bed\u8a00\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u53d1\u5c55\u65e9\u671f\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002</li> <li>C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u662f\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff08\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff09\u90fd\u5e7f\u6cdb\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul> <p>\u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u4ece\u800c\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u589e\u52a0\u989d\u5916\u4ee3\u7801\uff0c\u8fd9\u63d0\u9ad8\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c\u8c03\u8bd5\u96be\u5ea6\u3002</p> <p>\u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\u3002</p> <ul> <li>Python \u4e2d\u7684 <code>str</code> \u4f7f\u7528 Unicode \u7f16\u7801\uff0c\u5e76\u91c7\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u82e5\u5b57\u7b26\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u6709\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u4e2a\u5b57\u8282\u3002</li> <li>Go \u8bed\u8a00\u7684 <code>string</code> \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86 <code>rune</code> \u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> <li>Rust \u8bed\u8a00\u7684 str \u548c String \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 <code>char</code> \u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"<p>\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\uff0c\u5b83\u4eec\u53ef\u4ee5\u4ece\u201c\u903b\u8f91\u7ed3\u6784\u201d\u548c\u201c\u7269\u7406\u7ed3\u6784\u201d\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"<p>\u903b\u8f91\u7ed3\u6784\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u4e00\u5b9a\u987a\u5e8f\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002</p> <p>\u5982\u56fe 3-1 \u6240\u793a\uff0c\u903b\u8f91\u7ed3\u6784\u53ef\u5206\u4e3a\u201c\u7ebf\u6027\u201d\u548c\u201c\u975e\u7ebf\u6027\u201d\u4e24\u5927\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\u3002</p> <ul> <li>\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3002</li> <li>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u56fe\u3001\u54c8\u5e0c\u8868\u3002</li> </ul> <p></p> <p> \u56fe 3-1 \u00a0 \u7ebf\u6027\u6570\u636e\u7ed3\u6784\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784 </p> <p>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u6811\u5f62\u7ed3\u6784\u548c\u7f51\u72b6\u7ed3\u6784\u3002</p> <ul> <li>\u7ebf\u6027\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u961f\u5217\u3001\u6808\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u4e00\u7684\u987a\u5e8f\u5173\u7cfb\u3002</li> <li>\u6811\u5f62\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> <li>\u7f51\u72b6\u7ed3\u6784\uff1a\u56fe\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u591a\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> </ul>"},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2 \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u5206\u6563","text":"<p>\u5f53\u7b97\u6cd5\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\u4e3b\u8981\u88ab\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u56fe 3-2 \u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8\u4e00\u5b9a\u5927\u5c0f\u7684\u6570\u636e\u3002</p> <p>\u7cfb\u7edf\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u76ee\u6807\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u5982\u56fe 3-2 \u6240\u793a\uff0c\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002</p> <p></p> <p> \u56fe 3-2 \u00a0 \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740 </p> <p>Tip</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5c06\u5185\u5b58\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u4e2a\u7b80\u5316\u7684\u7c7b\u6bd4\uff0c\u5b9e\u9645\u5185\u5b58\u7684\u5de5\u4f5c\u673a\u5236\u6bd4\u8f83\u590d\u6742\uff0c\u6d89\u53ca\u5230\u5730\u5740\u7a7a\u95f4\u3001\u5185\u5b58\u7ba1\u7406\u3001\u7f13\u5b58\u673a\u5236\u3001\u865a\u62df\u548c\u7269\u7406\u5185\u5b58\u7b49\u6982\u5ff5\u3002</p> <p>\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u67d0\u5757\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5219\u65e0\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u7f3a\u5c11\u8fde\u7eed\u5927\u5757\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u5206\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002</p> <p>\u5982\u56fe 3-3 \u6240\u793a\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002</p> <p></p> <p> \u56fe 3-3 \u00a0 \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u5206\u6563\u7a7a\u95f4\u5b58\u50a8 </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u3002</li> </ul> <p>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u79f0\u201c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u611f\u89c9\u7269\u7406\u7ed3\u6784\u7406\u89e3\u8d77\u6765\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\uff0c\u7136\u540e\u518d\u56de\u987e\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"<p>Note</p> <p>\u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u6574\u6570\u7f16\u7801","text":"<p>\u5728\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\uff0c\u4f8b\u5982 <code>byte</code> \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002</p> <p>\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6570\u5b57\u662f\u4ee5\u201c\u8865\u7801\u201d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u5728\u5206\u6790\u8fd9\u6837\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\u3002</p> <ul> <li>\u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002</li> <li>\u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002</li> <li>\u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002</li> </ul> <p>\u56fe 3-4 \u5c55\u793a\u4e86\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u4e4b\u95f4\u7684\u8f6c\u6362\u65b9\u6cd5\u3002</p> <p></p> <p> \u56fe 3-4 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362 </p> <p>\u300c\u539f\u7801 sign\u2013magnitude\u300d\u867d\u7136\u6700\u76f4\u89c2\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u300c\u53cd\u7801 1's complement\u300d\u3002\u5982\u679c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u540e\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u6362\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u7801)} + 1000 \\; 0010 \\; \\text{(\u539f\u7801)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1101 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u3002\u6bd4\u5982\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u5219\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5224\u65ad\u7ed3\u679c\u51fa\u9519\u3002\u800c\u5982\u679c\u6211\u4eec\u60f3\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u8fd9\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>\u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\uff0c\u56e0\u6b64\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u300c\u8865\u7801 2's complement\u300d\u3002\u6211\u4eec\u5148\u6765\u89c2\u5bdf\u4e00\u4e0b\u8d1f\u96f6\u7684\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u8f6c\u6362\u8fc7\u7a0b\uff1a</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(\u539f\u7801)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(\u53cd\u7801)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline \\end{aligned} \\] <p>\u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u4f46 <code>byte</code> \u7c7b\u578b\u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\; 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u6b63\u8d1f\u96f6\u6b67\u4e49\u4ece\u800c\u5f97\u5230\u89e3\u51b3\u3002</p> <p>\u8fd8\u5269\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1a<code>byte</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u3002</p> <p>\u7136\u800c\uff0c\u8865\u7801 \\(1000 \\; 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\; 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u7801)} + 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u8865\u7801)} + 1111  \\; 1111 \\; \\text{(\u8865\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>\u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u8fd0\u7b97\u901f\u5ea6\u66f4\u5feb\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002</p> <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u987b\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002</p> <p>\u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5c31\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\uff0c\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5165\u4e86\u89e3\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"<p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1a<code>int</code> \u548c <code>float</code> \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes \uff0c\u4f46\u4e3a\u4ec0\u4e48 <code>float</code> \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4 <code>float</code> \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 <code>float</code> \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8bb0\u4e00\u4e2a 32-bit \u957f\u5ea6\u7684\u4e8c\u8fdb\u5236\u6570\u4e3a\uff1a</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684 <code>float</code> \u7531\u4ee5\u4e0b\u4e09\u4e2a\u90e8\u5206\u6784\u6210\u3002</p> <ul> <li>\u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff0c\u5bf9\u5e94 \\(b_{31}\\) \u3002</li> <li>\u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff0c\u5bf9\u5e94 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002</li> <li>\u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 23 bits \uff0c\u5bf9\u5e94 \\(b_{22} b_{21} \\ldots b_0\\) \u3002</li> </ul> <p>\u4e8c\u8fdb\u5236\u6570 <code>float</code> \u5bf9\u5e94\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e3a\uff1a</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>\u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>\u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a\uff1a</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> \u56fe 3-5 \u00a0 IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u7684\u8ba1\u7b97\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 3-5 \uff0c\u7ed9\u5b9a\u4e00\u4e2a\u793a\u4f8b\u6570\u636e \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5219\u6709\uff1a</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1a<code>float</code> \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0c<code>float</code> \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002</p> <p>\u5c3d\u7ba1\u6d6e\u70b9\u6570 <code>float</code> \u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b <code>int</code> \u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 <code>float</code> \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002</p> <p>\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002</p> <p> \u8868 3-2 \u00a0 \u6307\u6570\u4f4d\u542b\u4e49 </p> \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23}\\) \u3002</p> <p>\u53cc\u7cbe\u5ea6 <code>double</code> \u4e5f\u91c7\u7528\u7c7b\u4f3c\u4e8e <code>float</code> \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\u3002</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002</li> <li>\u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002</li> <li>\u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002</li> <li>\u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \uff0c\u6d6e\u70b9\u6570 <code>float</code>\u3001<code>double</code> \uff0c\u5b57\u7b26 <code>char</code> \u548c\u5e03\u5c14 <code>boolean</code> \u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002</li> <li>\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002</li> <li>\u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002</li> <li>\u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u56e0\u6b64\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002</li> <li>ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u5185\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002</li> <li>UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u6bd4 UTF-8 \u7684\u5360\u7528\u7a7a\u95f4\u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f</p> <p>\u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u94fe\u5f0f\u5730\u5740\u201d\uff08\u540e\u7eed\u54c8\u5e0c\u8868\u7ae0\u8282\u4f1a\u8bb2\uff09\uff1a\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u6876\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff0c\u5f53\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002 \u4ece\u5b58\u50a8\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u94fe\u8868\u6216\u6811\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\uff09\u6570\u636e\u7ed3\u6784\u3002</p> <p><code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 byte \u5417\uff1f</p> <p><code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u7531\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u51b3\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 char \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 bytes \u3002</p> <p>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u88ab\u79f0\u4e3a\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d \u662f\u5426\u6709\u6b67\u4e49\uff1f\u56e0\u4e3a\u6808\u4e5f\u53ef\u4ee5\u8fdb\u884c\u51fa\u6808\u548c\u5165\u6808\u7b49\u64cd\u4f5c\uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52a8\u6001\u201d\u7684\u3002</p> <p>\u6808\u786e\u5b9e\u53ef\u4ee5\u5b9e\u73b0\u52a8\u6001\u7684\u6570\u636e\u64cd\u4f5c\uff0c\u4f46\u6570\u636e\u7ed3\u6784\u4ecd\u7136\u662f\u201c\u9759\u6001\u201d\uff08\u957f\u5ea6\u4e0d\u53ef\u53d8\uff09\u7684\u3002\u5c3d\u7ba1\u57fa\u4e8e\u6570\u7ec4\u7684\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u52a8\u6001\u5730\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u4eec\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u6570\u636e\u91cf\u8d85\u51fa\u4e86\u9884\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u8001\u6570\u7ec4\u7684\u5185\u5bb9\u590d\u5236\u5230\u65b0\u6570\u7ec4\u4e2d\u3002</p> <p>\u5728\u6784\u5efa\u6808\uff08\u961f\u5217\uff09\u7684\u65f6\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u4e3a\u4ec0\u4e48\u5b83\u4eec\u662f\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\u5462\uff1f</p> <p>\u5728\u9ad8\u7ea7\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e0\u987b\u4eba\u5de5\u6307\u5b9a\u6808\uff08\u961f\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u8fd9\u4e2a\u5de5\u4f5c\u662f\u7531\u7c7b\u5185\u90e8\u81ea\u52a8\u5b8c\u6210\u7684\u3002\u4f8b\u5982\uff0cJava \u7684 ArrayList \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u4e3a 10 \u3002\u53e6\u5916\uff0c\u6269\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52a8\u5b9e\u73b0\u7684\u3002\u8be6\u89c1\u672c\u4e66\u7684\u201c\u5217\u8868\u201d\u7ae0\u8282\u3002</p>"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"<p>Abstract</p> <p>\u96be\u9898\u88ab\u9010\u5c42\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u53d8\u5f97\u66f4\u4e3a\u7b80\u5355\u3002</p> <p>\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u4e2a\u91cd\u8981\u7684\u4e8b\u5b9e\uff1a\u4ece\u7b80\u5355\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u590d\u6742\u3002</p>"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5</li> <li>12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565</li> <li>12.3 \u00a0 \u6784\u5efa\u6811\u95ee\u9898</li> <li>12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898</li> <li>12.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_divide_and_conquer/binary_search_recur/","title":"12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565","text":"<p>\u6211\u4eec\u5df2\u7ecf\u5b66\u8fc7\uff0c\u641c\u7d22\u7b97\u6cd5\u5206\u4e3a\u4e24\u5927\u7c7b\u3002</p> <ul> <li>\u66b4\u529b\u641c\u7d22\uff1a\u5b83\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u81ea\u9002\u5e94\u641c\u7d22\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f\u6216\u5148\u9a8c\u4fe1\u606f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u548c\u6811\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u6bcf\u4e00\u6b65\u90fd\u5c06\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\u5206\u89e3\u4e3a\u4e00\u4e2a\u5c0f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u4e00\u76f4\u6301\u7eed\u5230\u6570\u7ec4\u4e3a\u7a7a\u6216\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u6811\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u5806\u7b49\u6570\u636e\u7ed3\u6784\u4e2d\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u67e5\u627e\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\u5206\u89e3\u4e3a\u5b50\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\uff0c\u8fd9\u662f\u901a\u8fc7\u6bd4\u8f83\u4e2d\u95f4\u5143\u7d20\u548c\u76ee\u6807\u5143\u7d20\u6765\u5b9e\u73b0\u7684\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u5904\u7406\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u95ee\u9898\u7684\u5f71\u54cd\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u65e0\u987b\u5408\u5e76\uff1a\u4e8c\u5206\u67e5\u627e\u65e8\u5728\u67e5\u627e\u4e00\u4e2a\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u3002\u5f53\u5b50\u95ee\u9898\u5f97\u5230\u89e3\u51b3\u65f6\uff0c\u539f\u95ee\u9898\u4e5f\u4f1a\u540c\u65f6\u5f97\u5230\u89e3\u51b3\u3002</li> </ul> <p>\u5206\u6cbb\u80fd\u591f\u63d0\u5347\u641c\u7d22\u6548\u7387\uff0c\u672c\u8d28\u4e0a\u662f\u56e0\u4e3a\u66b4\u529b\u641c\u7d22\u6bcf\u8f6e\u53ea\u80fd\u6392\u9664\u4e00\u4e2a\u9009\u9879\uff0c\u800c\u5206\u6cbb\u641c\u7d22\u6bcf\u8f6e\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9009\u9879\u3002</p>"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u4e8e\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e","text":"<p>\u5728\u4e4b\u524d\u7684\u7ae0\u8282\u4e2d\uff0c\u4e8c\u5206\u67e5\u627e\u662f\u57fa\u4e8e\u9012\u63a8\uff08\u8fed\u4ee3\uff09\u5b9e\u73b0\u7684\u3002\u73b0\u5728\u6211\u4eec\u57fa\u4e8e\u5206\u6cbb\uff08\u9012\u5f52\uff09\u6765\u5b9e\u73b0\u5b83\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8bf7\u67e5\u627e\u5143\u7d20 <code>target</code> \u3002</p> <p>\u4ece\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u4eec\u5c06\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u8bb0\u4e3a \\(f(i, j)\\) \u3002</p> <p>\u4ee5\u539f\u95ee\u9898 \\(f(0, n-1)\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u8fdb\u884c\u4e8c\u5206\u67e5\u627e\u3002</p> <ol> <li>\u8ba1\u7b97\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u7684\u4e2d\u70b9 \\(m\\) \uff0c\u6839\u636e\u5b83\u6392\u9664\u4e00\u534a\u641c\u7d22\u533a\u95f4\u3002</li> <li>\u9012\u5f52\u6c42\u89e3\u89c4\u6a21\u51cf\u5c0f\u4e00\u534a\u7684\u5b50\u95ee\u9898\uff0c\u53ef\u80fd\u4e3a \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002</li> <li>\u5faa\u73af\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\uff0c\u76f4\u81f3\u627e\u5230 <code>target</code> \u6216\u533a\u95f4\u4e3a\u7a7a\u65f6\u8fd4\u56de\u3002</li> </ol> <p>\u56fe 12-4 \u5c55\u793a\u4e86\u5728\u6570\u7ec4\u4e2d\u4e8c\u5206\u67e5\u627e\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 12-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u8fc7\u7a0b </p> <p>\u5728\u5b9e\u73b0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \u6765\u6c42\u89e3\u95ee\u9898 \\(f(i, j)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j)\"\"\"\n# \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif i &gt; j:\nreturn -1\n# \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nm = (i + j) // 2\nif nums[m] &lt; target:\n# \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j)\nelif nums[m] &gt; target:\n# \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1)\nelse:\n# \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\ndef binary_search(nums: list[int], target: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\"\"\"\nn = len(nums)\n# \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif (i &gt; j) {\nreturn -1;\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nint m = (i + j) / 2;\nif (nums[m] &lt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j);\n} else if (nums[m] &gt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\nint n = nums.size();\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif (i &gt; j) {\nreturn -1;\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nint m = (i + j) / 2;\nif (nums[m] &lt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j);\n} else if (nums[m] &gt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int[] nums, int target) {\nint n = nums.length;\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif (i &gt; j) {\nreturn -1;\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nint m = (i + j) / 2;\nif (nums[m] &lt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn DFS(nums, target, m + 1, j);\n} else if (nums[m] &gt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn DFS(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nint BinarySearch(int[] nums, int target) {\nint n = nums.Length;\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n// \u5982\u679c\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u6ca1\u6709\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif i &gt; j {\nreturn -1\n}\n//    \u8ba1\u7b97\u7d22\u5f15\u4e2d\u70b9\nm := i + ((j - i) &gt;&gt; 1)\n//\u5224\u65ad\u4e2d\u70b9\u4e0e\u76ee\u6807\u5143\u7d20\u5927\u5c0f\nif nums[m] &lt; target {\n// \u5c0f\u4e8e\u5219\u9012\u5f52\u53f3\u534a\u6570\u7ec4\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m+1, j)\n} else if nums[m] &gt; target {\n// \u5c0f\u4e8e\u5219\u9012\u5f52\u5de6\u534a\u6570\u7ec4\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m-1)\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums []int, target int) int {\nn := len(nums)\nreturn dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif i &gt; j {\nreturn -1\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nlet m = (i + j) / 2\nif nums[m] &lt; target {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums: nums, target: target, i: m + 1, j: j)\n} else if nums[m] &gt; target {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums: nums, target: target, i: i, j: m - 1)\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\nlet n = nums.count\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums: nums, target: target, i: 0, j: n - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums, target, i, j) {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif (i &gt; j) {\nreturn -1;\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = i + ((j - i) &gt;&gt; 1);\nif (nums[m] &lt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j);\n} else if (nums[m] &gt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums, target) {\nconst n = nums.length;\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif (i &gt; j) {\nreturn -1;\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = i + ((j - i) &gt;&gt; 1);\nif (nums[m] &lt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j);\n} else if (nums[m] &gt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums: number[], target: number): number {\nconst n = nums.length;\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif (i &gt; j) {\nreturn -1;\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nint m = (i + j) ~/ 2;\nif (nums[m] &lt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j);\n} else if (nums[m] &gt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(List&lt;int&gt; nums, int target) {\nint n = nums.length;\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif i &gt; j { return -1; }\nlet m: i32 = (i + j) / 2;\nif nums[m as usize] &lt; target {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j);\n} else if nums[m as usize] &gt; target {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\nlet n = nums.len() as i32;\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\ndfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n// \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\nif (i &gt; j) {\nreturn -1;\n}\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nint m = (i + j) / 2;\nif (nums[m] &lt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\nreturn dfs(nums, target, m + 1, j);\n} else if (nums[m] &gt; target) {\n// \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\nreturn dfs(nums, target, i, m - 1);\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int nums[], int target, int numsSize) {\nint n = numsSize;\n// \u6c42\u89e3\u95ee\u9898 f(0, n-1)\nreturn dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.zig<pre><code>[class]{}-[func]{dfs}\n[class]{}-[func]{binarySearch}\n</code></pre>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \uff0c\u8bf7\u4ece\u4e2d\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u8fd4\u56de\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u3002\u5047\u8bbe\u4e8c\u53c9\u6811\u4e2d\u6ca1\u6709\u503c\u91cd\u590d\u7684\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 12-5 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65ad\u662f\u5426\u4e3a\u5206\u6cbb\u95ee\u9898","text":"<p>\u539f\u95ee\u9898\u5b9a\u4e49\u4e3a\u4ece <code>preorder</code> \u548c <code>inorder</code> \u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u662f\u4e00\u4e2a\u5178\u578b\u7684\u5206\u6cbb\u95ee\u9898\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4ece\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\u3001\u6784\u5efa\u53f3\u5b50\u6811\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u8282\u70b9\u3002\u800c\u5bf9\u4e8e\u6bcf\u68f5\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u590d\u7528\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u5c06\u5176\u5212\u5206\u4e3a\u66f4\u5c0f\u7684\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u7a7a\u5b50\u6811\uff09\u65f6\u7ec8\u6b62\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u4ea4\u96c6\u3002\u5728\u6784\u5efa\u5de6\u5b50\u6811\u65f6\uff0c\u6211\u4eec\u53ea\u9700\u5173\u6ce8\u4e2d\u5e8f\u904d\u5386\u548c\u524d\u5e8f\u904d\u5386\u4e2d\u4e0e\u5de6\u5b50\u6811\u5bf9\u5e94\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6811\u540c\u7406\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5b83\u4eec\u94fe\u63a5\u5230\u6839\u8282\u70b9\u4e0a\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ul>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","title":"2. \u00a0 \u5982\u4f55\u5212\u5206\u5b50\u6811","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u8fd9\u9053\u9898\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u6765\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u901a\u8fc7\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u6765\u5212\u5206\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5462\uff1f</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c<code>preorder</code> \u548c <code>inorder</code> \u90fd\u53ef\u4ee5\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u524d\u5e8f\u904d\u5386\uff1a<code>[ \u6839\u8282\u70b9 | \u5de6\u5b50\u6811 | \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> <li>\u4e2d\u5e8f\u904d\u5386\uff1a<code>[ \u5de6\u5b50\u6811 | \u6839\u8282\u70b9 \uff5c \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 9 | 3 | 1 2 7 ]</code> \u3002</li> </ul> <p>\u4ee5\u4e0a\u56fe\u6570\u636e\u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u56fe 12-6 \u6240\u793a\u7684\u6b65\u9aa4\u5f97\u5230\u5212\u5206\u7ed3\u679c\u3002</p> <ol> <li>\u524d\u5e8f\u904d\u5386\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u8282\u70b9\u7684\u503c\u3002</li> <li>\u67e5\u627e\u6839\u8282\u70b9 3 \u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8be5\u7d22\u5f15\u53ef\u5c06 <code>inorder</code> \u5212\u5206\u4e3a <code>[ 9 | 3 \uff5c 1 2 7 ]</code> \u3002</li> <li>\u6839\u636e <code>inorder</code> \u5212\u5206\u7ed3\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u5206\u522b\u4e3a 1 \u548c 3 \uff0c\u4ece\u800c\u53ef\u5c06 <code>preorder</code> \u5212\u5206\u4e3a <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> </ol> <p></p> <p> \u56fe 12-6 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u5212\u5206\u5b50\u6811 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u4e8e\u53d8\u91cf\u63cf\u8ff0\u5b50\u6811\u533a\u95f4","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u6839\u8282\u70b9\u3001\u5de6\u5b50\u6811\u3001\u53f3\u5b50\u6811\u5728 <code>preorder</code> \u548c <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u800c\u4e3a\u4e86\u63cf\u8ff0\u8fd9\u4e9b\u7d22\u5f15\u533a\u95f4\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u3002</p> <ul> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(i\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(m\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u8bb0\u4e3a \\([l, r]\\) \u3002</li> </ul> <p>\u5982\u8868 12-1 \u6240\u793a\uff0c\u901a\u8fc7\u4ee5\u4e0a\u53d8\u91cf\u5373\u53ef\u8868\u793a\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002</p> <p> \u8868 12-1 \u00a0 \u6839\u8282\u70b9\u548c\u5b50\u6811\u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u7d22\u5f15 </p> \u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15 \u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4 \u5f53\u524d\u6811 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6811 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6811 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>\u8bf7\u6ce8\u610f\uff0c\u53f3\u5b50\u6811\u6839\u8282\u70b9\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u4e49\u662f\u201c\u5de6\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u201d\uff0c\u5efa\u8bae\u914d\u5408\u56fe 12-7 \u7406\u89e3\u3002</p> <p></p> <p> \u56fe 12-7 \u00a0 \u6839\u8282\u70b9\u548c\u5de6\u53f3\u5b50\u6811\u7684\u7d22\u5f15\u533a\u95f4\u8868\u793a </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u63d0\u5347\u67e5\u8be2 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>hmap</code> \u6765\u5b58\u50a8\u6570\u7ec4 <code>inorder</code> \u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig build_tree.py<pre><code>def dfs(\npreorder: list[int],\ninorder_map: dict[int, int],\ni: int,\nl: int,\nr: int,\n) -&gt; TreeNode | None:\n\"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb\"\"\"\n# \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif r - l &lt; 0:\nreturn None\n# \u521d\u59cb\u5316\u6839\u8282\u70b9\nroot = TreeNode(preorder[i])\n# \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nm = inorder_map[preorder[i]]\n# \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n# \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n# \u8fd4\u56de\u6839\u8282\u70b9\nreturn root\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n\"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\"\"\"\n# \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\ninorder_map = {val: i for i, val in enumerate(inorder)}\nroot = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\nreturn root\n</code></pre> build_tree.cpp<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif (r - l &lt; 0)\nreturn NULL;\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nTreeNode *root = new TreeNode(preorder[i]);\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nint m = inorderMap[preorder[i]];\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root;\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nunordered_map&lt;int, int&gt; inorderMap;\nfor (int i = 0; i &lt; inorder.size(); i++) {\ninorderMap[inorder[i]] = i;\n}\nTreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\nreturn root;\n}\n</code></pre> build_tree.java<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif (r - l &lt; 0)\nreturn null;\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nTreeNode root = new TreeNode(preorder[i]);\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nint m = inorderMap.get(preorder[i]);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root;\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nMap&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\nfor (int i = 0; i &lt; inorder.length; i++) {\ninorderMap.put(inorder[i], i);\n}\nTreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\nreturn root;\n}\n</code></pre> build_tree.cs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif (r - l &lt; 0)\nreturn null;\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nTreeNode root = new(preorder[i]);\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nint m = inorderMap[preorder[i]];\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root;\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nDictionary&lt;int, int&gt; inorderMap = [];\nfor (int i = 0; i &lt; inorder.Length; i++) {\ninorderMap.TryAdd(inorder[i], i);\n}\nTreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\nreturn root;\n}\n</code></pre> build_tree.go<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif r-l &lt; 0 {\nreturn nil\n}\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nroot := NewTreeNode(preorder[i])\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nm := inorderMap[preorder[i]]\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\ninorderMap := make(map[int]int, len(inorder))\nfor i := 0; i &lt; len(inorder); i++ {\ninorderMap[inorder[i]] = i\n}\nroot := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\nreturn root\n}\n</code></pre> build_tree.swift<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif r - l &lt; 0 {\nreturn nil\n}\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nlet root = TreeNode(x: preorder[i])\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nlet m = inorderMap[preorder[i]]!\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nlet inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\nreturn dfs(preorder: preorder, inorderMap: inorderMap, i: 0, l: 0, r: inorder.count - 1)\n}\n</code></pre> build_tree.js<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif (r - l &lt; 0) return null;\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nconst root = new TreeNode(preorder[i]);\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nconst m = inorderMap.get(preorder[i]);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root;\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder, inorder) {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nlet inorderMap = new Map();\nfor (let i = 0; i &lt; inorder.length; i++) {\ninorderMap.set(inorder[i], i);\n}\nconst root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\nreturn root;\n}\n</code></pre> build_tree.ts<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(\npreorder: number[],\ninorderMap: Map&lt;number, number&gt;,\ni: number,\nl: number,\nr: number\n): TreeNode | null {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif (r - l &lt; 0) return null;\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nconst root: TreeNode = new TreeNode(preorder[i]);\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nconst m = inorderMap.get(preorder[i]);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root;\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nlet inorderMap = new Map&lt;number, number&gt;();\nfor (let i = 0; i &lt; inorder.length; i++) {\ninorderMap.set(inorder[i], i);\n}\nconst root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\nreturn root;\n}\n</code></pre> build_tree.dart<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? dfs(\nList&lt;int&gt; preorder,\nMap&lt;int, int&gt; inorderMap,\nint i,\nint l,\nint r,\n) {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif (r - l &lt; 0) {\nreturn null;\n}\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nTreeNode? root = TreeNode(preorder[i]);\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nint m = inorderMap[preorder[i]]!;\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root;\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nMap&lt;int, int&gt; inorderMap = {};\nfor (int i = 0; i &lt; inorder.length; i++) {\ninorderMap[inorder[i]] = i;\n}\nTreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\nreturn root;\n}\n</code></pre> build_tree.rs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfn dfs(preorder: &amp;[i32], inorder_map: &amp;HashMap&lt;i32, i32&gt;, i: i32, l: i32, r: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif r - l &lt; 0 { return None; }\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nlet root = TreeNode::new(preorder[i as usize]);\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nlet m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n// \u8fd4\u56de\u6839\u8282\u70b9\nSome(root)\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nlet mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\nfor i in 0..inorder.len() {\ninorder_map.insert(inorder[i], i as i32);\n}\nlet root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\nroot\n}\n</code></pre> build_tree.c<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n// \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\nif (r - l &lt; 0)\nreturn NULL;\n// \u521d\u59cb\u5316\u6839\u8282\u70b9\nTreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\nroot-&gt;val = preorder[i];\nroot-&gt;left = NULL;\nroot-&gt;right = NULL;\n// \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\nint m = inorderMap[preorder[i]];\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\nroot-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n// \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\nroot-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n// \u8fd4\u56de\u6839\u8282\u70b9\nreturn root;\n}\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n// \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\nint *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\nfor (int i = 0; i &lt; inorderSize; i++) {\ninorderMap[inorder[i]] = i;\n}\nTreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\nfree(inorderMap);\nreturn root;\n}\n</code></pre> build_tree.zig<pre><code>[class]{}-[func]{dfs}\n[class]{}-[func]{buildTree}\n</code></pre> <p>\u56fe 12-8 \u5c55\u793a\u4e86\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5404\u4e2a\u8282\u70b9\u662f\u5728\u5411\u4e0b\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u6761\u8fb9\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-8 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u5185\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u7684\u5212\u5206\u7ed3\u679c\u5982\u56fe 12-9 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 12-9 \u00a0 \u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u7684\u5212\u5206\u7ed3\u679c </p> <p>\u8bbe\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u4e2a\u8282\u70b9\uff08\u6267\u884c\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \uff09\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u54c8\u5e0c\u8868\u5b58\u50a8 <code>inorder</code> \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5","text":"<p>\u300c\u5206\u6cbb divide and conquer\u300d\uff0c\u5168\u79f0\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u79cd\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u89c1\u7684\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u4e24\u4e2a\u6b65\u9aa4\u3002</p> <ol> <li>\u5206\uff08\u5212\u5206\u9636\u6bb5\uff09\uff1a\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u6216\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898\u65f6\u7ec8\u6b62\u3002</li> <li>\u6cbb\uff08\u5408\u5e76\u9636\u6bb5\uff09\uff1a\u4ece\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ol> <p>\u5982\u56fe 12-1 \u6240\u793a\uff0c\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\u4e4b\u4e00\u3002</p> <ol> <li>\u5206\uff1a\u9012\u5f52\u5730\u5c06\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u5230\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u6cbb\uff1a\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u6709\u5e8f\u7684\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol> <p></p> <p> \u56fe 12-1 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65ad\u5206\u6cbb\u95ee\u9898","text":"<p>\u4e00\u4e2a\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u51b3\uff0c\u901a\u5e38\u53ef\u4ee5\u53c2\u8003\u4ee5\u4e0b\u51e0\u4e2a\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u89c4\u6a21\u66f4\u5c0f\u3001\u7c7b\u4f3c\u7684\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u80fd\u591f\u4ee5\u76f8\u540c\u65b9\u5f0f\u9012\u5f52\u5730\u8fdb\u884c\u5212\u5206\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5b50\u95ee\u9898\u4e4b\u95f4\u6ca1\u6709\u91cd\u53e0\uff0c\u4e92\u4e0d\u4f9d\u8d56\uff0c\u53ef\u4ee5\u72ec\u7acb\u89e3\u51b3\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u539f\u95ee\u9898\u7684\u89e3\u901a\u8fc7\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u6765\u3002</li> </ol> <p>\u663e\u7136\uff0c\u5f52\u5e76\u6392\u5e8f\u6ee1\u8db3\u4ee5\u4e0a\u4e09\u6761\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u9012\u5f52\u5730\u5c06\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u53ef\u4ee5\u72ec\u7acb\u5730\u8fdb\u884c\u6392\u5e8f\uff08\u5b50\u95ee\u9898\u53ef\u4ee5\u72ec\u7acb\u8fdb\u884c\u6c42\u89e3\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","title":"12.1.2 \u00a0 \u901a\u8fc7\u5206\u6cbb\u63d0\u5347\u6548\u7387","text":"<p>\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u76f8\u8f83\u4e8e\u9009\u62e9\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u4e3a\u5b83\u4eec\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u4e0d\u7981\u53d1\u95ee\uff1a\u4e3a\u4ec0\u4e48\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c42\u903b\u8f91\u662f\u4ec0\u4e48\uff1f\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c06\u5927\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\u3001\u89e3\u51b3\u5b50\u95ee\u9898\u3001\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u5408\u5e76\u4e3a\u539f\u95ee\u9898\u7684\u89e3\uff0c\u8fd9\u51e0\u6b65\u7684\u6548\u7387\u4e3a\u4ec0\u4e48\u6bd4\u76f4\u63a5\u89e3\u51b3\u539f\u95ee\u9898\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u4ece\u64cd\u4f5c\u6570\u91cf\u548c\u5e76\u884c\u8ba1\u7b97\u4e24\u65b9\u9762\u6765\u8ba8\u8bba\u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6570\u91cf\u4f18\u5316","text":"<p>\u4ee5\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u4e3a\u4f8b\uff0c\u5176\u5904\u7406\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u9700\u8981 \\(O(n^2)\\) \u65f6\u95f4\u3002\u5047\u8bbe\u6211\u4eec\u6309\u7167\u56fe 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u5219\u5212\u5206\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6bcf\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O((n / 2)^2)\\) \u65f6\u95f4\uff0c\u5408\u5e76\u4e24\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> \u56fe 12-2 \u00a0 \u5212\u5206\u6570\u7ec4\u524d\u540e\u7684\u5192\u6ce1\u6392\u5e8f </p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba1\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u8fb9\u548c\u53f3\u8fb9\u5206\u522b\u4e3a\u5212\u5206\u524d\u548c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u603b\u6570\uff1a</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>\u8fd9\u610f\u5473\u7740\u5f53 \\(n &gt; 4\\) \u65f6\uff0c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u6570\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u5e94\u8be5\u66f4\u9ad8\u3002\u8bf7\u6ce8\u610f\uff0c\u5212\u5206\u540e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u9636 \\(O(n^2)\\) \uff0c\u53ea\u662f\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\u4e86\u3002</p> <p>\u8fdb\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u4eec\u628a\u5b50\u6570\u7ec4\u4e0d\u65ad\u5730\u518d\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u505c\u6b62\u5212\u5206\u5462\uff1f\u8fd9\u79cd\u601d\u8def\u5b9e\u9645\u4e0a\u5c31\u662f\u201c\u5f52\u5e76\u6392\u5e8f\u201d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u4eec\u591a\u8bbe\u7f6e\u51e0\u4e2a\u5212\u5206\u70b9\uff0c\u5c06\u539f\u6570\u7ec4\u5e73\u5747\u5212\u5206\u4e3a \\(k\\) \u4e2a\u5b50\u6570\u7ec4\u5462\uff1f\u8fd9\u79cd\u60c5\u51b5\u4e0e\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u7c7b\u4f3c\uff0c\u5b83\u975e\u5e38\u9002\u5408\u6392\u5e8f\u6d77\u91cf\u6570\u636e\uff0c\u7406\u8bba\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n + k)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e76\u884c\u8ba1\u7b97\u4f18\u5316","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u5e76\u884c\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8fd8\u6709\u5229\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</p> <p>\u5e76\u884c\u4f18\u5316\u5728\u591a\u6838\u6216\u591a\u5904\u7406\u5668\u7684\u73af\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u4e3a\u7cfb\u7edf\u53ef\u4ee5\u540c\u65f6\u5904\u7406\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8ba1\u7b97\u8d44\u6e90\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u603b\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u3002</p> <p>\u6bd4\u5982\u5728\u56fe 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u4eec\u5c06\u6d77\u91cf\u7684\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u5219\u53ef\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52a1\u5206\u6563\u5230\u5404\u4e2a\u8ba1\u7b97\u5355\u5143\uff0c\u5b8c\u6210\u540e\u518d\u5408\u5e76\u7ed3\u679c\u3002</p> <p></p> <p> \u56fe 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e76\u884c\u8ba1\u7b97 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u89c1\u5e94\u7528","text":"<p>\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8bb8\u591a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff1a\u8be5\u7b97\u6cd5\u9996\u5148\u5c06\u70b9\u96c6\u5206\u6210\u4e24\u90e8\u5206\uff0c\u7136\u540e\u5206\u522b\u627e\u51fa\u4e24\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u70b9\u5bf9\uff0c\u6700\u540e\u627e\u51fa\u8de8\u8d8a\u4e24\u90e8\u5206\u7684\u6700\u8fd1\u70b9\u5bf9\u3002</li> <li>\u5927\u6574\u6570\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u6574\u6570\u4e58\u6cd5\u5206\u89e3\u4e3a\u51e0\u4e2a\u8f83\u5c0f\u7684\u6574\u6570\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u77e9\u9635\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u77e9\u9635\u4e58\u6cd5\u5206\u89e3\u4e3a\u591a\u4e2a\u5c0f\u77e9\u9635\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u6c49\u8bfa\u5854\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u89e3\u51b3\uff0c\u8fd9\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u5e94\u7528\u3002</li> <li>\u6c42\u89e3\u9006\u5e8f\u5bf9\uff1a\u5728\u4e00\u4e2a\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6570\u5b57\u5927\u4e8e\u540e\u9762\u7684\u6570\u5b57\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u6570\u5b57\u6784\u6210\u4e00\u4e2a\u9006\u5e8f\u5bf9\u3002\u6c42\u89e3\u9006\u5e8f\u5bf9\u95ee\u9898\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u501f\u52a9\u5f52\u5e76\u6392\u5e8f\u8fdb\u884c\u6c42\u89e3\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u4e2d\u5e94\u7528\u975e\u5e38\u5e7f\u6cdb\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\uff1a\u4e8c\u5206\u67e5\u627e\u662f\u5c06\u6709\u5e8f\u6570\u7ec4\u4ece\u4e2d\u70b9\u7d22\u5f15\u5904\u5206\u4e3a\u4e24\u90e8\u5206\uff0c\u7136\u540e\u6839\u636e\u76ee\u6807\u503c\u4e0e\u4e2d\u95f4\u5143\u7d20\u503c\u6bd4\u8f83\u7ed3\u679c\uff0c\u51b3\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u533a\u95f4\uff0c\u5e76\u5728\u5269\u4f59\u533a\u95f4\u6267\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u672c\u8282\u5f00\u5934\u5df2\u4ecb\u7ecd\uff0c\u4e0d\u518d\u8d58\u8ff0\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9009\u53d6\u4e00\u4e2a\u57fa\u51c6\u503c\uff0c\u7136\u540e\u628a\u6570\u7ec4\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5927\uff0c\u518d\u5bf9\u8fd9\u4e24\u90e8\u5206\u8fdb\u884c\u76f8\u540c\u7684\u5212\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002</li> <li>\u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u6570\u636e\u5206\u6563\u5230\u591a\u4e2a\u6876\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u5404\u4e2a\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002</li> <li>\u6811\uff1a\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u7ea2\u9ed1\u6811\u3001B \u6811\u3001B+ \u6811\u7b49\uff0c\u5b83\u4eec\u7684\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u89c6\u4e3a\u5206\u6cbb\u7b56\u7565\u7684\u5e94\u7528\u3002</li> <li>\u5806\uff1a\u5806\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u5176\u5404\u79cd\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u5220\u9664\u548c\u5806\u5316\uff0c\u5b9e\u9645\u4e0a\u90fd\u9690\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u867d\u7136\u54c8\u5e0c\u8868\u6765\u5e76\u4e0d\u76f4\u63a5\u5e94\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u54c8\u5e0c\u51b2\u7a81\u89e3\u51b3\u65b9\u6848\u95f4\u63a5\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u94fe\u5f0f\u5730\u5740\u4e2d\u7684\u957f\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u79cd\u201c\u6da6\u7269\u7ec6\u65e0\u58f0\u201d\u7684\u7b97\u6cd5\u601d\u60f3\uff0c\u9690\u542b\u5728\u5404\u79cd\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u4e2d\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898","text":"<p>\u5728\u5f52\u5e76\u6392\u5e8f\u548c\u6784\u5efa\u4e8c\u53c9\u6811\u4e2d\uff0c\u6211\u4eec\u90fd\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a\u539f\u95ee\u9898\u4e00\u534a\u7684\u5b50\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u6c49\u8bfa\u5854\u95ee\u9898\uff0c\u6211\u4eec\u91c7\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8bb0\u4e3a <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \u3002\u8d77\u59cb\u72b6\u6001\u4e0b\uff0c\u67f1\u5b50 <code>A</code> \u4e0a\u5957\u7740 \\(n\\) \u4e2a\u5706\u76d8\uff0c\u5b83\u4eec\u4ece\u4e0a\u5230\u4e0b\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002\u6211\u4eec\u7684\u4efb\u52a1\u662f\u8981\u628a\u8fd9 \\(n\\) \u4e2a\u5706\u76d8\u79fb\u5230\u67f1\u5b50 <code>C</code> \u4e0a\uff0c\u5e76\u4fdd\u6301\u5b83\u4eec\u7684\u539f\u6709\u987a\u5e8f\u4e0d\u53d8\uff08\u5982\u56fe 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52a8\u5706\u76d8\u7684\u8fc7\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u89c4\u5219\u3002</p> <ol> <li>\u5706\u76d8\u53ea\u80fd\u4ece\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u62ff\u51fa\uff0c\u4ece\u53e6\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u653e\u5165\u3002</li> <li>\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\u3002</li> <li>\u5c0f\u5706\u76d8\u5fc5\u987b\u65f6\u523b\u4f4d\u4e8e\u5927\u5706\u76d8\u4e4b\u4e0a\u3002</li> </ol> <p></p> <p> \u56fe 12-10 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u793a\u4f8b </p> <p>\u6211\u4eec\u5c06\u89c4\u6a21\u4e3a \\(i\\) \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u8bb0\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c06 \\(3\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u8651\u57fa\u672c\u60c5\u51b5","text":"<p>\u5982\u56fe 12-11 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(1)\\) \uff0c\u5373\u5f53\u53ea\u6709\u4e00\u4e2a\u5706\u76d8\u65f6\uff0c\u6211\u4eec\u5c06\u5b83\u76f4\u63a5\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u5373\u53ef\u3002</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> \u56fe 12-11 \u00a0 \u89c4\u6a21\u4e3a 1 \u95ee\u9898\u7684\u89e3 </p> <p>\u5982\u56fe 12-12 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(2)\\) \uff0c\u5373\u5f53\u6709\u4e24\u4e2a\u5706\u76d8\u65f6\uff0c\u7531\u4e8e\u8981\u65f6\u523b\u6ee1\u8db3\u5c0f\u5706\u76d8\u5728\u5927\u5706\u76d8\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u501f\u52a9 <code>B</code> \u6765\u5b8c\u6210\u79fb\u52a8\u3002</p> <ol> <li>\u5148\u5c06\u4e0a\u9762\u7684\u5c0f\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u518d\u5c06\u5927\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002</li> <li>\u6700\u540e\u5c06\u5c0f\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-12 \u00a0 \u89c4\u6a21\u4e3a 2 \u95ee\u9898\u7684\u89e3 </p> <p>\u89e3\u51b3\u95ee\u9898 \\(f(2)\\) \u7684\u8fc7\u7a0b\u53ef\u603b\u7ed3\u4e3a\uff1a\u5c06\u4e24\u4e2a\u5706\u76d8\u501f\u52a9 <code>B</code> \u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002\u5176\u4e2d\uff0c<code>C</code> \u79f0\u4e3a\u76ee\u6807\u67f1\u3001<code>B</code> \u79f0\u4e3a\u7f13\u51b2\u67f1\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u95ee\u9898\u5206\u89e3","text":"<p>\u5bf9\u4e8e\u95ee\u9898 \\(f(3)\\) \uff0c\u5373\u5f53\u6709\u4e09\u4e2a\u5706\u76d8\u65f6\uff0c\u60c5\u51b5\u53d8\u5f97\u7a0d\u5fae\u590d\u6742\u4e86\u4e00\u4e9b\u3002</p> <p>\u56e0\u4e3a\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ece\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c06 <code>A</code> \u9876\u90e8\u7684\u4e24\u4e2a\u5706\u76d8\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u6267\u884c\u56fe 12-13 \u6240\u793a\u7684\u6b65\u9aa4\u3002\u8fd9\u6837\u4e09\u4e2a\u5706\u76d8\u5c31\u88ab\u987a\u5229\u5730\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u4e86\u3002</p> <ol> <li>\u4ee4 <code>B</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>C</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06 <code>A</code> \u4e2d\u5269\u4f59\u7684\u4e00\u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u52a8\u81f3 <code>C</code> \u3002</li> <li>\u4ee4 <code>C</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>A</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-13 \u00a0 \u89c4\u6a21\u4e3a 3 \u95ee\u9898\u7684\u89e3 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u5c06\u95ee\u9898 \\(f(3)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(2)\\) \u548c\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u4e4b\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002\u8fd9\u8bf4\u660e\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u603b\u7ed3\u51fa\u56fe 12-14 \u6240\u793a\u7684\u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c06\u539f\u95ee\u9898 \\(f(n)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \uff0c\u5e76\u6309\u7167\u4ee5\u4e0b\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u3002</p> <ol> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>C</code> \u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06\u5269\u4f59 \\(1\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u81f3 <code>C</code> \u3002</li> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>A</code> \u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> <p>\u5bf9\u4e8e\u8fd9\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u901a\u8fc7\u76f8\u540c\u7684\u65b9\u5f0f\u8fdb\u884c\u9012\u5f52\u5212\u5206\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52a8\u64cd\u4f5c\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 12-14 \u00a0 \u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs(i, src, buf, tar)</code> \uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c06\u67f1 <code>src</code> \u9876\u90e8\u7684 \\(i\\) \u4e2a\u5706\u76d8\u501f\u52a9\u7f13\u51b2\u67f1 <code>buf</code> \u79fb\u52a8\u81f3\u76ee\u6807\u67f1 <code>tar</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n\"\"\"\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\"\"\"\n# \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\npan = src.pop()\n# \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.append(pan)\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n\"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i)\"\"\"\n# \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif i == 1:\nmove(src, tar)\nreturn\n# \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, tar, buf)\n# \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, tar)\n# \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, src, tar)\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n\"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898\"\"\"\nn = len(A)\n# \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nint pan = src.back();\nsrc.pop_back();\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.push_back(pan);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif (i == 1) {\nmove(src, tar);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, tar, buf);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, tar);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, src, tar);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\nint n = A.size();\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nInteger pan = src.remove(src.size() - 1);\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.add(pan);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif (i == 1) {\nmove(src, tar);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, tar, buf);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, tar);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, src, tar);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\nint n = A.size();\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nint pan = src[^1];\nsrc.RemoveAt(src.Count - 1);\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.Add(pan);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif (i == 1) {\nMove(src, tar);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\nDFS(i - 1, src, tar, buf);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nMove(src, tar);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\nDFS(i - 1, buf, src, tar);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\nint n = A.Count;\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\nDFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src, tar *list.List) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\npan := src.Back()\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.PushBack(pan.Value)\n// \u79fb\u9664 src \u9876\u90e8\u5706\u76d8\nsrc.Remove(pan)\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif i == 1 {\nmove(src, tar)\nreturn\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfsHanota(i-1, src, tar, buf)\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, tar)\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfsHanota(i-1, buf, src, tar)\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A, B, C *list.List) {\nn := A.Len()\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nlet pan = src.popLast()!\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.append(pan)\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif i == 1 {\nmove(src: &amp;src, tar: &amp;tar)\nreturn\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src: &amp;src, tar: &amp;tar)\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\nlet n = A.count\n// \u5217\u8868\u5c3e\u90e8\u662f\u67f1\u5b50\u9876\u90e8\n// \u5c06 src \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src, tar) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nconst pan = src.pop();\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.push(pan);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i, src, buf, tar) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif (i === 1) {\nmove(src, tar);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, tar, buf);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, tar);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, src, tar);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A, B, C) {\nconst n = A.length;\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src: number[], tar: number[]): void {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nconst pan = src.pop();\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.push(pan);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif (i === 1) {\nmove(src, tar);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, tar, buf);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, tar);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, src, tar);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\nconst n = A.length;\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nint pan = src.removeLast();\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.add(pan);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif (i == 1) {\nmove(src, tar);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, tar, buf);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, tar);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, src, tar);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\nint n = A.length;\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nlet pan = src.remove(src.len() - 1);\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar.push(pan);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif i == 1 {\nmove_pan(src, tar);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, tar, buf);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove_pan(src, tar);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, src, tar);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\nlet n = A.len() as i32;\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n// \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\nint pan = src[*srcSize - 1];\nsrc[*srcSize - 1] = 0;\n(*srcSize)--;\n// \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\ntar[*tarSize] = pan;\n(*tarSize)++;\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n// \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\nif (i == 1) {\nmove(src, srcSize, tar, tarSize);\nreturn;\n}\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\ndfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n// \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\nmove(src, srcSize, tar, tarSize);\n// \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\ndfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n// \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\ndfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.zig<pre><code>[class]{}-[func]{move}\n[class]{}-[func]{dfs}\n[class]{}-[func]{solveHanota}\n</code></pre> <p>\u5982\u56fe 12-15 \u6240\u793a\uff0c\u6c49\u8bfa\u5854\u95ee\u9898\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5f00\u542f\u7684 <code>dfs()</code> \u51fd\u6570\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 12-15 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u7684\u9012\u5f52\u6811 </p> <p>Quote</p> <p>\u6c49\u8bfa\u5854\u95ee\u9898\u6e90\u81ea\u4e00\u4e2a\u53e4\u8001\u7684\u4f20\u8bf4\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u4e2a\u5bfa\u5e99\u91cc\uff0c\u50e7\u4fa3\u4eec\u6709\u4e09\u6839\u9ad8\u5927\u7684\u94bb\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u4e2a\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5706\u76d8\u3002\u50e7\u4fa3\u4eec\u4e0d\u65ad\u5730\u79fb\u52a8\u5706\u76d8\uff0c\u4ed6\u4eec\u76f8\u4fe1\u5728\u6700\u540e\u4e00\u4e2a\u5706\u76d8\u88ab\u6b63\u786e\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u8fd9\u4e2a\u4e16\u754c\u5c31\u4f1a\u7ed3\u675f\u3002</p> <p>\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fa3\u4eec\u6bcf\u79d2\u949f\u79fb\u52a8\u4e00\u6b21\uff0c\u603b\u5171\u9700\u8981\u5927\u7ea6 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7ea6 \\(5850\\) \u4ebf\u5e74\uff0c\u8fdc\u8fdc\u8d85\u8fc7\u4e86\u73b0\u5728\u5bf9\u5b87\u5b99\u5e74\u9f84\u7684\u4f30\u8ba1\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u8fd9\u4e2a\u4f20\u8bf4\u662f\u771f\u7684\uff0c\u6211\u4eec\u5e94\u8be5\u4e0d\u9700\u8981\u62c5\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u6765\u3002</p>"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u5206\u6cbb\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5212\u5206\uff09\u548c\u6cbb\uff08\u5408\u5e76\uff09\u4e24\u4e2a\u9636\u6bb5\uff0c\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002</li> <li>\u5224\u65ad\u662f\u5426\u662f\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\u7684\u4f9d\u636e\u5305\u62ec\uff1a\u95ee\u9898\u80fd\u5426\u5206\u89e3\u3001\u5b50\u95ee\u9898\u662f\u5426\u72ec\u7acb\u3001\u5b50\u95ee\u9898\u80fd\u5426\u5408\u5e76\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\uff0c\u5176\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u7b49\u957f\u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u5f00\u59cb\u9010\u5c42\u5408\u5e76\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u51cf\u5c11\u4e86\u64cd\u4f5c\u6570\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u540e\u6709\u5229\u4e8e\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</li> <li>\u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u51b3\u8bb8\u591a\u7b97\u6cd5\u95ee\u9898\uff0c\u4e5f\u5e7f\u6cdb\u5e94\u7528\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u5904\u5904\u53ef\u89c1\u5176\u8eab\u5f71\u3002</li> <li>\u76f8\u8f83\u4e8e\u66b4\u529b\u641c\u7d22\uff0c\u81ea\u9002\u5e94\u641c\u7d22\u6548\u7387\u66f4\u9ad8\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u4e2a\u5178\u578b\u5e94\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u7684\u6b65\u9aa4\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e\u3002</li> <li>\u5728\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u95ee\u9898\u4e2d\uff0c\u6784\u5efa\u6811\uff08\u539f\u95ee\u9898\uff09\u53ef\u4ee5\u5212\u5206\u4e3a\u6784\u5efa\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5212\u5206\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u7684\u7d22\u5f15\u533a\u95f4\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u6c49\u8bfa\u5854\u95ee\u9898\u4e2d\uff0c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(n\\) \u7684\u95ee\u9898\u53ef\u4ee5\u5212\u5206\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(n-1\\) \u7684\u5b50\u95ee\u9898\u548c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(1\\) \u7684\u5b50\u95ee\u9898\u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/","title":"\u7b2c 14 \u7ae0 \u00a0 \u52a8\u6001\u89c4\u5212","text":"<p>Abstract</p> <p>\u5c0f\u6eaa\u6c47\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u6c47\u5165\u5927\u6d77\u3002</p> <p>\u52a8\u6001\u89c4\u5212\u5c06\u5c0f\u95ee\u9898\u7684\u89e3\u6c47\u96c6\u6210\u5927\u95ee\u9898\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9886\u6211\u4eec\u8d70\u5411\u89e3\u51b3\u95ee\u9898\u7684\u5f7c\u5cb8\u3002</p>"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212</li> <li>14.2 \u00a0 DP \u95ee\u9898\u7279\u6027</li> <li>14.3 \u00a0 DP \u89e3\u9898\u601d\u8def</li> <li>14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898</li> <li>14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898</li> <li>14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</li> <li>14.7 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"14.2 \u00a0 \u52a8\u6001\u89c4\u5212\u95ee\u9898\u7279\u6027","text":"<p>\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u5206\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u7684\u4fa7\u91cd\u70b9\u4e0d\u540c\u3002</p> <ul> <li>\u5206\u6cbb\u7b97\u6cd5\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u591a\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u95ee\u9898\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u7ec8\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u4e5f\u5bf9\u95ee\u9898\u8fdb\u884c\u9012\u5f52\u5206\u89e3\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u4f9d\u8d56\u7684\uff0c\u5728\u5206\u89e3\u8fc7\u7a0b\u4e2d\u4f1a\u51fa\u73b0\u8bb8\u591a\u91cd\u53e0\u5b50\u95ee\u9898\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u4e2a\u5b50\u95ee\u9898\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5b83\u4eec\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"<p>\u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002</p> <p>\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\uff08\u8d77\u59cb\u70b9\uff09\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f</p> <p>\u5982\u56fe 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002</p> <p></p> <p> \u56fe 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7 </p> <p>\u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff1a</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>\u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002</p> <p>\u672c\u9898\u663e\u7136\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002</p> <p>\u90a3\u4e48\uff0c\u4e0a\u4e00\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u201d\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u89e3\u91ca\u65b9\u5f0f\u6bd4\u8f83\u7075\u6d3b\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002</p> <p>\u6839\u636e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u52a8\u6001\u89c4\u5212\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n\"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(cost) - 1\nif n == 1 or n == 2:\nreturn cost[n]\n# \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [0] * (n + 1)\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1], dp[2] = cost[1], cost[2]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\nreturn dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\nint n = cost.size() - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector&lt;int&gt; dp(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDP(int[] cost) {\nint n = cost.Length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost []int) int {\nn := len(cost) - 1\nif n == 1 || n == 2 {\nreturn cost[n]\n}\nmin := func(a, b int) int {\nif a &lt; b {\nreturn a\n}\nreturn b\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp := make([]int, n+1)\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1]\ndp[2] = cost[2]\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i := 3; i &lt;= n; i++ {\ndp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n}\nreturn dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\nlet n = cost.count - 1\nif n == 1 || n == 2 {\nreturn cost[n]\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvar dp = Array(repeating: 0, count: n + 1)\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1]\ndp[2] = cost[2]\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in stride(from: 3, through: n, by: 1) {\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n}\nreturn dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost) {\nconst n = cost.length - 1;\nif (n === 1 || n === 2) {\nreturn cost[n];\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nconst dp = new Array(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (let i = 3; i &lt;= n; i++) {\ndp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\nconst n = cost.length - 1;\nif (n === 1 || n === 2) {\nreturn cost[n];\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nconst dp = new Array(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (let i = 3; i &lt;= n; i++) {\ndp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2) return cost[n];\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nList&lt;int&gt; dp = List.filled(n + 1, 0);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\nlet n = cost.len() - 1;\nif n == 1 || n == 2 { return cost[n]; }\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nlet mut dp = vec![-1; n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in 3..=n {\ndp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n}\ndp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\nint n = costSize - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint *dp = calloc(n + 1, sizeof(int));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n}\nint res = dp[n];\n// \u91ca\u653e\u5185\u5b58\nfree(dp);\nreturn res;\n}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\ncomptime var n = cost.len - 1;\nif (n == 1 or n == 2) {\nreturn cost[n];\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvar dp = [_]i32{-1} ** (n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = cost[1];\ndp[2] = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (3..n + 1) |i| {\ndp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n}\nreturn dp[n];\n}\n</code></pre> <p>\u56fe 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-7 \u00a0 \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u672c\u9898\u4e5f\u53ef\u4ee5\u8fdb\u884c\u7a7a\u95f4\u4f18\u5316\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n\"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(cost) - 1\nif n == 1 or n == 2:\nreturn cost[n]\na, b = cost[1], cost[2]\nfor i in range(3, n + 1):\na, b = b, min(a, b) + cost[i]\nreturn b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\nint n = cost.size() - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = Math.min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDPComp(int[] cost) {\nint n = cost.Length - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = Math.Min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\nn := len(cost) - 1\nif n == 1 || n == 2 {\nreturn cost[n]\n}\nmin := func(a, b int) int {\nif a &lt; b {\nreturn a\n}\nreturn b\n}\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\na, b := cost[1], cost[2]\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i := 3; i &lt;= n; i++ {\ntmp := b\nb = min(a, tmp) + cost[i]\na = tmp\n}\nreturn b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\nlet n = cost.count - 1\nif n == 1 || n == 2 {\nreturn cost[n]\n}\nvar (a, b) = (cost[1], cost[2])\nfor i in stride(from: 3, through: n, by: 1) {\n(a, b) = (b, min(a, b) + cost[i])\n}\nreturn b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost) {\nconst n = cost.length - 1;\nif (n === 1 || n === 2) {\nreturn cost[n];\n}\nlet a = cost[1],\nb = cost[2];\nfor (let i = 3; i &lt;= n; i++) {\nconst tmp = b;\nb = Math.min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\nconst n = cost.length - 1;\nif (n === 1 || n === 2) {\nreturn cost[n];\n}\nlet a = cost[1],\nb = cost[2];\nfor (let i = 3; i &lt;= n; i++) {\nconst tmp = b;\nb = Math.min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\nint n = cost.length - 1;\nif (n == 1 || n == 2) return cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\nlet n = cost.len() - 1;\nif n == 1 || n == 2 { return cost[n] };\nlet (mut a, mut b) = (cost[1], cost[2]);\nfor i in 3..=n {\nlet tmp = b;\nb = cmp::min(a, tmp) + cost[i];\na = tmp;\n}\nb\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\nint n = costSize - 1;\nif (n == 1 || n == 2)\nreturn cost[n];\nint a = cost[1], b = cost[2];\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = myMin(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\nvar n = cost.len - 1;\nif (n == 1 or n == 2) {\nreturn cost[n];\n}\nvar a = cost[1];\nvar b = cost[2];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (3..n + 1) |i| {\nvar tmp = b;\nb = @min(a, tmp) + cost[i];\na = tmp;\n}\nreturn b;\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u65e0\u540e\u6548\u6027","text":"<p>\u65e0\u540e\u6548\u6027\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002</p> <p>\u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u987b\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u7ed9\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002</p> <p>\u5e26\u7ea6\u675f\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002</p> <p></p> <p> \u56fe 14-8 \u00a0 \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u8f6e\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002</p> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u5c31\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u6765\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u4ece\u4f55\u800c\u6765\u7684\u3002</p> <ul> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u4ece \\(dp[i-1, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u4ece \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> </ul> <p>\u5982\u56fe 14-9 \u6240\u793a\uff0c\u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u6b64\u65f6\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> \u56fe 14-9 \u00a0 \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb </p> <p>\u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n\"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn 1\n# \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [[0] * 3 for _ in range(n + 1)]\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1], dp[1][2] = 1, 0\ndp[2][1], dp[2][2] = 0, 1\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i][1] = dp[i - 1][2]\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2]\nreturn dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[][] dp = new int[n + 1][3];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[,] dp = new int[n + 1, 3];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1, 1] = 1;\ndp[1, 2] = 0;\ndp[2, 1] = 0;\ndp[2, 2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i, 1] = dp[i - 1, 2];\ndp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n}\nreturn dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n int) int {\nif n == 1 || n == 2 {\nreturn 1\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp := make([][3]int, n+1)\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1\ndp[1][2] = 0\ndp[2][1] = 0\ndp[2][2] = 1\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i := 3; i &lt;= n; i++ {\ndp[i][1] = dp[i-1][2]\ndp[i][2] = dp[i-2][1] + dp[i-2][2]\n}\nreturn dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\nif n == 1 || n == 2 {\nreturn 1\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvar dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1\ndp[1][2] = 0\ndp[2][1] = 0\ndp[2][2] = 1\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in stride(from: 3, through: n, by: 1) {\ndp[i][1] = dp[i - 1][2]\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n}\nreturn dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n) {\nif (n === 1 || n === 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nconst dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (let i = 3; i &lt;= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n: number): number {\nif (n === 1 || n === 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nconst dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (let i = 3; i &lt;= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nList&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\nif n == 1 || n == 2 { return 1 };\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nlet mut dp = vec![vec![-1; 3]; n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in 3..=n {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\ndp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\nif (n == 1 || n == 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint **dp = malloc((n + 1) * sizeof(int *));\nfor (int i = 0; i &lt;= n; i++) {\ndp[i] = calloc(3, sizeof(int));\n}\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nint res = dp[n][1] + dp[n][2];\n// \u91ca\u653e\u5185\u5b58\nfor (int i = 0; i &lt;= n; i++) {\nfree(dp[i]);\n}\nfree(dp);\nreturn res;\n}\n</code></pre> climbing_stairs_constraint_dp.zig<pre><code>// \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\nif (n == 1 or n == 2) {\nreturn 1;\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvar dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1][1] = 1;\ndp[1][2] = 0;\ndp[2][1] = 0;\ndp[2][2] = 1;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (3..n + 1) |i| {\ndp[i][1] = dp[i - 1][2];\ndp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n}\nreturn dp[n][1] + dp[n][2];\n}\n</code></pre> <p>\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u91cd\u65b0\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\u3002</p> <p>\u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u5728\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u590d\u6742\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898\uff09\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u53ef\u7528\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52a8\u6001\u89c4\u5212\u89e3\u9898\u601d\u8def","text":"<p>\u4e0a\u4e24\u8282\u4ecb\u7ecd\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4e3b\u8981\u7279\u5f81\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u4e00\u8d77\u63a2\u7a76\u4e24\u4e2a\u66f4\u52a0\u5b9e\u7528\u7684\u95ee\u9898\u3002</p> <ol> <li>\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u95ee\u9898\u662f\u4e0d\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff1f</li> <li>\u6c42\u89e3\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8be5\u4ece\u4f55\u5904\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f</li> </ol>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","title":"14.3.1 \u00a0 \u95ee\u9898\u5224\u65ad","text":"<p>\u603b\u7684\u6765\u8bf4\uff0c\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u90a3\u4e48\u5b83\u901a\u5e38\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u4eec\u5f88\u96be\u4ece\u95ee\u9898\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u8fd9\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u653e\u5bbd\u6761\u4ef6\uff0c\u5148\u89c2\u5bdf\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7a77\u4e3e\uff09\u89e3\u51b3\u3002</p> <p>\u9002\u5408\u7528\u56de\u6eaf\u89e3\u51b3\u7684\u95ee\u9898\u901a\u5e38\u6ee1\u8db3\u201c\u51b3\u7b56\u6811\u6a21\u578b\u201d\uff0c\u8fd9\u79cd\u95ee\u9898\u53ef\u4ee5\u4f7f\u7528\u6811\u5f62\u7ed3\u6784\u6765\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\uff0c\u6bcf\u4e00\u6761\u8def\u5f84\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u95ee\u9898\u5305\u542b\u660e\u786e\u7684\u51b3\u7b56\u6982\u5ff5\uff0c\u5e76\u4e14\u89e3\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u51b3\u7b56\u4ea7\u751f\u7684\uff0c\u90a3\u4e48\u5b83\u5c31\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u6765\u89e3\u51b3\u3002</p> <p>\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8fd8\u6709\u4e00\u4e9b\u5224\u65ad\u7684\u201c\u52a0\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u4f18\u5316\u63cf\u8ff0\u3002</li> <li>\u95ee\u9898\u7684\u72b6\u6001\u80fd\u591f\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u3001\u591a\u7ef4\u77e9\u9635\u6216\u6811\u6765\u8868\u793a\uff0c\u5e76\u4e14\u4e00\u4e2a\u72b6\u6001\u4e0e\u5176\u5468\u56f4\u7684\u72b6\u6001\u5b58\u5728\u9012\u63a8\u5173\u7cfb\u3002</li> </ul> <p>\u76f8\u5e94\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u51cf\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u4f18\u89e3\u3002</li> <li>\u95ee\u9898\u63cf\u8ff0\u4e2d\u6709\u660e\u663e\u7684\u6392\u5217\u7ec4\u5408\u7684\u7279\u5f81\uff0c\u9700\u8981\u8fd4\u56de\u5177\u4f53\u7684\u591a\u4e2a\u65b9\u6848\u3002</li> </ul> <p>\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u5e76\u5177\u6709\u8f83\u4e3a\u660e\u663e\u7684\u201c\u52a0\u5206\u9879\u201c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5047\u8bbe\u5b83\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u5e76\u5728\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u9a8c\u8bc1\u5b83\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u95ee\u9898\u6c42\u89e3\u6b65\u9aa4","text":"<p>\u52a8\u6001\u89c4\u5212\u7684\u89e3\u9898\u6d41\u7a0b\u4f1a\u56e0\u95ee\u9898\u7684\u6027\u8d28\u548c\u96be\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9aa4\uff1a\u63cf\u8ff0\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5bfc\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u7b49\u3002</p> <p>\u4e3a\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u9898\u6b65\u9aa4\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\u201c\u6700\u5c0f\u8def\u5f84\u548c\u201d\u6765\u4e3e\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a \\(n \\times m\\) \u7684\u4e8c\u7ef4\u7f51\u683c <code>grid</code> \uff0c\u7f51\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5305\u542b\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u8be5\u5355\u5143\u683c\u7684\u4ee3\u4ef7\u3002\u673a\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u5355\u5143\u683c\u4e3a\u8d77\u59cb\u70b9\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52a8\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u8fbe\u53f3\u4e0b\u89d2\u5355\u5143\u683c\u3002\u8bf7\u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f84\u548c\u3002</p> <p>\u56fe 14-10 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u7f51\u683c\u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e3a \\(13\\) \u3002</p> <p></p> <p> \u56fe 14-10 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u793a\u4f8b\u6570\u636e </p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u672c\u9898\u7684\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u5c31\u662f\u4ece\u5f53\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8bbe\u5f53\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\([i, j]\\) \uff0c\u5219\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u540e\uff0c\u7d22\u5f15\u53d8\u4e3a \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72b6\u6001\u5e94\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u4e24\u4e2a\u53d8\u91cf\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u4ece\u8d77\u59cb\u70b9 \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\uff0c\u89e3\u8bb0\u4e3a \\(dp[i, j]\\) \u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u56fe 14-11 \u6240\u793a\u7684\u4e8c\u7ef4 \\(dp\\) \u77e9\u9635\uff0c\u5176\u5c3a\u5bf8\u4e0e\u8f93\u5165\u7f51\u683c \\(grid\\) \u76f8\u540c\u3002</p> <p></p> <p> \u56fe 14-11 \u00a0 \u72b6\u6001\u5b9a\u4e49\u4e0e dp \u8868 </p> <p>Note</p> <p>\u52a8\u6001\u89c4\u5212\u548c\u56de\u6eaf\u8fc7\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\uff0c\u800c\u72b6\u6001\u7531\u6240\u6709\u51b3\u7b56\u53d8\u91cf\u6784\u6210\u3002\u5b83\u5e94\u5f53\u5305\u542b\u63cf\u8ff0\u89e3\u9898\u8fdb\u5ea6\u7684\u6240\u6709\u53d8\u91cf\uff0c\u5176\u5305\u542b\u4e86\u8db3\u591f\u7684\u4fe1\u606f\uff0c\u80fd\u591f\u7528\u6765\u63a8\u5bfc\u51fa\u4e0b\u4e00\u4e2a\u72b6\u6001\u3002</p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5b9a\u4e49\u4e00\u4e2a \\(dp\\) \u8868\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u72b6\u6001\u7684\u6bcf\u4e2a\u72ec\u7acb\u53d8\u91cf\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72b6\u6001\u548c\u5b50\u95ee\u9898\u7684\u89e3\u4e4b\u95f4\u7684\u6620\u5c04\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5bf9\u4e8e\u72b6\u6001 \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u4ece\u4e0a\u8fb9\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u8fb9\u683c\u5b50 \\([i, j-1]\\) \u8f6c\u79fb\u800c\u6765\u3002\u56e0\u6b64\u6700\u4f18\u5b50\u7ed3\u6784\u4e3a\uff1a\u5230\u8fbe \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e0e \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\u51b3\u5b9a\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u56fe 14-12 \u6240\u793a\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> \u56fe 14-12 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b </p> <p>Note</p> <p>\u6839\u636e\u5b9a\u4e49\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u95ee\u9898\u548c\u5b50\u95ee\u9898\u7684\u5173\u7cfb\uff0c\u627e\u51fa\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u4f18\u5b50\u7ed3\u6784\u3002</p> <p>\u4e00\u65e6\u6211\u4eec\u627e\u5230\u4e86\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5728\u672c\u9898\u4e2d\uff0c\u5904\u5728\u9996\u884c\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u5de6\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u5904\u5728\u9996\u5217\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u4e0a\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u8fb9\u754c\u6761\u4ef6\u3002</p> <p>\u5982\u56fe 14-13 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u5faa\u73af\u6765\u904d\u5386\u77e9\u9635\uff0c\u5916\u5faa\u73af\u904d\u5386\u5404\u884c\uff0c\u5185\u5faa\u73af\u904d\u5386\u5404\u5217\u3002</p> <p></p> <p> \u56fe 14-13 \u00a0 \u8fb9\u754c\u6761\u4ef6\u4e0e\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f </p> <p>Note</p> <p>\u8fb9\u754c\u6761\u4ef6\u5728\u52a8\u6001\u89c4\u5212\u4e2d\u7528\u4e8e\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u7d22\u4e2d\u7528\u4e8e\u526a\u679d\u3002</p> <p>\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8bc1\u5728\u8ba1\u7b97\u5f53\u524d\u95ee\u9898\u7684\u89e3\u65f6\uff0c\u6240\u6709\u5b83\u4f9d\u8d56\u7684\u66f4\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u90fd\u5df2\u7ecf\u88ab\u6b63\u786e\u5730\u8ba1\u7b97\u51fa\u6765\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u5199\u51fa\u52a8\u6001\u89c4\u5212\u4ee3\u7801\u3002\u7136\u800c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u7d22 \\(\\rightarrow\\) \u8bb0\u5fc6\u5316\u641c\u7d22 \\(\\rightarrow\\) \u52a8\u6001\u89c4\u5212\u201d\u7684\u987a\u5e8f\u5b9e\u73b0\u66f4\u52a0\u7b26\u5408\u601d\u7ef4\u4e60\u60ef\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u4ece\u72b6\u6001 \\([i, j]\\) \u5f00\u59cb\u641c\u7d22\uff0c\u4e0d\u65ad\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u72b6\u6001 \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u9012\u5f52\u51fd\u6570\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, j]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u4ece \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c \\(dp[i, j]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53 \\(i = 0\\) \u4e14 \\(j = 0\\) \u65f6\uff0c\u8fd4\u56de\u4ee3\u4ef7 \\(grid[0, 0]\\) \u3002</li> <li>\u526a\u679d\uff1a\u5f53 \\(i &lt; 0\\) \u65f6\u6216 \\(j &lt; 0\\) \u65f6\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u65f6\u8fd4\u56de\u4ee3\u4ef7 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002</li> </ul> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n\"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n# \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0 and j == 0:\nreturn grid[0][0]\n# \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 or j &lt; 0:\nreturn inf\n# \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nup = min_path_sum_dfs(grid, i - 1, j)\nleft = min_path_sum_dfs(grid, i, j - 1)\n# \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn INT_MAX;\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFS(grid, i - 1, j);\nint left = minPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn Integer.MAX_VALUE;\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFS(grid, i - 1, j);\nint left = minPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn int.MaxValue;\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = MinPathSumDFS(grid, i - 1, j);\nint left = MinPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0 &amp;&amp; j == 0 {\nreturn grid[0][0]\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 || j &lt; 0 {\nreturn math.MaxInt\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nup := minPathSumDFS(grid, i-1, j)\nleft := minPathSumDFS(grid, i, j-1)\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0, j == 0 {\nreturn grid[0][0]\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 || j &lt; 0 {\nreturn .max\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nlet up = minPathSumDFS(grid: grid, i: i - 1, j: j)\nlet left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(grid, i, j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i === 0 &amp;&amp; j === 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn Infinity;\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nconst up = minPathSumDFS(grid, i - 1, j);\nconst left = minPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(\ngrid: Array&lt;Array&lt;number&gt;&gt;,\ni: number,\nj: number\n): number {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i === 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn Infinity;\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nconst up = minPathSumDFS(grid, i - 1, j);\nconst left = minPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\n// \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\nreturn BigInt.from(2).pow(31).toInt();\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFS(grid, i - 1, j);\nint left = minPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0 &amp;&amp; j == 0 {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 || j &lt; 0 {\nreturn i32::MAX;\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nlet up = min_path_sum_dfs(grid, i - 1, j);\nlet left = min_path_sum_dfs(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nstd::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn INT_MAX;\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFS(grid, i - 1, j);\nint left = minPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 and j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 or j &lt; 0) {\nreturn std.math.maxInt(i32);\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nvar up = minPathSumDFS(grid, i - 1, j);\nvar left = minPathSumDFS(grid, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nreturn @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> <p>\u56fe 14-14 \u7ed9\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u4e3a\u6839\u8282\u70b9\u7684\u9012\u5f52\u6811\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5176\u6570\u91cf\u4f1a\u968f\u7740\u7f51\u683c <code>grid</code> \u7684\u5c3a\u5bf8\u53d8\u5927\u800c\u6025\u5267\u589e\u591a\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u539f\u56e0\u4e3a\uff1a\u5b58\u5728\u591a\u6761\u8def\u5f84\u53ef\u4ee5\u4ece\u5de6\u4e0a\u89d2\u5230\u8fbe\u67d0\u4e00\u5355\u5143\u683c\u3002</p> <p></p> <p> \u56fe 14-14 \u00a0 \u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u4e24\u79cd\u9009\u62e9\uff0c\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u603b\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^{m + n})\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u8fd9\u79cd\u8ba1\u7b97\u65b9\u5f0f\u672a\u8003\u8651\u4e34\u8fd1\u7f51\u683c\u8fb9\u754c\u7684\u60c5\u51b5\uff0c\u5f53\u5230\u8fbe\u7f51\u7edc\u8fb9\u754c\u65f6\u53ea\u5269\u4e0b\u4e00\u79cd\u9009\u62e9\uff0c\u56e0\u6b64\u5b9e\u9645\u7684\u8def\u5f84\u6570\u91cf\u4f1a\u5c11\u4e00\u4e9b\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u6211\u4eec\u5f15\u5165\u4e00\u4e2a\u548c\u7f51\u683c <code>grid</code> \u76f8\u540c\u5c3a\u5bf8\u7684\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5404\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u8fdb\u884c\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\ngrid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n\"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0 and j == 0:\nreturn grid[0][0]\n# \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 or j &lt; 0:\nreturn inf\n# \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][j] != -1:\nreturn mem[i][j]\n# \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nup = min_path_sum_dfs_mem(grid, mem, i - 1, j)\nleft = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n# \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = min(left, up) + grid[i][j]\nreturn mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn INT_MAX;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][j] != -1) {\nreturn mem[i][j];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFSMem(grid, mem, i - 1, j);\nint left = minPathSumDFSMem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\nreturn mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn Integer.MAX_VALUE;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][j] != -1) {\nreturn mem[i][j];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFSMem(grid, mem, i - 1, j);\nint left = minPathSumDFSMem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = Math.min(left, up) + grid[i][j];\nreturn mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn int.MaxValue;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][j] != -1) {\nreturn mem[i][j];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = MinPathSumDFSMem(grid, mem, i - 1, j);\nint left = MinPathSumDFSMem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = Math.Min(left, up) + grid[i][j];\nreturn mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0 &amp;&amp; j == 0 {\nreturn grid[0][0]\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 || j &lt; 0 {\nreturn math.MaxInt\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][j] != -1 {\nreturn mem[i][j]\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nup := minPathSumDFSMem(grid, mem, i-1, j)\nleft := minPathSumDFSMem(grid, mem, i, j-1)\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\nreturn mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0, j == 0 {\nreturn grid[0][0]\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 || j &lt; 0 {\nreturn .max\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][j] != -1 {\nreturn mem[i][j]\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nlet up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\nlet left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = min(left, up) + grid[i][j]\nreturn mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i === 0 &amp;&amp; j === 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn Infinity;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][j] !== -1) {\nreturn mem[i][j];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nconst up = minPathSumDFSMem(grid, mem, i - 1, j);\nconst left = minPathSumDFSMem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = Math.min(left, up) + grid[i][j];\nreturn mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(\ngrid: Array&lt;Array&lt;number&gt;&gt;,\nmem: Array&lt;Array&lt;number&gt;&gt;,\ni: number,\nj: number\n): number {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i === 0 &amp;&amp; j === 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn Infinity;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][j] != -1) {\nreturn mem[i][j];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nconst up = minPathSumDFSMem(grid, mem, i - 1, j);\nconst left = minPathSumDFSMem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = Math.min(left, up) + grid[i][j];\nreturn mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\n// \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\nreturn BigInt.from(2).pow(31).toInt();\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][j] != -1) {\nreturn mem[i][j];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFSMem(grid, mem, i - 1, j);\nint left = minPathSumDFSMem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = min(left, up) + grid[i][j];\nreturn mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif i == 0 &amp;&amp; j == 0 {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif i &lt; 0 || j &lt; 0 {\nreturn i32::MAX;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i as usize][j as usize] != -1 {\nreturn mem[i as usize][j as usize];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nlet up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\nlet left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\nmem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 &amp;&amp; j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 || j &lt; 0) {\nreturn INT_MAX;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][j] != -1) {\nreturn mem[i][j];\n}\n// \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nint up = minPathSumDFSMem(grid, mem, i - 1, j);\nint left = minPathSumDFSMem(grid, mem, i, j - 1);\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\nreturn mem[i][j];\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n// \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\nif (i == 0 and j == 0) {\nreturn grid[0][0];\n}\n// \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\nif (i &lt; 0 or j &lt; 0) {\nreturn std.math.maxInt(i32);\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\nreturn mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n// \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nvar up = minPathSumDFSMem(grid, mem, i - 1, j);\nvar left = minPathSumDFSMem(grid, mem, i, j - 1);\n// \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\nmem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\nreturn mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> <p>\u5982\u56fe 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8bb0\u5fc6\u5316\u540e\uff0c\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u72b6\u6001\u603b\u6570\uff0c\u5373\u7f51\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002</p> <p></p> <p> \u56fe 14-15 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n\"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn, m = len(grid), len(grid[0])\n# \u521d\u59cb\u5316 dp \u8868\ndp = [[0] * m for _ in range(n)]\ndp[0][0] = grid[0][0]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j in range(1, m):\ndp[0][j] = dp[0][j - 1] + grid[0][j]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor i in range(1, n):\ndp[i][0] = dp[i - 1][0] + grid[i][0]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in range(1, n):\nfor j in range(1, m):\ndp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\nreturn dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\nint n = grid.size(), m = grid[0].size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt; m; j++) {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (int i = 1; i &lt; n; i++) {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt; n; i++) {\nfor (int j = 1; j &lt; m; j++) {\ndp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\nreturn dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int[][] grid) {\nint n = grid.length, m = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nint[][] dp = new int[n][m];\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt; m; j++) {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (int i = 1; i &lt; n; i++) {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt; n; i++) {\nfor (int j = 1; j &lt; m; j++) {\ndp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\nreturn dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDP(int[][] grid) {\nint n = grid.Length, m = grid[0].Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[,] dp = new int[n, m];\ndp[0, 0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt; m; j++) {\ndp[0, j] = dp[0, j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (int i = 1; i &lt; n; i++) {\ndp[i, 0] = dp[i - 1, 0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt; n; i++) {\nfor (int j = 1; j &lt; m; j++) {\ndp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n}\n}\nreturn dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid [][]int) int {\nn, m := len(grid), len(grid[0])\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([][]int, n)\nfor i := 0; i &lt; n; i++ {\ndp[i] = make([]int, m)\n}\ndp[0][0] = grid[0][0]\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j := 1; j &lt; m; j++ {\ndp[0][j] = dp[0][j-1] + grid[0][j]\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor i := 1; i &lt; n; i++ {\ndp[i][0] = dp[i-1][0] + grid[i][0]\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i := 1; i &lt; n; i++ {\nfor j := 1; j &lt; m; j++ {\ndp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n}\n}\nreturn dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\nlet n = grid.count\nlet m = grid[0].count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: Array(repeating: 0, count: m), count: n)\ndp[0][0] = grid[0][0]\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j in stride(from: 1, to: m, by: 1) {\ndp[0][j] = dp[0][j - 1] + grid[0][j]\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor i in stride(from: 1, to: n, by: 1) {\ndp[i][0] = dp[i - 1][0] + grid[i][0]\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in stride(from: 1, to: n, by: 1) {\nfor j in stride(from: 1, to: m, by: 1) {\ndp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n}\n}\nreturn dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid) {\nconst n = grid.length,\nm = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n }, () =&gt;\nArray.from({ length: m }, () =&gt; 0)\n);\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (let j = 1; j &lt; m; j++) {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (let i = 1; i &lt; n; i++) {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (let i = 1; i &lt; n; i++) {\nfor (let j = 1; j &lt; m; j++) {\ndp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\nreturn dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\nconst n = grid.length,\nm = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n }, () =&gt;\nArray.from({ length: m }, () =&gt; 0)\n);\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (let j = 1; j &lt; m; j++) {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (let i = 1; i &lt; n; i++) {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (let i = 1; i &lt; n; i++) {\nfor (let j: number = 1; j &lt; m; j++) {\ndp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\nreturn dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\nint n = grid.length, m = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt; m; j++) {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (int i = 1; i &lt; n; i++) {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt; n; i++) {\nfor (int j = 1; j &lt; m; j++) {\ndp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\nreturn dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\nlet (n, m) = (grid.len(), grid[0].len());\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![vec![0; m]; n];\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j in 1..m {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor i in 1..n {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in 1..n {\nfor j in 1..m {\ndp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\ndp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n// \u521d\u59cb\u5316 dp \u8868\nint **dp = malloc(n * sizeof(int *));\nfor (int i = 0; i &lt; n; i++) {\ndp[i] = calloc(m, sizeof(int));\n}\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt; m; j++) {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (int i = 1; i &lt; n; i++) {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt; n; i++) {\nfor (int j = 1; j &lt; m; j++) {\ndp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\nint res = dp[n - 1][m - 1];\n// \u91ca\u653e\u5185\u5b58\nfor (int i = 0; i &lt; n; i++) {\nfree(dp[i]);\n}\nreturn res;\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\nfn minPathSumDP(comptime grid: anytype) i32 {\ncomptime var n = grid.len;\ncomptime var m = grid[0].len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_][m]i32{[_]i32{0} ** m} ** n;\ndp[0][0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (1..m) |j| {\ndp[0][j] = dp[0][j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nfor (1..n) |i| {\ndp[i][0] = dp[i - 1][0] + grid[i][0];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (1..n) |i| {\nfor (1..m) |j| {\ndp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n}\n}\nreturn dp[n - 1][m - 1];\n}\n</code></pre> <p>\u56fe 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f84\u548c\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\uff0c\u5176\u904d\u5386\u4e86\u6574\u4e2a\u7f51\u683c\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> <p>\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u4e3a \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-16 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u53ea\u4e0e\u5176\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u683c\u5b50\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u53ea\u7528\u4e00\u4e2a\u5355\u884c\u6570\u7ec4\u6765\u5b9e\u73b0 \\(dp\\) \u8868\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u6570\u7ec4 <code>dp</code> \u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72b6\u6001\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72b6\u6001\uff0c\u800c\u662f\u5728\u904d\u5386\u6bcf\u884c\u65f6\u66f4\u65b0\u5b83\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n\"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn, m = len(grid), len(grid[0])\n# \u521d\u59cb\u5316 dp \u8868\ndp = [0] * m\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0]\nfor j in range(1, m):\ndp[j] = dp[j - 1] + grid[0][j]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor i in range(1, n):\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j in range(1, m):\ndp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\nreturn dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\nint n = grid.size(), m = grid[0].size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;int&gt; dp(m);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0];\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt; n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nreturn dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int[][] grid) {\nint n = grid.length, m = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[m];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0];\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt; n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nreturn dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDPComp(int[][] grid) {\nint n = grid.Length, m = grid[0].Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[m];\ndp[0] = grid[0][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt; n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nreturn dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid [][]int) int {\nn, m := len(grid), len(grid[0])\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([]int, m)\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0]\nfor j := 1; j &lt; m; j++ {\ndp[j] = dp[j-1] + grid[0][j]\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i := 1; i &lt; n; i++ {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0]\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j := 1; j &lt; m; j++ {\ndp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n}\n}\nreturn dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\nlet n = grid.count\nlet m = grid[0].count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: 0, count: m)\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0]\nfor j in stride(from: 1, to: m, by: 1) {\ndp[j] = dp[j - 1] + grid[0][j]\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor i in stride(from: 1, to: n, by: 1) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0]\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j in stride(from: 1, to: m, by: 1) {\ndp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n}\n}\nreturn dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid) {\nconst n = grid.length,\nm = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = new Array(m);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0];\nfor (let j = 1; j &lt; m; j++) {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (let i = 1; i &lt; n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (let j = 1; j &lt; m; j++) {\ndp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nreturn dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\nconst n = grid.length,\nm = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = new Array(m);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0];\nfor (let j = 1; j &lt; m; j++) {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (let i = 1; i &lt; n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (let j = 1; j &lt; m; j++) {\ndp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nreturn dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\nint n = grid.length, m = grid[0].length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;int&gt; dp = List.filled(m, 0);\ndp[0] = grid[0][0];\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt; n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nreturn dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\nlet (n, m) = (grid.len(), grid[0].len());\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![0; m];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0];\nfor j in 1..m {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor i in 1..n {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j in 1..m {\ndp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\ndp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n// \u521d\u59cb\u5316 dp \u8868\nint *dp = calloc(m, sizeof(int));\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0];\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt; n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt; m; j++) {\ndp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nint res = dp[m - 1];\n// \u91ca\u653e\u5185\u5b58\nfree(dp);\nreturn res;\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minPathSumDPComp(comptime grid: anytype) i32 {\ncomptime var n = grid.len;\ncomptime var m = grid[0].len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_]i32{0} ** m;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\ndp[0] = grid[0][0];\nfor (1..m) |j| {\ndp[j] = dp[j - 1] + grid[0][j];\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (1..n) |i| {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\ndp[0] = dp[0] + grid[i][0];\nfor (1..m) |j| {\ndp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n}\n}\nreturn dp[m - 1];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898","text":"<p>\u7f16\u8f91\u8ddd\u79bb\uff0c\u4e5f\u79f0 Levenshtein \u8ddd\u79bb\uff0c\u6307\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u4e92\u76f8\u8f6c\u6362\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6570\uff0c\u901a\u5e38\u7528\u4e8e\u5728\u4fe1\u606f\u68c0\u7d22\u548c\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u5ea6\u91cf\u4e24\u4e2a\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002</p> <p>Question</p> <p>\u8f93\u5165\u4e24\u4e2a\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c06 \\(s\\) \u8f6c\u6362\u4e3a \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u4f60\u53ef\u4ee5\u5728\u4e00\u4e2a\u5b57\u7b26\u4e32\u4e2d\u8fdb\u884c\u4e09\u79cd\u7f16\u8f91\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u4e2a\u5b57\u7b26\u3001\u5220\u9664\u4e00\u4e2a\u5b57\u7b26\u3001\u5c06\u5b57\u7b26\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u4e2a\u5b57\u7b26\u3002</p> <p>\u5982\u56fe 14-27 \u6240\u793a\uff0c\u5c06 <code>kitten</code> \u8f6c\u6362\u4e3a <code>sitting</code> \u9700\u8981\u7f16\u8f91 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u4e0e 1 \u6b21\u6dfb\u52a0\u64cd\u4f5c\uff1b\u5c06 <code>hello</code> \u8f6c\u6362\u4e3a <code>algo</code> \u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u548c 1 \u6b21\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 14-27 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u51b3\u7b56\u6811\u6a21\u578b\u6765\u89e3\u91ca\u3002\u5b57\u7b26\u4e32\u5bf9\u5e94\u6811\u8282\u70b9\uff0c\u4e00\u8f6e\u51b3\u7b56\uff08\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\uff09\u5bf9\u5e94\u6811\u7684\u4e00\u6761\u8fb9\u3002</p> <p>\u5982\u56fe 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u51b5\u4e0b\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8bb8\u591a\u6761\u8fb9\uff0c\u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u79cd\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u4ece <code>hello</code> \u8f6c\u6362\u5230 <code>algo</code> \u6709\u8bb8\u591a\u79cd\u53ef\u80fd\u7684\u8def\u5f84\u3002</p> <p>\u4ece\u51b3\u7b56\u6811\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u8282\u70b9 <code>hello</code> \u548c\u8282\u70b9 <code>algo</code> \u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002</p> <p></p> <p> \u56fe 14-28 \u00a0 \u57fa\u4e8e\u51b3\u7b56\u6811\u6a21\u578b\u8868\u793a\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u662f\u5bf9\u5b57\u7b26\u4e32 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\u3002</p> <p>\u6211\u4eec\u5e0c\u671b\u5728\u7f16\u8f91\u64cd\u4f5c\u7684\u8fc7\u7a0b\u4e2d\uff0c\u95ee\u9898\u7684\u89c4\u6a21\u9010\u6e10\u7f29\u5c0f\uff0c\u8fd9\u6837\u624d\u80fd\u6784\u5efa\u5b50\u95ee\u9898\u3002\u8bbe\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \uff0c\u6211\u4eec\u5148\u8003\u8651\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002</p> <ul> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u4eec\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u76f4\u63a5\u8003\u8651 \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002</li> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u4eec\u9700\u8981\u5bf9 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\uff08\u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\uff09\uff0c\u4f7f\u5f97\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26\u76f8\u540c\uff0c\u4ece\u800c\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u8003\u8651\u89c4\u6a21\u66f4\u5c0f\u7684\u95ee\u9898\u3002</li> </ul> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5728\u5b57\u7b26\u4e32 \\(s\\) \u4e2d\u8fdb\u884c\u7684\u6bcf\u4e00\u8f6e\u51b3\u7b56\uff08\u7f16\u8f91\u64cd\u4f5c\uff09\uff0c\u90fd\u4f1a\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u4f59\u7684\u5f85\u5339\u914d\u5b57\u7b26\u53d1\u751f\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u72b6\u6001\u4e3a\u5f53\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u8651\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u4e2a\u5b57\u7b26\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\uff1a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u8003\u8651\u5b50\u95ee\u9898 \\(dp[i, j]\\) \uff0c\u5176\u5bf9\u5e94\u7684\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5c3e\u90e8\u5b57\u7b26\u4e3a \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u636e\u4e0d\u540c\u7f16\u8f91\u64cd\u4f5c\u5206\u4e3a\u56fe 14-29 \u6240\u793a\u7684\u4e09\u79cd\u60c5\u51b5\u3002</p> <ol> <li>\u5728 \\(s[i-1]\\) \u4e4b\u540e\u6dfb\u52a0 \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i, j-1]\\) \u3002</li> <li>\u5220\u9664 \\(s[i-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j]\\) \u3002</li> <li>\u5c06 \\(s[i-1]\\) \u66ff\u6362\u4e3a \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 14-29 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u72b6\u6001\u8f6c\u79fb </p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7f16\u8f91\u6b65\u6570 \\(1\\) \u3002\u5bf9\u5e94\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5f53 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u4e24\u5b57\u7b26\u4e32\u90fd\u4e3a\u7a7a\u65f6\uff0c\u7f16\u8f91\u6b65\u6570\u4e3a \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u5f53 \\(s\\) \u4e3a\u7a7a\u4f46 \\(t\\) \u4e0d\u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(t\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u5f53 \\(s\\) \u4e0d\u4e3a\u7a7a\u4f46 \\(t\\) \u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(s\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002</p> <p>\u89c2\u5bdf\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8d56\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n\"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn, m = len(s), len(t)\ndp = [[0] * (m + 1) for _ in range(n + 1)]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor i in range(1, n + 1):\ndp[i][0] = i\nfor j in range(1, m + 1):\ndp[0][j] = j\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in range(1, n + 1):\nfor j in range(1, m + 1):\nif s[i - 1] == t[j - 1]:\n# \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1]\nelse:\n# \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\nreturn dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(string s, string t) {\nint n = s.length(), m = t.length();\nvector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int i = 1; i &lt;= n; i++) {\ndp[i][0] = i;\n}\nfor (int j = 1; j &lt;= m; j++) {\ndp[0][j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int j = 1; j &lt;= m; j++) {\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n}\n}\n}\nreturn dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\nint n = s.length(), m = t.length();\nint[][] dp = new int[n + 1][m + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int i = 1; i &lt;= n; i++) {\ndp[i][0] = i;\n}\nfor (int j = 1; j &lt;= m; j++) {\ndp[0][j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int j = 1; j &lt;= m; j++) {\nif (s.charAt(i - 1) == t.charAt(j - 1)) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n}\n}\n}\nreturn dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDP(string s, string t) {\nint n = s.Length, m = t.Length;\nint[,] dp = new int[n + 1, m + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int i = 1; i &lt;= n; i++) {\ndp[i, 0] = i;\n}\nfor (int j = 1; j &lt;= m; j++) {\ndp[0, j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int j = 1; j &lt;= m; j++) {\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i, j] = dp[i - 1, j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n}\n}\n}\nreturn dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s string, t string) int {\nn := len(s)\nm := len(t)\ndp := make([][]int, n+1)\nfor i := 0; i &lt;= n; i++ {\ndp[i] = make([]int, m+1)\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor i := 1; i &lt;= n; i++ {\ndp[i][0] = i\n}\nfor j := 1; j &lt;= m; j++ {\ndp[0][j] = j\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i := 1; i &lt;= n; i++ {\nfor j := 1; j &lt;= m; j++ {\nif s[i-1] == t[j-1] {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i-1][j-1]\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n}\n}\n}\nreturn dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\nlet n = s.utf8CString.count\nlet m = t.utf8CString.count\nvar dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor i in stride(from: 1, through: n, by: 1) {\ndp[i][0] = i\n}\nfor j in stride(from: 1, through: m, by: 1) {\ndp[0][j] = j\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in stride(from: 1, through: n, by: 1) {\nfor j in stride(from: 1, through: m, by: 1) {\nif s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1]\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n}\n}\n}\nreturn dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s, t) {\nconst n = s.length,\nm = t.length;\nconst dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (let i = 1; i &lt;= n; i++) {\ndp[i][0] = i;\n}\nfor (let j = 1; j &lt;= m; j++) {\ndp[0][j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (let i = 1; i &lt;= n; i++) {\nfor (let j = 1; j &lt;= m; j++) {\nif (s.charAt(i - 1) === t.charAt(j - 1)) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] =\nMath.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n}\n}\n}\nreturn dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s: string, t: string): number {\nconst n = s.length,\nm = t.length;\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: m + 1 }, () =&gt; 0)\n);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (let i = 1; i &lt;= n; i++) {\ndp[i][0] = i;\n}\nfor (let j = 1; j &lt;= m; j++) {\ndp[0][j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (let i = 1; i &lt;= n; i++) {\nfor (let j = 1; j &lt;= m; j++) {\nif (s.charAt(i - 1) === t.charAt(j - 1)) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] =\nMath.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n}\n}\n}\nreturn dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\nint n = s.length, m = t.length;\nList&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int i = 1; i &lt;= n; i++) {\ndp[i][0] = i;\n}\nfor (int j = 1; j &lt;= m; j++) {\ndp[0][j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int j = 1; j &lt;= m; j++) {\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n}\n}\n}\nreturn dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\nlet (n, m) = (s.len(), t.len());\nlet mut dp = vec![vec![0; m + 1]; n + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor i in 1..= n {\ndp[i][0] = i as i32;\n}\nfor j in 1..m {\ndp[0][j] = j as i32;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in 1..=n {\nfor j in 1..=m {\nif s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n}\n}\n}\ndp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(char *s, char *t, int n, int m) {\nint **dp = malloc((n + 1) * sizeof(int *));\nfor (int i = 0; i &lt;= n; i++) {\ndp[i] = calloc(m + 1, sizeof(int));\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int i = 1; i &lt;= n; i++) {\ndp[i][0] = i;\n}\nfor (int j = 1; j &lt;= m; j++) {\ndp[0][j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int j = 1; j &lt;= m; j++) {\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n}\n}\n}\nint res = dp[n][m];\n// \u91ca\u653e\u5185\u5b58\nfor (int i = 0; i &lt;= n; i++) {\nfree(dp[i]);\n}\nreturn res;\n}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\ncomptime var n = s.len;\ncomptime var m = t.len;\nvar dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (1..n + 1) |i| {\ndp[i][0] = @intCast(i);\n}\nfor (1..m + 1) |j| {\ndp[0][j] = @intCast(j);\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (1..n + 1) |i| {\nfor (1..m + 1) |j| {\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[i][j] = dp[i - 1][j - 1];\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n}\n}\n}\nreturn dp[n][m];\n}\n</code></pre> <p>\u5982\u56fe 14-30 \u6240\u793a\uff0c\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\u4e0e\u80cc\u5305\u95ee\u9898\u975e\u5e38\u7c7b\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5199\u4e00\u4e2a\u4e8c\u7ef4\u7f51\u683c\u7684\u8fc7\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-30 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f6c\u79fb\u800c\u6765\u7684\uff0c\u800c\u6b63\u5e8f\u904d\u5386\u4f1a\u4e22\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u904d\u5386\u65e0\u6cd5\u63d0\u524d\u6784\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u4e24\u79cd\u904d\u5386\u987a\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>leftup</code> \u6765\u6682\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u4ece\u800c\u53ea\u9700\u8003\u8651\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u65f6\u7684\u60c5\u51b5\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u904d\u5386\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n\"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn, m = len(s), len(t)\ndp = [0] * (m + 1)\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j in range(1, m + 1):\ndp[j] = j\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor i in range(1, n + 1):\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nleftup = dp[0]  # \u6682\u5b58 dp[i-1, j-1]\ndp[0] += 1\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j in range(1, m + 1):\ntemp = dp[j]\nif s[i - 1] == t[j - 1]:\n# \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup\nelse:\n# \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = min(dp[j - 1], dp[j], leftup) + 1\nleftup = temp  # \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\nreturn dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(string s, string t) {\nint n = s.length(), m = t.length();\nvector&lt;int&gt; dp(m + 1, 0);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt;= m; j++) {\ndp[j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt;= n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nint leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt;= m; j++) {\nint temp = dp[j];\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\nint n = s.length(), m = t.length();\nint[] dp = new int[m + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt;= m; j++) {\ndp[j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt;= n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nint leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt;= m; j++) {\nint temp = dp[j];\nif (s.charAt(i - 1) == t.charAt(j - 1)) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDPComp(string s, string t) {\nint n = s.Length, m = t.Length;\nint[] dp = new int[m + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt;= m; j++) {\ndp[j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt;= n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nint leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt;= m; j++) {\nint temp = dp[j];\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s string, t string) int {\nn := len(s)\nm := len(t)\ndp := make([]int, m+1)\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j := 1; j &lt;= m; j++ {\ndp[j] = j\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor i := 1; i &lt;= n; i++ {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nleftUp := dp[0] // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j := 1; j &lt;= m; j++ {\ntemp := dp[j]\nif s[i-1] == t[j-1] {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftUp\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n}\nleftUp = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\nlet n = s.utf8CString.count\nlet m = t.utf8CString.count\nvar dp = Array(repeating: 0, count: m + 1)\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j in stride(from: 1, through: m, by: 1) {\ndp[j] = j\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor i in stride(from: 1, through: n, by: 1) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nvar leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j in stride(from: 1, through: m, by: 1) {\nlet temp = dp[j]\nif s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n}\nleftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s, t) {\nconst n = s.length,\nm = t.length;\nconst dp = new Array(m + 1).fill(0);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (let j = 1; j &lt;= m; j++) {\ndp[j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (let i = 1; i &lt;= n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nlet leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (let j = 1; j &lt;= m; j++) {\nconst temp = dp[j];\nif (s.charAt(i - 1) === t.charAt(j - 1)) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s: string, t: string): number {\nconst n = s.length,\nm = t.length;\nconst dp = new Array(m + 1).fill(0);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (let j = 1; j &lt;= m; j++) {\ndp[j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (let i = 1; i &lt;= n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nlet leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (let j = 1; j &lt;= m; j++) {\nconst temp = dp[j];\nif (s.charAt(i - 1) === t.charAt(j - 1)) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\nint n = s.length, m = t.length;\nList&lt;int&gt; dp = List.filled(m + 1, 0);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt;= m; j++) {\ndp[j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt;= n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nint leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt;= m; j++) {\nint temp = dp[j];\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\nlet (n, m) = (s.len(), t.len());\nlet mut dp = vec![0; m + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor j in 1..m {\ndp[j] = j as i32;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor i in 1..=n {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nlet mut leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i as i32;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor j in 1..=m {\nlet temp = dp[j];\nif s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\ndp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\nint *dp = calloc(m + 1, sizeof(int));\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (int j = 1; j &lt;= m; j++) {\ndp[j] = j;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (int i = 1; i &lt;= n; i++) {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nint leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = i;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (int j = 1; j &lt;= m; j++) {\nint temp = dp[j];\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nint res = dp[m];\n// \u91ca\u653e\u5185\u5b58\nfree(dp);\nreturn res;\n}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\ncomptime var n = s.len;\ncomptime var m = t.len;\nvar dp = [_]i32{0} ** (m + 1);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\nfor (1..m + 1) |j| {\ndp[j] = @intCast(j);\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\nfor (1..n + 1) |i| {\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\nvar leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\ndp[0] = @intCast(i);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\nfor (1..m + 1) |j| {\nvar temp = dp[j];\nif (s[i - 1] == t[j - 1]) {\n// \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\ndp[j] = leftup;\n} else {\n// \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\ndp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n}\nleftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n}\n}\nreturn dp[m];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"<p>\u300c\u52a8\u6001\u89c4\u5212 dynamic programming\u300d\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u7b97\u6cd5\u8303\u5f0f\uff0c\u5b83\u5c06\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u5927\u5e45\u63d0\u5347\u65f6\u95f4\u6548\u7387\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u4ece\u4e00\u4e2a\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u9010\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p> <p>\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-1 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002</p> <p></p> <p> \u56fe 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n\"\"\"\u56de\u6eaf\"\"\"\n# \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif state == n:\nres[0] += 1\n# \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices:\n# \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif state + choice &gt; n:\ncontinue\n# \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res)\n# \u56de\u9000\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n\"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\nchoices = [1, 2]  # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nstate = 0  # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nres = [0]  # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res)\nreturn res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* \u56de\u6eaf */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres[0]++;\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (auto &amp;choice : choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n)\ncontinue;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nvector&lt;int&gt; choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nint state = 0;                // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nvector&lt;int&gt; res = {0};        // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres.set(0, res.get(0) + 1);\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (Integer choice : choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n)\ncontinue;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nList&lt;Integer&gt; choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nint state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nList&lt;Integer&gt; res = new ArrayList&lt;&gt;();\nres.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* \u56de\u6eaf */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres[0]++;\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nforeach (int choice in choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n)\ncontinue;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nBacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\nList&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nint state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nList&lt;int&gt; res = [0]; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nBacktrack(choices, state, n, res);\nreturn res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif state == n {\nres[0] = res[0] + 1\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor _, choice := range choices {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif state+choice &gt; n {\ncontinue\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state+choice, n, res)\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n// \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nchoices := []int{1, 2}\n// \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nstate := 0\nres := make([]int, 1)\n// \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nres[0] = 0\nbacktrack(choices, state, n, res)\nreturn res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif state == n {\nres[0] += 1\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor choice in choices {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif state + choice &gt; n {\ncontinue\n}\nbacktrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\nlet choices = [1, 2] // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nlet state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nvar res: [Int] = []\nres.append(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices: choices, state: state, n: n, res: &amp;res)\nreturn res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state === n) res.set(0, res.get(0) + 1);\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n) continue;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\nconst choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nconst state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nconst res = new Map();\nres.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* \u56de\u6eaf */\nfunction backtrack(\nchoices: number[],\nstate: number,\nn: number,\nres: Map&lt;0, any&gt;\n): void {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state === n) res.set(0, res.get(0) + 1);\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (const choice of choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n) continue;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\nconst choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nconst state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nconst res = new Map();\nres.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n) {\nres[0]++;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int choice in choices) {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n) continue;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nList&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nint state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nList&lt;int&gt; res = [];\nres.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(choices, state, n, res);\nreturn res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* \u56de\u6eaf */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif state == n { res[0] = res[0] + 1; }\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor &amp;choice in choices {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif state + choice &gt; n { continue; }\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\nlet choices = vec![ 1, 2 ]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nlet state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nlet mut res = Vec::new();\nres.push(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(&amp;choices, state, n as i32, &amp;mut res);\nres[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n)\nres[0]++;\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (int i = 0; i &lt; len; i++) {\nint choice = choices[i];\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n)\ncontinue;\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res, len);\n// \u56de\u9000\n}\n}\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\nint choices[2] = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nint state = 0;           // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nint *res = (int *)malloc(sizeof(int));\n*res = 0; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nint len = sizeof(choices) / sizeof(int);\nbacktrack(choices, state, n, res, len);\nint result = *res;\nfree(res);\nreturn result;\n}\n</code></pre> climbing_stairs_backtrack.zig<pre><code>// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n// \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\nif (state == n) {\nres.items[0] = res.items[0] + 1;\n}\n// \u904d\u5386\u6240\u6709\u9009\u62e9\nfor (choices) |choice| {\n// \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\nif (state + choice &gt; n) {\ncontinue;\n}\n// \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\nbacktrack(choices, state + choice, n, res);\n// \u56de\u9000\n}\n}\n// \u722c\u697c\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\nvar choices = [_]i32{ 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\nvar state: i32 = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\nvar res = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer res.deinit();\ntry res.append(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\nbacktrack(&amp;choices, state, @intCast(n), res);\nreturn res.items[0];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u6c42\u89e3\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u4ece\u95ee\u9898\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u8fd9\u9053\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec\uff1a</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>\u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u8fc8\u5411\u7b2c \\(i\\) \u9636\u3002</p> <p>\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u63a8\u8bba\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\u3002\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>\u8fd9\u610f\u5473\u7740\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u5b58\u5728\u9012\u63a8\u5173\u7cfb\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u5efa\u5f97\u6765\u3002\u56fe 14-2 \u5c55\u793a\u4e86\u8be5\u9012\u63a8\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 14-2 \u00a0 \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb </p> <p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9012\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u7d22\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u9012\u5f52\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u9636\u5206\u522b\u6709 \\(1\\)\u3001\\(2\\) \u79cd\u65b9\u6848\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u548c\u6807\u51c6\u56de\u6eaf\u4ee3\u7801\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u66f4\u52a0\u7b80\u6d01\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n\"\"\"\u641c\u7d22\"\"\"\n# \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 or i == 2:\nreturn i\n# dp[i] = dp[i-1] + dp[i-2]\ncount = dfs(i - 1) + dfs(i - 2)\nreturn count\ndef climbing_stairs_dfs(n: int) -&gt; int:\n\"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\nreturn dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* \u641c\u7d22 */\nint DFS(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = DFS(i - 1) + DFS(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint ClimbingStairsDFS(int n) {\nreturn DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* \u641c\u7d22 */\nfunc dfs(i int) int {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 || i == 2 {\nreturn i\n}\n// dp[i] = dp[i-1] + dp[i-2]\ncount := dfs(i-1) + dfs(i-2)\nreturn count\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n int) int {\nreturn dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* \u641c\u7d22 */\nfunc dfs(i: Int) -&gt; Int {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 || i == 2 {\nreturn i\n}\n// dp[i] = dp[i-1] + dp[i-2]\nlet count = dfs(i: i - 1) + dfs(i: i - 2)\nreturn count\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\ndfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* \u641c\u7d22 */\nfunction dfs(i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i === 1 || i === 2) return i;\n// dp[i] = dp[i-1] + dp[i-2]\nconst count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n) {\nreturn dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* \u641c\u7d22 */\nfunction dfs(i: number): number {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i === 1 || i === 2) return i;\n// dp[i] = dp[i-1] + dp[i-2]\nconst count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n: number): number {\nreturn dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2) return i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* \u641c\u7d22 */\nfn dfs(i: usize) -&gt; i32 {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 || i == 2 { return i as i32; }\n// dp[i] = dp[i-1] + dp[i-2]\nlet count = dfs(i - 1) + dfs(i - 2);\ncount\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\ndfs(n) }\n</code></pre> climbing_stairs_dfs.c<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\nreturn dfs(n);\n}\n</code></pre> climbing_stairs_dfs.zig<pre><code>// \u641c\u7d22\nfn dfs(i: usize) i32 {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 or i == 2) {\nreturn @intCast(i);\n}\n// dp[i] = dp[i-1] + dp[i-2]\nvar count = dfs(i - 1) + dfs(i - 2);\nreturn count;\n}\n// \u722c\u697c\u68af\uff1a\u641c\u7d22\nfn climbingStairsDFS(comptime n: usize) i32 {\nreturn dfs(n);\n}\n</code></pre> <p>\u56fe 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u7d22\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u5176\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u5c5e\u4e8e\u7206\u70b8\u5f0f\u589e\u957f\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002</p> <p></p> <p> \u56fe 14-3 \u00a0 \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811 </p> <p>\u89c2\u5bdf\u56fe 14-3 \uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u91cd\u53e0\u5b50\u95ee\u9898\u201d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \u3002</p> <p>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5b50\u95ee\u9898\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\u3002\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u95ee\u9898\u4e0a\u3002</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u6570\u7ec4 <code>mem</code> \u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\u3002</p> <ol> <li>\u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3 <code>mem[i]</code> \uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\u3002</li> <li>\u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece <code>mem[i]</code> \u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u8be5\u5b50\u95ee\u9898\u3002</li> </ol> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n\"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 or i == 2:\nreturn i\n# \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif mem[i] != -1:\nreturn mem[i]\n# dp[i] = dp[i-1] + dp[i-2]\ncount = dfs(i - 1, mem) + dfs(i - 2, mem)\n# \u8bb0\u5f55 dp[i]\nmem[i] = count\nreturn count\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n\"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nmem = [-1] * (n + 1)\nreturn dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nvector&lt;int&gt; mem(n + 1, -1);\nreturn dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nint[] mem = new int[n + 1];\nArrays.fill(mem, -1);\nreturn dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint DFS(int i, int[] mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = DFS(i - 1, mem) + DFS(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint ClimbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nint[] mem = new int[n + 1];\nArray.Fill(mem, -1);\nreturn DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfsMem(i int, mem []int) int {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 || i == 2 {\nreturn i\n}\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif mem[i] != -1 {\nreturn mem[i]\n}\n// dp[i] = dp[i-1] + dp[i-2]\ncount := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n// \u8bb0\u5f55 dp[i]\nmem[i] = count\nreturn count\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n int) int {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nmem := make([]int, n+1)\nfor i := range mem {\nmem[i] = -1\n}\nreturn dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 || i == 2 {\nreturn i\n}\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif mem[i] != -1 {\nreturn mem[i]\n}\n// dp[i] = dp[i-1] + dp[i-2]\nlet count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n// \u8bb0\u5f55 dp[i]\nmem[i] = count\nreturn count\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nvar mem = Array(repeating: -1, count: n + 1)\nreturn dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i, mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i === 1 || i === 2) return i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1) return mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nconst count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nconst mem = new Array(n + 1).fill(-1);\nreturn dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i: number, mem: number[]): number {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i === 1 || i === 2) return i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1) return mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nconst count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n: number): number {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nconst mem = new Array(n + 1).fill(-1);\nreturn dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, List&lt;int&gt; mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2) return i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1) return mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nList&lt;int&gt; mem = List.filled(n + 1, -1);\nreturn dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif i == 1 || i == 2 { return i as i32; }\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif mem[i] != -1 { return mem[i]; }\n// dp[i] = dp[i-1] + dp[i-2]\nlet count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\ncount\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nlet mut mem = vec![-1; n + 1];\ndfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int *mem) {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 || i == 2)\nreturn i;\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1)\nreturn mem[i];\n// dp[i] = dp[i-1] + dp[i-2]\nint count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nint *mem = (int *)malloc((n + 1) * sizeof(int));\nfor (int i = 0; i &lt;= n; i++) {\nmem[i] = -1;\n}\nint result = dfs(n, mem);\nfree(mem);\nreturn result;\n}\n</code></pre> climbing_stairs_dfs_mem.zig<pre><code>// \u8bb0\u5fc6\u5316\u641c\u7d22\nfn dfs(i: usize, mem: []i32) i32 {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (i == 1 or i == 2) {\nreturn @intCast(i);\n}\n// \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\nif (mem[i] != -1) {\nreturn mem[i];\n}\n// dp[i] = dp[i-1] + dp[i-2]\nvar count = dfs(i - 1, mem) + dfs(i - 2, mem);\n// \u8bb0\u5f55 dp[i]\nmem[i] = count;\nreturn count;\n}\n// \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n// mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\nvar mem = [_]i32{ -1 } ** (n + 1);\nreturn dfs(n, &amp;mem);\n}\n</code></pre> <p>\u89c2\u5bdf\u56fe 14-4 \uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002</p> <p></p> <p> \u56fe 14-4 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\u3002\u4e4b\u540e\uff0c\u901a\u8fc7\u56de\u6eaf\u9010\u5c42\u6536\u96c6\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u4e0e\u4e4b\u76f8\u53cd\uff0c\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u201c\u4ece\u5e95\u81f3\u9876\u201d\u7684\u65b9\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6784\u5efa\u66f4\u5927\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u5faa\u73af\u8fed\u4ee3\u5b9e\u73b0\uff0c\u65e0\u987b\u4f7f\u7528\u9012\u5f52\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u4e0e\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4 <code>mem</code> \u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n\"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\n# \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp = [0] * (n + 1)\n# \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1], dp[2] = 1, 2\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in range(3, n + 1):\ndp[i] = dp[i - 1] + dp[i - 2]\nreturn dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvector&lt;int&gt; dp(n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint[] dp = new int[n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n int) int {\nif n == 1 || n == 2 {\nreturn n\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\ndp := make([]int, n+1)\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1\ndp[2] = 2\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i := 3; i &lt;= n; i++ {\ndp[i] = dp[i-1] + dp[i-2]\n}\nreturn dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\nif n == 1 || n == 2 {\nreturn n\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvar dp = Array(repeating: 0, count: n + 1)\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1\ndp[2] = 2\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in stride(from: 3, through: n, by: 1) {\ndp[i] = dp[i - 1] + dp[i - 2]\n}\nreturn dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n) {\nif (n === 1 || n === 2) return n;\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nconst dp = new Array(n + 1).fill(-1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (let i = 3; i &lt;= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n: number): number {\nif (n === 1 || n === 2) return n;\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nconst dp = new Array(n + 1).fill(-1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (let i = 3; i &lt;= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2) return n;\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nList&lt;int&gt; dp = List.filled(n + 1, 0);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif n == 1 || n == 2 { return n as i32; }\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nlet mut dp = vec![-1; n + 1];\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i in 3..=n {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\ndp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\nif (n == 1 || n == 2)\nreturn n;\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nint *dp = (int *)malloc((n + 1) * sizeof(int));\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (int i = 3; i &lt;= n; i++) {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nint result = dp[n];\nfree(dp);\nreturn result;\n}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsDP(comptime n: usize) i32 {\n// \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\nif (n == 1 or n == 2) {\nreturn @intCast(n);\n}\n// \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\nvar dp = [_]i32{-1} ** (n + 1);\n// \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\ndp[1] = 1;\ndp[2] = 2;\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor (3..n + 1) |i| {\ndp[i] = dp[i - 1] + dp[i - 2];\n}\nreturn dp[n];\n}\n</code></pre> <p>\u56fe 14-5 \u6a21\u62df\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-5 \u00a0 \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u697c\u68af\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570 \\(i\\) \u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5185\u5bb9\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u5c06\u6570\u7ec4 <code>dp</code> \u79f0\u4e3a\u300c\\(dp\\) \u8868\u300d\uff0c\\(dp[i]\\) \u8868\u793a\u72b6\u6001 \\(i\\) \u5bf9\u5e94\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u5c06\u6700\u5c0f\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u7b2c \\(1\\) \u9636\u548c\u7b2c \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u300c\u521d\u59cb\u72b6\u6001\u300d\u3002</li> <li>\u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","title":"14.1.4 \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u987b\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n\"\"\"\u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nif n == 1 or n == 2:\nreturn n\na, b = 1, 2\nfor _ in range(3, n + 1):\na, b = b, a + b\nreturn b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n int) int {\nif n == 1 || n == 2 {\nreturn n\n}\na, b := 1, 2\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\nfor i := 3; i &lt;= n; i++ {\na, b = b, a+b\n}\nreturn b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\nif n == 1 || n == 2 {\nreturn n\n}\nvar a = 1\nvar b = 2\nfor _ in stride(from: 3, through: n, by: 1) {\n(a, b) = (b, a + b)\n}\nreturn b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n) {\nif (n === 1 || n === 2) return n;\nlet a = 1,\nb = 2;\nfor (let i = 3; i &lt;= n; i++) {\nconst tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n: number): number {\nif (n === 1 || n === 2) return n;\nlet a = 1,\nb = 2;\nfor (let i = 3; i &lt;= n; i++) {\nconst tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2) return n;\nint a = 1, b = 2;\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\nif n == 1 || n == 2 { return n as i32; }\nlet (mut a, mut b) = (1, 2);\nfor _ in 3..=n {\nlet tmp = b;\nb = a + b;\na = tmp;\n}\nb\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\nif (n == 1 || n == 2)\nreturn n;\nint a = 1, b = 2;\nfor (int i = 3; i &lt;= n; i++) {\nint tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn climbingStairsDPComp(comptime n: usize) i32 {\nif (n == 1 or n == 2) {\nreturn @intCast(n);\n}\nvar a: i32 = 1;\nvar b: i32 = 2;\nfor (3..n + 1) |_| {\nvar tmp = b;\nb = a + b;\na = tmp;\n}\nreturn b;\n}\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4 <code>dp</code> \u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</p> <p>\u5728\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u5f80\u5f80\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u88ab\u79f0\u4e3a\u201c\u6eda\u52a8\u53d8\u91cf\u201d\u6216\u201c\u6eda\u52a8\u6570\u7ec4\u201d\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u52a8\u6001\u89c4\u5212\u5165\u95e8\u9898\u76ee\uff0c\u662f\u52a8\u6001\u89c4\u5212\u4e2d\u6700\u5e38\u89c1\u7684\u95ee\u9898\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u95ee\u9898\u3001\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898\u7b49\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6765\u6c42\u89e3\u6700\u5e38\u89c1\u7684 0-1 \u80cc\u5305\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u89c2\u5bdf\u56fe 14-17 \uff0c\u7531\u4e8e\u7269\u54c1\u7f16\u53f7 \\(i\\) \u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u6570\u7ec4\u7d22\u5f15\u4ece \\(0\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5bf9\u5e94\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u4ef7\u503c \\(val[i-1]\\) \u3002</p> <p></p> <p> \u56fe 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06 0-1 \u80cc\u5305\u95ee\u9898\u770b\u4f5c\u4e00\u4e2a\u7531 \\(n\\) \u8f6e\u51b3\u7b56\u7ec4\u6210\u7684\u8fc7\u7a0b\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u4f53\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u4e24\u79cd\u51b3\u7b56\uff0c\u56e0\u6b64\u8be5\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\u3002</p> <p>\u8be5\u95ee\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u201d\uff0c\u56e0\u6b64\u8f83\u5927\u6982\u7387\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\u6765\u8bf4\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72b6\u6001\u5b9a\u4e49\uff1a\u5f53\u524d\u7269\u54c1\u7f16\u53f7 \\(i\\) \u548c\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8bb0\u4e3a \\([i, c]\\) \u3002</p> <p>\u72b6\u6001 \\([i, c]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\uff0c\u8bb0\u4e3a \\(dp[i, c]\\) \u3002</p> <p>\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5f53\u6211\u4eec\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u51b3\u7b56\u540e\uff0c\u5269\u4f59\u7684\u662f\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u7684\u51b3\u7b56\uff0c\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u51cf\u5c11 \\(wgt[i-1]\\) \uff0c\u4ef7\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u4eec\u63ed\u793a\u4e86\u672c\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6700\u5927\u4ef7\u503c \\(dp[i, c]\\) \u7b49\u4e8e\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\u3002\u7531\u6b64\u53ef\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u65e0\u7269\u54c1\u6216\u65e0\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u65f6\u6700\u5927\u4ef7\u503c\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u524d\u72b6\u6001 \\([i, c]\\) \u4ece\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c-wgt[i-1]]\\) \u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u6309\u987a\u5e8f\u5b9e\u73b0\u66b4\u529b\u641c\u7d22\u3001\u8bb0\u5fc6\u5316\u641c\u7d22\u3001\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u641c\u7d22\u4ee3\u7801\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, c]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5b50\u95ee\u9898\u7684\u89e3 \\(dp[i, c]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53\u7269\u54c1\u7f16\u53f7\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\u4e3a \\(0\\) \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5e76\u8fd4\u56de\u4ef7\u503c \\(0\\) \u3002</li> <li>\u526a\u679d\uff1a\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n\"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n# \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 or c == 0:\nreturn 0\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] &gt; c:\nreturn knapsack_dfs(wgt, val, i - 1, c)\n# \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nno = knapsack_dfs(wgt, val, i - 1, c)\nyes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n# \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFS(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFS(wgt, val, i - 1, c);\nint yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFS(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFS(wgt, val, i - 1, c);\nint yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (weight[i - 1] &gt; c) {\nreturn KnapsackDFS(weight, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = KnapsackDFS(weight, val, i - 1, c);\nint yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 || c == 0 {\nreturn 0\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i-1] &gt; c {\nreturn knapsackDFS(wgt, val, i-1, c)\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nno := knapsackDFS(wgt, val, i-1, c)\nyes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 || c == 0 {\nreturn 0\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] &gt; c {\nreturn knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nlet no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\nlet yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(wgt, val, i, c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i === 0 || c === 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFS(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nconst no = knapsackDFS(wgt, val, i - 1, c);\nconst yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(\nwgt: Array&lt;number&gt;,\nval: Array&lt;number&gt;,\ni: number,\nc: number\n): number {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i === 0 || c === 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFS(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nconst no = knapsackDFS(wgt, val, i - 1, c);\nconst yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFS(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFS(wgt, val, i - 1, c);\nint yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 || c == 0 {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] &gt; c as i32 {\nreturn knapsack_dfs(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nlet no = knapsack_dfs(wgt, val, i - 1, c);\nlet yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nstd::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFS(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFS(wgt, val, i - 1, c);\nint yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn myMax(no, yes);\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 or c == 0) {\nreturn 0;\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFS(wgt, val, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nvar no = knapsackDFS(wgt, val, i - 1, c);\nvar yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nreturn @max(no, yes);\n}\n</code></pre> <p>\u5982\u56fe 14-18 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u7269\u54c1\u90fd\u4f1a\u4ea7\u751f\u4e0d\u9009\u548c\u9009\u4e24\u6761\u641c\u7d22\u5206\u652f\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002</p> <p>\u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5bb9\u6613\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u5f53\u7269\u54c1\u8f83\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f83\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f83\u591a\u65f6\uff0c\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u6570\u91cf\u5c06\u4f1a\u5927\u5e45\u589e\u591a\u3002</p> <p></p> <p> \u56fe 14-18 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u6211\u4eec\u501f\u52a9\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \u6765\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5176\u4e2d <code>mem[i][c]</code> \u5bf9\u5e94 \\(dp[i, c]\\) \u3002</p> <p>\u5f15\u5165\u8bb0\u5fc6\u5316\u4e4b\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5b50\u95ee\u9898\u6570\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dfs_mem(\nwgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n\"\"\"0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n# \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 or c == 0:\nreturn 0\n# \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][c] != -1:\nreturn mem[i][c]\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] &gt; c:\nreturn knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n# \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nno = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\nyes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n# \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = max(no, yes)\nreturn mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] != -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFSMem(wgt, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFSMem(wgt, val, mem, i - 1, c);\nint yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = max(no, yes);\nreturn mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] != -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFSMem(wgt, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFSMem(wgt, val, mem, i - 1, c);\nint yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = Math.max(no, yes);\nreturn mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] != -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (weight[i - 1] &gt; c) {\nreturn KnapsackDFSMem(weight, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = KnapsackDFSMem(weight, val, mem, i - 1, c);\nint yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = Math.Max(no, yes);\nreturn mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 || c == 0 {\nreturn 0\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][c] != -1 {\nreturn mem[i][c]\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i-1] &gt; c {\nreturn knapsackDFSMem(wgt, val, mem, i-1, c)\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nno := knapsackDFSMem(wgt, val, mem, i-1, c)\nyes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n// \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = int(math.Max(float64(no), float64(yes)))\nreturn mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 || c == 0 {\nreturn 0\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][c] != -1 {\nreturn mem[i][c]\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] &gt; c {\nreturn knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nlet no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\nlet yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = max(no, yes)\nreturn mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i === 0 || c === 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] !== -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFSMem(wgt, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nconst no = knapsackDFSMem(wgt, val, mem, i - 1, c);\nconst yes =\nknapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = Math.max(no, yes);\nreturn mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(\nwgt: Array&lt;number&gt;,\nval: Array&lt;number&gt;,\nmem: Array&lt;Array&lt;number&gt;&gt;,\ni: number,\nc: number\n): number {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i === 0 || c === 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] !== -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFSMem(wgt, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nconst no = knapsackDFSMem(wgt, val, mem, i - 1, c);\nconst yes =\nknapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = Math.max(no, yes);\nreturn mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(\nList&lt;int&gt; wgt,\nList&lt;int&gt; val,\nList&lt;List&lt;int&gt;&gt; mem,\nint i,\nint c,\n) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] != -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFSMem(wgt, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFSMem(wgt, val, mem, i - 1, c);\nint yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = max(no, yes);\nreturn mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif i == 0 || c == 0 {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif mem[i][c] != -1 {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif wgt[i - 1] &gt; c as i32 {\nreturn knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nlet no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\nlet yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = std::cmp::max(no, yes);\nmem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 || c == 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] != -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nint no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\nint yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = myMax(no, yes);\nreturn mem[i][c];\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n// \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\nif (i == 0 or c == 0) {\nreturn 0;\n}\n// \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (mem[i][c] != -1) {\nreturn mem[i][c];\n}\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\nif (wgt[i - 1] &gt; c) {\nreturn knapsackDFSMem(wgt, val, mem, i - 1, c);\n}\n// \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\nvar no = knapsackDFSMem(wgt, val, mem, i - 1, c);\nvar yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n// \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\nmem[i][c] = @max(no, yes);\nreturn mem[i][c];\n}\n</code></pre> <p>\u56fe 14-19 \u5c55\u793a\u4e86\u5728\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u88ab\u526a\u6389\u7684\u641c\u7d22\u5206\u652f\u3002</p> <p></p> <p> \u56fe 14-19 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u52a8\u6001\u89c4\u5212\u5b9e\u8d28\u4e0a\u5c31\u662f\u5728\u72b6\u6001\u8f6c\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u8fc7\u7a0b\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n\"\"\"0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(wgt)\n# \u521d\u59cb\u5316 dp \u8868\ndp = [[0] * (cap + 1) for _ in range(n + 1)]\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\nfor c in range(1, cap + 1):\nif wgt[i - 1] &gt; c:\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c]\nelse:\n# \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\nreturn dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\nint n = wgt.size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nint[][] dp = new int[n + 1][cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\nint n = weight.Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[,] dp = new int[n + 1, cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (weight[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i, c] = dp[i - 1, c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n}\n}\n}\nreturn dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt, val []int, cap int) int {\nn := len(wgt)\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([][]int, n+1)\nfor i := 0; i &lt;= n; i++ {\ndp[i] = make([]int, cap+1)\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor i := 1; i &lt;= n; i++ {\nfor c := 1; c &lt;= cap; c++ {\nif wgt[i-1] &gt; c {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i-1][c]\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n}\n}\n}\nreturn dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\nlet n = wgt.count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n// \u72b6\u6001\u8f6c\u79fb\nfor i in stride(from: 1, through: n, by: 1) {\nfor c in stride(from: 1, through: cap, by: 1) {\nif wgt[i - 1] &gt; c {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c]\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n}\n}\n}\nreturn dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(wgt, val, cap) {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array(n + 1)\n.fill(0)\n.map(() =&gt; Array(cap + 1).fill(0));\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = Math.max(\ndp[i - 1][c],\ndp[i - 1][c - wgt[i - 1]] + val[i - 1]\n);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(\nwgt: Array&lt;number&gt;,\nval: Array&lt;number&gt;,\ncap: number\n): number {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: cap + 1 }, () =&gt; 0)\n);\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = Math.max(\ndp[i - 1][c],\ndp[i - 1][c - wgt[i - 1]] + val[i - 1]\n);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\nlet n = wgt.len();\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![vec![0; cap + 1]; n + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor i in 1..=n {\nfor c in 1..=cap {\nif wgt[i - 1] &gt; c as i32 {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = std::cmp::max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1]);\n}\n}\n}\ndp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\nint n = wgtSize;\n// \u521d\u59cb\u5316 dp \u8868\nint **dp = malloc((n + 1) * sizeof(int *));\nfor (int i = 0; i &lt;= n; i++) {\ndp[i] = calloc(cap + 1, sizeof(int));\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nint res = dp[n][cap];\n// \u91ca\u653e\u5185\u5b58\nfor (int i = 0; i &lt;= n; i++) {\nfree(dp[i]);\n}\nreturn res;\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\ncomptime var n = wgt.len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n// \u72b6\u6001\u8f6c\u79fb\nfor (1..n + 1) |i| {\nfor (1..cap + 1) |c| {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> <p>\u5982\u56fe 14-20 \u6240\u793a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u7531\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u51b3\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-20 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u90fd\u53ea\u4e0e\u5176\u4e0a\u4e00\u884c\u7684\u72b6\u6001\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6570\u7ec4\u6eda\u52a8\u524d\u8fdb\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\uff0c\u6211\u4eec\u80fd\u5426\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u5b9e\u73b0\u7a7a\u95f4\u4f18\u5316\u5462\uff1f\u89c2\u5bdf\u53ef\u77e5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f6c\u79fb\u8fc7\u6765\u7684\u3002\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u6570\u7ec4\uff0c\u5f53\u5f00\u59cb\u904d\u5386\u7b2c \\(i\\) \u884c\u65f6\uff0c\u8be5\u6570\u7ec4\u5b58\u50a8\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72b6\u6001\u3002</p> <ul> <li>\u5982\u679c\u91c7\u53d6\u6b63\u5e8f\u904d\u5386\uff0c\u90a3\u4e48\u904d\u5386\u5230 \\(dp[i, j]\\) \u65f6\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7ecf\u88ab\u8986\u76d6\uff0c\u6b64\u65f6\u5c31\u65e0\u6cd5\u5f97\u5230\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u7ed3\u679c\u3002</li> <li>\u5982\u679c\u91c7\u53d6\u5012\u5e8f\u904d\u5386\uff0c\u5219\u4e0d\u4f1a\u53d1\u751f\u8986\u76d6\u95ee\u9898\uff0c\u72b6\u6001\u8f6c\u79fb\u53ef\u4ee5\u6b63\u786e\u8fdb\u884c\u3002</li> </ul> <p>\u56fe 14-21 \u5c55\u793a\u4e86\u5728\u5355\u4e2a\u6570\u7ec4\u4e0b\u4ece\u7b2c \\(i = 1\\) \u884c\u8f6c\u6362\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u8fc7\u7a0b\u3002\u8bf7\u601d\u8003\u6b63\u5e8f\u904d\u5386\u548c\u5012\u5e8f\u904d\u5386\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4 \\(i\\) \u76f4\u63a5\u5220\u9664\uff0c\u5e76\u4e14\u628a\u5185\u5faa\u73af\u66f4\u6539\u4e3a\u5012\u5e8f\u904d\u5386\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n\"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(wgt)\n# \u521d\u59cb\u5316 dp \u8868\ndp = [0] * (cap + 1)\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\n# \u5012\u5e8f\u904d\u5386\nfor c in range(cap, 0, -1):\nif wgt[i - 1] &gt; c:\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c]\nelse:\n# \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\nreturn dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\nint n = wgt.size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;int&gt; dp(cap + 1, 0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\n// \u5012\u5e8f\u904d\u5386\nfor (int c = cap; c &gt;= 1; c--) {\nif (wgt[i - 1] &lt;= c) {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\n// \u5012\u5e8f\u904d\u5386\nfor (int c = cap; c &gt;= 1; c--) {\nif (wgt[i - 1] &lt;= c) {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\nint n = weight.Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\n// \u5012\u5e8f\u904d\u5386\nfor (int c = cap; c &gt; 0; c--) {\nif (weight[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\nn := len(wgt)\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([]int, cap+1)\n// \u72b6\u6001\u8f6c\u79fb\nfor i := 1; i &lt;= n; i++ {\n// \u5012\u5e8f\u904d\u5386\nfor c := cap; c &gt;= 1; c-- {\nif wgt[i-1] &lt;= c {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n}\n}\n}\nreturn dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\nlet n = wgt.count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: 0, count: cap + 1)\n// \u72b6\u6001\u8f6c\u79fb\nfor i in stride(from: 1, through: n, by: 1) {\n// \u5012\u5e8f\u904d\u5386\nfor c in stride(from: cap, through: 1, by: -1) {\nif wgt[i - 1] &lt;= c {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n}\n}\n}\nreturn dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(wgt, val, cap) {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array(cap + 1).fill(0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\n// \u5012\u5e8f\u904d\u5386\nfor (let c = cap; c &gt;= 1; c--) {\nif (wgt[i - 1] &lt;= c) {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(\nwgt: Array&lt;number&gt;,\nval: Array&lt;number&gt;,\ncap: number\n): number {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array(cap + 1).fill(0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\n// \u5012\u5e8f\u904d\u5386\nfor (let c = cap; c &gt;= 1; c--) {\nif (wgt[i - 1] &lt;= c) {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;int&gt; dp = List.filled(cap + 1, 0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\n// \u5012\u5e8f\u904d\u5386\nfor (int c = cap; c &gt;= 1; c--) {\nif (wgt[i - 1] &lt;= c) {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\nlet n = wgt.len();\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![0; cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor i in 1..=n {\n// \u5012\u5e8f\u904d\u5386\nfor c in (1..=cap).rev() {\nif wgt[i - 1] &lt;= c as i32 {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n}\n}\n}\ndp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\nint n = wgtSize;\n// \u521d\u59cb\u5316 dp \u8868\nint *dp = calloc(cap + 1, sizeof(int));\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\n// \u5012\u5e8f\u904d\u5386\nfor (int c = cap; c &gt;= 1; c--) {\nif (wgt[i - 1] &lt;= c) {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nint res = dp[cap];\n// \u91ca\u653e\u5185\u5b58\nfree(dp);\nreturn res;\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\nvar n = wgt.len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_]i32{0} ** (cap + 1);\n// \u72b6\u6001\u8f6c\u79fb\nfor (1..n + 1) |i| {\n// \u5012\u5e8f\u904d\u5386\nvar c = cap;\nwhile (c &gt; 0) : (c -= 1) {\nif (wgt[i - 1] &lt; c) {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u52a8\u6001\u89c4\u5212\u5bf9\u95ee\u9898\u8fdb\u884c\u5206\u89e3\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u89c4\u907f\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u9ad8 \u8ba1\u7b97\u6548\u7387\u3002</li> <li>\u4e0d\u8003\u8651\u65f6\u95f4\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52a8\u6001\u89c4\u5212\u95ee\u9898\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u7d22\uff09\u8fdb\u884c\u6c42\u89e3\uff0c\u4f46\u9012\u5f52\u6811\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6548\u7387\u6781\u4f4e\u3002\u901a\u8fc7\u5f15\u5165\u8bb0\u5fc6\u5316\u5217\u8868\uff0c\u53ef\u4ee5\u5b58\u50a8\u6240\u6709\u8ba1\u7b97\u8fc7\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u800c\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002</li> <li>\u8bb0\u5fc6\u5316\u9012\u5f52\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u9012\u5f52\u5f0f\u89e3\u6cd5\uff0c\u800c\u4e0e\u4e4b\u5bf9\u5e94\u7684\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u4ece\u5e95\u81f3\u9876\u7684\u9012\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5199\u8868\u683c\u201d\u4e00\u6837\u3002\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u4ec5\u4f9d\u8d56\u67d0\u4e9b\u5c40\u90e8\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff0c\u4ece\u800c\u964d\u4f4e\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8d28\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u95ee\u9898\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</li> <li>\u5982\u679c\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\uff0c\u5219\u5b83\u5c31\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u3002</li> <li>\u65e0\u540e\u6548\u6027\u6307\u5bf9\u4e8e\u4e00\u4e2a\u72b6\u6001\uff0c\u5176\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u8be5\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u4e0d\u5177\u6709\u65e0\u540e\u6548\u6027\uff0c\u65e0\u6cd5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u5feb\u901f\u6c42\u89e3\u3002</li> </ul> <p>\u80cc\u5305\u95ee\u9898</p> <ul> <li>\u80cc\u5305\u95ee\u9898\u662f\u6700\u5178\u578b\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u53d8\u79cd\u3002</li> <li>0-1 \u80cc\u5305\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u3002\u6839\u636e\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u4e24\u79cd\u51b3\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\uff0c\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u904d\u5386\u5217\u8868\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72b6\u6001\u88ab\u8986\u76d6\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u6bcf\u79cd\u7269\u54c1\u7684\u9009\u53d6\u6570\u91cf\u65e0\u9650\u5236\uff0c\u56e0\u6b64\u9009\u62e9\u653e\u5165\u7269\u54c1\u7684\u72b6\u6001\u8f6c\u79fb\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\u3002\u7531\u4e8e\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\u5e94\u5f53\u6b63\u5e8f\u904d\u5386\u3002</li> <li>\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u4e2a\u53d8\u79cd\u3002\u5b83\u4ece\u6c42\u201c\u6700\u5927\u201d\u4ef7\u503c\u53d8\u4e3a\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e01\u6570\u91cf\uff0c\u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u5e94\u6539\u4e3a \\(\\min()\\) \u3002\u4ece\u8ffd\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u6765\u8868\u793a\u201c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u201d\u7684\u65e0\u6548\u89e3\u3002</li> <li>\u96f6\u94b1\u5151\u6362 II \u95ee\u9898\u4ece\u6c42\u201c\u6700\u5c11\u786c\u5e01\u6570\u91cf\u201d\u6539\u4e3a\u6c42\u201c\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u201d\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u76f8\u5e94\u5730\u4ece \\(\\min()\\) \u6539\u4e3a\u6c42\u548c\u8fd0\u7b97\u7b26\u3002</li> </ul> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</p> <ul> <li>\u7f16\u8f91\u8ddd\u79bb\uff08Levenshtein \u8ddd\u79bb\uff09\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u4ece\u4e00\u4e2a\u5b57\u7b26\u4e32\u5230\u53e6\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u7f16\u8f91\u64cd\u4f5c\u5305\u62ec\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\u3002</li> <li>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002\u5f53 \\(s[i] \\ne t[j]\\) \u65f6\uff0c\u5177\u6709\u4e09\u79cd\u51b3\u7b56\uff1a\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\uff0c\u5b83\u4eec\u90fd\u6709\u76f8\u5e94\u7684\u5269\u4f59\u5b50\u95ee\u9898\u3002\u636e\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u6784\u5efa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u800c\u5f53 \\(s[i] = t[j]\\) \u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\u3002</li> <li>\u5728\u7f16\u8f91\u8ddd\u79bb\u4e2d\uff0c\u72b6\u6001\u4f9d\u8d56\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u6b63\u5e8f\u6216\u5012\u5e8f\u904d\u5386\u90fd\u65e0\u6cd5\u6b63\u786e\u5730\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u53d8\u91cf\u6682\u5b58\u5de6\u4e0a\u65b9\u72b6\u6001\uff0c\u4ece\u800c\u8f6c\u5316\u5230\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7b49\u4ef7\u7684\u60c5\u51b5\uff0c\u53ef\u4ee5\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","title":"14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898","text":"<p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6c42\u89e3\u53e6\u4e00\u4e2a\u5e38\u89c1\u7684\u80cc\u5305\u95ee\u9898\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u4e86\u89e3\u5b83\u7684\u4e00\u79cd\u7279\u4f8b\uff1a\u96f6\u94b1\u5151\u6362\u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 14-22 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\uff0c\u533a\u522b\u4ec5\u5728\u4e8e\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9009\u62e9\u6b21\u6570\u3002</p> <ul> <li>\u5728 0-1 \u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u53ea\u6709\u4e00\u4e2a\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u53ea\u80fd\u4ece\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> <li>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u7684\u6570\u91cf\u662f\u65e0\u9650\u7684\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u4ecd\u53ef\u4ee5\u4ece\u524d \\(i\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> </ul> <p>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u89c4\u5b9a\u4e0b\uff0c\u72b6\u6001 \\([i, c]\\) \u7684\u53d8\u5316\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4ece\u800c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u53d8\u4e3a\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5bf9\u6bd4\u4e24\u9053\u9898\u76ee\u7684\u4ee3\u7801\uff0c\u72b6\u6001\u8f6c\u79fb\u4e2d\u6709\u4e00\u5904\u4ece \\(i-1\\) \u53d8\u4e3a \\(i\\) \uff0c\u5176\u4f59\u5b8c\u5168\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n\"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(wgt)\n# \u521d\u59cb\u5316 dp \u8868\ndp = [[0] * (cap + 1) for _ in range(n + 1)]\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\nfor c in range(1, cap + 1):\nif wgt[i - 1] &gt; c:\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c]\nelse:\n# \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\nreturn dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\nint n = wgt.size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nint[][] dp = new int[n + 1][cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\nint n = wgt.Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[,] dp = new int[n + 1, cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i, c] = dp[i - 1, c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\nn := len(wgt)\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([][]int, n+1)\nfor i := 0; i &lt;= n; i++ {\ndp[i] = make([]int, cap+1)\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor i := 1; i &lt;= n; i++ {\nfor c := 1; c &lt;= cap; c++ {\nif wgt[i-1] &gt; c {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i-1][c]\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n}\n}\n}\nreturn dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\nlet n = wgt.count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n// \u72b6\u6001\u8f6c\u79fb\nfor i in stride(from: 1, through: n, by: 1) {\nfor c in stride(from: 1, through: cap, by: 1) {\nif wgt[i - 1] &gt; c {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c]\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n}\n}\n}\nreturn dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: cap + 1 }, () =&gt; 0)\n);\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = Math.max(\ndp[i - 1][c],\ndp[i][c - wgt[i - 1]] + val[i - 1]\n);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(\nwgt: Array&lt;number&gt;,\nval: Array&lt;number&gt;,\ncap: number\n): number {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: cap + 1 }, () =&gt; 0)\n);\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = Math.max(\ndp[i - 1][c],\ndp[i][c - wgt[i - 1]] + val[i - 1]\n);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\nlet n = wgt.len();\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![vec![0; cap + 1]; n + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor i in 1..=n {\nfor c in 1..=cap {\nif wgt[i - 1] &gt; c as i32 {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\nint n = wgtSize;\n// \u521d\u59cb\u5316 dp \u8868\nint **dp = malloc((n + 1) * sizeof(int *));\nfor (int i = 0; i &lt;= n; i++) {\ndp[i] = calloc(cap + 1, sizeof(int));\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nint res = dp[n][cap];\n// \u91ca\u653e\u5185\u5b58\nfor (int i = 0; i &lt;= n; i++) {\nfree(dp[i]);\n}\nreturn res;\n}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\ncomptime var n = wgt.len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n// \u72b6\u6001\u8f6c\u79fb\nfor (1..n + 1) |i| {\nfor (1..cap + 1) |c| {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[i][c] = dp[i - 1][c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n}\n}\n}\nreturn dp[n][cap];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u662f\u4ece\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u72b6\u6001\u8f6c\u79fb\u800c\u6765\u7684\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u5e94\u8be5\u5bf9 \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</p> <p>\u8fd9\u4e2a\u904d\u5386\u987a\u5e8f\u4e0e 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8bf7\u501f\u52a9\u56fe 14-23 \u6765\u7406\u89e3\u4e24\u8005\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4ee3\u7801\u5b9e\u73b0\u6bd4\u8f83\u7b80\u5355\uff0c\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4\u5220\u9664\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n\"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(wgt)\n# \u521d\u59cb\u5316 dp \u8868\ndp = [0] * (cap + 1)\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\n# \u6b63\u5e8f\u904d\u5386\nfor c in range(1, cap + 1):\nif wgt[i - 1] &gt; c:\n# \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c]\nelse:\n# \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\nreturn dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\nint n = wgt.size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;int&gt; dp(cap + 1, 0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\nint n = wgt.Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\nn := len(wgt)\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([]int, cap+1)\n// \u72b6\u6001\u8f6c\u79fb\nfor i := 1; i &lt;= n; i++ {\nfor c := 1; c &lt;= cap; c++ {\nif wgt[i-1] &gt; c {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c]\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n}\n}\n}\nreturn dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\nlet n = wgt.count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: 0, count: cap + 1)\n// \u72b6\u6001\u8f6c\u79fb\nfor i in stride(from: 1, through: n, by: 1) {\nfor c in stride(from: 1, through: cap, by: 1) {\nif wgt[i - 1] &gt; c {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c]\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n}\n}\n}\nreturn dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(\nwgt: Array&lt;number&gt;,\nval: Array&lt;number&gt;,\ncap: number\n): number {\nconst n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\nint n = wgt.length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;int&gt; dp = List.filled(cap + 1, 0);\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\nlet n = wgt.len();\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![0; cap + 1];\n// \u72b6\u6001\u8f6c\u79fb\nfor i in 1..=n {\nfor c in 1..=cap {\nif wgt[i - 1] &gt; c as i32 {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n}\n}\n}\ndp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\nint n = wgtSize;\n// \u521d\u59cb\u5316 dp \u8868\nint *dp = calloc(cap + 1, sizeof(int));\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int c = 1; c &lt;= cap; c++) {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n}\n}\n}\nint res = dp[cap];\n// \u91ca\u653e\u5185\u5b58\nfree(dp);\nreturn res;\n}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\ncomptime var n = wgt.len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_]i32{0} ** (cap + 1);\n// \u72b6\u6001\u8f6c\u79fb\nfor (1..n + 1) |i| {\nfor (1..cap + 1) |c| {\nif (wgt[i - 1] &gt; c) {\n// \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\ndp[c] = dp[c];\n} else {\n// \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\ndp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n}\n}\n}\nreturn dp[cap];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u5927\u7c7b\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4ee3\u8868\uff0c\u5176\u62e5\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 14-24 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-24 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u96f6\u94b1\u5151\u6362\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\uff0c\u4e24\u8005\u5177\u6709\u4ee5\u4e0b\u8054\u7cfb\u4e0e\u4e0d\u540c\u70b9\u3002</p> <ul> <li>\u4e24\u9053\u9898\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u201c\u7269\u54c1\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5bf9\u5e94\u201c\u76ee\u6807\u91d1\u989d\u201d\u3002</li> <li>\u4f18\u5316\u76ee\u6807\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u4ef7\u503c\uff0c\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e01\u6570\u91cf\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u94b1\u5151\u6362\u662f\u6c42\u201c\u6070\u597d\u201d\u51d1\u5230\u76ee\u6807\u91d1\u989d\u7684\u89e3\u3002</li> </ul> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u72b6\u6001 \\([i, a]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\uff0c\u8bb0\u4e3a \\(dp[i, a]\\) \u3002</p> <p>\u4e8c\u7ef4 \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt+1)\\) \u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u672c\u9898\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u4e24\u70b9\u5dee\u5f02\u3002</p> <ul> <li>\u672c\u9898\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c06\u8fd0\u7b97\u7b26 \\(\\max()\\) \u66f4\u6539\u4e3a \\(\\min()\\) \u3002</li> <li>\u4f18\u5316\u4e3b\u4f53\u662f\u786c\u5e01\u6570\u91cf\u800c\u975e\u5546\u54c1\u4ef7\u503c\uff0c\u56e0\u6b64\u5728\u9009\u4e2d\u786c\u5e01\u65f6\u6267\u884c \\(+1\\) \u5373\u53ef\u3002</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u51d1\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u610f \\(&gt; 0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u5373\u662f\u65e0\u6548\u89e3\u3002\u4e3a\u4f7f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u6570\u80fd\u591f\u8bc6\u522b\u5e76\u8fc7\u6ee4\u65e0\u6548\u89e3\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528 \\(+ \\infty\\) \u6765\u8868\u793a\u5b83\u4eec\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(+ \\infty\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u5e76\u672a\u63d0\u4f9b \\(+ \\infty\\) \u53d8\u91cf\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b <code>int</code> \u7684\u6700\u5927\u503c\u6765\u4ee3\u66ff\u3002\u800c\u8fd9\u53c8\u4f1a\u5bfc\u81f4\u5927\u6570\u8d8a\u754c\uff1a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u53d1\u751f\u6ea2\u51fa\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u6570\u5b57 \\(amt + 1\\) \u6765\u8868\u793a\u65e0\u6548\u89e3\uff0c\u56e0\u4e3a\u51d1\u51fa \\(amt\\) \u7684\u786c\u5e01\u6570\u91cf\u6700\u591a\u4e3a \\(amt\\) \u3002\u6700\u540e\u8fd4\u56de\u524d\uff0c\u5224\u65ad \\(dp[n, amt]\\) \u662f\u5426\u7b49\u4e8e \\(amt + 1\\) \uff0c\u82e5\u662f\u5219\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n\"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(coins)\nMAX = amt + 1\n# \u521d\u59cb\u5316 dp \u8868\ndp = [[0] * (amt + 1) for _ in range(n + 1)]\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor a in range(1, amt + 1):\ndp[0][a] = MAX\n# \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in range(1, n + 1):\nfor a in range(1, amt + 1):\nif coins[i - 1] &gt; a:\n# \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a]\nelse:\n# \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\nreturn dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\nint n = coins.size();\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int a = 1; a &lt;= amt; a++) {\ndp[0][a] = MAX;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int[] coins, int amt) {\nint n = coins.length;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nint[][] dp = new int[n + 1][amt + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int a = 1; a &lt;= amt; a++) {\ndp[0][a] = MAX;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDP(int[] coins, int amt) {\nint n = coins.Length;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nint[,] dp = new int[n + 1, amt + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int a = 1; a &lt;= amt; a++) {\ndp[0, a] = MAX;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i, a] = dp[i - 1, a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins []int, amt int) int {\nn := len(coins)\nmax := amt + 1\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([][]int, n+1)\nfor i := 0; i &lt;= n; i++ {\ndp[i] = make([]int, amt+1)\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor a := 1; a &lt;= amt; a++ {\ndp[0][a] = max\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i := 1; i &lt;= n; i++ {\nfor a := 1; a &lt;= amt; a++ {\nif coins[i-1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i-1][a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n}\n}\n}\nif dp[n][amt] != max {\nreturn dp[n][amt]\n}\nreturn -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\nlet n = coins.count\nlet MAX = amt + 1\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor a in stride(from: 1, through: amt, by: 1) {\ndp[0][a] = MAX\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in stride(from: 1, through: n, by: 1) {\nfor a in stride(from: 1, through: amt, by: 1) {\nif coins[i - 1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n}\n}\n}\nreturn dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins, amt) {\nconst n = coins.length;\nconst MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: amt + 1 }, () =&gt; 0)\n);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (let a = 1; a &lt;= amt; a++) {\ndp[0][a] = MAX;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\nconst n = coins.length;\nconst MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: amt + 1 }, () =&gt; 0)\n);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (let a = 1; a &lt;= amt; a++) {\ndp[0][a] = MAX;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\nint n = coins.length;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int a = 1; a &lt;= amt; a++) {\ndp[0][a] = MAX;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\nlet n = coins.len();\nlet max = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![vec![0; amt + 1]; n + 1];\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor a in 1..= amt {\ndp[0][a] = max;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i in 1..=n {\nfor a in 1..=amt {\nif coins[i - 1] &gt; a as i32 {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n}\n}\n}\nif dp[n][amt] != max { return dp[n][amt] as i32; } else { -1 }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\nint n = coinsSize;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nint **dp = malloc((n + 1) * sizeof(int *));\nfor (int i = 0; i &lt;= n; i++) {\ndp[i] = calloc(amt + 1, sizeof(int));\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (int a = 1; a &lt;= amt; a++) {\ndp[0][a] = MAX;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n}\n}\n}\nint res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n// \u91ca\u653e\u5185\u5b58\nfor (int i = 0; i &lt;= n; i++) {\nfree(dp[i]);\n}\nfree(dp);\nreturn res;\n}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\ncomptime var n = coins.len;\ncomptime var max = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\nfor (1..amt + 1) |a| {\ndp[0][a] = max;\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor (1..n + 1) |i| {\nfor (1..amt + 1) |a| {\nif (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n}\n}\n}\nif (dp[n][amt] != max) {\nreturn @intCast(dp[n][amt]);\n} else {\nreturn -1;\n}\n}\n</code></pre> <p>\u56fe 14-25 \u5c55\u793a\u4e86\u96f6\u94b1\u5151\u6362\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-25 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u96f6\u94b1\u5151\u6362\u7684\u7a7a\u95f4\u4f18\u5316\u7684\u5904\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n\"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(coins)\nMAX = amt + 1\n# \u521d\u59cb\u5316 dp \u8868\ndp = [MAX] * (amt + 1)\ndp[0] = 0\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\n# \u6b63\u5e8f\u904d\u5386\nfor a in range(1, amt + 1):\nif coins[i - 1] &gt; a:\n# \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a]\nelse:\n# \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\nreturn dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\nint n = coins.size();\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;int&gt; dp(amt + 1, MAX);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int[] coins, int amt) {\nint n = coins.length;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[amt + 1];\nArrays.fill(dp, MAX);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDPComp(int[] coins, int amt) {\nint n = coins.Length;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[amt + 1];\nArray.Fill(dp, MAX);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins []int, amt int) int {\nn := len(coins)\nmax := amt + 1\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([]int, amt+1)\nfor i := 1; i &lt;= amt; i++ {\ndp[i] = max\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor i := 1; i &lt;= n; i++ {\n// \u5012\u5e8f\u904d\u5386\nfor a := 1; a &lt;= amt; a++ {\nif coins[i-1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n}\n}\n}\nif dp[amt] != max {\nreturn dp[amt]\n}\nreturn -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\nlet n = coins.count\nlet MAX = amt + 1\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: MAX, count: amt + 1)\ndp[0] = 0\n// \u72b6\u6001\u8f6c\u79fb\nfor i in stride(from: 1, through: n, by: 1) {\nfor a in stride(from: 1, through: amt, by: 1) {\nif coins[i - 1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n}\n}\n}\nreturn dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins, amt) {\nconst n = coins.length;\nconst MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\nconst n = coins.length;\nconst MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\nint n = coins.length;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;int&gt; dp = List.filled(amt + 1, MAX);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n}\n}\n}\nreturn dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\nlet n = coins.len();\nlet max = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![0; amt + 1];\ndp.fill(max);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor i in 1..=n {\nfor a in 1..=amt {\nif coins[i - 1] &gt; a as i32 {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n}\n}\n}\nif dp[amt] != max { return dp[amt] as i32; } else { -1 }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\nint n = coinsSize;\nint MAX = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nint *dp = calloc(amt + 1, sizeof(int));\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n}\n}\n}\nint res = dp[amt] != MAX ? dp[amt] : -1;\n// \u91ca\u653e\u5185\u5b58\nfree(dp);\nreturn res;\n}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\ncomptime var n = coins.len;\ncomptime var max = amt + 1;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_]i32{0} ** (amt + 1);\n@memset(&amp;dp, max);\ndp[0] = 0;\n// \u72b6\u6001\u8f6c\u79fb\nfor (1..n + 1) |i| {\nfor (1..amt + 1) |a| {\nif (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n}\n}\n}\nif (dp[amt] != max) {\nreturn @intCast(dp[amt]);\n} else {\nreturn -1;\n}\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u3002\u793a\u4f8b\u5982\u56fe 14-26 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-26 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u76f8\u6bd4\u4e8e\u4e0a\u4e00\u9898\uff0c\u672c\u9898\u76ee\u6807\u662f\u6c42\u7ec4\u5408\u6570\u91cf\uff0c\u56e0\u6b64\u5b50\u95ee\u9898\u53d8\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u7ec4\u5408\u6570\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7ef4\u77e9\u9635\u3002</p> <p>\u5f53\u524d\u72b6\u6001\u7684\u7ec4\u5408\u6570\u91cf\u7b49\u4e8e\u4e0d\u9009\u5f53\u524d\u786c\u5e01\u4e0e\u9009\u5f53\u524d\u786c\u5e01\u8fd9\u4e24\u79cd\u51b3\u7b56\u7684\u7ec4\u5408\u6570\u91cf\u4e4b\u548c\u3002\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u65e0\u987b\u9009\u62e9\u4efb\u4f55\u786c\u5e01\u5373\u53ef\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u5e94\u5c06\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u4e3a \\(1\\) \u3002\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u4f55 \\(&gt;0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n\"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(coins)\n# \u521d\u59cb\u5316 dp \u8868\ndp = [[0] * (amt + 1) for _ in range(n + 1)]\n# \u521d\u59cb\u5316\u9996\u5217\nfor i in range(n + 1):\ndp[i][0] = 1\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\nfor a in range(1, amt + 1):\nif coins[i - 1] &gt; a:\n# \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a]\nelse:\n# \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\nreturn dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\nint n = coins.size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n// \u521d\u59cb\u5316\u9996\u5217\nfor (int i = 0; i &lt;= n; i++) {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n}\n}\n}\nreturn dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int[] coins, int amt) {\nint n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nint[][] dp = new int[n + 1][amt + 1];\n// \u521d\u59cb\u5316\u9996\u5217\nfor (int i = 0; i &lt;= n; i++) {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n}\n}\n}\nreturn dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDP(int[] coins, int amt) {\nint n = coins.Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[,] dp = new int[n + 1, amt + 1];\n// \u521d\u59cb\u5316\u9996\u5217\nfor (int i = 0; i &lt;= n; i++) {\ndp[i, 0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i, a] = dp[i - 1, a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n}\n}\n}\nreturn dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins []int, amt int) int {\nn := len(coins)\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([][]int, n+1)\nfor i := 0; i &lt;= n; i++ {\ndp[i] = make([]int, amt+1)\n}\n// \u521d\u59cb\u5316\u9996\u5217\nfor i := 0; i &lt;= n; i++ {\ndp[i][0] = 1\n}\n// \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\nfor i := 1; i &lt;= n; i++ {\nfor a := 1; a &lt;= amt; a++ {\nif coins[i-1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i-1][a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n}\n}\n}\nreturn dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\nlet n = coins.count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n// \u521d\u59cb\u5316\u9996\u5217\nfor i in stride(from: 0, through: n, by: 1) {\ndp[i][0] = 1\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor i in stride(from: 1, through: n, by: 1) {\nfor a in stride(from: 1, through: amt, by: 1) {\nif coins[i - 1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n}\n}\n}\nreturn dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins, amt) {\nconst n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: amt + 1 }, () =&gt; 0)\n);\n// \u521d\u59cb\u5316\u9996\u5217\nfor (let i = 0; i &lt;= n; i++) {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n}\n}\n}\nreturn dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\nconst n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: n + 1 }, () =&gt;\nArray.from({ length: amt + 1 }, () =&gt; 0)\n);\n// \u521d\u59cb\u5316\u9996\u5217\nfor (let i = 0; i &lt;= n; i++) {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n}\n}\n}\nreturn dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\nint n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n// \u521d\u59cb\u5316\u9996\u5217\nfor (int i = 0; i &lt;= n; i++) {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n}\n}\n}\nreturn dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\nlet n = coins.len();\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![vec![0; amt + 1]; n + 1];\n// \u521d\u59cb\u5316\u9996\u5217\nfor i in 0..= n {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor i in 1..=n {\nfor a in 1..=amt {\nif coins[i - 1] &gt; a as i32 {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n}\n}\n}\ndp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\nint n = coinsSize;\n// \u521d\u59cb\u5316 dp \u8868\nint **dp = malloc((n + 1) * sizeof(int *));\nfor (int i = 0; i &lt;= n; i++) {\ndp[i] = calloc(amt + 1, sizeof(int));\n}\n// \u521d\u59cb\u5316\u9996\u5217\nfor (int i = 0; i &lt;= n; i++) {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n}\n}\n}\nint res = dp[n][amt];\n// \u91ca\u653e\u5185\u5b58\nfor (int i = 0; i &lt;= n; i++) {\nfree(dp[i]);\n}\nfree(dp);\nreturn res;\n}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\ncomptime var n = coins.len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n// \u521d\u59cb\u5316\u9996\u5217\nfor (0..n + 1) |i| {\ndp[i][0] = 1;\n}\n// \u72b6\u6001\u8f6c\u79fb\nfor (1..n + 1) |i| {\nfor (1..amt + 1) |a| {\nif (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[i][a] = dp[i - 1][a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n}\n}\n}\nreturn dp[n][amt];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7a7a\u95f4\u4f18\u5316\u5904\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u5220\u9664\u786c\u5e01\u7ef4\u5ea6\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n\"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\nn = len(coins)\n# \u521d\u59cb\u5316 dp \u8868\ndp = [0] * (amt + 1)\ndp[0] = 1\n# \u72b6\u6001\u8f6c\u79fb\nfor i in range(1, n + 1):\n# \u6b63\u5e8f\u904d\u5386\nfor a in range(1, amt + 1):\nif coins[i - 1] &gt; a:\n# \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a]\nelse:\n# \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]]\nreturn dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\nint n = coins.size();\n// \u521d\u59cb\u5316 dp \u8868\nvector&lt;int&gt; dp(amt + 1, 0);\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]];\n}\n}\n}\nreturn dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int[] coins, int amt) {\nint n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[amt + 1];\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]];\n}\n}\n}\nreturn dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\nint n = coins.Length;\n// \u521d\u59cb\u5316 dp \u8868\nint[] dp = new int[amt + 1];\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]];\n}\n}\n}\nreturn dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\nn := len(coins)\n// \u521d\u59cb\u5316 dp \u8868\ndp := make([]int, amt+1)\ndp[0] = 1\n// \u72b6\u6001\u8f6c\u79fb\nfor i := 1; i &lt;= n; i++ {\n// \u5012\u5e8f\u904d\u5386\nfor a := 1; a &lt;= amt; a++ {\nif coins[i-1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a-coins[i-1]]\n}\n}\n}\nreturn dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\nlet n = coins.count\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = Array(repeating: 0, count: amt + 1)\ndp[0] = 1\n// \u72b6\u6001\u8f6c\u79fb\nfor i in stride(from: 1, through: n, by: 1) {\nfor a in stride(from: 1, through: amt, by: 1) {\nif coins[i - 1] &gt; a {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a]\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]]\n}\n}\n}\nreturn dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins, amt) {\nconst n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: amt + 1 }, () =&gt; 0);\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]];\n}\n}\n}\nreturn dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\nconst n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nconst dp = Array.from({ length: amt + 1 }, () =&gt; 0);\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (let i = 1; i &lt;= n; i++) {\nfor (let a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]];\n}\n}\n}\nreturn dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\nint n = coins.length;\n// \u521d\u59cb\u5316 dp \u8868\nList&lt;int&gt; dp = List.filled(amt + 1, 0);\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]];\n}\n}\n}\nreturn dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\nlet n = coins.len();\n// \u521d\u59cb\u5316 dp \u8868\nlet mut dp = vec![0; amt + 1];\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor i in 1..=n {\nfor a in 1..=amt {\nif coins[i - 1] &gt; a as i32 {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n}\n}\n}\ndp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\nint n = coinsSize;\n// \u521d\u59cb\u5316 dp \u8868\nint *dp = calloc(amt + 1, sizeof(int));\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (int i = 1; i &lt;= n; i++) {\nfor (int a = 1; a &lt;= amt; a++) {\nif (coins[i - 1] &gt; a) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\ndp[a] = dp[a] + dp[a - coins[i - 1]];\n}\n}\n}\nint res = dp[amt];\n// \u91ca\u653e\u5185\u5b58\nfree(dp);\nreturn res;\n}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\ncomptime var n = coins.len;\n// \u521d\u59cb\u5316 dp \u8868\nvar dp = [_]i32{0} ** (amt + 1);\ndp[0] = 1;\n// \u72b6\u6001\u8f6c\u79fb\nfor (1..n + 1) |i| {\nfor (1..amt + 1) |a| {\nif (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n// \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\ndp[a] = dp[a];\n} else {\n// \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\ndp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n}\n}\n}\nreturn dp[amt];\n}\n</code></pre>"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u56fe","text":"<p>Abstract</p> <p>\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u4eec\u5c31\u50cf\u662f\u4e00\u4e2a\u4e2a\u8282\u70b9\uff0c\u88ab\u65e0\u6570\u770b\u4e0d\u89c1\u7684\u8fb9\u76f8\u8fde\u3002</p> <p>\u6bcf\u4e00\u6b21\u7684\u76f8\u8bc6\u4e0e\u76f8\u79bb\uff0c\u90fd\u5728\u8fd9\u5f20\u5de8\u5927\u7684\u7f51\u7edc\u56fe\u4e2d\u7559\u4e0b\u72ec\u7279\u7684\u5370\u8bb0\u3002</p>"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>9.1 \u00a0 \u56fe</li> <li>9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c</li> <li>9.3 \u00a0 \u56fe\u7684\u904d\u5386</li> <li>9.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 \u56fe","text":"<p>\u300c\u56fe graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 vertex\u300d\u548c\u300c\u8fb9 edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>\u5982\u679c\u5c06\u9876\u70b9\u770b\u4f5c\u8282\u70b9\uff0c\u5c06\u8fb9\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u56fe\u770b\u4f5c\u4e00\u79cd\u4ece\u94fe\u8868\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u5982\u56fe 9-1 \u6240\u793a\uff0c\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</p> <p></p> <p> \u56fe 9-1 \u00a0 \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb </p>"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b\u4e0e\u672f\u8bed","text":"<p>\u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u300c\u65e0\u5411\u56fe undirected graph\u300d\u548c\u300c\u6709\u5411\u56fe directed graph\u300d\uff0c\u5982\u56fe 9-2 \u6240\u793a\u3002</p> <ul> <li>\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\u3002</li> <li>\u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\u3002</li> </ul> <p></p> <p> \u56fe 9-2 \u00a0 \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe </p> <p>\u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u300c\u8fde\u901a\u56fe connected graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe disconnected graph\u300d\uff0c\u5982\u56fe 9-3 \u6240\u793a\u3002</p> <ul> <li>\u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\u3002</li> <li>\u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002</li> </ul> <p></p> <p> \u56fe 9-3 \u00a0 \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe </p> <p>\u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u5982\u56fe 9-4 \u6240\u793a\u7684\u300c\u6709\u6743\u56fe weighted graph\u300d\u3002\u4f8b\u5982\u5728\u201c\u738b\u8005\u8363\u8000\u201d\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002</p> <p></p> <p> \u56fe 9-4 \u00a0 \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe </p> <p>\u56fe\u6570\u636e\u7ed3\u6784\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u300c\u90bb\u63a5 adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002</li> <li>\u300c\u8def\u5f84 path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002</li> <li>\u300c\u5ea6 degree\u300d\uff1a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 in-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 out-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002</li> </ul>"},{"location":"chapter_graph/graph/#912","title":"9.1.2 \u00a0 \u56fe\u7684\u8868\u793a","text":"<p>\u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002</p>"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u90bb\u63a5\u77e9\u9635","text":"<p>\u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 adjacency matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002</p> <p>\u5982\u56fe 9-5 \u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\)\u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002</p> <p></p> <p> \u56fe 9-5 \u00a0 \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a </p> <p>\u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002</li> <li>\u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002</li> <li>\u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) \u548c \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002</li> </ul> <p>\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002</p>"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u90bb\u63a5\u8868","text":"<p>\u300c\u90bb\u63a5\u8868 adjacency list\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002\u56fe 9-6 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f7f\u7528\u90bb\u63a5\u8868\u5b58\u50a8\u7684\u56fe\u7684\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 9-6 \u00a0 \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a </p> <p>\u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002</p> <p>\u89c2\u5bdf\u56fe 9-6 \uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u201c\u94fe\u5f0f\u5730\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u8fd8\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002</p>"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"<p>\u5982\u8868 9-1 \u6240\u793a\uff0c\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002</p> <p> \u8868 9-1 \u00a0 \u73b0\u5b9e\u751f\u6d3b\u4e2d\u5e38\u89c1\u7684\u56fe </p> \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"<p>\u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u201c\u8fb9\u201d\u7684\u64cd\u4f5c\u548c\u5bf9\u201c\u9876\u70b9\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002</p>"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u5f0f\u5982\u56fe 9-7 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 <code>vertices</code> \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 <code>adjMat</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-7 \u00a0 \u90bb\u63a5\u77e9\u9635\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n\"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\ndef __init__(self, vertices: list[int], edges: list[list[int]]):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n# \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nself.vertices: list[int] = []\n# \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nself.adj_mat: list[list[int]] = []\n# \u6dfb\u52a0\u9876\u70b9\nfor val in vertices:\nself.add_vertex(val)\n# \u6dfb\u52a0\u8fb9\n# \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges:\nself.add_edge(e[0], e[1])\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\nreturn len(self.vertices)\ndef add_vertex(self, val: int):\n\"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\nn = self.size()\n# \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nself.vertices.append(val)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnew_row = [0] * n\nself.adj_mat.append(new_row)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor row in self.adj_mat:\nrow.append(0)\ndef remove_vertex(self, index: int):\n\"\"\"\u5220\u9664\u9876\u70b9\"\"\"\nif index &gt;= self.size():\nraise IndexError()\n# \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nself.vertices.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nself.adj_mat.pop(index)\n# \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor row in self.adj_mat:\nrow.pop(index)\ndef add_edge(self, i: int, j: int):\n\"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\nraise IndexError()\n# \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nself.adj_mat[i][j] = 1\nself.adj_mat[j][i] = 1\ndef remove_edge(self, i: int, j: int):\n\"\"\"\u5220\u9664\u8fb9\"\"\"\n# \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n# \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\nraise IndexError()\nself.adj_mat[i][j] = 0\nself.adj_mat[j][i] = 0\ndef print(self):\n\"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\nprint(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\nprint(\"\u90bb\u63a5\u77e9\u9635 =\")\nprint_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvector&lt;int&gt; vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvector&lt;vector&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const vector&lt;int&gt; &amp;edge : edges) {\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() const {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.push_back(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nadjMat.emplace_back(vector&lt;int&gt;(n, 0));\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (vector&lt;int&gt; &amp;row : adjMat) {\nrow.push_back(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index &gt;= size()) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.erase(vertices.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.erase(adjMat.begin() + index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (vector&lt;int&gt; &amp;row : adjMat) {\nrow.erase(row.begin() + index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\nthrow out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid print() {\ncout &lt;&lt; \"\u9876\u70b9\u5217\u8868 = \";\nprintVector(vertices);\ncout &lt;&lt; \"\u90bb\u63a5\u77e9\u9635 =\" &lt;&lt; endl;\nprintVectorMatrix(adjMat);\n}\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList&lt;Integer&gt; vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList&lt;List&lt;Integer&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = new ArrayList&lt;&gt;();\nthis.adjMat = new ArrayList&lt;&gt;();\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val : vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (int[] e : edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn vertices.size();\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\nfor (int j = 0; j &lt; n; j++) {\nnewRow.add(0);\n}\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List&lt;Integer&gt; row : adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(int index) {\nif (index &gt;= size())\nthrow new IndexOutOfBoundsException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List&lt;Integer&gt; row : adjMat) {\nrow.remove(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\nthrow new IndexOutOfBoundsException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat.get(i).set(j, 1);\nadjMat.get(j).set(i, 1);\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\nthrow new IndexOutOfBoundsException();\nadjMat.get(i).set(j, 0);\nadjMat.get(j).set(i, 0);\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void print() {\nSystem.out.print(\"\u9876\u70b9\u5217\u8868 = \");\nSystem.out.println(vertices);\nSystem.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.printMatrix(adjMat);\n}\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList&lt;int&gt; vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList&lt;List&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjMat(int[] vertices, int[][] edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nforeach (int val in vertices) {\nAddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nforeach (int[] e in edges) {\nAddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint Size() {\nreturn vertices.Count;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void AddVertex(int val) {\nint n = Size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.Add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList&lt;int&gt; newRow = new(n);\nfor (int j = 0; j &lt; n; j++) {\nnewRow.Add(0);\n}\nadjMat.Add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nforeach (List&lt;int&gt; row in adjMat) {\nrow.Add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void RemoveVertex(int index) {\nif (index &gt;= Size())\nthrow new IndexOutOfRangeException();\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.RemoveAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nforeach (List&lt;int&gt; row in adjMat) {\nrow.RemoveAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void AddEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\nthrow new IndexOutOfRangeException();\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npublic void RemoveEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\nthrow new IndexOutOfRangeException();\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npublic void Print() {\nConsole.Write(\"\u9876\u70b9\u5217\u8868 = \");\nPrintUtil.PrintList(vertices);\nConsole.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\nPrintUtil.PrintMatrix(adjMat);\n}\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n// \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nvertices []int\n// \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat [][]int\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n// \u6dfb\u52a0\u9876\u70b9\nn := len(vertices)\nadjMat := make([][]int, n)\nfor i := range adjMat {\nadjMat[i] = make([]int, n)\n}\n// \u521d\u59cb\u5316\u56fe\ng := &amp;graphAdjMat{\nvertices: vertices,\nadjMat:   adjMat,\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor i := range edges {\ng.addEdge(edges[i][0], edges[i][1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\nreturn len(g.vertices)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\nn := g.size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\ng.vertices = append(g.vertices, val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nnewRow := make([]int, n)\ng.adjMat = append(g.adjMat, newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i], 0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\nif index &gt;= g.size() {\nreturn\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\ng.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\ng.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i := range g.adjMat {\ng.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\ng.adjMat[i][j] = 1\ng.adjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\nfmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n}\ng.adjMat[i][j] = 0\ng.adjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\nfmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\nfmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\nfor i := range g.adjMat {\nfmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n}\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nprivate var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nprivate var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(vertices: [Int], edges: [[Int]]) {\nself.vertices = []\nadjMat = []\n// \u6dfb\u52a0\u9876\u70b9\nfor val in vertices {\naddVertex(val: val)\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor e in edges {\naddEdge(i: e[0], j: e[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc size() -&gt; Int {\nvertices.count\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc addVertex(val: Int) {\nlet n = size()\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.append(val)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nlet newRow = Array(repeating: 0, count: n)\nadjMat.append(newRow)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor i in adjMat.indices {\nadjMat[i].append(0)\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc removeVertex(index: Int) {\nif index &gt;= size() {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.remove(at: index)\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor i in adjMat.indices {\nadjMat[i].remove(at: index)\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc addEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1\nadjMat[j][i] = 1\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc removeEdge(i: Int, j: Int) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\nfatalError(\"\u8d8a\u754c\")\n}\nadjMat[i][j] = 0\nadjMat[j][i] = 0\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc print() {\nSwift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\nSwift.print(vertices)\nSwift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\nPrintUtil.printMatrix(matrix: adjMat)\n}\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices, edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val) {\nconst n = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow = [];\nfor (let j = 0; j &lt; n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index) {\nif (index &gt;= this.size()) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i, j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint() {\nconsole.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\nconsole.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n}\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nvertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nadjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u51fd\u6570 */\nconstructor(vertices: number[], edges: number[][]) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (const val of vertices) {\nthis.addVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (const e of edges) {\nthis.addEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(val: number): void {\nconst n: number = this.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nthis.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nconst newRow: number[] = [];\nfor (let j: number = 0; j &lt; n; j++) {\nnewRow.push(0);\n}\nthis.adjMat.push(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (const row of this.adjMat) {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(index: number): void {\nif (index &gt;= this.size()) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nthis.vertices.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nthis.adjMat.splice(index, 1);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (const row of this.adjMat) {\nrow.splice(index, 1);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\naddEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\nthis.adjMat[i][j] = 1;\nthis.adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nremoveEdge(i: number, j: number): void {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\nthrow new RangeError('Index Out Of Bounds Exception');\n}\nthis.adjMat[i][j] = 0;\nthis.adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nprint(): void {\nconsole.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\nconsole.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n}\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\nList&lt;int&gt; vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\nList&lt;List&lt;int&gt;&gt; adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\nthis.vertices = [];\nthis.adjMat = [];\n// \u6dfb\u52a0\u9876\u70b9\nfor (int val in vertices) {\naddVertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor (List&lt;int&gt; e in edges) {\naddEdge(e[0], e[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn vertices.length;\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(int val) {\nint n = size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nvertices.add(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nList&lt;int&gt; newRow = List.filled(n, 0, growable: true);\nadjMat.add(newRow);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor (List&lt;int&gt; row in adjMat) {\nrow.add(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(int index) {\nif (index &gt;= size()) {\nthrow IndexError;\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nvertices.removeAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nadjMat.removeAt(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (List&lt;int&gt; row in adjMat) {\nrow.removeAt(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\nthrow IndexError;\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nadjMat[i][j] = 1;\nadjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(int i, int j) {\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\nthrow IndexError;\n}\nadjMat[i][j] = 0;\nadjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printAdjMat() {\nprint(\"\u9876\u70b9\u5217\u8868 = $vertices\");\nprint(\"\u90bb\u63a5\u77e9\u9635 = \");\nprintMatrix(adjMat);\n}\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n// \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\npub vertices: Vec&lt;i32&gt;,\n// \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\npub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\nimpl GraphAdjMat {\n/* \u6784\u9020\u65b9\u6cd5 */\npub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\nlet mut graph = GraphAdjMat {\nvertices: vec![],\nadj_mat: vec![],\n};\n// \u6dfb\u52a0\u9876\u70b9\nfor val in vertices {\ngraph.add_vertex(val);\n}\n// \u6dfb\u52a0\u8fb9\n// \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfor edge in edges {\ngraph.add_edge(edge[0], edge[1])\n}\ngraph\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npub fn size(&amp;self) -&gt; usize {\nself.vertices.len()\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npub fn add_vertex(&amp;mut self, val: i32) {\nlet n = self.size();\n// \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\nself.vertices.push(val);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\nself.adj_mat.push(vec![0; n]);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\nfor row in &amp;mut self.adj_mat {\nrow.push(0);\n}\n}\n/* \u5220\u9664\u9876\u70b9 */\npub fn remove_vertex(&amp;mut self, index: usize) {\nif index &gt;= self.size() {\npanic!(\"index error\")\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nself.vertices.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nself.adj_mat.remove(index);\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor row in &amp;mut self.adj_mat {\nrow.remove(index);\n}\n}\n/* \u6dfb\u52a0\u8fb9 */\npub fn add_edge(&amp;mut self, i: usize, j: usize) {\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &gt;= self.size() || j &gt;= self.size() || i == j {\npanic!(\"index error\")\n}\n// \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\nself.adj_mat[i][j] = 1;\nself.adj_mat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\npub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n// \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\nif i &gt;= self.size() || j &gt;= self.size() || i == j {\npanic!(\"index error\")\n}\nself.adj_mat[i][j] = 0;\nself.adj_mat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\npub fn print(&amp;self) {\nprintln!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\nprintln!(\"\u90bb\u63a5\u77e9\u9635 =\");\nprintln!(\"[\");\nfor row in &amp;self.adj_mat {\nprintln!(\"  {:?},\", row);\n}\nprintln!(\"]\")\n}\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\nint vertices[MAX_SIZE];\nint adjMat[MAX_SIZE][MAX_SIZE];\nint size;\n} GraphAdjMat;\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\nGraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\ngraph-&gt;size = 0;\nfor (int i = 0; i &lt; MAX_SIZE; i++) {\nfor (int j = 0; j &lt; MAX_SIZE; j++) {\ngraph-&gt;adjMat[i][j] = 0;\n}\n}\nreturn graph;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\nfree(graph);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\nif (graph-&gt;size == MAX_SIZE) {\nfprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\nreturn;\n}\n// \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\nint n = graph-&gt;size;\ngraph-&gt;vertices[n] = val;\nfor (int i = 0; i &lt;= n; i++) {\ngraph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n}\ngraph-&gt;size++;\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\nif (index &lt; 0 || index &gt;= graph-&gt;size) {\nfprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\nreturn;\n}\n// \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\nfor (int i = index; i &lt; graph-&gt;size - 1; i++) {\ngraph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n}\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\nfor (int i = index; i &lt; graph-&gt;size - 1; i++) {\nfor (int j = 0; j &lt; graph-&gt;size; j++) {\ngraph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n}\n}\n// \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\nfor (int i = 0; i &lt; graph-&gt;size; i++) {\nfor (int j = index; j &lt; graph-&gt;size - 1; j++) {\ngraph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n}\n}\ngraph-&gt;size--;\n}\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\nif (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\nfprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\nreturn;\n}\ngraph-&gt;adjMat[i][j] = 1;\ngraph-&gt;adjMat[j][i] = 1;\n}\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\nif (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\nfprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\nreturn;\n}\ngraph-&gt;adjMat[i][j] = 0;\ngraph-&gt;adjMat[j][i] = 0;\n}\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\nprintf(\"\u9876\u70b9\u5217\u8868 = \");\nprintArray(graph-&gt;vertices, graph-&gt;size);\nprintf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\nfor (int i = 0; i &lt; graph-&gt;size; i++) {\nprintArray(graph-&gt;adjMat[i], graph-&gt;size);\n}\n}\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre>"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"<p>\u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\)\u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u53ef\u6839\u636e\u56fe 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5b9e\u73b0\u5404\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-8 \u00a0 \u90bb\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u90bb\u63a5\u8868\u7684\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u6bd4\u56fe 9-8 \uff0c\u5b9e\u9645\u4ee3\u7801\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002</p> <ul> <li>\u4e3a\u4e86\u65b9\u4fbf\u6dfb\u52a0\u4e0e\u5220\u9664\u9876\u70b9\uff0c\u4ee5\u53ca\u7b80\u5316\u4ee3\u7801\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u6765\u4ee3\u66ff\u94fe\u8868\u3002</li> <li>\u4f7f\u7528\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u90bb\u63a5\u8868\uff0c<code>key</code> \u4e3a\u9876\u70b9\u5b9e\u4f8b\uff0c<code>value</code> \u4e3a\u8be5\u9876\u70b9\u7684\u90bb\u63a5\u9876\u70b9\u5217\u8868\uff08\u94fe\u8868\uff09\u3002</li> </ul> <p>\u53e6\u5916\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 <code>Vertex</code> \u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u4e0e\u90bb\u63a5\u77e9\u9635\u4e00\u6837\uff0c\u7528\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u5047\u8bbe\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u6240\u6709\u5927\u4e8e \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684 <code>Vertex</code> \u5b9e\u4f8b\uff0c\u5220\u9664\u67d0\u4e00\u9876\u70b9\u4e4b\u540e\u5c31\u65e0\u987b\u6539\u52a8\u5176\u4ed6\u9876\u70b9\u4e86\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n\"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\ndef __init__(self, edges: list[list[Vertex]]):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n# \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nself.adj_list = dict[Vertex, list[Vertex]]()\n# \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges:\nself.add_vertex(edge[0])\nself.add_vertex(edge[1])\nself.add_edge(edge[0], edge[1])\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\nreturn len(self.adj_list)\ndef add_edge(self, vet1: Vertex, vet2: Vertex):\n\"\"\"\u6dfb\u52a0\u8fb9\"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError()\n# \u6dfb\u52a0\u8fb9 vet1 - vet2\nself.adj_list[vet1].append(vet2)\nself.adj_list[vet2].append(vet1)\ndef remove_edge(self, vet1: Vertex, vet2: Vertex):\n\"\"\"\u5220\u9664\u8fb9\"\"\"\nif vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\nraise ValueError()\n# \u5220\u9664\u8fb9 vet1 - vet2\nself.adj_list[vet1].remove(vet2)\nself.adj_list[vet2].remove(vet1)\ndef add_vertex(self, vet: Vertex):\n\"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\nif vet in self.adj_list:\nreturn\n# \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nself.adj_list[vet] = []\ndef remove_vertex(self, vet: Vertex):\n\"\"\"\u5220\u9664\u9876\u70b9\"\"\"\nif vet not in self.adj_list:\nraise ValueError()\n# \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nself.adj_list.pop(vet)\n# \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor vertex in self.adj_list:\nif vet in self.adj_list[vertex]:\nself.adj_list[vertex].remove(vet)\ndef print(self):\n\"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\nprint(\"\u90bb\u63a5\u8868 =\")\nfor vertex in self.adj_list:\ntmp = [v.val for v in self.adj_list[vertex]]\nprint(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\npublic:\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nunordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n/* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\nvoid remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\nfor (int i = 0; i &lt; vec.size(); i++) {\nif (vec[i] == vet) {\nvec.erase(vec.begin() + i);\nbreak;\n}\n}\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const vector&lt;Vertex *&gt; &amp;edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex *vet1, Vertex *vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].push_back(vet2);\nadjList[vet2].push_back(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex *vet1, Vertex *vet2) {\nif (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5220\u9664\u8fb9 vet1 - vet2\nremove(adjList[vet1], vet2);\nremove(adjList[vet2], vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex *vet) {\nif (adjList.count(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = vector&lt;Vertex *&gt;();\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex *vet) {\nif (!adjList.count(vet))\nthrow invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.erase(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (auto &amp;adj : adjList) {\nremove(adj.second, vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid print() {\ncout &lt;&lt; \"\u90bb\u63a5\u8868 =\" &lt;&lt; endl;\nfor (auto &amp;adj : adjList) {\nconst auto &amp;key = adj.first;\nconst auto &amp;vec = adj.second;\ncout &lt;&lt; key-&gt;val &lt;&lt; \": \";\nprintVector(vetsToVals(vec));\n}\n}\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n/* \u6784\u9020\u65b9\u6cd5 */\npublic GraphAdjList(Vertex[][] edges) {\nthis.adjList = new HashMap&lt;&gt;();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (Vertex[] edge : edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic int size() {\nreturn adjList.size();\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList.get(vet1).add(vet2);\nadjList.get(vet2).add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\nthrow new IllegalArgumentException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList.get(vet1).remove(vet2);\nadjList.get(vet2).remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void addVertex(Vertex vet) {\nif (adjList.containsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.put(vet, new ArrayList&lt;&gt;());\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet))\nthrow new IllegalArgumentException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (List&lt;Vertex&gt; list : adjList.values()) {\nlist.remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void print() {\nSystem.out.println(\"\u90bb\u63a5\u8868 =\");\nfor (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\nList&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\nfor (Vertex vertex : pair.getValue())\ntmp.add(vertex.val);\nSystem.out.println(pair.getKey().val + \": \" + tmp + \",\");\n}\n}\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n/* \u6784\u9020\u51fd\u6570 */\npublic GraphAdjList(Vertex[][] edges) {\nadjList = [];\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nforeach (Vertex[] edge in edges) {\nAddVertex(edge[0]);\nAddVertex(edge[1]);\nAddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint Size() {\nreturn adjList.Count;\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic void AddEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1].Add(vet2);\nadjList[vet2].Add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\npublic void RemoveEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\nthrow new InvalidOperationException();\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1].Remove(vet2);\nadjList[vet2].Remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic void AddVertex(Vertex vet) {\nif (adjList.ContainsKey(vet))\nreturn;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList.Add(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic void RemoveVertex(Vertex vet) {\nif (!adjList.ContainsKey(vet))\nthrow new InvalidOperationException();\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.Remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nforeach (List&lt;Vertex&gt; list in adjList.Values) {\nlist.Remove(vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic void Print() {\nConsole.WriteLine(\"\u90bb\u63a5\u8868 =\");\nforeach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\nList&lt;int&gt; tmp = [];\nforeach (Vertex vertex in pair.Value)\ntmp.Add(vertex.val);\nConsole.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n}\n}\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList map[Vertex][]Vertex\n}\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\ng := &amp;graphAdjList{\nadjList: make(map[Vertex][]Vertex),\n}\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor _, edge := range edges {\ng.addVertex(edge[0])\ng.addVertex(edge[1])\ng.addEdge(edge[0], edge[1])\n}\nreturn g\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\nreturn len(g.adjList)\n}\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\ng.adjList[vet1] = append(g.adjList[vet1], vet2)\ng.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n_, ok1 := g.adjList[vet1]\n_, ok2 := g.adjList[vet2]\nif !ok1 || !ok2 || vet1 == vet2 {\npanic(\"error\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\ng.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\ng.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif ok {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\ng.adjList[vet] = make([]Vertex, 0)\n}\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n_, ok := g.adjList[vet]\nif !ok {\npanic(\"error\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\ndelete(g.adjList, vet)\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor v, list := range g.adjList {\ng.adjList[v] = DeleteSliceElms(list, vet)\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\nvar builder strings.Builder\nfmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\nfor k, v := range g.adjList {\nbuilder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\nfor _, vet := range v {\nbuilder.WriteString(strconv.Itoa(vet.Val) + \" \")\n}\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npublic private(set) var adjList: [Vertex: [Vertex]]\n/* \u6784\u9020\u65b9\u6cd5 */\npublic init(edges: [[Vertex]]) {\nadjList = [:]\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges {\naddVertex(vet: edge[0])\naddVertex(vet: edge[1])\naddEdge(vet1: edge[0], vet2: edge[1])\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\npublic func size() -&gt; Int {\nadjList.count\n}\n/* \u6dfb\u52a0\u8fb9 */\npublic func addEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]?.append(vet2)\nadjList[vet2]?.append(vet1)\n}\n/* \u5220\u9664\u8fb9 */\npublic func removeEdge(vet1: Vertex, vet2: Vertex) {\nif adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]?.removeAll(where: { $0 == vet2 })\nadjList[vet2]?.removeAll(where: { $0 == vet1 })\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npublic func addVertex(vet: Vertex) {\nif adjList[vet] != nil {\nreturn\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = []\n}\n/* \u5220\u9664\u9876\u70b9 */\npublic func removeVertex(vet: Vertex) {\nif adjList[vet] == nil {\nfatalError(\"\u53c2\u6570\u9519\u8bef\")\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.removeValue(forKey: vet)\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor key in adjList.keys {\nadjList[key]?.removeAll(where: { $0 == vet })\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npublic func print() {\nSwift.print(\"\u90bb\u63a5\u8868 =\")\nfor pair in adjList {\nvar tmp: [Int] = []\nfor vertex in pair.value {\ntmp.append(vertex.val)\n}\nSwift.print(\"\\(pair.key.val): \\(tmp),\")\n}\n}\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize() {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1, vet2) {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1, vet2) {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet) {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet) {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (const set of this.adjList.values()) {\nconst index = set.indexOf(vet);\nif (index &gt; -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint() {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nadjList: Map&lt;Vertex, Vertex[]&gt;;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(edges: Vertex[][]) {\nthis.adjList = new Map();\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor (const edge of edges) {\nthis.addVertex(edge[0]);\nthis.addVertex(edge[1]);\nthis.addEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.adjList.size;\n}\n/* \u6dfb\u52a0\u8fb9 */\naddEdge(vet1: Vertex, vet2: Vertex): void {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).push(vet2);\nthis.adjList.get(vet2).push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nremoveEdge(vet1: Vertex, vet2: Vertex): void {\nif (\n!this.adjList.has(vet1) ||\n!this.adjList.has(vet2) ||\nvet1 === vet2\n) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nthis.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\nthis.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\naddVertex(vet: Vertex): void {\nif (this.adjList.has(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nthis.adjList.set(vet, []);\n}\n/* \u5220\u9664\u9876\u70b9 */\nremoveVertex(vet: Vertex): void {\nif (!this.adjList.has(vet)) {\nthrow new Error('Illegal Argument Exception');\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nthis.adjList.delete(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (const set of this.adjList.values()) {\nconst index: number = set.indexOf(vet);\nif (index &gt; -1) {\nset.splice(index, 1);\n}\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nprint(): void {\nconsole.log('\u90bb\u63a5\u8868 =');\nfor (const [key, value] of this.adjList.entries()) {\nconst tmp = [];\nfor (const vertex of value) {\ntmp.push(vertex.val);\n}\nconsole.log(key.val + ': ' + tmp.join());\n}\n}\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nMap&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n/* \u6784\u9020\u65b9\u6cd5 */\nGraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\nfor (List&lt;Vertex&gt; edge in edges) {\naddVertex(edge[0]);\naddVertex(edge[1]);\naddEdge(edge[0], edge[1]);\n}\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nint size() {\nreturn adjList.length;\n}\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) ||\n!adjList.containsKey(vet2) ||\nvet1 == vet2) {\nthrow ArgumentError;\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nadjList[vet1]!.add(vet2);\nadjList[vet2]!.add(vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(Vertex vet1, Vertex vet2) {\nif (!adjList.containsKey(vet1) ||\n!adjList.containsKey(vet2) ||\nvet1 == vet2) {\nthrow ArgumentError;\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nadjList[vet1]!.remove(vet2);\nadjList[vet2]!.remove(vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(Vertex vet) {\nif (adjList.containsKey(vet)) return;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nadjList[vet] = [];\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(Vertex vet) {\nif (!adjList.containsKey(vet)) {\nthrow ArgumentError;\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nadjList.remove(vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nadjList.forEach((key, value) {\nvalue.remove(vet);\n});\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nvoid printAdjList() {\nprint(\"\u90bb\u63a5\u8868 =\");\nadjList.forEach((key, value) {\nList&lt;int&gt; tmp = [];\nfor (Vertex vertex in value) {\ntmp.add(vertex.val);\n}\nprint(\"${key.val}: $tmp,\");\n});\n}\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n// \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\npub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\nimpl GraphAdjList {\n/* \u6784\u9020\u65b9\u6cd5 */\npub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\nlet mut graph = GraphAdjList {\nadj_list: HashMap::new(),\n};\n// \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\nfor edge in edges {\ngraph.add_vertex(edge[0]);\ngraph.add_vertex(edge[1]);\ngraph.add_edge(edge[0], edge[1]);\n}\ngraph\n}\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n#[allow(unused)]\npub fn size(&amp;self) -&gt; usize {\nself.adj_list.len()\n}\n/* \u6dfb\u52a0\u8fb9 */\npub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\nif !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n{\npanic!(\"value error\");\n}\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\nself.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\nself.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n}\n/* \u5220\u9664\u8fb9 */\n#[allow(unused)]\npub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\nif !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n{\npanic!(\"value error\");\n}\n// \u5220\u9664\u8fb9 vet1 - vet2\nself.adj_list\n.get_mut(&amp;vet1)\n.unwrap()\n.retain(|&amp;vet| vet != vet2);\nself.adj_list\n.get_mut(&amp;vet2)\n.unwrap()\n.retain(|&amp;vet| vet != vet1);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\npub fn add_vertex(&amp;mut self, vet: Vertex) {\nif self.adj_list.contains_key(&amp;vet) {\nreturn;\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\nself.adj_list.insert(vet, vec![]);\n}\n/* \u5220\u9664\u9876\u70b9 */\n#[allow(unused)]\npub fn remove_vertex(&amp;mut self, vet: Vertex) {\nif !self.adj_list.contains_key(&amp;vet) {\npanic!(\"value error\");\n}\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nself.adj_list.remove(&amp;vet);\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor list in self.adj_list.values_mut() {\nlist.retain(|&amp;v| v != vet);\n}\n}\n/* \u6253\u5370\u90bb\u63a5\u8868 */\npub fn print(&amp;self) {\nprintln!(\"\u90bb\u63a5\u8868 =\");\nfor (vertex, list) in &amp;self.adj_list {\nlet list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\nprintln!(\"{}: {:?},\", vertex.val, list);\n}\n}\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\nVertex *vertex;           // \u9876\u70b9\nstruct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\nfor (int i = 0; i &lt; graph-&gt;size; i++) {\nif (graph-&gt;heads[i]-&gt;vertex == vet) {\nreturn graph-&gt;heads[i];\n}\n}\nreturn NULL;\n}\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\nAdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\nnode-&gt;vertex = vet;\n// \u5934\u63d2\u6cd5\nnode-&gt;next = head-&gt;next;\nhead-&gt;next = node;\n}\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\nAdjListNode *pre = head;\nAdjListNode *cur = head-&gt;next;\n// \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\nwhile (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\npre = cur;\ncur = cur-&gt;next;\n}\nif (cur == NULL)\nreturn;\n// \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\npre-&gt;next = cur-&gt;next;\n// \u91ca\u653e\u5185\u5b58\nfree(cur);\n}\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\nAdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\nint size;                     // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\nGraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\nif (!graph) {\nreturn NULL;\n}\ngraph-&gt;size = 0;\nfor (int i = 0; i &lt; MAX_SIZE; i++) {\ngraph-&gt;heads[i] = NULL;\n}\nreturn graph;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\nfor (int i = 0; i &lt; graph-&gt;size; i++) {\nAdjListNode *cur = graph-&gt;heads[i];\nwhile (cur != NULL) {\nAdjListNode *next = cur-&gt;next;\nif (cur != graph-&gt;heads[i]) {\nfree(cur);\n}\ncur = next;\n}\nfree(graph-&gt;heads[i]-&gt;vertex);\nfree(graph-&gt;heads[i]);\n}\nfree(graph);\n}\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\nfor (int i = 0; i &lt; graph-&gt;size; i++) {\nif (graph-&gt;heads[i]-&gt;vertex == vet) {\nreturn graph-&gt;heads[i];\n}\n}\nreturn NULL;\n}\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\nAdjListNode *head1 = findNode(graph, vet1);\nAdjListNode *head2 = findNode(graph, vet2);\nassert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n// \u6dfb\u52a0\u8fb9 vet1 - vet2\naddEdgeHelper(head1, vet2);\naddEdgeHelper(head2, vet1);\n}\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\nAdjListNode *head1 = findNode(graph, vet1);\nAdjListNode *head2 = findNode(graph, vet2);\nassert(head1 != NULL &amp;&amp; head2 != NULL);\n// \u5220\u9664\u8fb9 vet1 - vet2\nremoveEdgeHelper(head1, head2-&gt;vertex);\nremoveEdgeHelper(head2, head1-&gt;vertex);\n}\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\nassert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\nAdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\nhead-&gt;vertex = vet;\nhead-&gt;next = NULL;\n// \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\ngraph-&gt;heads[graph-&gt;size++] = head;\n}\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\nAdjListNode *node = findNode(graph, vet);\nassert(node != NULL);\n// \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\nAdjListNode *cur = node, *pre = NULL;\nwhile (cur) {\npre = cur;\ncur = cur-&gt;next;\nfree(pre);\n}\n// \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\nfor (int i = 0; i &lt; graph-&gt;size; i++) {\ncur = graph-&gt;heads[i];\npre = NULL;\nwhile (cur) {\npre = cur;\ncur = cur-&gt;next;\nif (cur &amp;&amp; cur-&gt;vertex == vet) {\npre-&gt;next = cur-&gt;next;\nfree(cur);\nbreak;\n}\n}\n}\n// \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\nint i;\nfor (i = 0; i &lt; graph-&gt;size; i++) {\nif (graph-&gt;heads[i] == node)\nbreak;\n}\nfor (int j = i; j &lt; graph-&gt;size - 1; j++) {\ngraph-&gt;heads[j] = graph-&gt;heads[j + 1];\n}\ngraph-&gt;size--;\nfree(vet);\n}\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre>"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"<p>\u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u8868 9-2 \u5bf9\u6bd4\u4e86\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u3002</p> <p> \u8868 9-2 \u00a0 \u90bb\u63a5\u77e9\u9635\u4e0e\u90bb\u63a5\u8868\u5bf9\u6bd4 </p> \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>\u89c2\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u56fe\u7684\u904d\u5386","text":"<p>\u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\u3002</p> <p>\u56fe\u548c\u6811\u90fd\u9700\u8981\u5e94\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal\u300d\u3002\u5b83\u4eec\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search\u300d\uff0c\u7b80\u79f0 BFS \u548c DFS \u3002</p>"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u67d0\u4e2a\u8282\u70b9\u51fa\u53d1\uff0c\u59cb\u7ec8\u4f18\u5148\u8bbf\u95ee\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5982\u56fe 9-9 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u9996\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002</p> <p></p> <p> \u56fe 9-9 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>BFS \u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002</p> <ol> <li>\u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 <code>startVet</code> \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\u3002</li> <li>\u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\u3002</li> <li>\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6bd5\u540e\u7ed3\u675f\u3002</li> </ol> <p>\u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n\"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS\"\"\"\n# \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set[Vertex]([start_vet])\n# \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nque = deque[Vertex]([start_vet])\n# \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile len(que) &gt; 0:\nvet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adj_vet in graph.adj_list[vet]:\nif adj_vet in visited:\ncontinue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\nque.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector&lt;Vertex *&gt; res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set&lt;Vertex *&gt; visited = {startVet};\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nqueue&lt;Vertex *&gt; que;\nque.push(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.empty()) {\nVertex *vet = que.front();\nque.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (auto adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\nque.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\nque.offer(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!que.isEmpty()) {\nVertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\nque.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList&lt;Vertex&gt; res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nHashSet&lt;Vertex&gt; visited = [startVet];\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue&lt;Vertex&gt; que = new();\nque.Enqueue(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.Count &gt; 0) {\nVertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nforeach (Vertex adjVet in graph.adjList[vet]) {\nif (visited.Contains(adjVet)) {\ncontinue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\nque.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\nvisited[startVet] = struct{}{}\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\nqueue := make([]Vertex, 0)\nqueue = append(queue, startVet)\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nfor len(queue) &gt; 0 {\n// \u961f\u9996\u9876\u70b9\u51fa\u961f\nvet := queue[0]\nqueue = queue[1:]\n// \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nres = append(res, vet)\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nif !isExist {\nqueue = append(queue, adjVet)\nvisited[adjVet] = struct{}{}\n}\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set&lt;Vertex&gt; = [startVet]\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nvar que: [Vertex] = [startVet]\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile !que.isEmpty {\nlet vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\nque.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set&lt;Vertex&gt; = new Set();\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nconst que = [startVet];\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.length) {\nconst vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet) ?? []) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\nque.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList&lt;Vertex&gt; res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet&lt;Vertex&gt; visited = {};\nvisited.add(startVet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue&lt;Vertex&gt; que = Queue();\nque.add(startVet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (que.isNotEmpty) {\nVertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet in graph.adjList[vet]!) {\nif (visited.contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\nque.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nlet mut res = vec![];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nlet mut visited = HashSet::new();\nvisited.insert(start_vet);\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nlet mut que = VecDeque::new();\nque.push_back(start_vet);\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile !que.is_empty() {\nlet vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nif let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\nfor &amp;adj_vet in adj_vets {\nif visited.contains(&amp;adj_vet) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\nque.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\n}\n}\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nres\n}\n</code></pre> graph_bfs.c<pre><code>/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\nVertex *vertices[MAX_SIZE];\nint front, rear, size;\n} Queue;\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\nQueue *q = (Queue *)malloc(sizeof(Queue));\nq-&gt;front = q-&gt;rear = q-&gt;size = 0;\nreturn q;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\nreturn q-&gt;size == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\nq-&gt;vertices[q-&gt;rear] = vet;\nq-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\nq-&gt;size++;\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\nVertex *vet = q-&gt;vertices[q-&gt;front];\nq-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\nq-&gt;size--;\nreturn vet;\n}\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n// \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\nfor (int i = 0; i &lt; size; i++) {\nif (visited[i] == vet)\nreturn 1;\n}\nreturn 0;\n}\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n// \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\nQueue *queue = newQueue();\nenqueue(queue, startVet);\nvisited[(*visitedSize)++] = startVet;\n// \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\nwhile (!isEmpty(queue)) {\nVertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\nres[(*resSize)++] = vet;      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nAdjListNode *node = findNode(graph, vet);\nwhile (node != NULL) {\n// \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\nif (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\nenqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\nvisited[(*visitedSize)++] = node-&gt;vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n}\nnode = node-&gt;next;\n}\n}\n// \u91ca\u653e\u5185\u5b58\nfree(queue);\n}\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> <p>\u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u56fe 9-10 \u6765\u52a0\u6df1\u7406\u89e3\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-10 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u56fe 9-10 \u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\)\u3001\\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\uff0c\u9876\u70b9 \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \uff0c\u961f\u5217 <code>que</code> \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5982\u56fe 9-11 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002</p> <p></p> <p> \u56fe 9-11 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>\u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934\u518d\u8fd4\u56de\u201d\u7684\u7b97\u6cd5\u8303\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u5728\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u4e2d\uff0c\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n\"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570\"\"\"\nres.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n# \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adj_list[vet]:\nif adjVet in visited:\ncontinue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n# \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet)\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n\"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS\"\"\"\n# \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n# \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres = []\n# \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited = set[Vertex]()\ndfs(graph, visited, res, start_vet)\nreturn res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\nres.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex *adjVet : graph.adjList[vet]) {\nif (visited.count(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvector&lt;Vertex *&gt; res;\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nunordered_set&lt;Vertex *&gt; visited;\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\nres.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet : graph.adjList.get(vet)) {\nif (visited.contains(adjVet))\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\nres.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nforeach (Vertex adjVet in graph.adjList[vet]) {\nif (visited.Contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9                             \n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\nDFS(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList&lt;Vertex&gt; res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nHashSet&lt;Vertex&gt; visited = [];\nDFS(graph, visited, res, startVet);\nreturn res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n// append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n*res = append(*res, vet)\nvisited[vet] = struct{}{}\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor _, adjVet := range g.adjList[vet] {\n_, isExist := visited[adjVet]\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\nif !isExist {\ndfs(g, visited, res, adjVet)\n}\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nres := make([]Vertex, 0)\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvisited := make(map[Vertex]struct{})\ndfs(g, visited, &amp;res, startVet)\n// \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\nreturn res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\nres.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor adjVet in graph.adjList[vet] ?? [] {\nif visited.contains(adjVet) {\ncontinue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nvar res: [Vertex] = []\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nvar visited: Set&lt;Vertex&gt; = []\ndfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\nreturn res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\ngraph: GraphAdjList,\nvisited: Set&lt;Vertex&gt;,\nres: Vertex[],\nvet: Vertex\n): void {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (const adjVet of graph.adjList.get(vet)) {\nif (visited.has(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nconst res: Vertex[] = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nconst visited: Set&lt;Vertex&gt; = new Set();\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\nGraphAdjList graph,\nSet&lt;Vertex&gt; visited,\nList&lt;Vertex&gt; res,\nVertex vet,\n) {\nres.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfor (Vertex adjVet in graph.adjList[vet]!) {\nif (visited.contains(adjVet)) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adjVet);\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nList&lt;Vertex&gt; res = [];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nSet&lt;Vertex&gt; visited = {};\ndfs(graph, visited, res, startVet);\nreturn res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\nres.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nvisited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nif let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\nfor &amp;adj_vet in adj_vets {\nif visited.contains(&amp;adj_vet) {\ncontinue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n}\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, visited, res, adj_vet);\n}\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n// \u9876\u70b9\u904d\u5386\u5e8f\u5217\nlet mut res = vec![];\n// \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\nlet mut visited = HashSet::new();\ndfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\nres\n}\n</code></pre> graph_dfs.c<pre><code>/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n// \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\nfor (int i = 0; i &lt; size; i++) {\nif (res[i] == vet) {\nreturn 1;\n}\n}\nreturn 0;\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n// \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\nres[(*resSize)++] = vet;\n// \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nAdjListNode *node = findNode(graph, vet);\nwhile (node != NULL) {\n// \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\nif (!isVisited(res, *resSize, node-&gt;vertex)) {\n// \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\ndfs(graph, res, resSize, node-&gt;vertex);\n}\nnode = node-&gt;next;\n}\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\ndfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n[class]{}-[func]{graphDFS}\n</code></pre> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 9-12 \u6240\u793a\u3002</p> <ul> <li>\u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\u3002</li> <li>\u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002</li> </ul> <p>\u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe 9-12 \u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\u6a21\u62df\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-12 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p> <p>\u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002</li> <li>\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</li> <li>\u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002</li> <li>\u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002</li> <li>\u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> <li>\u4ece\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</li> <li>\u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002</li> <li>\u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002</li> <li>\u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002</li> <li>\u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f</p> <p>\u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. \u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u89c6\u4e3a\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002</p> <p>\u975e\u8fde\u901a\u56fe\u4e2d\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f</p> <p>\u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002</p> <p>\u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f</p> <p>\u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\uff0c\u8fd9\u6837\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002</p>"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8d2a\u5fc3","text":"<p>Abstract</p> <p>\u5411\u65e5\u8475\u671d\u7740\u592a\u9633\u8f6c\u52a8\uff0c\u65f6\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u957f\u7684\u6700\u5927\u53ef\u80fd\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u5728\u4e00\u8f6e\u8f6e\u7684\u7b80\u5355\u9009\u62e9\u4e2d\uff0c\u9010\u6b65\u5bfc\u5411\u6700\u4f73\u7b54\u6848\u3002</p>"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5</li> <li>15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898</li> <li>15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898</li> <li>15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898</li> <li>15.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_greedy/fractional_knapsack_problem/","title":"15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u4ef7\u503c\u6839\u636e\u9009\u62e9\u7684\u91cd\u91cf\u6bd4\u4f8b\u8ba1\u7b97\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 15-3 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-3 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5206\u6570\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u6574\u4f53\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72b6\u6001\u5305\u542b\u5f53\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6807\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u4e0d\u540c\u70b9\u5728\u4e8e\uff0c\u672c\u9898\u5141\u8bb8\u53ea\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u56fe 15-4 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u7269\u54c1\u4efb\u610f\u5730\u8fdb\u884c\u5207\u5206\uff0c\u5e76\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u6765\u8ba1\u7b97\u76f8\u5e94\u4ef7\u503c\u3002</p> <ol> <li>\u5bf9\u4e8e\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c\u4e3a \\(val[i-1] / wgt[i-1]\\) \uff0c\u7b80\u79f0\u5355\u4f4d\u4ef7\u503c\u3002</li> <li>\u5047\u8bbe\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u4e3a \\(w\\) \uff0c\u5219\u80cc\u5305\u589e\u52a0\u7684\u4ef7\u503c\u4e3a \\(w \\times val[i-1] / wgt[i-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 15-4 \u00a0 \u7269\u54c1\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6700\u5927\u5316\u80cc\u5305\u5185\u7269\u54c1\u603b\u4ef7\u503c\uff0c\u672c\u8d28\u4e0a\u662f\u6700\u5927\u5316\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u4ef7\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u56fe 15-5 \u6240\u793a\u7684\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\u3002</li> <li>\u904d\u5386\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f6e\u8d2a\u5fc3\u5730\u9009\u62e9\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002</li> <li>\u82e5\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u4f7f\u7528\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6ee1\u80cc\u5305\u3002</li> </ol> <p></p> <p> \u56fe 15-5 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565 </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5efa\u7acb\u4e86\u4e00\u4e2a\u7269\u54c1\u7c7b <code>Item</code> \uff0c\u4ee5\u4fbf\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u8fdb\u884c\u6392\u5e8f\u3002\u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u5f53\u80cc\u5305\u5df2\u6ee1\u65f6\u8df3\u51fa\u5e76\u8fd4\u56de\u89e3\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig fractional_knapsack.py<pre><code>class Item:\n\"\"\"\u7269\u54c1\"\"\"\ndef __init__(self, w: int, v: int):\nself.w = w  # \u7269\u54c1\u91cd\u91cf\nself.v = v  # \u7269\u54c1\u4ef7\u503c\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n\"\"\"\u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3\"\"\"\n# \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nitems = [Item(w, v) for w, v in zip(wgt, val)]\n# \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nitems.sort(key=lambda item: item.v / item.w, reverse=True)\n# \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\nres = 0\nfor item in items:\nif item.w &lt;= cap:\n# \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v\ncap -= item.w\nelse:\n# \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += (item.v / item.w) * cap\n# \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak\nreturn res\n</code></pre> fractional_knapsack.cpp<pre><code>/* \u7269\u54c1 */\nclass Item {\npublic:\nint w; // \u7269\u54c1\u91cd\u91cf\nint v; // \u7269\u54c1\u4ef7\u503c\nItem(int w, int v) : w(w), v(v) {\n}\n};\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nvector&lt;Item&gt; items;\nfor (int i = 0; i &lt; wgt.size(); i++) {\nitems.push_back(Item(wgt[i], val[i]));\n}\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nsort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\ndouble res = 0;\nfor (auto &amp;item : items) {\nif (item.w &lt;= cap) {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v;\ncap -= item.w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += (double)item.v / item.w * cap;\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\nreturn res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* \u7269\u54c1 */\nclass Item {\nint w; // \u7269\u54c1\u91cd\u91cf\nint v; // \u7269\u54c1\u4ef7\u503c\npublic Item(int w, int v) {\nthis.w = w;\nthis.v = v;\n}\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nItem[] items = new Item[wgt.length];\nfor (int i = 0; i &lt; wgt.length; i++) {\nitems[i] = new Item(wgt[i], val[i]);\n}\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nArrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\ndouble res = 0;\nfor (Item item : items) {\nif (item.w &lt;= cap) {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v;\ncap -= item.w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += (double) item.v / item.w * cap;\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\nreturn res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* \u7269\u54c1 */\nclass Item(int w, int v) {\npublic int w = w; // \u7269\u54c1\u91cd\u91cf\npublic int v = v; // \u7269\u54c1\u4ef7\u503c\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nItem[] items = new Item[wgt.Length];\nfor (int i = 0; i &lt; wgt.Length; i++) {\nitems[i] = new Item(wgt[i], val[i]);\n}\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nArray.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\ndouble res = 0;\nforeach (Item item in items) {\nif (item.w &lt;= cap) {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v;\ncap -= item.w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += (double)item.v / item.w * cap;\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\nreturn res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* \u7269\u54c1 */\ntype Item struct {\nw int // \u7269\u54c1\u91cd\u91cf\nv int // \u7269\u54c1\u4ef7\u503c\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nitems := make([]Item, len(wgt))\nfor i := 0; i &lt; len(wgt); i++ {\nitems[i] = Item{wgt[i], val[i]}\n}\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nsort.Slice(items, func(i, j int) bool {\nreturn float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n})\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\nres := 0.0\nfor _, item := range items {\nif item.w &lt;= cap {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += float64(item.v)\ncap -= item.w\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += float64(item.v) / float64(item.w) * float64(cap)\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak\n}\n}\nreturn res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* \u7269\u54c1 */\nclass Item {\nvar w: Int // \u7269\u54c1\u91cd\u91cf\nvar v: Int // \u7269\u54c1\u4ef7\u503c\ninit(w: Int, v: Int) {\nself.w = w\nself.v = v\n}\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nvar items = zip(wgt, val).map { Item(w: $0, v: $1) }\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nitems.sort(by: { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) })\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\nvar res = 0.0\nvar cap = cap\nfor item in items {\nif item.w &lt;= cap {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += Double(item.v)\ncap -= item.w\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += Double(item.v) / Double(item.w) * Double(cap)\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak\n}\n}\nreturn res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* \u7269\u54c1 */\nclass Item {\nconstructor(w, v) {\nthis.w = w; // \u7269\u54c1\u91cd\u91cf\nthis.v = v; // \u7269\u54c1\u4ef7\u503c\n}\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt, val, cap) {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nconst items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nitems.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\nlet res = 0;\nfor (const item of items) {\nif (item.w &lt;= cap) {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v;\ncap -= item.w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += (item.v / item.w) * cap;\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\nreturn res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* \u7269\u54c1 */\nclass Item {\nw: number; // \u7269\u54c1\u91cd\u91cf\nv: number; // \u7269\u54c1\u4ef7\u503c\nconstructor(w: number, v: number) {\nthis.w = w;\nthis.v = v;\n}\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nconst items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nitems.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\nlet res = 0;\nfor (const item of items) {\nif (item.w &lt;= cap) {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v;\ncap -= item.w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += (item.v / item.w) * cap;\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\nreturn res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* \u7269\u54c1 */\nclass Item {\nint w; // \u7269\u54c1\u91cd\u91cf\nint v; // \u7269\u54c1\u4ef7\u503c\nItem(this.w, this.v);\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nList&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nitems.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\ndouble res = 0;\nfor (Item item in items) {\nif (item.w &lt;= cap) {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v;\ncap -= item.w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += item.v / item.w * cap;\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\nreturn res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* \u7269\u54c1 */\nstruct Item {\nw: i32, // \u7269\u54c1\u91cd\u91cf\nv: i32, // \u7269\u54c1\u4ef7\u503c\n}\nimpl Item {\nfn new(w: i32, v: i32) -&gt; Self {\nSelf { w, v }\n}\n}\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nlet mut items = wgt\n.iter()\n.zip(val.iter())\n.map(|(&amp;w, &amp;v)| Item::new(w, v))\n.collect::&lt;Vec&lt;Item&gt;&gt;();\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nitems.sort_by(|a, b| {\n(b.v as f64 / b.w as f64)\n.partial_cmp(&amp;(a.v as f64 / a.w as f64))\n.unwrap()\n});\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\nlet mut res = 0.0;\nfor item in &amp;items {\nif item.w &lt;= cap {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += item.v as f64;\ncap -= item.w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += item.v as f64 / item.w as f64 * cap as f64;\n// \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\nres\n}\n</code></pre> fractional_knapsack.c<pre><code>/* \u7269\u54c1 */\ntypedef struct {\nint w; // \u7269\u54c1\u91cd\u91cf\nint v; // \u7269\u54c1\u4ef7\u503c\n} Item;\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n// \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\nItem *items = malloc(sizeof(Item) * itemCount);\nfor (int i = 0; i &lt; itemCount; i++) {\nitems[i] = (Item){.w = wgt[i], .v = val[i]};\n}\n// \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\nqsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\nfloat res = 0.0;\nfor (int i = 0; i &lt; itemCount; i++) {\nif (items[i].w &lt;= cap) {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\nres += items[i].v;\ncap -= items[i].w;\n} else {\n// \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\nres += (float)cap / items[i].w * items[i].v;\ncap = 0;\nbreak;\n}\n}\nfree(items);\nreturn res;\n}\n</code></pre> fractional_knapsack.zig<pre><code>[class]{Item}-[func]{}\n[class]{}-[func]{fractionalKnapsack}\n</code></pre> <p>\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u9700\u8981\u904d\u5386\u6574\u4e2a\u7269\u54c1\u5217\u8868\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7269\u54c1\u6570\u91cf\u3002</p> <p>\u7531\u4e8e\u521d\u59cb\u5316\u4e86\u4e00\u4e2a <code>Item</code> \u5bf9\u8c61\u5217\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u91c7\u7528\u53cd\u8bc1\u6cd5\u3002\u5047\u8bbe\u7269\u54c1 \\(x\\) \u662f\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u4ef7\u503c\u4e3a <code>res</code> \uff0c\u4f46\u8be5\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u73b0\u5728\u4ece\u80cc\u5305\u4e2d\u62ff\u51fa\u5355\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u5e76\u66ff\u6362\u4e3a\u5355\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u4e8e\u7269\u54c1 \\(x\\) \u7684\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u6362\u540e\u7684\u603b\u4ef7\u503c\u4e00\u5b9a\u5927\u4e8e <code>res</code> \u3002\u8fd9\u4e0e <code>res</code> \u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u8bf4\u660e\u6700\u4f18\u89e3\u4e2d\u5fc5\u987b\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u5bf9\u4e8e\u8be5\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u6784\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u5355\u4f4d\u4ef7\u503c\u66f4\u5927\u7684\u7269\u54c1\u603b\u662f\u66f4\u4f18\u9009\u62e9\uff0c\u8fd9\u8bf4\u660e\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p> <p>\u5982\u56fe 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c06\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u5355\u4f4d\u4ef7\u503c\u5206\u522b\u770b\u4f5c\u4e00\u5f20\u4e8c\u7ef4\u56fe\u8868\u7684\u6a2a\u8f74\u548c\u7eb5\u8f74\uff0c\u5219\u5206\u6570\u80cc\u5305\u95ee\u9898\u53ef\u8f6c\u5316\u4e3a\u201c\u6c42\u5728\u6709\u9650\u6a2a\u8f74\u533a\u95f4\u4e0b\u7684\u6700\u5927\u56f4\u6210\u9762\u79ef\u201d\u3002\u8fd9\u4e2a\u7c7b\u6bd4\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</p> <p></p> <p> \u56fe 15-6 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u51e0\u4f55\u8868\u793a </p>"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5","text":"<p>\u300c\u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm\u300d\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u95ee\u9898\u7684\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\uff0c\u90fd\u9009\u62e9\u5f53\u524d\u770b\u8d77\u6765\u6700\u4f18\u7684\u9009\u62e9\uff0c\u5373\u8d2a\u5fc3\u5730\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002\u8d2a\u5fc3\u7b97\u6cd5\u7b80\u6d01\u4e14\u9ad8\u6548\uff0c\u5728\u8bb8\u591a\u5b9e\u9645\u95ee\u9898\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002</p> <p>\u8d2a\u5fc3\u7b97\u6cd5\u548c\u52a8\u6001\u89c4\u5212\u90fd\u5e38\u7528\u4e8e\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u3002\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u5904\uff0c\u6bd4\u5982\u90fd\u4f9d\u8d56\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002</p> <ul> <li>\u52a8\u6001\u89c4\u5212\u4f1a\u6839\u636e\u4e4b\u524d\u9636\u6bb5\u7684\u6240\u6709\u51b3\u7b56\u6765\u8003\u8651\u5f53\u524d\u51b3\u7b56\uff0c\u5e76\u4f7f\u7528\u8fc7\u53bb\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6784\u5efa\u5f53\u524d\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4f1a\u8003\u8651\u8fc7\u53bb\u7684\u51b3\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u4e0d\u65ad\u7f29\u5c0f\u95ee\u9898\u8303\u56f4\uff0c\u76f4\u81f3\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> </ul> <p>\u6211\u4eec\u5148\u901a\u8fc7\u4f8b\u9898\u201c\u96f6\u94b1\u5151\u6362\u201d\u4e86\u89e3\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u8fd9\u9053\u9898\u5df2\u7ecf\u5728\u201c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u76f8\u4fe1\u4f60\u5bf9\u5b83\u5e76\u4e0d\u964c\u751f\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u672c\u9898\u91c7\u53d6\u7684\u8d2a\u5fc3\u7b56\u7565\u5982\u56fe 15-1 \u6240\u793a\u3002\u7ed9\u5b9a\u76ee\u6807\u91d1\u989d\uff0c\u6211\u4eec\u8d2a\u5fc3\u5730\u9009\u62e9\u4e0d\u5927\u4e8e\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e01\uff0c\u4e0d\u65ad\u5faa\u73af\u8be5\u6b65\u9aa4\uff0c\u76f4\u81f3\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u4e3a\u6b62\u3002</p> <p></p> <p> \u56fe 15-1 \u00a0 \u96f6\u94b1\u5151\u6362\u7684\u8d2a\u5fc3\u7b56\u7565 </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u4f60\u53ef\u80fd\u4f1a\u4e0d\u7531\u5730\u53d1\u51fa\u611f\u53f9\uff1aSo clean \uff01\u8d2a\u5fc3\u7b97\u6cd5\u4ec5\u7528\u7ea6\u5341\u884c\u4ee3\u7801\u5c31\u89e3\u51b3\u4e86\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n\"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3\"\"\"\n# \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\ni = len(coins) - 1\ncount = 0\n# \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile amt &gt; 0:\n# \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile i &gt; 0 and coins[i] &gt; amt:\ni -= 1\n# \u9009\u62e9 coins[i]\namt -= coins[i]\ncount += 1\n# \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\nint i = coins.size() - 1;\nint count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile (amt &gt; 0) {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\ni--;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount++;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int[] coins, int amt) {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\nint i = coins.length - 1;\nint count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile (amt &gt; 0) {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\ni--;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount++;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint CoinChangeGreedy(int[] coins, int amt) {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\nint i = coins.Length - 1;\nint count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile (amt &gt; 0) {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\ni--;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount++;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\ni := len(coins) - 1\ncount := 0\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nfor amt &gt; 0 {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nfor i &gt; 0 &amp;&amp; coins[i] &gt; amt {\ni--\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i]\ncount++\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nif amt != 0 {\nreturn -1\n}\nreturn count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\nvar i = coins.count - 1\nvar count = 0\nvar amt = amt\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile amt &gt; 0 {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile i &gt; 0 &amp;&amp; coins[i] &gt; amt {\ni -= 1\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i]\ncount += 1\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins, amt) {\n// \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\nlet i = coins.length - 1;\nlet count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile (amt &gt; 0) {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\ni--;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount++;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n// \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\nlet i = coins.length - 1;\nlet count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile (amt &gt; 0) {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\ni--;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount++;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\nint i = coins.length - 1;\nint count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile (amt &gt; 0) {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\ni--;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount++;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\nlet mut i = coins.len() - 1;\nlet mut count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile amt &gt; 0 {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile i &gt; 0 &amp;&amp; coins[i] &gt; amt {\ni -= 1;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount += 1;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nif amt == 0 {\ncount\n} else {\n-1\n}\n}\n</code></pre> coin_change_greedy.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n// \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\nint i = size - 1;\nint count = 0;\n// \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\nwhile (amt &gt; 0) {\n// \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\nwhile (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\ni--;\n}\n// \u9009\u62e9 coins[i]\namt -= coins[i];\ncount++;\n}\n// \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\nreturn amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.zig<pre><code>[class]{}-[func]{coinChangeGreedy}\n</code></pre>"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8d2a\u5fc3\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u64cd\u4f5c\u76f4\u63a5\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u4ee3\u7801\u4e2d\uff0c\u8bb0\u786c\u5e01\u6700\u5c0f\u9762\u503c\u4e3a \\(\\min(coins)\\) \uff0c\u5219\u8d2a\u5fc3\u9009\u62e9\u6700\u591a\u5faa\u73af \\(amt / \\min(coins)\\) \u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(amt / \\min(coins))\\) \u3002\u8fd9\u6bd4\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\times amt)\\) \u63d0\u5347\u4e86\u4e00\u4e2a\u6570\u91cf\u7ea7\u3002</p> <p>\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u9762\u503c\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u80fd\u627e\u5230\u6700\u4f18\u89e3\u3002\u56fe 15-2 \u7ed9\u51fa\u4e86\u4e24\u4e2a\u793a\u4f8b\u3002</p> <ul> <li>\u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8be5\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u7ed9\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8bbe \\(amt = 60\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(11\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(20 + 20 + 20\\) \uff0c\u4ec5\u9700 \\(3\\) \u679a\u786c\u5e01\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8bbe \\(amt = 98\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(49\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(49 + 49\\) \uff0c\u4ec5\u9700 \\(2\\) \u679a\u786c\u5e01\u3002</li> </ul> <p></p> <p> \u56fe 15-2 \u00a0 \u8d2a\u5fc3\u65e0\u6cd5\u627e\u51fa\u6700\u4f18\u89e3\u7684\u793a\u4f8b </p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u4fdd\u8bc1\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\uff0c\u5e76\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\u3002</p> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9002\u7528\u60c5\u51b5\u5206\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ol> <li>\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u5f80\u5f80\u662f\u6700\u4f18\u9009\u62e9\uff0c\u56e0\u4e3a\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u66f4\u9ad8\u6548\u3002</li> <li>\u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5bf9\u4e8e\u5f88\u591a\u590d\u6742\u95ee\u9898\u6765\u8bf4\uff0c\u5bfb\u627e\u5168\u5c40\u6700\u4f18\u89e3\u975e\u5e38\u56f0\u96be\uff0c\u80fd\u4ee5\u8f83\u9ad8\u6548\u7387\u627e\u5230\u6b21\u4f18\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u9519\u7684\u3002</li> </ol>"},{"location":"chapter_greedy/greedy_algorithm/#1512","title":"15.1.2 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u7279\u6027","text":"<p>\u90a3\u4e48\u95ee\u9898\u6765\u4e86\uff0c\u4ec0\u4e48\u6837\u7684\u95ee\u9898\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8bf4\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1f</p> <p>\u76f8\u8f83\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f7f\u7528\u6761\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u5173\u6ce8\u95ee\u9898\u7684\u4e24\u4e2a\u6027\u8d28\u3002</p> <ul> <li>\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff1a\u53ea\u6709\u5f53\u5c40\u90e8\u6700\u4f18\u9009\u62e9\u59cb\u7ec8\u53ef\u4ee5\u5bfc\u81f4\u5168\u5c40\u6700\u4f18\u89e3\u65f6\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u624d\u80fd\u4fdd\u8bc1\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u6700\u4f18\u5b50\u7ed3\u6784\u5df2\u7ecf\u5728\u201c\u52a8\u6001\u89c4\u5212\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u8fd9\u91cc\u4e0d\u518d\u8d58\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\u5e76\u4e0d\u660e\u663e\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u3002</p> <p>\u6211\u4eec\u4e3b\u8981\u63a2\u7a76\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u5224\u65ad\u65b9\u6cd5\u3002\u867d\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f83\u7b80\u5355\uff0c\u4f46\u5b9e\u9645\u4e0a\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\uff0c\u8bc1\u660e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u5e76\u975e\u6613\u4e8b\u3002</p> <p>\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u6211\u4eec\u867d\u7136\u80fd\u591f\u5bb9\u6613\u5730\u4e3e\u51fa\u53cd\u4f8b\uff0c\u5bf9\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u8fdb\u884c\u8bc1\u4f2a\uff0c\u4f46\u8bc1\u5b9e\u7684\u96be\u5ea6\u8f83\u5927\u3002\u5982\u679c\u95ee\uff1a\u6ee1\u8db3\u4ec0\u4e48\u6761\u4ef6\u7684\u786c\u5e01\u7ec4\u5408\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u4eec\u5f80\u5f80\u53ea\u80fd\u51ed\u501f\u76f4\u89c9\u6216\u4e3e\u4f8b\u5b50\u6765\u7ed9\u51fa\u4e00\u4e2a\u6a21\u68f1\u4e24\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96be\u4ee5\u7ed9\u51fa\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\u3002</p> <p>Quote</p> <p>\u6709\u4e00\u7bc7\u8bba\u6587\u7ed9\u51fa\u4e86\u4e00\u4e2a \\(O(n^3)\\) \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u7b97\u6cd5\uff0c\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u786c\u5e01\u7ec4\u5408\u80fd\u5426\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u989d\u7684\u6700\u4f18\u89e3\u3002</p> <p>Pearson, David. A polynomial-time algorithm for the change-making problem. Operations Research Letters 33.3 (2005): 231-234.</p>"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8d2a\u5fc3\u89e3\u9898\u6b65\u9aa4","text":"<p>\u8d2a\u5fc3\u95ee\u9898\u7684\u89e3\u51b3\u6d41\u7a0b\u5927\u4f53\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e09\u6b65\u3002</p> <ol> <li>\u95ee\u9898\u5206\u6790\uff1a\u68b3\u7406\u4e0e\u7406\u89e3\u95ee\u9898\u7279\u6027\uff0c\u5305\u62ec\u72b6\u6001\u5b9a\u4e49\u3001\u4f18\u5316\u76ee\u6807\u548c\u7ea6\u675f\u6761\u4ef6\u7b49\u3002\u8fd9\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52a8\u6001\u89c4\u5212\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</li> <li>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\uff1a\u786e\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8d2a\u5fc3\u9009\u62e9\u3002\u8fd9\u4e2a\u7b56\u7565\u80fd\u591f\u5728\u6bcf\u4e00\u6b65\u51cf\u5c0f\u95ee\u9898\u7684\u89c4\u6a21\uff0c\u5e76\u6700\u7ec8\u89e3\u51b3\u6574\u4e2a\u95ee\u9898\u3002</li> <li>\u6b63\u786e\u6027\u8bc1\u660e\uff1a\u901a\u5e38\u9700\u8981\u8bc1\u660e\u95ee\u9898\u5177\u6709\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8fd9\u4e2a\u6b65\u9aa4\u53ef\u80fd\u9700\u8981\u7528\u5230\u6570\u5b66\u8bc1\u660e\uff0c\u4f8b\u5982\u5f52\u7eb3\u6cd5\u6216\u53cd\u8bc1\u6cd5\u7b49\u3002</li> </ol> <p>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6c42\u89e3\u95ee\u9898\u7684\u6838\u5fc3\u6b65\u9aa4\uff0c\u4f46\u5b9e\u65bd\u8d77\u6765\u53ef\u80fd\u5e76\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e0d\u540c\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\u7684\u5dee\u5f02\u8f83\u5927\u3002\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\u6765\u8bf4\uff0c\u8d2a\u5fc3\u7b56\u7565\u6bd4\u8f83\u6d45\u663e\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u4e0e\u5c1d\u8bd5\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u9690\u853d\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u975e\u5e38\u8003\u9a8c\u4e2a\u4eba\u7684\u89e3\u9898\u7ecf\u9a8c\u4e0e\u7b97\u6cd5\u80fd\u529b\u4e86\u3002</li> <li>\u67d0\u4e9b\u8d2a\u5fc3\u7b56\u7565\u5177\u6709\u8f83\u5f3a\u7684\u8ff7\u60d1\u6027\u3002\u5f53\u6211\u4eec\u6ee1\u6000\u4fe1\u5fc3\u8bbe\u8ba1\u597d\u8d2a\u5fc3\u7b56\u7565\uff0c\u5199\u51fa\u89e3\u9898\u4ee3\u7801\u5e76\u63d0\u4ea4\u8fd0\u884c\uff0c\u5f88\u53ef\u80fd\u53d1\u73b0\u90e8\u5206\u6d4b\u8bd5\u6837\u4f8b\u65e0\u6cd5\u901a\u8fc7\u3002\u8fd9\u662f\u56e0\u4e3a\u8bbe\u8ba1\u7684\u8d2a\u5fc3\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u786e\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7ecd\u7684\u96f6\u94b1\u5151\u6362\u5c31\u662f\u4e00\u4e2a\u5178\u578b\u6848\u4f8b\u3002</li> </ul> <p>\u4e3a\u4e86\u4fdd\u8bc1\u6b63\u786e\u6027\uff0c\u6211\u4eec\u5e94\u8be5\u5bf9\u8d2a\u5fc3\u7b56\u7565\u8fdb\u884c\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8bc1\u6cd5\u6216\u6570\u5b66\u5f52\u7eb3\u6cd5\u3002</p> <p>\u7136\u800c\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6ca1\u6709\u5934\u7eea\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u9762\u5411\u6d4b\u8bd5\u7528\u4f8b\u8fdb\u884c\u4ee3\u7801\u8c03\u8bd5\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u4e0e\u9a8c\u8bc1\u8d2a\u5fc3\u7b56\u7565\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8d2a\u5fc3\u5178\u578b\u4f8b\u9898","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u5e38\u5e38\u5e94\u7528\u5728\u6ee1\u8db3\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u4ee5\u4e0b\u5217\u4e3e\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8d2a\u5fc3\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u786c\u5e01\u627e\u96f6\u95ee\u9898\uff1a\u5728\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u603b\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u5047\u8bbe\u4f60\u6709\u4e00\u4e9b\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u5728\u4e00\u6bb5\u65f6\u95f4\u5185\u8fdb\u884c\uff0c\u4f60\u7684\u76ee\u6807\u662f\u5b8c\u6210\u5c3d\u53ef\u80fd\u591a\u7684\u4efb\u52a1\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4efb\u52a1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u8f7d\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6807\u662f\u9009\u62e9\u4e00\u7ec4\u7269\u54c1\uff0c\u4f7f\u5f97\u603b\u91cd\u91cf\u4e0d\u8d85\u8fc7\u8f7d\u91cd\u91cf\uff0c\u4e14\u603b\u4ef7\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u6027\u4ef7\u6bd4\u6700\u9ad8\uff08\u4ef7\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u80a1\u7968\u4e70\u5356\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u80a1\u7968\u7684\u5386\u53f2\u4ef7\u683c\uff0c\u4f60\u53ef\u4ee5\u8fdb\u884c\u591a\u6b21\u4e70\u5356\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7ecf\u6301\u6709\u80a1\u7968\uff0c\u90a3\u4e48\u5728\u5356\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u4e70\uff0c\u76ee\u6807\u662f\u83b7\u53d6\u6700\u5927\u5229\u6da6\u3002</li> <li>\u970d\u592b\u66fc\u7f16\u7801\uff1a\u970d\u592b\u66fc\u7f16\u7801\u662f\u4e00\u79cd\u7528\u4e8e\u65e0\u635f\u6570\u636e\u538b\u7f29\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002\u901a\u8fc7\u6784\u5efa\u970d\u592b\u66fc\u6811\uff0c\u6bcf\u6b21\u9009\u62e9\u51fa\u73b0\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u8282\u70b9\u5408\u5e76\uff0c\u6700\u540e\u5f97\u5230\u7684\u970d\u592b\u66fc\u6811\u7684\u5e26\u6743\u8def\u5f84\u957f\u5ea6\uff08\u7f16\u7801\u957f\u5ea6\uff09\u6700\u5c0f\u3002</li> <li>Dijkstra \u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u79cd\u89e3\u51b3\u7ed9\u5b9a\u6e90\u9876\u70b9\u5230\u5176\u4f59\u5404\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u95ee\u9898\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_greedy/max_capacity_problem/","title":"15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6570\u7ec4 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4ee3\u8868\u4e00\u4e2a\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u7a7a\u95f4\u53ef\u4ee5\u7ec4\u6210\u4e00\u4e2a\u5bb9\u5668\u3002</p> <p>\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u548c\u5bbd\u5ea6\u7684\u4e58\u79ef\uff08\u9762\u79ef\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f83\u77ed\u7684\u9694\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u4e2a\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002</p> <p>\u8bf7\u5728\u6570\u7ec4\u4e2d\u9009\u62e9\u4e24\u4e2a\u9694\u677f\uff0c\u4f7f\u5f97\u7ec4\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u56fe 15-7 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5bb9\u5668\u7531\u4efb\u610f\u4e24\u4e2a\u9694\u677f\u56f4\u6210\uff0c\u56e0\u6b64\u672c\u9898\u7684\u72b6\u6001\u4e3a\u4e24\u4e2a\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u6839\u636e\u9898\u610f\uff0c\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u4e58\u4ee5\u5bbd\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002\u8bbe\u5bb9\u91cf\u4e3a \\(cap[i, j]\\) \uff0c\u5219\u53ef\u5f97\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>\u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u4e24\u4e2a\u9694\u677f\u7684\u7ec4\u5408\u6570\u91cf\uff08\u72b6\u6001\u603b\u6570\uff09\u4e3a \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u4e2a\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u7a77\u4e3e\u6240\u6709\u72b6\u6001\uff0c\u4ece\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u8fd9\u9053\u9898\u8fd8\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u56fe 15-8 \u6240\u793a\uff0c\u73b0\u9009\u53d6\u4e00\u4e2a\u72b6\u6001 \\([i, j]\\) \uff0c\u5176\u6ee1\u8db3\u7d22\u5f15 \\(i &lt; j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] &lt; ht[j]\\) \uff0c\u5373 \\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002</p> <p></p> <p> \u56fe 15-8 \u00a0 \u521d\u59cb\u72b6\u6001 </p> <p>\u5982\u56fe 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u65f6\u5c06\u957f\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5219\u5bb9\u91cf\u4e00\u5b9a\u53d8\u5c0f\u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u5728\u79fb\u52a8\u957f\u677f \\(j\\) \u540e\uff0c\u5bbd\u5ea6 \\(j-i\\) \u80af\u5b9a\u53d8\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u53d8\uff08 \\(i\\) \u4ecd\u4e3a\u77ed\u677f\uff09\u6216\u53d8\u5c0f\uff08\u79fb\u52a8\u540e\u7684 \\(j\\) \u6210\u4e3a\u77ed\u677f\uff09\u3002</p> <p></p> <p> \u56fe 15-9 \u00a0 \u5411\u5185\u79fb\u52a8\u957f\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u53cd\u5411\u601d\u8003\uff0c\u6211\u4eec\u53ea\u6709\u5411\u5185\u6536\u7f29\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u53d8\u5927\u3002\u56e0\u4e3a\u867d\u7136\u5bbd\u5ea6\u4e00\u5b9a\u53d8\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u53d8\u5927\uff08\u79fb\u52a8\u540e\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u4f1a\u53d8\u957f\uff09\u3002\u4f8b\u5982\u5728\u56fe 15-10 \u4e2d\uff0c\u79fb\u52a8\u77ed\u677f\u540e\u9762\u79ef\u53d8\u5927\u3002</p> <p></p> <p> \u56fe 15-10 \u00a0 \u5411\u5185\u79fb\u52a8\u77ed\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\uff1a\u521d\u59cb\u5316\u4e24\u6307\u9488\u5206\u5217\u5bb9\u5668\u4e24\u7aef\uff0c\u6bcf\u8f6e\u5411\u5185\u6536\u7f29\u77ed\u677f\u5bf9\u5e94\u7684\u6307\u9488\uff0c\u76f4\u81f3\u4e24\u6307\u9488\u76f8\u9047\u3002</p> <p>\u56fe 15-11 \u5c55\u793a\u4e86\u8d2a\u5fc3\u7b56\u7565\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u5217\u4e0e\u6570\u7ec4\u4e24\u7aef\u3002</li> <li>\u8ba1\u7b97\u5f53\u524d\u72b6\u6001\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u5e76\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002</li> <li>\u6bd4\u8f83\u677f \\(i\\) \u548c \u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u5e76\u5c06\u77ed\u677f\u5411\u5185\u79fb\u52a8\u4e00\u683c\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u65f6\u7ed3\u675f\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u8d2a\u5fc3\u8fc7\u7a0b </p>"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4ee3\u7801\u5faa\u73af\u6700\u591a \\(n\\) \u8f6e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u53d8\u91cf \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n\"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3\"\"\"\n# \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\ni, j = 0, len(ht) - 1\n# \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nres = 0\n# \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile i &lt; j:\n# \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\ncap = min(ht[i], ht[j]) * (j - i)\nres = max(res, cap)\n# \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif ht[i] &lt; ht[j]:\ni += 1\nelse:\nj -= 1\nreturn res\n</code></pre> max_capacity.cpp<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nint i = 0, j = ht.size() - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nint res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile (i &lt; j) {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nint cap = min(ht[i], ht[j]) * (j - i);\nres = max(res, cap);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif (ht[i] &lt; ht[j]) {\ni++;\n} else {\nj--;\n}\n}\nreturn res;\n}\n</code></pre> max_capacity.java<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int[] ht) {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nint i = 0, j = ht.length - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nint res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile (i &lt; j) {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nint cap = Math.min(ht[i], ht[j]) * (j - i);\nres = Math.max(res, cap);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif (ht[i] &lt; ht[j]) {\ni++;\n} else {\nj--;\n}\n}\nreturn res;\n}\n</code></pre> max_capacity.cs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint MaxCapacity(int[] ht) {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nint i = 0, j = ht.Length - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nint res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile (i &lt; j) {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nint cap = Math.Min(ht[i], ht[j]) * (j - i);\nres = Math.Max(res, cap);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif (ht[i] &lt; ht[j]) {\ni++;\n} else {\nj--;\n}\n}\nreturn res;\n}\n</code></pre> max_capacity.go<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht []int) int {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\ni, j := 0, len(ht)-1\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nres := 0\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nfor i &lt; j {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\ncapacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\nres = int(math.Max(float64(res), float64(capacity)))\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif ht[i] &lt; ht[j] {\ni++\n} else {\nj--\n}\n}\nreturn res\n}\n</code></pre> max_capacity.swift<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nvar i = 0, j = ht.count - 1\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nvar res = 0\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile i &lt; j {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nlet cap = min(ht[i], ht[j]) * (j - i)\nres = max(res, cap)\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif ht[i] &lt; ht[j] {\ni += 1\n} else {\nj -= 1\n}\n}\nreturn res\n}\n</code></pre> max_capacity.js<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht) {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nlet i = 0,\nj = ht.length - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nlet res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile (i &lt; j) {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nconst cap = Math.min(ht[i], ht[j]) * (j - i);\nres = Math.max(res, cap);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif (ht[i] &lt; ht[j]) {\ni += 1;\n} else {\nj -= 1;\n}\n}\nreturn res;\n}\n</code></pre> max_capacity.ts<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht: number[]): number {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nlet i = 0,\nj = ht.length - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nlet res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile (i &lt; j) {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nconst cap: number = Math.min(ht[i], ht[j]) * (j - i);\nres = Math.max(res, cap);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif (ht[i] &lt; ht[j]) {\ni += 1;\n} else {\nj -= 1;\n}\n}\nreturn res;\n}\n</code></pre> max_capacity.dart<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(List&lt;int&gt; ht) {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nint i = 0, j = ht.length - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nint res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile (i &lt; j) {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nint cap = min(ht[i], ht[j]) * (j - i);\nres = max(res, cap);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif (ht[i] &lt; ht[j]) {\ni++;\n} else {\nj--;\n}\n}\nreturn res;\n}\n</code></pre> max_capacity.rs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nlet mut i = 0;\nlet mut j = ht.len() - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nlet mut res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile i &lt; j {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nlet cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\nres = std::cmp::max(res, cap);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif ht[i] &lt; ht[j] {\ni += 1;\n} else {\nj -= 1;\n}\n}\nres\n}\n</code></pre> max_capacity.c<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int ht[], int htLength) {\n// \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\nint i = 0;\nint j = htLength - 1;\n// \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\nint res = 0;\n// \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\nwhile (i &lt; j) {\n// \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\nint capacity = myMin(ht[i], ht[j]) * (j - i);\nres = myMax(res, capacity);\n// \u5411\u5185\u79fb\u52a8\u77ed\u677f\nif (ht[i] &lt; ht[j]) {\ni++;\n} else {\nj--;\n}\n}\nreturn res;\n}\n</code></pre> max_capacity.zig<pre><code>[class]{}-[func]{maxCapacity}\n</code></pre>"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4e4b\u6240\u4ee5\u8d2a\u5fc3\u6bd4\u7a77\u4e3e\u66f4\u5feb\uff0c\u662f\u56e0\u4e3a\u6bcf\u8f6e\u7684\u8d2a\u5fc3\u9009\u62e9\u90fd\u4f1a\u201c\u8df3\u8fc7\u201d\u4e00\u4e9b\u72b6\u6001\u3002</p> <p>\u6bd4\u5982\u5728\u72b6\u6001 \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002\u82e5\u8d2a\u5fc3\u5730\u5c06\u77ed\u677f \\(i\\) \u5411\u5185\u79fb\u52a8\u4e00\u683c\uff0c\u4f1a\u5bfc\u81f4\u56fe 15-12 \u6240\u793a\u7684\u72b6\u6001\u88ab\u201c\u8df3\u8fc7\u201d\u3002\u8fd9\u610f\u5473\u7740\u4e4b\u540e\u65e0\u6cd5\u9a8c\u8bc1\u8fd9\u4e9b\u72b6\u6001\u7684\u5bb9\u91cf\u5927\u5c0f\u3002</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> \u56fe 15-12 \u00a0 \u79fb\u52a8\u77ed\u677f\u5bfc\u81f4\u88ab\u8df3\u8fc7\u7684\u72b6\u6001 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u8fd9\u4e9b\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u5b9e\u9645\u4e0a\u5c31\u662f\u5c06\u957f\u677f \\(j\\) \u5411\u5185\u79fb\u52a8\u7684\u6240\u6709\u72b6\u6001\u3002\u524d\u9762\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u5185\u79fb\u957f\u677f\u4e00\u5b9a\u4f1a\u5bfc\u81f4\u5bb9\u91cf\u53d8\u5c0f\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u4f18\u89e3\uff0c\u8df3\u8fc7\u5b83\u4eec\u4e0d\u4f1a\u5bfc\u81f4\u9519\u8fc7\u6700\u4f18\u89e3\u3002</p> <p>\u4ee5\u4e0a\u5206\u6790\u8bf4\u660e\uff0c\u79fb\u52a8\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5c06\u5176\u5207\u5206\u4e3a\u81f3\u5c11\u4e24\u4e2a\u6b63\u6574\u6570\u7684\u548c\uff0c\u6c42\u5207\u5206\u540e\u6240\u6709\u6574\u6570\u7684\u4e58\u79ef\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u56fe 15-13 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u95ee\u9898\u5b9a\u4e49 </p> <p>\u5047\u8bbe\u6211\u4eec\u5c06 \\(n\\) \u5207\u5206\u4e3a \\(m\\) \u4e2a\u6574\u6570\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u4e2a\u56e0\u5b50\u8bb0\u4e3a \\(n_i\\) \uff0c\u5373</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u5f97\u6240\u6709\u6574\u6570\u56e0\u5b50\u7684\u6700\u5927\u4e58\u79ef\uff0c\u5373</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>\u6211\u4eec\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6570\u91cf \\(m\\) \u5e94\u8be5\u591a\u5927\uff0c\u6bcf\u4e2a \\(n_i\\) \u5e94\u8be5\u662f\u591a\u5c11\uff1f</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6839\u636e\u7ecf\u9a8c\uff0c\u4e24\u4e2a\u6574\u6570\u7684\u4e58\u79ef\u5f80\u5f80\u6bd4\u5b83\u4eec\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8bbe\u4ece \\(n\\) \u4e2d\u5206\u51fa\u4e00\u4e2a\u56e0\u5b50 \\(2\\) \uff0c\u5219\u5b83\u4eec\u7684\u4e58\u79ef\u4e3a \\(2(n-2)\\) \u3002\u6211\u4eec\u5c06\u8be5\u4e58\u79ef\u4e0e \\(n\\) \u4f5c\u6bd4\u8f83\uff1a</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>\u5982\u56fe 15-14 \u6240\u793a\uff0c\u5f53 \\(n \\geq 4\\) \u65f6\uff0c\u5207\u5206\u51fa\u4e00\u4e2a \\(2\\) \u540e\u4e58\u79ef\u4f1a\u53d8\u5927\uff0c\u8fd9\u8bf4\u660e\u5927\u4e8e\u7b49\u4e8e \\(4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u88ab\u5207\u5206\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u4e48\u5b83\u5c31\u5e94\u8be5\u88ab\u7ee7\u7eed\u5207\u5206\u3002\u6700\u7ec8\u7684\u5207\u5206\u65b9\u6848\u53ea\u5e94\u51fa\u73b0 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u79cd\u56e0\u5b50\u3002</p> <p></p> <p> \u56fe 15-14 \u00a0 \u5207\u5206\u5bfc\u81f4\u4e58\u79ef\u53d8\u5927 </p> <p>\u63a5\u4e0b\u6765\u601d\u8003\u54ea\u4e2a\u56e0\u5b50\u662f\u6700\u4f18\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u4e2a\u56e0\u5b50\u4e2d\uff0c\u663e\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u4e3a \\(1 \\times (n-1) &lt; n\\) \u6052\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u4f1a\u5bfc\u81f4\u4e58\u79ef\u51cf\u5c0f\u3002</p> <p>\u5982\u56fe 15-15 \u6240\u793a\uff0c\u5f53 \\(n = 6\\) \u65f6\uff0c\u6709 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\) \u3002\u8fd9\u610f\u5473\u7740\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u4f18\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u53ea\u5e94\u5b58\u5728\u4e24\u4e2a \\(2\\) \u3002\u56e0\u4e3a\u4e09\u4e2a \\(2\\) \u603b\u662f\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002</p> <p></p> <p> \u56fe 15-15 \u00a0 \u6700\u4f18\u5207\u5206\u56e0\u5b50 </p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u8f93\u5165\u6574\u6570 \\(n\\) \uff0c\u4ece\u5176\u4e0d\u65ad\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u4f59\u6570\u4e3a \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(0\\) \u65f6\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6570\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u5904\u7406\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(2\\) \u65f6\uff0c\u4e0d\u7ee7\u7eed\u5212\u5206\uff0c\u4fdd\u7559\u4e4b\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(1\\) \u65f6\uff0c\u7531\u4e8e \\(2 \\times 2 &gt; 1 \\times 3\\) \uff0c\u56e0\u6b64\u5e94\u5c06\u6700\u540e\u4e00\u4e2a \\(3\\) \u66ff\u6362\u4e3a \\(2\\) \u3002</li> </ol>"},{"location":"chapter_greedy/max_product_cutting_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5982\u56fe 15-16 \u6240\u793a\uff0c\u6211\u4eec\u65e0\u987b\u901a\u8fc7\u5faa\u73af\u6765\u5207\u5206\u6574\u6570\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u8fd0\u7b97\u5f97\u5230 \\(3\\) \u7684\u4e2a\u6570 \\(a\\) \uff0c\u7528\u53d6\u6a21\u8fd0\u7b97\u5f97\u5230\u4f59\u6570 \\(b\\) \uff0c\u6b64\u65f6\u6709\uff1a</p> \\[ n = 3 a + b \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e \\(n \\leq 3\\) \u7684\u8fb9\u754c\u60c5\u51b5\uff0c\u5fc5\u987b\u62c6\u5206\u51fa\u4e00\u4e2a \\(1\\) \uff0c\u4e58\u79ef\u4e3a \\(1 \\times (n - 1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n\"\"\"\u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3\"\"\"\n# \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif n &lt;= 3:\nreturn 1 * (n - 1)\n# \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\na, b = n // 3, n % 3\nif b == 1:\n# \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn int(math.pow(3, a - 1)) * 2 * 2\nif b == 2:\n# \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn int(math.pow(3, a)) * 2\n# \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif (n &lt;= 3) {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nint a = n / 3;\nint b = n % 3;\nif (b == 1) {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn (int)pow(3, a - 1) * 2 * 2;\n}\nif (b == 2) {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn (int)pow(3, a) * 2;\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif (n &lt;= 3) {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nint a = n / 3;\nint b = n % 3;\nif (b == 1) {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn (int) Math.pow(3, a - 1) * 2 * 2;\n}\nif (b == 2) {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn (int) Math.pow(3, a) * 2;\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint MaxProductCutting(int n) {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif (n &lt;= 3) {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nint a = n / 3;\nint b = n % 3;\nif (b == 1) {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn (int)Math.Pow(3, a - 1) * 2 * 2;\n}\nif (b == 2) {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn (int)Math.Pow(3, a) * 2;\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n int) int {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif n &lt;= 3 {\nreturn 1 * (n - 1)\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\na := n / 3\nb := n % 3\nif b == 1 {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn int(math.Pow(3, float64(a-1))) * 2 * 2\n}\nif b == 2 {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn int(math.Pow(3, float64(a))) * 2\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n: Int) -&gt; Int {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif n &lt;= 3 {\nreturn 1 * (n - 1)\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nlet a = n / 3\nlet b = n % 3\nif b == 1 {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn pow(3, a - 1) * 2 * 2\n}\nif b == 2 {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn pow(3, a) * 2\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n) {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif (n &lt;= 3) {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nlet a = Math.floor(n / 3);\nlet b = n % 3;\nif (b === 1) {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn Math.pow(3, a - 1) * 2 * 2;\n}\nif (b === 2) {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn Math.pow(3, a) * 2;\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n: number): number {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif (n &lt;= 3) {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nlet a: number = Math.floor(n / 3);\nlet b: number = n % 3;\nif (b === 1) {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn Math.pow(3, a - 1) * 2 * 2;\n}\nif (b === 2) {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn Math.pow(3, a) * 2;\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif (n &lt;= 3) {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nint a = n ~/ 3;\nint b = n % 3;\nif (b == 1) {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn (pow(3, a - 1) * 2 * 2).toInt();\n}\nif (b == 2) {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn (pow(3, a) * 2).toInt();\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfn max_product_cutting(n: i32) -&gt; i32 {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif n &lt;= 3 {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nlet a = n / 3;\nlet b = n % 3;\nif b == 1 {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n3_i32.pow(a as u32 - 1) * 2 * 2\n} else if b == 2 {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n3_i32.pow(a as u32) * 2\n} else {\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n3_i32.pow(a as u32)\n}\n}\n</code></pre> max_product_cutting.c<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n// \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\nif (n &lt;= 3) {\nreturn 1 * (n - 1);\n}\n// \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\nint a = n / 3;\nint b = n % 3;\nif (b == 1) {\n// \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\nreturn pow(3, a - 1) * 2 * 2;\n}\nif (b == 2) {\n// \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn pow(3, a) * 2;\n}\n// \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\nreturn pow(3, a);\n}\n</code></pre> max_product_cutting.zig<pre><code>[class]{}-[func]{maxProductCutting}\n</code></pre> <p></p> <p> \u56fe 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u8ba1\u7b97\u65b9\u6cd5 </p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u5e42\u8fd0\u7b97\u7684\u5b9e\u73b0\u65b9\u6cd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5e38\u7528\u7684\u5e42\u8ba1\u7b97\u51fd\u6570\u6709\u4e09\u79cd\u3002</p> <ul> <li>\u8fd0\u7b97\u7b26 <code>**</code> \u548c\u51fd\u6570 <code>pow()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log\u2061 a)\\) \u3002</li> <li>\u51fd\u6570 <code>math.pow()</code> \u5185\u90e8\u8c03\u7528 C \u8bed\u8a00\u5e93\u7684 <code>pow()</code> \u51fd\u6570\uff0c\u5176\u6267\u884c\u6d6e\u70b9\u53d6\u5e42\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> </ul> <p>\u53d8\u91cf \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u51b5\u3002</p> <ol> <li>\u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u5c06\u5176\u7ee7\u7eed\u5212\u5206\u4e3a \\(2(x-2)\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u4e2a\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u5e76\u5165\u53e6\u5916\u4e00\u4e2a\u56e0\u5b50\u4e2d\uff0c\u4ee5\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u4e24\u4e2a \\(2\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u4e2a \\(2\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4e58\u79ef\u66f4\u5927\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> </ol>"},{"location":"chapter_greedy/summary/","title":"15.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u8d2a\u5fc3\u7b97\u6cd5\u901a\u5e38\u7528\u4e8e\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\u90fd\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4f1a\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u4e2a\u53c8\u4e00\u4e2a\u7684\u8d2a\u5fc3\u9009\u62e9\uff0c\u6bcf\u8f6e\u90fd\u5c06\u95ee\u9898\u8f6c\u5316\u6210\u4e00\u4e2a\u89c4\u6a21\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u5b9e\u73b0\u7b80\u5355\uff0c\u8fd8\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u9898\u6548\u7387\u3002\u76f8\u6bd4\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002</li> <li>\u5728\u96f6\u94b1\u5151\u6362\u95ee\u9898\u4e2d\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1b\u5bf9\u4e8e\u53e6\u5916\u4e00\u4e9b\u786c\u5e01\u7ec4\u5408\u5219\u4e0d\u7136\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002</li> <li>\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u7684\u95ee\u9898\u5177\u6709\u4e24\u5927\u6027\u8d28\uff1a\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u4ee3\u8868\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u8bc1\u660e\u5e76\u4e0d\u7b80\u5355\u3002\u76f8\u5bf9\u6765\u8bf4\uff0c\u8bc1\u4f2a\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</li> <li>\u6c42\u89e3\u8d2a\u5fc3\u95ee\u9898\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\uff1a\u95ee\u9898\u5206\u6790\u3001\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u3001\u6b63\u786e\u6027\u8bc1\u660e\u3002\u5176\u4e2d\uff0c\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9aa4\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u5f80\u5f80\u662f\u96be\u70b9\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\u5728 0-1 \u80cc\u5305\u7684\u57fa\u7840\u4e0a\uff0c\u5141\u8bb8\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u3002\u8d2a\u5fc3\u7b56\u7565\u7684\u6b63\u786e\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\u3002</li> <li>\u6700\u5927\u5bb9\u91cf\u95ee\u9898\u53ef\u4f7f\u7528\u7a77\u4e3e\u6cd5\u6c42\u89e3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u901a\u8fc7\u8bbe\u8ba1\u8d2a\u5fc3\u7b56\u7565\uff0c\u6bcf\u8f6e\u5411\u5185\u79fb\u52a8\u77ed\u677f\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u5728\u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u63a8\u7406\u51fa\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u7ee7\u7eed\u5207\u5206\u3001\u6700\u4f18\u5207\u5206\u56e0\u5b50\u4e3a \\(3\\) \u3002\u4ee3\u7801\u4e2d\u5305\u542b\u5e42\u8fd0\u7b97\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5e42\u8fd0\u7b97\u5b9e\u73b0\u65b9\u6cd5\uff0c\u901a\u5e38\u4e3a \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002</li> </ul>"},{"location":"chapter_hashing/","title":"\u7b2c 6 \u7ae0 \u00a0 \u54c8\u5e0c\u8868","text":"<p>Abstract</p> <p>\u5728\u8ba1\u7b97\u673a\u4e16\u754c\u4e2d\uff0c\u54c8\u5e0c\u8868\u5982\u540c\u4e00\u4f4d\u806a\u6167\u7684\u56fe\u4e66\u7ba1\u7406\u5458\u3002</p> <p>\u4ed6\u77e5\u9053\u5982\u4f55\u8ba1\u7b97\u7d22\u4e66\u53f7\uff0c\u4ece\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6807\u56fe\u4e66\u3002</p>"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>6.1 \u00a0 \u54c8\u5e0c\u8868</li> <li>6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81</li> <li>6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5</li> <li>6.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u524d\u4e24\u8282\u4ecb\u7ecd\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5f0f\u5730\u5740\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002</p> <p>\u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u5982\u56fe 6-8 \u6240\u793a\uff0c\u5bf9\u4e8e\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u4f73\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u6240\u6709\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 6-8 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u60c5\u51b5\u4e0e\u6700\u5dee\u60c5\u51b5 </p> <p>\u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u60c5\u51b5\u3002</p> <p>\u8fd9\u610f\u5473\u7740\uff0c\u4e3a\u4e86\u964d\u4f4e\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u5e94\u5f53\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u7684\u8bbe\u8ba1\u4e0a\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5177\u5907\u4ee5\u4e0b\u7279\u70b9\u3002</p> <ul> <li>\u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002</li> <li>\u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002</li> <li>\u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5747\u5300\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\u3002</p> <ul> <li>\u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002</li> <li>\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u3002</li> </ul> <p>\u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u4e3a\u4e86\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u5177\u5907\u66f4\u9ad8\u7b49\u7ea7\u7684\u5b89\u5168\u7279\u6027\u3002</p> <ul> <li>\u5355\u5411\u6027\uff1a\u65e0\u6cd5\u901a\u8fc7\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u5173\u4e8e\u8f93\u5165\u6570\u636e\u7684\u4efb\u4f55\u4fe1\u606f\u3002</li> <li>\u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002</li> <li>\u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165 <code>key</code> \u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570 <code>key % 100</code> \u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684 <code>key</code> \u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 <code>key</code> \uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"<p>\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u590d\u6742\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>\u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002</li> <li>\u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n\"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash += ord(c)\nreturn hash % modulus\ndef mul_hash(key: str) -&gt; int:\n\"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash = 31 * hash + ord(c)\nreturn hash % modulus\ndef xor_hash(key: str) -&gt; int:\n\"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash ^= ord(c)\nreturn hash % modulus\ndef rot_hash(key: str) -&gt; int:\n\"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\nhash = 0\nmodulus = 1000000007\nfor c in key:\nhash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\nreturn hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = (hash + (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = (31 * hash + (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\nint hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash ^= (int)c;\n}\nreturn hash &amp; MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (unsigned char c : key) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n}\nreturn (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = (hash + (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = (31 * hash + (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\nint hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash ^= (int) c;\n}\nreturn hash &amp; MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\nlong hash = 0;\nfinal int MODULUS = 1000000007;\nfor (char c : key.toCharArray()) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n}\nreturn (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = (hash + c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = (31 * hash + c) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\nint hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash ^= c;\n}\nreturn hash &amp; MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\nlong hash = 0;\nconst int MODULUS = 1000000007;\nforeach (char c in key) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n}\nreturn (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = (hash + int64(b)) % modulus\n}\nreturn int(hash)\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = (31*hash + int64(b)) % modulus\n}\nreturn int(hash)\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\nhash := 0\nmodulus := 1000000007\nfor _, b := range []byte(key) {\nfmt.Println(int(b))\nhash ^= int(b)\nhash = (31*hash + int(b)) % modulus\n}\nreturn hash &amp; modulus\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\nvar hash int64\nvar modulus int64\nmodulus = 1000000007\nfor _, b := range []byte(key) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n}\nreturn int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -&gt; Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = (hash + Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -&gt; Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = (31 * hash + Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -&gt; Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash ^= Int(scalar.value)\n}\n}\nreturn hash &amp; MODULUS\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -&gt; Int {\nvar hash = 0\nlet MODULUS = 1_000_000_007\nfor c in key {\nfor scalar in c.unicodeScalars {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n}\n}\nreturn hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash = (hash + c.charCodeAt(0)) % MODULUS;\n}\nreturn hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n}\nreturn hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash ^= c.charCodeAt(0);\n}\nreturn hash &amp; MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n}\nreturn hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash = (hash + c.charCodeAt(0)) % MODULUS;\n}\nreturn hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n}\nreturn hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash ^= c.charCodeAt(0);\n}\nreturn hash &amp; MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\nlet hash = 0;\nconst MODULUS = 1000000007;\nfor (const c of key) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n}\nreturn hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\nint hash = 0;\nfinal int MODULUS = 1000000007;\nfor (int i = 0; i &lt; key.length; i++) {\nhash = (hash + key.codeUnitAt(i)) % MODULUS;\n}\nreturn hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\nint hash = 0;\nfinal int MODULUS = 1000000007;\nfor (int i = 0; i &lt; key.length; i++) {\nhash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n}\nreturn hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\nint hash = 0;\nfinal int MODULUS = 1000000007;\nfor (int i = 0; i &lt; key.length; i++) {\nhash ^= key.codeUnitAt(i);\n}\nreturn hash &amp; MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\nint hash = 0;\nfinal int MODULUS = 1000000007;\nfor (int i = 0; i &lt; key.length; i++) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n}\nreturn hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &amp;str) -&gt; i32 {\nlet mut hash = 0_i64;\nconst MODULUS: i64 = 1000000007;\nfor c in key.chars() {\nhash = (hash + c as i64) % MODULUS;\n}\nhash as i32\n} /* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &amp;str) -&gt; i32 {\nlet mut hash = 0_i64;\nconst MODULUS: i64 = 1000000007;\nfor c in key.chars() {\nhash = (31 * hash + c as i64) % MODULUS;\n}\nhash as i32\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &amp;str) -&gt; i32 {\nlet mut hash = 0_i64;\nconst MODULUS: i64 = 1000000007;\nfor c in key.chars() {\nhash ^= c as i64;\n}\n(hash &amp; MODULUS) as i32\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &amp;str) -&gt; i32 {\nlet mut hash = 0_i64;\nconst MODULUS: i64 = 1000000007;\nfor c in key.chars() {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n}\nhash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (int i = 0; i &lt; strlen(key); i++) {\nhash = (hash + (unsigned char)key[i]) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (int i = 0; i &lt; strlen(key); i++) {\nhash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n}\nreturn (int)hash;\n}\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\nint hash = 0;\nconst int MODULUS = 1000000007;\nfor (int i = 0; i &lt; strlen(key); i++) {\nhash ^= (unsigned char)key[i];\n}\nreturn hash &amp; MODULUS;\n}\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\nlong long hash = 0;\nconst int MODULUS = 1000000007;\nfor (int i = 0; i &lt; strlen(key); i++) {\nhash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n}\nreturn (int)hash;\n}\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n[class]{}-[func]{mulHash}\n[class]{}-[func]{xorHash}\n[class]{}-[func]{rotHash}\n</code></pre> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u6216\u8005\u8bf4\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002</p> <p>\u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>\u5982\u679c\u8f93\u5165 <code>key</code> \u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06 <code>modulus</code> \u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8e <code>key</code> \u548c <code>modulus</code> \u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u56e0\u6b64\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1 <code>key</code> \u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u53ef\u4ee5\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53 <code>key</code> \u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002</p> <p>\u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u8db3\u591f\u5927\uff0c\u4ee5\u5c3d\u53ef\u80fd\u6d88\u9664\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2\u3001SHA-3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002</p> <p>\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002</li> <li>SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002</li> <li>SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002</li> </ul> <p> \u8868 6-2 \u00a0 \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5 </p> MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bits 160 bits 256/512 bits 224/256/384/512 bits \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684 <code>key</code> \u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528 <code>hash()</code> \u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u3002</p> <ul> <li>\u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002</li> <li>\u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8bf7\u81ea\u884c\u5b66\u4e60\u3002</li> <li>\u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002</li> <li>\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002</li> </ul> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 &lt;ListNode object at 0x1058fd810&gt; \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 468167534\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 568126464209439262\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 4952851536318644461\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\nprintln!(\"\u5c0f\u6570 {} \u7684\u54c8\u5e0c\u503c\u4e3a {}\", dec, hash_dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2566941990314602357\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 16092673739211250988\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7ec4 (12836, \"\u5c0f\u54c8\") \u7684\u54c8\u5e0c\u503c\u4e3a 1885128010422702749\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// \u8282\u70b9\u5bf9\u8c61 RefCell { value: ListNode { val: 42, next: None } } \u7684\u54c8\u5e0c\u503c\u4e3a15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> <p>\u5728\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684 <code>key</code> \u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3a <code>key</code> \uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684 <code>value</code> \u4e86\u3002</p> <p>\u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u6bd4\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002</p> <p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\u5728\u4e0d\u540c\u63a7\u5236\u53f0\u4e2d\u8fd0\u884c\u7a0b\u5e8f\u65f6\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u662f\u4e0d\u540c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08Salt\uff09\u503c\u3002\u8fd9\u79cd\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"<p>\u4e0a\u4e00\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u7406\u8bba\u4e0a\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002</p> <p>\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u7b56\u7565\u3002</p> <ol> <li>\u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002</li> <li>\u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u624d\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002</li> </ol> <p>\u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u94fe\u5f0f\u5730\u5740\u201d\u548c\u201c\u5f00\u653e\u5bfb\u5740\u201d\u3002</p>"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u94fe\u5f0f\u5730\u5740","text":"<p>\u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u300c\u94fe\u5f0f\u5730\u5740 separate chaining\u300d\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002\u56fe 6-5 \u5c55\u793a\u4e86\u4e00\u4e2a\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 6-5 \u00a0 \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 </p> <p>\u57fa\u4e8e\u94fe\u5f0f\u5730\u5740\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u53d1\u751f\u4e86\u4ee5\u4e0b\u53d8\u5316\u3002</p> <ul> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 <code>key</code> \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 <code>key</code> \u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002</li> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u9996\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\u5e76\u5c06\u5176\u5220\u9664\u3002</li> </ul> <p>\u94fe\u5f0f\u5730\u5740\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\u589e\u5927\uff0c\u94fe\u8868\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u5b83\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\u3002</li> </ul> <p>\u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\u4e24\u70b9\u3002</p> <ul> <li>\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\uff0c\u4ece\u800c\u7b80\u5316\u4ee3\u7801\u3002\u5728\u8fd9\u79cd\u8bbe\u5b9a\u4e0b\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002</li> <li>\u4ee5\u4e0b\u5b9e\u73b0\u5305\u542b\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(\\frac{2}{3}\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map_chaining.py<pre><code>class HashMapChaining:\n\"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\nself.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\nself.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nself.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\nself.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\ndef hash_func(self, key: int) -&gt; int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nreturn key % self.capacity\ndef load_factor(self) -&gt; float:\n\"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\nreturn self.size / self.capacity\ndef get(self, key: int) -&gt; str | None:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor pair in bucket:\nif pair.key == key:\nreturn pair.val\n# \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\nreturn None\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n# \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() &gt; self.load_thres:\nself.extend()\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor pair in bucket:\nif pair.key == key:\npair.val = val\nreturn\n# \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\npair = Pair(key, val)\nbucket.append(pair)\nself.size += 1\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex = self.hash_func(key)\nbucket = self.buckets[index]\n# \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor pair in bucket:\nif pair.key == key:\nbucket.remove(pair)\nself.size -= 1\nbreak\ndef extend(self):\n\"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n# \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nbuckets = self.buckets\n# \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio\nself.buckets = [[] for _ in range(self.capacity)]\nself.size = 0\n# \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor bucket in buckets:\nfor pair in bucket:\nself.put(pair.key, pair.val)\ndef print(self):\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor bucket in self.buckets:\nres = []\nfor pair in bucket:\nres.append(str(pair.key) + \" -&gt; \" + pair.val)\nprint(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nprivate:\nint size;                       // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio;                // \u6269\u5bb9\u500d\u6570\nvector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u6570\u7ec4\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\nbuckets.resize(capacity);\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~HashMapChaining() {\nfor (auto &amp;bucket : buckets) {\nfor (Pair *pair : bucket) {\n// \u91ca\u653e\u5185\u5b58\ndelete pair;\n}\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn (double)size / (double)capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair *pair : buckets[index]) {\nif (pair-&gt;key == key) {\nreturn pair-&gt;val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\";\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() &gt; loadThres) {\nextend();\n}\nint index = hashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair *pair : buckets[index]) {\nif (pair-&gt;key == key) {\npair-&gt;val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nbuckets[index].push_back(new Pair(key, val));\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nauto &amp;bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (int i = 0; i &lt; bucket.size(); i++) {\nif (bucket[i]-&gt;key == key) {\nPair *tmp = bucket[i];\nbucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\ndelete tmp;                       // \u91ca\u653e\u5185\u5b58\nsize--;\nreturn;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nvector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets.clear();\nbuckets.resize(capacity);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (auto &amp;bucket : bucketsTmp) {\nfor (Pair *pair : bucket) {\nput(pair-&gt;key, pair-&gt;val);\n// \u91ca\u653e\u5185\u5b58\ndelete pair;\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (auto &amp;bucket : buckets) {\ncout &lt;&lt; \"[\";\nfor (Pair *pair : bucket) {\ncout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n}\ncout &lt;&lt; \"]\\n\";\n}\n}\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nList&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = new ArrayList&lt;&gt;(capacity);\nfor (int i = 0; i &lt; capacity; i++) {\nbuckets.add(new ArrayList&lt;&gt;());\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn (double) size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString get(int key) {\nint index = hashFunc(key);\nList&lt;Pair&gt; bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair pair : bucket) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() &gt; loadThres) {\nextend();\n}\nint index = hashFunc(key);\nList&lt;Pair&gt; bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair pair : bucket) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nPair pair = new Pair(key, val);\nbucket.add(pair);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nList&lt;Pair&gt; bucket = buckets.get(index);\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (Pair pair : bucket) {\nif (pair.key == key) {\nbucket.remove(pair);\nsize--;\nbreak;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new ArrayList&lt;&gt;(capacity);\nfor (int i = 0; i &lt; capacity; i++) {\nbuckets.add(new ArrayList&lt;&gt;());\n}\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (List&lt;Pair&gt; bucket : bucketsTmp) {\nfor (Pair pair : bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (List&lt;Pair&gt; bucket : buckets) {\nList&lt;String&gt; res = new ArrayList&lt;&gt;();\nfor (Pair pair : bucket) {\nres.add(pair.key + \" -&gt; \" + pair.val);\n}\nSystem.out.println(res);\n}\n}\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio; // \u6269\u5bb9\u500d\u6570\nList&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\nfor (int i = 0; i &lt; capacity; i++) {\nbuckets.Add([]);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint HashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble LoadFactor() {\nreturn (double)size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string? Get(int key) {\nint index = HashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nforeach (Pair pair in buckets[index]) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void Put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (LoadFactor() &gt; loadThres) {\nExtend();\n}\nint index = HashFunc(key);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nforeach (Pair pair in buckets[index]) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nbuckets[index].Add(new Pair(key, val));\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void Remove(int key) {\nint index = HashFunc(key);\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nforeach (Pair pair in buckets[index].ToList()) {\nif (pair.key == key) {\nbuckets[index].Remove(pair);\nsize--;\nbreak;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid Extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\nfor (int i = 0; i &lt; capacity; i++) {\nbuckets.Add([]);\n}\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nforeach (List&lt;Pair&gt; bucket in bucketsTmp) {\nforeach (Pair pair in bucket) {\nPut(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void Print() {\nforeach (List&lt;Pair&gt; bucket in buckets) {\nList&lt;string&gt; res = [];\nforeach (Pair pair in bucket) {\nres.Add(pair.key + \" -&gt; \" + pair.val);\n}\nforeach (string kv in res) {\nConsole.WriteLine(kv);\n}\n}\n}\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\nsize        int      // \u952e\u503c\u5bf9\u6570\u91cf\ncapacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\nloadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nextendRatio int      // \u6269\u5bb9\u500d\u6570\nbuckets     [][]pair // \u6876\u6570\u7ec4\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\nbuckets := make([][]pair, 4)\nfor i := 0; i &lt; 4; i++ {\nbuckets[i] = make([]pair, 0)\n}\nreturn &amp;hashMapChaining{\nsize:        0,\ncapacity:    4,\nloadThres:   2.0 / 3.0,\nextendRatio: 2,\nbuckets:     buckets,\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\nreturn key % m.capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\nreturn float64(m.size / m.capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\nidx := m.hashFunc(key)\nbucket := m.buckets[idx]\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor _, p := range bucket {\nif p.key == key {\nreturn p.val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\"\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif m.loadFactor() &gt; m.loadThres {\nm.extend()\n}\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor _, p := range m.buckets[idx] {\nif p.key == key {\np.val = val\nreturn\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\np := pair{\nkey: key,\nval: val,\n}\nm.buckets[idx] = append(m.buckets[idx], p)\nm.size += 1\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor i, p := range m.buckets[idx] {\nif p.key == key {\n// \u5207\u7247\u5220\u9664\nm.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\nm.size -= 1\nbreak\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\ntmpBuckets := make([][]pair, len(m.buckets))\nfor i := 0; i &lt; len(m.buckets); i++ {\ntmpBuckets[i] = make([]pair, len(m.buckets[i]))\ncopy(tmpBuckets[i], m.buckets[i])\n}\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nm.capacity *= m.extendRatio\nm.buckets = make([][]pair, m.capacity)\nfor i := 0; i &lt; m.capacity; i++ {\nm.buckets[i] = make([]pair, 0)\n}\nm.size = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor _, bucket := range tmpBuckets {\nfor _, p := range bucket {\nm.put(p.key, p.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\nvar builder strings.Builder\nfor _, bucket := range m.buckets {\nbuilder.WriteString(\"[\")\nfor _, p := range bucket {\nbuilder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n}\nbuilder.WriteString(\"]\")\nfmt.Println(builder.String())\nbuilder.Reset()\n}\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nvar size: Int // \u952e\u503c\u5bf9\u6570\u91cf\nvar capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\nvar loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nvar extendRatio: Int // \u6269\u5bb9\u500d\u6570\nvar buckets: [[Pair]] // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nsize = 0\ncapacity = 4\nloadThres = 2.0 / 3.0\nextendRatio = 2\nbuckets = Array(repeating: [], count: capacity)\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc hashFunc(key: Int) -&gt; Int {\nkey % capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc loadFactor() -&gt; Double {\nDouble(size / capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -&gt; String? {\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor pair in bucket {\nif pair.key == key {\nreturn pair.val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nil\nreturn nil\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif loadFactor() &gt; loadThres {\nextend()\n}\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor pair in bucket {\nif pair.key == key {\npair.val = val\nreturn\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nlet pair = Pair(key: key, val: val)\nbuckets[index].append(pair)\nsize += 1\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\nlet bucket = buckets[index]\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (pairIndex, pair) in bucket.enumerated() {\nif pair.key == key {\nbuckets[index].remove(at: pairIndex)\n}\n}\nsize -= 1\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet bucketsTmp = buckets\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio\nbuckets = Array(repeating: [], count: capacity)\nsize = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor bucket in bucketsTmp {\nfor pair in bucket {\nput(key: pair.key, val: pair.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor bucket in buckets {\nlet res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\nSwift.print(res)\n}\n}\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n#size; // \u952e\u503c\u5bf9\u6570\u91cf\n#capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n#loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n#extendRatio; // \u6269\u5bb9\u500d\u6570\n#buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#size = 0;\nthis.#capacity = 4;\nthis.#loadThres = 2.0 / 3.0;\nthis.#extendRatio = 2;\nthis.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key) {\nreturn key % this.#capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\n#loadFactor() {\nreturn this.#size / this.#capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key) {\nconst index = this.#hashFunc(key);\nconst bucket = this.#buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (const pair of bucket) {\nif (pair.key === key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nput(key, val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (this.#loadFactor() &gt; this.#loadThres) {\nthis.#extend();\n}\nconst index = this.#hashFunc(key);\nconst bucket = this.#buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (const pair of bucket) {\nif (pair.key === key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nconst pair = new Pair(key, val);\nbucket.push(pair);\nthis.#size++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nremove(key) {\nconst index = this.#hashFunc(key);\nlet bucket = this.#buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (let i = 0; i &lt; bucket.length; i++) {\nif (bucket[i].key === key) {\nbucket.splice(i, 1);\nthis.#size--;\nbreak;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n#extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nconst bucketsTmp = this.#buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nthis.#capacity *= this.#extendRatio;\nthis.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\nthis.#size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (const bucket of bucketsTmp) {\nfor (const pair of bucket) {\nthis.put(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint() {\nfor (const bucket of this.#buckets) {\nlet res = [];\nfor (const pair of bucket) {\nres.push(pair.key + ' -&gt; ' + pair.val);\n}\nconsole.log(res);\n}\n}\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n#size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n#capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n#loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n#extendRatio: number; // \u6269\u5bb9\u500d\u6570\n#buckets: Pair[][]; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#size = 0;\nthis.#capacity = 4;\nthis.#loadThres = 2.0 / 3.0;\nthis.#extendRatio = 2;\nthis.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key: number): number {\nreturn key % this.#capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\n#loadFactor(): number {\nreturn this.#size / this.#capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key: number): string | null {\nconst index = this.#hashFunc(key);\nconst bucket = this.#buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (const pair of bucket) {\nif (pair.key === key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nput(key: number, val: string): void {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (this.#loadFactor() &gt; this.#loadThres) {\nthis.#extend();\n}\nconst index = this.#hashFunc(key);\nconst bucket = this.#buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (const pair of bucket) {\nif (pair.key === key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nconst pair = new Pair(key, val);\nbucket.push(pair);\nthis.#size++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nremove(key: number): void {\nconst index = this.#hashFunc(key);\nlet bucket = this.#buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (let i = 0; i &lt; bucket.length; i++) {\nif (bucket[i].key === key) {\nbucket.splice(i, 1);\nthis.#size--;\nbreak;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n#extend(): void {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nconst bucketsTmp = this.#buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nthis.#capacity *= this.#extendRatio;\nthis.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\nthis.#size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (const bucket of bucketsTmp) {\nfor (const pair of bucket) {\nthis.put(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint(): void {\nfor (const bucket of this.#buckets) {\nlet res = [];\nfor (const pair of bucket) {\nres.push(pair.key + ' -&gt; ' + pair.val);\n}\nconsole.log(res);\n}\n}\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\nlate int size; // \u952e\u503c\u5bf9\u6570\u91cf\nlate int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nlate double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nlate int extendRatio; // \u6269\u5bb9\u500d\u6570\nlate List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapChaining() {\nsize = 0;\ncapacity = 4;\nloadThres = 2.0 / 3.0;\nextendRatio = 2;\nbuckets = List.generate(capacity, (_) =&gt; []);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn size / capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nint index = hashFunc(key);\nList&lt;Pair&gt; bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor (Pair pair in bucket) {\nif (pair.key == key) {\nreturn pair.val;\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() &gt; loadThres) {\nextend();\n}\nint index = hashFunc(key);\nList&lt;Pair&gt; bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor (Pair pair in bucket) {\nif (pair.key == key) {\npair.val = val;\nreturn;\n}\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nPair pair = Pair(key, val);\nbucket.add(pair);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\nList&lt;Pair&gt; bucket = buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor (Pair pair in bucket) {\nif (pair.key == key) {\nbucket.remove(pair);\nsize--;\nbreak;\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = List.generate(capacity, (_) =&gt; []);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (List&lt;Pair&gt; bucket in bucketsTmp) {\nfor (Pair pair in bucket) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (List&lt;Pair&gt; bucket in buckets) {\nList&lt;String&gt; res = [];\nfor (Pair pair in bucket) {\nres.add(\"${pair.key} -&gt; ${pair.val}\");\n}\nprint(res);\n}\n}\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\nsize: i32,\ncapacity: i32,\nload_thres: f32,\nextend_ratio: i32,\nbuckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\nimpl HashMapChaining {\n/* \u6784\u9020\u65b9\u6cd5 */\nfn new() -&gt; Self {\nSelf {\nsize: 0,\ncapacity: 4,\nload_thres: 2.0 / 3.0,\nextend_ratio: 2,\nbuckets: vec![vec![]; 4],\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfn hash_func(&amp;self, key: i32) -&gt; usize {\nkey as usize % self.capacity as usize\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfn load_factor(&amp;self) -&gt; f32 {\nself.size as f32 / self.capacity as f32\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\nlet index = self.hash_func(key);\nlet bucket = &amp;mut self.buckets[index];\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nfor i in 0..bucket.len() {\nif bucket[i].key == key {\nlet pair = bucket.remove(i);\nself.size -= 1;\nreturn Some(pair.val);\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\nNone\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfn extend(&amp;mut self) {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio;\nself.buckets = vec![Vec::new(); self.capacity as usize];\nself.size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor bucket in buckets_tmp {\nfor pair in bucket {\nself.put(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfn print(&amp;self) {\nfor bucket in &amp;self.buckets {\nlet mut res = Vec::new();\nfor pair in bucket {\nres.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n}\nprintln!(\"{:?}\", res);\n}\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfn put(&amp;mut self, key: i32, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() &gt; self.load_thres {\nself.extend();\n}\nlet index = self.hash_func(key);\nlet bucket = &amp;mut self.buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nfor pair in bucket {\nif pair.key == key {\npair.val = val.clone();\nreturn;\n}\n}\nlet bucket = &amp;mut self.buckets[index];\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\nlet pair = Pair {\nkey,\nval: val.clone(),\n};\nbucket.push(pair);\nself.size += 1;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\nlet index = self.hash_func(key);\nlet bucket = &amp;self.buckets[index];\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nfor pair in bucket {\nif pair.key == key {\nreturn Some(&amp;pair.val);\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\nNone\n}\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\nPair *pair;\nstruct Node *next;\n} Node;\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\nint size;         // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio;  // \u6269\u5bb9\u500d\u6570\nNode **buckets;   // \u6876\u6570\u7ec4\n} HashMapChaining;\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\nHashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\nhashMap-&gt;size = 0;\nhashMap-&gt;capacity = 4;\nhashMap-&gt;loadThres = 2.0 / 3.0;\nhashMap-&gt;extendRatio = 2;\nhashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\nfor (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\nhashMap-&gt;buckets[i] = NULL;\n}\nreturn hashMap;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\nfor (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\nNode *cur = hashMap-&gt;buckets[i];\nwhile (cur) {\nNode *tmp = cur;\ncur = cur-&gt;next;\nfree(tmp-&gt;pair);\nfree(tmp);\n}\n}\nfree(hashMap-&gt;buckets);\nfree(hashMap);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\nreturn key % hashMap-&gt;capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\nreturn (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\nint index = hashFunc(hashMap, key);\n// \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\nNode *cur = hashMap-&gt;buckets[index];\nwhile (cur) {\nif (cur-&gt;pair-&gt;key == key) {\nreturn cur-&gt;pair-&gt;val;\n}\ncur = cur-&gt;next;\n}\nreturn \"\"; // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\nextend(hashMap);\n}\nint index = hashFunc(hashMap, key);\n// \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nNode *cur = hashMap-&gt;buckets[index];\nwhile (cur) {\nif (cur-&gt;pair-&gt;key == key) {\nstrcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\nreturn;\n}\ncur = cur-&gt;next;\n}\n// \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nPair *newPair = (Pair *)malloc(sizeof(Pair));\nnewPair-&gt;key = key;\nstrcpy(newPair-&gt;val, val);\nNode *newNode = (Node *)malloc(sizeof(Node));\nnewNode-&gt;pair = newPair;\nnewNode-&gt;next = hashMap-&gt;buckets[index];\nhashMap-&gt;buckets[index] = newNode;\nhashMap-&gt;size++;\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nint oldCapacity = hashMap-&gt;capacity;\nNode **oldBuckets = hashMap-&gt;buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nhashMap-&gt;capacity *= hashMap-&gt;extendRatio;\nhashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\nfor (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\nhashMap-&gt;buckets[i] = NULL;\n}\nhashMap-&gt;size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (int i = 0; i &lt; oldCapacity; i++) {\nNode *cur = oldBuckets[i];\nwhile (cur) {\nput(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\nNode *temp = cur;\ncur = cur-&gt;next;\n// \u91ca\u653e\u5185\u5b58\nfree(temp-&gt;pair);\nfree(temp);\n}\n}\nfree(oldBuckets);\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\nint index = hashFunc(hashMap, key);\nNode *cur = hashMap-&gt;buckets[index];\nNode *pre = NULL;\nwhile (cur) {\nif (cur-&gt;pair-&gt;key == key) {\n// \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\nif (pre) {\npre-&gt;next = cur-&gt;next;\n} else {\nhashMap-&gt;buckets[index] = cur-&gt;next;\n}\n// \u91ca\u653e\u5185\u5b58\nfree(cur-&gt;pair);\nfree(cur);\nhashMap-&gt;size--;\nreturn;\n}\npre = cur;\ncur = cur-&gt;next;\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\nfor (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\nNode *cur = hashMap-&gt;buckets[i];\nprintf(\"[\");\nwhile (cur) {\nprintf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\ncur = cur-&gt;next;\n}\nprintf(\"]\\n\");\n}\n}\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5f53\u94fe\u8868\u5f88\u957f\u65f6\uff0c\u67e5\u8be2\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u65f6\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u201cAVL \u6811\u201d\u6216\u201c\u7ea2\u9ed1\u6811\u201d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u5f00\u653e\u5bfb\u5740","text":"<p>\u300c\u5f00\u653e\u5bfb\u5740 open addressing\u300d\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u7b49\u3002</p> <p>\u4e0b\u9762\u4ee5\u7ebf\u6027\u63a2\u6d4b\u4e3a\u4f8b\uff0c\u4ecb\u7ecd\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002</p>"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7ebf\u6027\u63a2\u6d4b","text":"<p>\u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u641c\u7d22\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u4e0e\u666e\u901a\u54c8\u5e0c\u8868\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de <code>value</code> \u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u76ee\u6807\u5143\u7d20\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de \\(\\text{None}\\) \u3002</li> </ul> <p>\u56fe 6-6 \u5c55\u793a\u4e86\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03\u3002\u6839\u636e\u6b64\u54c8\u5e0c\u51fd\u6570\uff0c\u6700\u540e\u4e24\u4f4d\u76f8\u540c\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u901a\u8fc7\u7ebf\u6027\u63a2\u6d4b\uff0c\u5b83\u4eec\u88ab\u4f9d\u6b21\u5b58\u50a8\u5728\u8be5\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 6-6 \u00a0 \u5f00\u653e\u5bfb\u5740\u548c\u7ebf\u6027\u63a2\u6d4b </p> <p>\u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5bb9\u6613\u4ea7\u751f\u201c\u805a\u96c6\u73b0\u8c61\u201d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6570\u7ec4\u4e2d\u8fde\u7eed\u88ab\u5360\u7528\u7684\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8be5\u4f4d\u7f6e\u7684\u805a\u5806\u751f\u957f\uff0c\u5f62\u6210\u6076\u6027\u5faa\u73af\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u4eec\u4e0d\u80fd\u5728\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u4e2d\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u8fd9\u662f\u56e0\u4e3a\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u6876 \\(\\text{None}\\) \uff0c\u800c\u5f53\u67e5\u8be2\u5143\u7d20\u65f6\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230\u8be5\u7a7a\u6876\u5c31\u4f1a\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8be5\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u65e0\u6cd5\u518d\u88ab\u8bbf\u95ee\u5230\uff0c\u7a0b\u5e8f\u53ef\u80fd\u8bef\u5224\u8fd9\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u56fe 6-7 \u00a0 \u5728\u5f00\u653e\u5bfb\u5740\u4e2d\u5220\u9664\u5143\u7d20\u5bfc\u81f4\u7684\u67e5\u8be2\u95ee\u9898 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u300c\u61d2\u5220\u9664 lazy deletion\u300d\u673a\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u4ece\u54c8\u5e0c\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u4e2a\u5e38\u91cf <code>TOMBSTONE</code> \u6765\u6807\u8bb0\u8fd9\u4e2a\u6876\u3002\u5728\u8be5\u673a\u5236\u4e0b\uff0c\\(\\text{None}\\) \u548c <code>TOMBSTONE</code> \u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u952e\u503c\u5bf9\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230 <code>TOMBSTONE</code> \u65f6\u5e94\u8be5\u7ee7\u7eed\u904d\u5386\uff0c\u56e0\u4e3a\u5176\u4e4b\u4e0b\u53ef\u80fd\u8fd8\u5b58\u5728\u952e\u503c\u5bf9\u3002</p> <p>\u7136\u800c\uff0c\u61d2\u5220\u9664\u53ef\u80fd\u4f1a\u52a0\u901f\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u9000\u5316\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6b21\u5220\u9664\u64cd\u4f5c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5220\u9664\u6807\u8bb0\uff0c\u968f\u7740 <code>TOMBSTONE</code> \u7684\u589e\u52a0\uff0c\u641c\u7d22\u65f6\u95f4\u4e5f\u4f1a\u589e\u52a0\uff0c\u56e0\u4e3a\u7ebf\u6027\u63a2\u6d4b\u53ef\u80fd\u9700\u8981\u8df3\u8fc7\u591a\u4e2a <code>TOMBSTONE</code> \u624d\u80fd\u627e\u5230\u76ee\u6807\u5143\u7d20\u3002</p> <p>\u4e3a\u6b64\uff0c\u8003\u8651\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a <code>TOMBSTONE</code> \u7684\u7d22\u5f15\uff0c\u5e76\u5c06\u641c\u7d22\u5230\u7684\u76ee\u6807\u5143\u7d20\u4e0e\u8be5 <code>TOMBSTONE</code> \u4ea4\u6362\u4f4d\u7f6e\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u5f53\u6bcf\u6b21\u67e5\u8be2\u6216\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u81f3\u8ddd\u79bb\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6d4b\u8d77\u59cb\u70b9\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u4ece\u800c\u4f18\u5316\u67e5\u8be2\u6548\u7387\u3002</p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u5305\u542b\u61d2\u5220\u9664\u7684\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u3002\u4e3a\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u770b\u4f5c\u4e00\u4e2a\u201c\u73af\u5f62\u6570\u7ec4\u201d\uff0c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n\"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\nself.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\nself.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nself.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\nself.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\nself.TOMBSTONE = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\ndef hash_func(self, key: int) -&gt; int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nreturn key % self.capacity\ndef load_factor(self) -&gt; float:\n\"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\nreturn self.size / self.capacity\ndef find_bucket(self, key: int) -&gt; int:\n\"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\nindex = self.hash_func(key)\nfirst_tombstone = -1\n# \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile self.buckets[index] is not None:\n# \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif self.buckets[index].key == key:\n# \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif first_tombstone != -1:\nself.buckets[first_tombstone] = self.buckets[index]\nself.buckets[index] = self.TOMBSTONE\nreturn first_tombstone  # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\nreturn index  # \u8fd4\u56de\u6876\u7d22\u5f15\n# \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\nfirst_tombstone = index\n# \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % self.capacity\n# \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn index if first_tombstone == -1 else first_tombstone\ndef get(self, key: int) -&gt; str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n# \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nindex = self.find_bucket(key)\n# \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif self.buckets[index] not in [None, self.TOMBSTONE]:\nreturn self.buckets[index].val\n# \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\nreturn None\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n# \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() &gt; self.load_thres:\nself.extend()\n# \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nindex = self.find_bucket(key)\n# \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif self.buckets[index] not in [None, self.TOMBSTONE]:\nself.buckets[index].val = val\nreturn\n# \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nself.buckets[index] = Pair(key, val)\nself.size += 1\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n# \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nindex = self.find_bucket(key)\n# \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif self.buckets[index] not in [None, self.TOMBSTONE]:\nself.buckets[index] = self.TOMBSTONE\nself.size -= 1\ndef extend(self):\n\"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n# \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nbuckets_tmp = self.buckets\n# \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio\nself.buckets = [None] * self.capacity\nself.size = 0\n# \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor pair in buckets_tmp:\nif pair not in [None, self.TOMBSTONE]:\nself.put(pair.key, pair.val)\ndef print(self):\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor pair in self.buckets:\nif pair is None:\nprint(\"None\")\nelif pair is self.TOMBSTONE:\nprint(\"TOMBSTONE\")\nelse:\nprint(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nprivate:\nint size;                             // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity = 4;                     // \u54c8\u5e0c\u8868\u5bb9\u91cf\nconst double loadThres = 2.0 / 3.0;     // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nconst int extendRatio = 2;            // \u6269\u5bb9\u500d\u6570\nvector&lt;Pair *&gt; buckets;               // \u6876\u6570\u7ec4\nPair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~HashMapOpenAddressing() {\nfor (Pair *pair : buckets) {\nif (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\ndelete pair;\n}\n}\ndelete TOMBSTONE;\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn (double)size / capacity;\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(int key) {\nint index = hashFunc(key);\nint firstTombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile (buckets[index] != nullptr) {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif (buckets[index]-&gt;key == key) {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif (firstTombstone != -1) {\nbuckets[firstTombstone] = buckets[index];\nbuckets[index] = TOMBSTONE;\nreturn firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index; // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\nfirstTombstone = index;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone == -1 ? index : firstTombstone;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\nreturn buckets[index]-&gt;val;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\";\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() &gt; loadThres) {\nextend();\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\nbuckets[index]-&gt;val = val;\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nbuckets[index] = new Pair(key, val);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\ndelete buckets[index];\nbuckets[index] = TOMBSTONE;\nsize--;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nvector&lt;Pair *&gt; bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = vector&lt;Pair *&gt;(capacity, nullptr);\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair *pair : bucketsTmp) {\nif (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\nput(pair-&gt;key, pair-&gt;val);\ndelete pair;\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (Pair *pair : buckets) {\nif (pair == nullptr) {\ncout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n} else if (pair == TOMBSTONE) {\ncout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n} else {\ncout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n}\n}\n}\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nprivate int size; // \u952e\u503c\u5bf9\u6570\u91cf\nprivate int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nprivate final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nprivate final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\nprivate Pair[] buckets; // \u6876\u6570\u7ec4\nprivate final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapOpenAddressing() {\nsize = 0;\nbuckets = new Pair[capacity];\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nprivate double loadFactor() {\nreturn (double) size / capacity;\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nprivate int findBucket(int key) {\nint index = hashFunc(key);\nint firstTombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile (buckets[index] != null) {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif (buckets[index].key == key) {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif (firstTombstone != -1) {\nbuckets[firstTombstone] = buckets[index];\nbuckets[index] = TOMBSTONE;\nreturn firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index; // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\nfirstTombstone = index;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone == -1 ? index : firstTombstone;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\nreturn buckets[index].val;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() &gt; loadThres) {\nextend();\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\nbuckets[index].val = val;\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nbuckets[index] = new Pair(key, val);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\nbuckets[index] = TOMBSTONE;\nsize--;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nprivate void extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nPair[] bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new Pair[capacity];\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair pair : bucketsTmp) {\nif (pair != null &amp;&amp; pair != TOMBSTONE) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Pair pair : buckets) {\nif (pair == null) {\nSystem.out.println(\"null\");\n} else if (pair == TOMBSTONE) {\nSystem.out.println(\"TOMBSTONE\");\n} else {\nSystem.out.println(pair.key + \" -&gt; \" + pair.val);\n}\n}\n}\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nint size; // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio = 2; // \u6269\u5bb9\u500d\u6570\nPair[] buckets; // \u6876\u6570\u7ec4\nPair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\npublic HashMapOpenAddressing() {\nsize = 0;\nbuckets = new Pair[capacity];\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint HashFunc(int key) {\nreturn key % capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble LoadFactor() {\nreturn (double)size / capacity;\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint FindBucket(int key) {\nint index = HashFunc(key);\nint firstTombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile (buckets[index] != null) {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif (buckets[index].key == key) {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif (firstTombstone != -1) {\nbuckets[firstTombstone] = buckets[index];\nbuckets[index] = TOMBSTONE;\nreturn firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index; // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\nfirstTombstone = index;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone == -1 ? index : firstTombstone;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string? Get(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = FindBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\nreturn buckets[index].val;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void Put(int key, string val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (LoadFactor() &gt; loadThres) {\nExtend();\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = FindBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\nbuckets[index].val = val;\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nbuckets[index] = new Pair(key, val);\nsize++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void Remove(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = FindBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\nbuckets[index] = TOMBSTONE;\nsize--;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid Extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nPair[] bucketsTmp = buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio;\nbuckets = new Pair[capacity];\nsize = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nforeach (Pair pair in bucketsTmp) {\nif (pair != null &amp;&amp; pair != TOMBSTONE) {\nPut(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void Print() {\nforeach (Pair pair in buckets) {\nif (pair == null) {\nConsole.WriteLine(\"null\");\n} else if (pair == TOMBSTONE) {\nConsole.WriteLine(\"TOMBSTONE\");\n} else {\nConsole.WriteLine(pair.key + \" -&gt; \" + pair.val);\n}\n}\n}\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\nsize        int     // \u952e\u503c\u5bf9\u6570\u91cf\ncapacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\nloadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nextendRatio int     // \u6269\u5bb9\u500d\u6570\nbuckets     []pair  // \u6876\u6570\u7ec4\nremoved     pair    // \u5220\u9664\u6807\u8bb0\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\nbuckets := make([]pair, 4)\nreturn &amp;hashMapOpenAddressing{\nsize:        0,\ncapacity:    4,\nloadThres:   2.0 / 3.0,\nextendRatio: 2,\nbuckets:     buckets,\nremoved: pair{\nkey: -1,\nval: \"-1\",\n},\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapOpenAddressing) hashFunc(key int) int {\nreturn key % m.capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapOpenAddressing) loadFactor() float64 {\nreturn float64(m.size) / float64(m.capacity)\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) get(key int) string {\nidx := m.hashFunc(key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i &lt; m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + i) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\nif m.buckets[j] == (pair{}) {\nreturn \"\"\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif m.buckets[j].key == key &amp;&amp; m.buckets[j] != m.removed {\nreturn m.buckets[j].val\n}\n}\n// \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\"\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) put(key int, val string) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif m.loadFactor() &gt; m.loadThres {\nm.extend()\n}\nidx := m.hashFunc(key)\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i &lt; m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + i) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\nif m.buckets[j] == (pair{}) || m.buckets[j] == m.removed {\nm.buckets[j] = pair{\nkey: key,\nval: val,\n}\nm.size += 1\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\nif m.buckets[j].key == key {\nm.buckets[j].val = val\n}\n}\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) remove(key int) {\nidx := m.hashFunc(key)\n// \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\nfor i := 0; i &lt; m.capacity; i++ {\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nj := (idx + i) % m.capacity\n// \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif m.buckets[j] == (pair{}) {\nreturn\n}\n// \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\nif m.buckets[j].key == key {\nm.buckets[j] = m.removed\nm.size -= 1\n}\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\ntmpBuckets := make([]pair, len(m.buckets))\ncopy(tmpBuckets, m.buckets)\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nm.capacity *= m.extendRatio\nm.buckets = make([]pair, m.capacity)\nm.size = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor _, p := range tmpBuckets {\nif p != (pair{}) &amp;&amp; p != m.removed {\nm.put(p.key, p.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) print() {\nfor _, p := range m.buckets {\nif p != (pair{}) {\nfmt.Println(strconv.Itoa(p.key) + \" -&gt; \" + p.val)\n} else {\nfmt.Println(\"nil\")\n}\n}\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nvar size: Int // \u952e\u503c\u5bf9\u6570\u91cf\nvar capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\nvar loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nvar extendRatio: Int // \u6269\u5bb9\u500d\u6570\nvar buckets: [Pair?] // \u6876\u6570\u7ec4\nvar TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\ninit() {\nsize = 0\ncapacity = 4\nloadThres = 2.0 / 3.0\nextendRatio = 2\nbuckets = Array(repeating: nil, count: capacity)\nTOMBSTONE = Pair(key: -1, val: \"-1\")\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc hashFunc(key: Int) -&gt; Int {\nkey % capacity\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc loadFactor() -&gt; Double {\nDouble(size / capacity)\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nfunc findBucket(key: Int) -&gt; Int {\nvar index = hashFunc(key: key)\nvar firstTombstone = -1\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile buckets[index] != nil {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif buckets[index]!.key == key {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif firstTombstone != -1 {\nbuckets[firstTombstone] = buckets[index]\nbuckets[index] = TOMBSTONE\nreturn firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\nfirstTombstone = index\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % capacity\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone == -1 ? index : firstTombstone\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -&gt; String? {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nlet index = findBucket(key: key)\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif buckets[index] != nil, buckets[index] != TOMBSTONE {\nreturn buckets[index]!.val\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\nreturn nil\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif loadFactor() &gt; loadThres {\nextend()\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nlet index = findBucket(key: key)\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif buckets[index] != nil, buckets[index] != TOMBSTONE {\nbuckets[index]!.val = val\nreturn\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nbuckets[index] = Pair(key: key, val: val)\nsize += 1\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nlet index = findBucket(key: key)\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif buckets[index] != nil, buckets[index] != TOMBSTONE {\nbuckets[index] = TOMBSTONE\nsize -= 1\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet bucketsTmp = buckets\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\ncapacity *= extendRatio\nbuckets = Array(repeating: nil, count: capacity)\nsize = 0\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor pair in bucketsTmp {\nif let pair, pair != TOMBSTONE {\nput(key: pair.key, val: pair.val)\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor pair in buckets {\nif pair == nil {\nSwift.print(\"null\")\n} else if pair == TOMBSTONE {\nSwift.print(\"TOMBSTONE\")\n} else {\nSwift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n}\n}\n}\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n#size; // \u952e\u503c\u5bf9\u6570\u91cf\n#capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n#loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n#extendRatio; // \u6269\u5bb9\u500d\u6570\n#buckets; // \u6876\u6570\u7ec4\n#TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\nthis.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nthis.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nthis.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\nthis.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\nthis.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key) {\nreturn key % this.#capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\n#loadFactor() {\nreturn this.#size / this.#capacity;\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n#findBucket(key) {\nlet index = this.#hashFunc(key);\nlet firstTombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile (this.#buckets[index] !== null) {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif (this.#buckets[index].key === key) {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif (firstTombstone !== -1) {\nthis.#buckets[firstTombstone] = this.#buckets[index];\nthis.#buckets[index] = this.#TOMBSTONE;\nreturn firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index; // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif (\nfirstTombstone === -1 &amp;&amp;\nthis.#buckets[index] === this.#TOMBSTONE\n) {\nfirstTombstone = index;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % this.#capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone === -1 ? index : firstTombstone;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nconst index = this.#findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (\nthis.#buckets[index] !== null &amp;&amp;\nthis.#buckets[index] !== this.#TOMBSTONE\n) {\nreturn this.#buckets[index].val;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nput(key, val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (this.#loadFactor() &gt; this.#loadThres) {\nthis.#extend();\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nconst index = this.#findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif (\nthis.#buckets[index] !== null &amp;&amp;\nthis.#buckets[index] !== this.#TOMBSTONE\n) {\nthis.#buckets[index].val = val;\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nthis.#buckets[index] = new Pair(key, val);\nthis.#size++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nremove(key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nconst index = this.#findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif (\nthis.#buckets[index] !== null &amp;&amp;\nthis.#buckets[index] !== this.#TOMBSTONE\n) {\nthis.#buckets[index] = this.#TOMBSTONE;\nthis.#size--;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n#extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nconst bucketsTmp = this.#buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nthis.#capacity *= this.#extendRatio;\nthis.#buckets = Array(this.#capacity).fill(null);\nthis.#size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (const pair of bucketsTmp) {\nif (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\nthis.put(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint() {\nfor (const pair of this.#buckets) {\nif (pair === null) {\nconsole.log('null');\n} else if (pair === this.#TOMBSTONE) {\nconsole.log('TOMBSTONE');\n} else {\nconsole.log(pair.key + ' -&gt; ' + pair.val);\n}\n}\n}\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nprivate size: number; // \u952e\u503c\u5bf9\u6570\u91cf\nprivate capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nprivate loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nprivate extendRatio: number; // \u6269\u5bb9\u500d\u6570\nprivate buckets: Array&lt;Pair | null&gt;; // \u6876\u6570\u7ec4\nprivate TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor() {\nthis.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\nthis.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\nthis.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nthis.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\nthis.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\nthis.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate hashFunc(key: number): number {\nreturn key % this.capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nprivate loadFactor(): number {\nreturn this.size / this.capacity;\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nprivate findBucket(key: number): number {\nlet index = this.hashFunc(key);\nlet firstTombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile (this.buckets[index] !== null) {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif (this.buckets[index]!.key === key) {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif (firstTombstone !== -1) {\nthis.buckets[firstTombstone] = this.buckets[index];\nthis.buckets[index] = this.TOMBSTONE;\nreturn firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index; // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif (\nfirstTombstone === -1 &amp;&amp;\nthis.buckets[index] === this.TOMBSTONE\n) {\nfirstTombstone = index;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % this.capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone === -1 ? index : firstTombstone;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key: number): string | null {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nconst index = this.findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (\nthis.buckets[index] !== null &amp;&amp;\nthis.buckets[index] !== this.TOMBSTONE\n) {\nreturn this.buckets[index]!.val;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nput(key: number, val: string): void {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (this.loadFactor() &gt; this.loadThres) {\nthis.extend();\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nconst index = this.findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif (\nthis.buckets[index] !== null &amp;&amp;\nthis.buckets[index] !== this.TOMBSTONE\n) {\nthis.buckets[index]!.val = val;\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nthis.buckets[index] = new Pair(key, val);\nthis.size++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nremove(key: number): void {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nconst index = this.findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif (\nthis.buckets[index] !== null &amp;&amp;\nthis.buckets[index] !== this.TOMBSTONE\n) {\nthis.buckets[index] = this.TOMBSTONE;\nthis.size--;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nprivate extend(): void {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nconst bucketsTmp = this.buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nthis.capacity *= this.extendRatio;\nthis.buckets = Array(this.capacity).fill(null);\nthis.size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (const pair of bucketsTmp) {\nif (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\nthis.put(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint(): void {\nfor (const pair of this.buckets) {\nif (pair === null) {\nconsole.log('null');\n} else if (pair === this.TOMBSTONE) {\nconsole.log('TOMBSTONE');\n} else {\nconsole.log(pair.key + ' -&gt; ' + pair.val);\n}\n}\n}\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\nlate int _size; // \u952e\u503c\u5bf9\u6570\u91cf\nint _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\nlate List&lt;Pair?&gt; _buckets; // \u6876\u6570\u7ec4\nPair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n/* \u6784\u9020\u65b9\u6cd5 */\nHashMapOpenAddressing() {\n_size = 0;\n_buckets = List.generate(_capacity, (index) =&gt; null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nreturn key % _capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor() {\nreturn _size / _capacity;\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(int key) {\nint index = hashFunc(key);\nint firstTombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile (_buckets[index] != null) {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif (_buckets[index]!.key == key) {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif (firstTombstone != -1) {\n_buckets[firstTombstone] = _buckets[index];\n_buckets[index] = _TOMBSTONE;\nreturn firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index; // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\nfirstTombstone = index;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % _capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone == -1 ? index : firstTombstone;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\nreturn _buckets[index]!.val;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\nreturn null;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor() &gt; _loadThres) {\nextend();\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n_buckets[index]!.val = val;\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n_buckets[index] = new Pair(key, val);\n_size++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n_buckets[index] = _TOMBSTONE;\n_size--;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend() {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nList&lt;Pair?&gt; bucketsTmp = _buckets;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n_capacity *= _extendRatio;\n_buckets = List.generate(_capacity, (index) =&gt; null);\n_size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (Pair? pair in bucketsTmp) {\nif (pair != null &amp;&amp; pair != _TOMBSTONE) {\nput(pair.key, pair.val);\n}\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (Pair? pair in _buckets) {\nif (pair == null) {\nprint(\"null\");\n} else if (pair == _TOMBSTONE) {\nprint(\"TOMBSTONE\");\n} else {\nprint(\"${pair.key} -&gt; ${pair.val}\");\n}\n}\n}\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\nsize: usize,                    // \u952e\u503c\u5bf9\u6570\u91cf\ncapacity: usize,                // \u54c8\u5e0c\u8868\u5bb9\u91cf\nload_thres: f64,                // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nextend_ratio: usize,            // \u6269\u5bb9\u500d\u6570\nbuckets: Vec&lt;Option&lt;Pair&gt;&gt;,     // \u6876\u6570\u7ec4\nTOMBSTONE: Option&lt;Pair&gt;,        // \u5220\u9664\u6807\u8bb0\n}\nimpl HashMapOpenAddressing {\n/* \u6784\u9020\u65b9\u6cd5 */\nfn new() -&gt; Self {\nSelf {\nsize: 0,\ncapacity: 4,\nload_thres: 2.0 / 3.0,\nextend_ratio: 2,\nbuckets: vec![None; 4],\nTOMBSTONE: Some(Pair {key: -1, val: \"-1\".to_string()}),\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfn hash_func(&amp;self, key: i32) -&gt; usize {\n(key % self.capacity as i32) as usize\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfn load_factor(&amp;self) -&gt; f64 {\nself.size as f64 / self.capacity as f64\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nfn find_bucket(&amp;mut self, key: i32) -&gt; usize {\nlet mut index = self.hash_func(key);\nlet mut first_tombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile self.buckets[index].is_some() {\n// \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nif self.buckets[index].as_ref().unwrap().key == key {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif first_tombstone != -1 {\nself.buckets[first_tombstone as usize] = self.buckets[index].take();\nself.buckets[index] = self.TOMBSTONE.clone();\nreturn first_tombstone as usize;    // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index;   // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\nfirst_tombstone = index as i32;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % self.capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nif first_tombstone == -1 { index } else { first_tombstone as usize }\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nlet index = self.find_bucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\nreturn self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\nNone\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfn put(&amp;mut self, key: i32, val: String) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif self.load_factor() &gt; self.load_thres {\nself.extend();\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nlet index = self.find_bucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\nself.buckets[index].as_mut().unwrap().val = val;\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nself.buckets[index] = Some(Pair { key, val });\nself.size += 1;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfn remove(&amp;mut self, key: i32) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nlet index = self.find_bucket(key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\nself.buckets[index] = self.TOMBSTONE.clone();\nself.size -= 1;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfn extend(&amp;mut self) {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nlet buckets_tmp = self.buckets.clone();\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nself.capacity *= self.extend_ratio;\nself.buckets = vec![None; self.capacity];\nself.size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor pair in buckets_tmp {\nif pair.is_none() || pair == self.TOMBSTONE {\ncontinue;\n}\nlet pair = pair.unwrap();\nself.put(pair.key, pair.val);\n}\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfn print(&amp;self) {\nfor pair in &amp;self.buckets {\nif pair.is_none() {\nprintln!(\"null\");\n} else if pair == &amp;self.TOMBSTONE {\nprintln!(\"TOMBSTONE\");\n} else {\nlet pair = pair.as_ref().unwrap();\nprintln!(\"{} -&gt; {}\", pair.key, pair.val);\n}\n}\n}\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\nint size;         // \u952e\u503c\u5bf9\u6570\u91cf\nint capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\ndouble loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\nint extendRatio;  // \u6269\u5bb9\u500d\u6570\nPair **buckets;   // \u6876\u6570\u7ec4\nPair *TOMBSTONE;  // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\nHashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\nhashMap-&gt;size = 0;\nhashMap-&gt;capacity = 4;\nhashMap-&gt;loadThres = 2.0 / 3.0;\nhashMap-&gt;extendRatio = 2;\nhashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\nhashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\nhashMap-&gt;TOMBSTONE-&gt;key = -1;\nhashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\nreturn hashMap;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\nfor (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\nPair *pair = hashMap-&gt;buckets[i];\nif (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\nfree(pair-&gt;val);\nfree(pair);\n}\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\nreturn key % hashMap-&gt;capacity;\n}\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\nreturn (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\nint index = hashFunc(hashMap, key);\nint firstTombstone = -1;\n// \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\nwhile (hashMap-&gt;buckets[index] != NULL) {\n// \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\nif (hashMap-&gt;buckets[index]-&gt;key == key) {\n// \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\nif (firstTombstone != -1) {\nhashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\nhashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\nreturn firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n}\nreturn index; // \u8fd4\u56de\u6876\u7d22\u5f15\n}\n// \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\nif (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\nfirstTombstone = index;\n}\n// \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\nindex = (index + 1) % hashMap-&gt;capacity;\n}\n// \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\nreturn firstTombstone == -1 ? index : firstTombstone;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(hashMap, key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\nif (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\nreturn hashMap-&gt;buckets[index]-&gt;val;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\nreturn \"\";\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n// \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\nif (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\nextend(hashMap);\n}\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(hashMap, key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\nif (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\nfree(hashMap-&gt;buckets[index]-&gt;val);\nhashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val + 1)));\nstrcpy(hashMap-&gt;buckets[index]-&gt;val, val);\nhashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\nreturn;\n}\n// \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\nPair *pair = (Pair *)malloc(sizeof(Pair));\npair-&gt;key = key;\npair-&gt;val = (char *)malloc(sizeof(strlen(val + 1)));\nstrcpy(pair-&gt;val, val);\npair-&gt;val[strlen(val)] = '\\0';\nhashMap-&gt;buckets[index] = pair;\nhashMap-&gt;size++;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n// \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\nint index = findBucket(hashMap, key);\n// \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\nif (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\nPair *pair = hashMap-&gt;buckets[index];\nfree(pair-&gt;val);\nfree(pair);\nhashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\nhashMap-&gt;size--;\n}\n}\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n// \u6682\u5b58\u539f\u54c8\u5e0c\u8868\nPair **bucketsTmp = hashMap-&gt;buckets;\nint oldCapacity = hashMap-&gt;capacity;\n// \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\nhashMap-&gt;capacity *= hashMap-&gt;extendRatio;\nhashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\nhashMap-&gt;size = 0;\n// \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\nfor (int i = 0; i &lt; oldCapacity; i++) {\nPair *pair = bucketsTmp[i];\nif (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\nput(hashMap, pair-&gt;key, pair-&gt;val);\nfree(pair-&gt;val);\nfree(pair);\n}\n}\nfree(bucketsTmp);\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\nfor (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\nPair *pair = hashMap-&gt;buckets[i];\nif (pair == NULL) {\nprintf(\"NULL\\n\");\n} else if (pair == hashMap-&gt;TOMBSTONE) {\nprintf(\"TOMBSTONE\\n\");\n} else {\nprintf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n}\n}\n}\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6d4b","text":"<p>\u5e73\u65b9\u63a2\u6d4b\u4e0e\u7ebf\u6027\u63a2\u6d4b\u7c7b\u4f3c\uff0c\u90fd\u662f\u5f00\u653e\u5bfb\u5740\u7684\u5e38\u89c1\u7b56\u7565\u4e4b\u4e00\u3002\u5f53\u53d1\u751f\u51b2\u7a81\u65f6\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e0d\u662f\u7b80\u5355\u5730\u8df3\u8fc7\u4e00\u4e2a\u56fa\u5b9a\u7684\u6b65\u6570\uff0c\u800c\u662f\u8df3\u8fc7\u201c\u63a2\u6d4b\u6b21\u6570\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6570\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002</p> <p>\u5e73\u65b9\u63a2\u6d4b\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u4f18\u52bf\u3002</p> <ul> <li>\u5e73\u65b9\u63a2\u6d4b\u901a\u8fc7\u8df3\u8fc7\u5e73\u65b9\u7684\u8ddd\u79bb\uff0c\u8bd5\u56fe\u7f13\u89e3\u7ebf\u6027\u63a2\u6d4b\u7684\u805a\u96c6\u6548\u5e94\u3002</li> <li>\u5e73\u65b9\u63a2\u6d4b\u4f1a\u8df3\u8fc7\u66f4\u5927\u7684\u8ddd\u79bb\u6765\u5bfb\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u4e8e\u6570\u636e\u5206\u5e03\u5f97\u66f4\u52a0\u5747\u5300\u3002</li> </ul> <p>\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u5e76\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002</p> <ul> <li>\u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73b0\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u5360\u7528\u3002</li> <li>\u7531\u4e8e\u5e73\u65b9\u7684\u589e\u957f\uff0c\u5e73\u65b9\u63a2\u6d4b\u53ef\u80fd\u4e0d\u4f1a\u63a2\u6d4b\u6574\u4e2a\u54c8\u5e0c\u8868\uff0c\u8fd9\u610f\u5473\u7740\u5373\u4f7f\u54c8\u5e0c\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u53ef\u80fd\u65e0\u6cd5\u8bbf\u95ee\u5230\u5b83\u3002</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3","title":"3. \u00a0 \u591a\u6b21\u54c8\u5e0c","text":"<p>\u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u8fdb\u884c\u63a2\u6d4b\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u6876\u540e\u63d2\u5165\u5143\u7d20\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u6876\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(\\text{None}\\) \u3002</li> </ul> <p>\u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u5e26\u6765\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\uff09\u54c8\u5e0c\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u201d\u7684\u95ee\u9898\u3002</p>"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9","text":"<p>\u5404\u4e2a\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u7b56\u7565\uff0c\u4ee5\u4e0b\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002</li> <li>Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 HashMap \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002</li> <li>Go \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\u3002\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 \u54c8\u5e0c\u8868","text":"<p>\u300c\u54c8\u5e0c\u8868 hash table\u300d\uff0c\u53c8\u79f0\u300c\u6563\u5217\u8868\u300d\uff0c\u5176\u901a\u8fc7\u5efa\u7acb\u952e <code>key</code> \u4e0e\u503c <code>value</code> \u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u8f93\u5165\u4e00\u4e2a\u952e <code>key</code> \uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684\u503c <code>value</code> \u3002</p> <p>\u5982\u56fe 6-1 \u6240\u793a\uff0c\u7ed9\u5b9a \\(n\\) \u4e2a\u5b66\u751f\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u56fe 6-1 \u6240\u793a\u7684\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</p> <p></p> <p> \u56fe 6-1 \u00a0 \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a </p> <p>\u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u4e5f\u53ef\u4ee5\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\uff0c\u5b83\u4eec\u7684\u6548\u7387\u5bf9\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u4ec5\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\uff08\u94fe\u8868\uff09\u662f\u4e71\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8be2\u5230\u5143\u7d20\uff0c\u518d\u4ece\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> </ul> <p> \u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8be2\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"<p>\u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\uff0c\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py<pre><code># \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nhmap: dict = {}\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map&lt;int, string&gt; map;\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   map.put(15937, \"\u5c0f\u5570\");   map.put(16750, \"\u5c0f\u7b97\");   map.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary&lt;int, string&gt; map = new() {\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\n{ 12836, \"\u5c0f\u54c8\" },\n{ 15937, \"\u5c0f\u5570\" },\n{ 16750, \"\u5c0f\u7b97\" },\n{ 13276, \"\u5c0f\u6cd5\" },\n{ 10583, \"\u5c0f\u9e2d\" }\n};\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nhmap := make(map[int]string)\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map&lt;number, string&gt;();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;int, String&gt; map = {};\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u5570\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9e2d\".to_string());\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.zig<pre><code>\n</code></pre> <p>\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py<pre><code># \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value in hmap.items():\nprint(key, \"-&gt;\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in hmap.keys():\nprint(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in hmap.values():\nprint(value)\n</code></pre> hash_map.cpp<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (auto kv: map) {\ncout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u904d\u5386 key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\ncout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\nSystem.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\nSystem.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\nSystem.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nforeach (var kv in map) {\nConsole.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys) {\nConsole.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (string val in map.Values) {\nConsole.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value := range hmap {\nfmt.Println(key, \"-&gt;\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range hmap {\nfmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range hmap {\nfmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in map {\nprint(\"\\(key) -&gt; \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\nprint(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\nprint(value)\n}\n</code></pre> hash_map.js<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\nconsole.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\nconsole.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\nconsole.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nmap.forEach((key, value) {\nprint('$key -&gt; $value');\n});\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\nprint(key);\n});\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\nprint(value);\n});\n</code></pre> hash_map.rs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in &amp;map {\nprintln!(\"{key} -&gt; {value}\");\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys() {\nprintln!(\"{key}\"); }\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values() {\nprintln!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.zig<pre><code>\n</code></pre>"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 bucket\u300d\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230 <code>key</code> \u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e <code>key</code> \u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u300c\u54c8\u5e0c\u51fd\u6570 hash function\u300d\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709 <code>key</code> \uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002</p> <p>\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u901a\u8fc7\u67d0\u79cd\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\u3002</li> <li>\u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09<code>capacity</code> \u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15 <code>index</code> \u3002</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528 <code>index</code> \u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u8bbe\u6570\u7ec4\u957f\u5ea6 <code>capacity = 100</code>\u3001\u54c8\u5e0c\u7b97\u6cd5 <code>hash(key) = key</code> \uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3a <code>key % 100</code> \u3002\u56fe 6-2 \u4ee5 <code>key</code> \u5b66\u53f7\u548c <code>value</code> \u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p></p> <p> \u56fe 6-2 \u00a0 \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406 </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06 <code>key</code> \u548c <code>value</code> \u5c01\u88c5\u6210\u4e00\u4e2a\u7c7b <code>Pair</code> \uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_hash_map.py<pre><code>class Pair:\n\"\"\"\u952e\u503c\u5bf9\"\"\"\ndef __init__(self, key: int, val: str):\nself.key = key\nself.val = val\nclass ArrayHashMap:\n\"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n# \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nself.buckets: list[Pair | None] = [None] * 100\ndef hash_func(self, key: int) -&gt; int:\n\"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\nindex = key % 100\nreturn index\ndef get(self, key: int) -&gt; str:\n\"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\nindex: int = self.hash_func(key)\npair: Pair = self.buckets[index]\nif pair is None:\nreturn None\nreturn pair.val\ndef put(self, key: int, val: str):\n\"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\npair = Pair(key, val)\nindex: int = self.hash_func(key)\nself.buckets[index] = pair\ndef remove(self, key: int):\n\"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\nindex: int = self.hash_func(key)\n# \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets[index] = None\ndef entry_set(self) -&gt; list[Pair]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\nresult: list[Pair] = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair)\nreturn result\ndef key_set(self) -&gt; list[int]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\nresult = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.key)\nreturn result\ndef value_set(self) -&gt; list[str]:\n\"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\nresult = []\nfor pair in self.buckets:\nif pair is not None:\nresult.append(pair.val)\nreturn result\ndef print(self):\n\"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\nfor pair in self.buckets:\nif pair is not None:\nprint(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u952e\u503c\u5bf9 */\nstruct Pair {\npublic:\nint key;\nstring val;\nPair(int key, string val) {\nthis-&gt;key = key;\nthis-&gt;val = val;\n}\n};\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate:\nvector&lt;Pair *&gt; buckets;\npublic:\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = vector&lt;Pair *&gt;(100);\n}\n~ArrayHashMap() {\n// \u91ca\u653e\u5185\u5b58\nfor (const auto &amp;bucket : buckets) {\ndelete bucket;\n}\nbuckets.clear();\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nstring get(int key) {\nint index = hashFunc(key);\nPair *pair = buckets[index];\nif (pair == nullptr)\nreturn \"\";\nreturn pair-&gt;val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, string val) {\nPair *pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nint index = hashFunc(key);\n// \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\ndelete buckets[index];\nbuckets[index] = nullptr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvector&lt;Pair *&gt; pairSet() {\nvector&lt;Pair *&gt; pairSet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\npairSet.push_back(pair);\n}\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvector&lt;int&gt; keySet() {\nvector&lt;int&gt; keySet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\nkeySet.push_back(pair-&gt;key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvector&lt;string&gt; valueSet() {\nvector&lt;string&gt; valueSet;\nfor (Pair *pair : buckets) {\nif (pair != nullptr) {\nvalueSet.push_back(pair-&gt;val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print() {\nfor (Pair *kv : pairSet()) {\ncout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n}\n}\n};\n</code></pre> array_hash_map.java<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\npublic int key;\npublic String val;\npublic Pair(int key, String val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate List&lt;Pair&gt; buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = new ArrayList&lt;&gt;();\nfor (int i = 0; i &lt; 100; i++) {\nbuckets.add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate int hashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic String get(int key) {\nint index = hashFunc(key);\nPair pair = buckets.get(index);\nif (pair == null)\nreturn null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void put(int key, String val) {\nPair pair = new Pair(key, val);\nint index = hashFunc(key);\nbuckets.set(index, pair);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void remove(int key) {\nint index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets.set(index, null);\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List&lt;Pair&gt; pairSet() {\nList&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\nfor (Pair pair : buckets) {\nif (pair != null)\npairSet.add(pair);\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List&lt;Integer&gt; keySet() {\nList&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\nfor (Pair pair : buckets) {\nif (pair != null)\nkeySet.add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List&lt;String&gt; valueSet() {\nList&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\nfor (Pair pair : buckets) {\nif (pair != null)\nvalueSet.add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void print() {\nfor (Pair kv : pairSet()) {\nSystem.out.println(kv.key + \" -&gt; \" + kv.val);\n}\n}\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\nclass Pair(int key, string val) {\npublic int key = key;\npublic string val = val;\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nList&lt;Pair?&gt; buckets;\npublic ArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets = [];\nfor (int i = 0; i &lt; 100; i++) {\nbuckets.Add(null);\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint HashFunc(int key) {\nint index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic string? Get(int key) {\nint index = HashFunc(key);\nPair? pair = buckets[index];\nif (pair == null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic void Put(int key, string val) {\nPair pair = new(key, val);\nint index = HashFunc(key);\nbuckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic void Remove(int key) {\nint index = HashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic List&lt;Pair&gt; PairSet() {\nList&lt;Pair&gt; pairSet = [];\nforeach (Pair? pair in buckets) {\nif (pair != null)\npairSet.Add(pair);\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic List&lt;int&gt; KeySet() {\nList&lt;int&gt; keySet = [];\nforeach (Pair? pair in buckets) {\nif (pair != null)\nkeySet.Add(pair.key);\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic List&lt;string&gt; ValueSet() {\nList&lt;string&gt; valueSet = [];\nforeach (Pair? pair in buckets) {\nif (pair != null)\nvalueSet.Add(pair.val);\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic void Print() {\nforeach (Pair kv in PairSet()) {\nConsole.WriteLine(kv.key + \" -&gt; \" + kv.val);\n}\n}\n}\n</code></pre> array_hash_map.go<pre><code>/* \u952e\u503c\u5bf9 */\ntype pair struct {\nkey int\nval string\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\nbuckets []*pair\n}\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nbuckets := make([]*pair, 100)\nreturn &amp;arrayHashMap{buckets: buckets}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\nindex := key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\nindex := a.hashFunc(key)\npair := a.buckets[index]\nif pair == nil {\nreturn \"Not Found\"\n}\nreturn pair.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\npair := &amp;pair{key: key, val: val}\nindex := a.hashFunc(key)\na.buckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\nindex := a.hashFunc(key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\na.buckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\nvar pairs []*pair\nfor _, pair := range a.buckets {\nif pair != nil {\npairs = append(pairs, pair)\n}\n}\nreturn pairs\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\nvar keys []int\nfor _, pair := range a.buckets {\nif pair != nil {\nkeys = append(keys, pair.key)\n}\n}\nreturn keys\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\nvar values []string\nfor _, pair := range a.buckets {\nif pair != nil {\nvalues = append(values, pair.val)\n}\n}\nreturn values\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\nfor _, pair := range a.buckets {\nif pair != nil {\nfmt.Println(pair.key, \"-&gt;\", pair.val)\n}\n}\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\npublic var key: Int\npublic var val: String\npublic init(key: Int, val: String) {\nself.key = key\nself.val = val\n}\npublic static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\nlhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate var buckets: [Pair?] = []\ninit() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nfor _ in 0 ..&lt; 100 {\nbuckets.append(nil)\n}\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate func hashFunc(key: Int) -&gt; Int {\nlet index = key % 100\nreturn index\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc get(key: Int) -&gt; String? {\nlet index = hashFunc(key: key)\nlet pair = buckets[index]\nreturn pair?.val\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc put(key: Int, val: String) {\nlet pair = Pair(key: key, val: val)\nlet index = hashFunc(key: key)\nbuckets[index] = pair\n}\n/* \u5220\u9664\u64cd\u4f5c */\nfunc remove(key: Int) {\nlet index = hashFunc(key: key)\n// \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\nbuckets[index] = nil\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nfunc pairSet() -&gt; [Pair] {\nvar pairSet: [Pair] = []\nfor pair in buckets {\nif let pair = pair {\npairSet.append(pair)\n}\n}\nreturn pairSet\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc keySet() -&gt; [Int] {\nvar keySet: [Int] = []\nfor pair in buckets {\nif let pair = pair {\nkeySet.append(pair.key)\n}\n}\nreturn keySet\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc valueSet() -&gt; [String] {\nvar valueSet: [String] = []\nfor pair in buckets {\nif let pair = pair {\nvalueSet.append(pair.val)\n}\n}\nreturn valueSet\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc print() {\nfor pair in pairSet() {\nSwift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n}\n}\n}\n</code></pre> array_hash_map.js<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\nconstructor(key, val) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n#buckets;\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.#buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\n#hashFunc(key) {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nget(key) {\nlet index = this.#hashFunc(key);\nlet pair = this.#buckets[index];\nif (pair === null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nset(key, val) {\nlet index = this.#hashFunc(key);\nthis.#buckets[index] = new Pair(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\ndelete(key) {\nlet index = this.#hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.#buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nentries() {\nlet arr = [];\nfor (let i = 0; i &lt; this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nkeys() {\nlet arr = [];\nfor (let i = 0; i &lt; this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i].key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvalues() {\nlet arr = [];\nfor (let i = 0; i &lt; this.#buckets.length; i++) {\nif (this.#buckets[i]) {\narr.push(this.#buckets[i].val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nprint() {\nlet pairSet = this.entries();\nfor (const pair of pairSet) {\nconsole.info(`${pair.key} -&gt; ${pair.val}`);\n}\n}\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\npublic key: number;\npublic val: string;\nconstructor(key: number, val: string) {\nthis.key = key;\nthis.val = val;\n}\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nprivate readonly buckets: (Pair | null)[];\nconstructor() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nthis.buckets = new Array(100).fill(null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nprivate hashFunc(key: number): number {\nreturn key % 100;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npublic get(key: number): string | null {\nlet index = this.hashFunc(key);\nlet pair = this.buckets[index];\nif (pair === null) return null;\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npublic set(key: number, val: string) {\nlet index = this.hashFunc(key);\nthis.buckets[index] = new Pair(key, val);\n}\n/* \u5220\u9664\u64cd\u4f5c */\npublic delete(key: number) {\nlet index = this.hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nthis.buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npublic entries(): (Pair | null)[] {\nlet arr: (Pair | null)[] = [];\nfor (let i = 0; i &lt; this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i]);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npublic keys(): (number | undefined)[] {\nlet arr: (number | undefined)[] = [];\nfor (let i = 0; i &lt; this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i].key);\n}\n}\nreturn arr;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npublic values(): (string | undefined)[] {\nlet arr: (string | undefined)[] = [];\nfor (let i = 0; i &lt; this.buckets.length; i++) {\nif (this.buckets[i]) {\narr.push(this.buckets[i].val);\n}\n}\nreturn arr;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npublic print() {\nlet pairSet = this.entries();\nfor (const pair of pairSet) {\nconsole.info(`${pair.key} -&gt; ${pair.val}`);\n}\n}\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\nint key;\nString val;\nPair(this.key, this.val);\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\nlate List&lt;Pair?&gt; _buckets;\nArrayHashMap() {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n_buckets = List.filled(100, null);\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nint _hashFunc(int key) {\nfinal int index = key % 100;\nreturn index;\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\nString? get(int key) {\nfinal int index = _hashFunc(key);\nfinal Pair? pair = _buckets[index];\nif (pair == null) {\nreturn null;\n}\nreturn pair.val;\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(int key, String val) {\nfinal Pair pair = Pair(key, val);\nfinal int index = _hashFunc(key);\n_buckets[index] = pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid remove(int key) {\nfinal int index = _hashFunc(key);\n_buckets[index] = null;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nList&lt;Pair&gt; pairSet() {\nList&lt;Pair&gt; pairSet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\npairSet.add(pair);\n}\n}\nreturn pairSet;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nList&lt;int&gt; keySet() {\nList&lt;int&gt; keySet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\nkeySet.add(pair.key);\n}\n}\nreturn keySet;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nList&lt;String&gt; values() {\nList&lt;String&gt; valueSet = [];\nfor (final Pair? pair in _buckets) {\nif (pair != null) {\nvalueSet.add(pair.val);\n}\n}\nreturn valueSet;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid printHashMap() {\nfor (final Pair kv in pairSet()) {\nprint(\"${kv.key} -&gt; ${kv.val}\");\n}\n}\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\npub key: i32,\npub val: String,\n}\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\nbuckets: Vec&lt;Option&lt;Pair&gt;&gt;\n}\nimpl ArrayHashMap {\npub fn new() -&gt; ArrayHashMap {\n// \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\nSelf { buckets: vec![None; 100] }\n}\n/* \u54c8\u5e0c\u51fd\u6570 */\nfn hash_func(&amp;self, key: i32) -&gt; usize {\nkey as usize % 100\n}\n/* \u67e5\u8be2\u64cd\u4f5c */\npub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\nlet index = self.hash_func(key);\nself.buckets[index].as_ref().map(|pair| &amp;pair.val)\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\npub fn put(&amp;mut self, key: i32, val: &amp;str) {\nlet index = self.hash_func(key);\nself.buckets[index] = Some(Pair {\nkey,\nval: val.to_string(),\n});\n}\n/* \u5220\u9664\u64cd\u4f5c */\npub fn remove(&amp;mut self, key: i32) {\nlet index = self.hash_func(key);\n// \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\nself.buckets[index] = None;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\npub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\nself.buckets.iter().filter_map(|pair| pair.as_ref()).collect()\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\npub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\nself.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key)).collect()\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\npub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\nself.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val)).collect()\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\npub fn print(&amp;self) {\nfor pair in self.entry_set() {\nprintln!(\"{} -&gt; {}\", pair.key, pair.val);\n}\n}\n}\n</code></pre> array_hash_map.c<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\ntypedef struct {\nint key;\nchar *val;\n} Pair;\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\nPair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\nArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\nreturn hmap;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\nfor (int i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\nif (hmap-&gt;buckets[i] != NULL) {\nfree(hmap-&gt;buckets[i]-&gt;val);\nfree(hmap-&gt;buckets[i]);\n}\n}\nfree(hmap);\n}\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\nPair *Pair = malloc(sizeof(Pair));\nPair-&gt;key = key;\nPair-&gt;val = malloc(strlen(val) + 1);\nstrcpy(Pair-&gt;val, val);\nint index = hashFunc(key);\nhmap-&gt;buckets[index] = Pair;\n}\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\nint index = hashFunc(key);\nfree(hmap-&gt;buckets[index]-&gt;val);\nfree(hmap-&gt;buckets[index]);\nhmap-&gt;buckets[index] = NULL;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\nPair *entries;\nint i = 0, index = 0;\nint total = 0;\n/* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\nfor (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\nif (hmap-&gt;buckets[i] != NULL) {\ntotal++;\n}\n}\nentries = malloc(sizeof(Pair) * total);\nfor (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\nif (hmap-&gt;buckets[i] != NULL) {\nentries[index].key = hmap-&gt;buckets[i]-&gt;key;\nentries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\nstrcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\nindex++;\n}\n}\nset-&gt;set = entries;\nset-&gt;len = total;\n}\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\nint *keys;\nint i = 0, index = 0;\nint total = 0;\n/* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\nfor (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\nif (hmap-&gt;buckets[i] != NULL) {\ntotal++;\n}\n}\nkeys = malloc(total * sizeof(int));\nfor (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\nif (hmap-&gt;buckets[i] != NULL) {\nkeys[index] = hmap-&gt;buckets[i]-&gt;key;\nindex++;\n}\n}\nset-&gt;set = keys;\nset-&gt;len = total;\n}\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\nchar **vals;\nint i = 0, index = 0;\nint total = 0;\n/* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\nfor (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\nif (hmap-&gt;buckets[i] != NULL) {\ntotal++;\n}\n}\nvals = malloc(total * sizeof(char *));\nfor (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\nif (hmap-&gt;buckets[i] != NULL) {\nvals[index] = hmap-&gt;buckets[i]-&gt;val;\nindex++;\n}\n}\nset-&gt;set = vals;\nset-&gt;len = total;\n}\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\nint i;\nMapSet set;\npairSet(hmap, &amp;set);\nPair *entries = (Pair *)set.set;\nfor (i = 0; i &lt; set.len; i++) {\nprintf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n}\nfree(set.set);\n}\n</code></pre> array_hash_map.zig<pre><code>// \u952e\u503c\u5bf9\nconst Pair = struct {\nkey: usize = undefined,\nval: []const u8 = undefined,\npub fn init(key: usize, val: []const u8) Pair {\nreturn Pair {\n.key = key,\n.val = val,\n};\n}\n};\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\nreturn struct {\nbucket: ?std.ArrayList(?T) = null,\nmem_allocator: std.mem.Allocator = undefined,\nconst Self = @This();\n// \u6784\u9020\u51fd\u6570\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nself.mem_allocator = allocator;\n// \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\nself.bucket = std.ArrayList(?T).init(self.mem_allocator);\nvar i: i32 = 0;\nwhile (i &lt; 100) : (i += 1) {\ntry self.bucket.?.append(null);\n}\n}\n// \u6790\u6784\u51fd\u6570\npub fn deinit(self: *Self) void {\nif (self.bucket != null) self.bucket.?.deinit();\n}\n// \u54c8\u5e0c\u51fd\u6570\nfn hashFunc(key: usize) usize {\nvar index = key % 100;\nreturn index;\n}\n// \u67e5\u8be2\u64cd\u4f5c\npub fn get(self: *Self, key: usize) []const u8 {\nvar index = hashFunc(key);\nvar pair = self.bucket.?.items[index];\nreturn pair.?.val;\n}\n// \u6dfb\u52a0\u64cd\u4f5c\npub fn put(self: *Self, key: usize, val: []const u8) !void {\nvar pair = Pair.init(key, val);\nvar index = hashFunc(key);\nself.bucket.?.items[index] = pair;\n}\n// \u5220\u9664\u64cd\u4f5c\npub fn remove(self: *Self, key: usize) !void {\nvar index = hashFunc(key);\n// \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\nself.bucket.?.items[index] = null;\n}       // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\npub fn pairSet(self: *Self) !std.ArrayList(T) {\nvar entry_set = std.ArrayList(T).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry entry_set.append(item.?);\n}\nreturn entry_set;\n}  // \u83b7\u53d6\u6240\u6709\u952e\npub fn keySet(self: *Self) !std.ArrayList(usize) {\nvar key_set = std.ArrayList(usize).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry key_set.append(item.?.key);\n}\nreturn key_set;\n}  // \u83b7\u53d6\u6240\u6709\u503c\npub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\nvar value_set = std.ArrayList([]const u8).init(self.mem_allocator);\nfor (self.bucket.?.items) |item| {\nif (item == null) continue;\ntry value_set.append(item.?.val);\n}\nreturn value_set;\n}\n// \u6253\u5370\u54c8\u5e0c\u8868\npub fn print(self: *Self) !void {\nvar entry_set = try self.pairSet();\ndefer entry_set.deinit();\nfor (entry_set.items) |item| {\nstd.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n}\n}\n};\n}\n</code></pre>"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"<p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u6240\u6709 <code>key</code> \u6784\u6210\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u6570\u7ec4\u6240\u6709\u7d22\u5f15\u6784\u6210\u7684\u8f93\u51fa\u7a7a\u95f4\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002</p> <p>\u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684 <code>key</code> \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>\u5982\u56fe 6-3 \u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 hash collision\u300d\u3002</p> <p></p> <p> \u56fe 6-3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b </p> <p>\u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a <code>key</code> \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u5982\u56fe 6-4 \u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9 <code>(136, A)</code> \u548c <code>(236, D)</code> \u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002</p> <p></p> <p> \u56fe 6-4 \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9 </p> <p>\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\uff1b\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u6765\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fd9\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u4e3a\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002</p> <p>\u300c\u8d1f\u8f7d\u56e0\u5b50 load factor\u300d\u662f\u54c8\u5e0c\u8868\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u8f93\u5165 <code>key</code> \uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230 <code>value</code> \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002</li> <li>\u54c8\u5e0c\u51fd\u6570\u5c06 <code>key</code> \u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6 <code>value</code> \u3002</li> <li>\u4e24\u4e2a\u4e0d\u540c\u7684 <code>key</code> \u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002</li> <li>\u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u901a\u8fc7\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002</li> <li>\u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002</li> <li>\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684 <code>HashMap</code> \u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684 <code>Dict</code> \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002</li> <li>\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002</li> <li>\u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u4e8e\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002</li> <li>\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u4e0d\u662f \\(O(n)\\) \uff1f</p> <p>\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u5f97\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002</p> <p>\u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86</p> <p>\u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002</p> <p>\u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u4eec\u66f4\u9ad8\u5462\uff1f</p> <p>\u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u5185\u5b58\u672a\u4f7f\u7528\u3002</p> <p>\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002</p> <p>\u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002</p> <p>\u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f</p> <p>\u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002</p> <p>\u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f</p> <p>\u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0 <code>key</code> \u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002</p> <p>\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f</p> <p>\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u4f59\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\u5185\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c <code>key</code> \u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2a <code>key</code> \uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002</p>"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806","text":"<p>Abstract</p> <p>\u5806\u5c31\u50cf\u662f\u5c71\u5cb3\u5cf0\u5ce6\uff0c\u5c42\u53e0\u8d77\u4f0f\u3001\u5f62\u6001\u5404\u5f02\u3002</p> <p>\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u9519\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u603b\u662f\u6700\u5148\u6620\u5165\u773c\u5e18\u3002</p>"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>8.1 \u00a0 \u5806</li> <li>8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c</li> <li>8.3 \u00a0 Top-K \u95ee\u9898</li> <li>8.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c","text":"<p>\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u6765\u6784\u5efa\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u3002</p>"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u501f\u52a9\u5165\u5806\u64cd\u4f5c\u5b9e\u73b0","text":"<p>\u6211\u4eec\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u904d\u5386\u5217\u8868\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u201c\u5165\u5806\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u5806\u7684\u5c3e\u90e8\uff0c\u518d\u5bf9\u8be5\u5143\u7d20\u6267\u884c\u201c\u4ece\u5e95\u81f3\u9876\u201d\u5806\u5316\u3002</p> <p>\u6bcf\u5f53\u4e00\u4e2a\u5143\u7d20\u5165\u5806\uff0c\u5806\u7684\u957f\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u4e8e\u8282\u70b9\u662f\u4ece\u9876\u5230\u5e95\u4f9d\u6b21\u88ab\u6dfb\u52a0\u8fdb\u4e8c\u53c9\u6811\u7684\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u6784\u5efa\u7684\u3002</p> <p>\u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6bcf\u4e2a\u5143\u7d20\u7684\u5165\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u65f6\u95f4\uff0c\u56e0\u6b64\u8be5\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p>"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u901a\u8fc7\u904d\u5386\u5806\u5316\u5b9e\u73b0","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u4e00\u79cd\u66f4\u4e3a\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5171\u5206\u4e3a\u4e24\u6b65\u3002</p> <ol> <li>\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u6b64\u65f6\u5806\u7684\u6027\u8d28\u5c1a\u672a\u5f97\u5230\u6ee1\u8db3\u3002</li> <li>\u5012\u5e8f\u904d\u5386\u5806\uff08\u5373\u5c42\u5e8f\u904d\u5386\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u975e\u53f6\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002</li> </ol> <p>\u6bcf\u5f53\u5806\u5316\u4e00\u4e2a\u8282\u70b9\u540e\uff0c\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\u5c31\u5f62\u6210\u4e00\u4e2a\u5408\u6cd5\u7684\u5b50\u5806\u3002\u800c\u7531\u4e8e\u662f\u5012\u5e8f\u904d\u5386\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u6784\u5efa\u7684\u3002</p> <p>\u4e4b\u6240\u4ee5\u9009\u62e9\u5012\u5e8f\u904d\u5386\uff0c\u662f\u56e0\u4e3a\u8fd9\u6837\u80fd\u591f\u4fdd\u8bc1\u5f53\u524d\u8282\u70b9\u4e4b\u4e0b\u7684\u5b50\u6811\u5df2\u7ecf\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u8fd9\u6837\u5806\u5316\u5f53\u524d\u8282\u70b9\u624d\u662f\u6709\u6548\u7684\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u53f6\u8282\u70b9\u6ca1\u6709\u5b50\u8282\u70b9\uff0c\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u56e0\u6b64\u65e0\u987b\u5806\u5316\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6700\u540e\u4e00\u4e2a\u975e\u53f6\u8282\u70b9\u662f\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u6211\u4eec\u4ece\u5b83\u5f00\u59cb\u5012\u5e8f\u904d\u5386\u5e76\u6267\u884c\u5806\u5316\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n\"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n# \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nself.max_heap = nums\n# \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in range(self.parent(self.size() - 1), -1, -1):\nself.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector&lt;int&gt; nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = parent(size() - 1); i &gt;= 0; i--) {\nsiftDown(i);\n}\n}\n</code></pre> my_heap.java<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;Integer&gt; nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new ArrayList&lt;&gt;(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = parent(size() - 1); i &gt;= 0; i--) {\nsiftDown(i);\n}\n}\n</code></pre> my_heap.cs<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = new List&lt;int&gt;(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nvar size = Parent(this.Size() - 1);\nfor (int i = size; i &gt;= 0; i--) {\nSiftDown(i);\n}\n}\n</code></pre> my_heap.go<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nh := &amp;maxHeap{data: nums}\nfor i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nh.siftDown(i)\n}\nreturn h\n}\n</code></pre> my_heap.swift<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nmaxHeap = nums\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\nsiftDown(i: i)\n}\n}\n</code></pre> my_heap.js<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.#maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\nthis.#siftDown(i);\n}\n}\n</code></pre> my_heap.ts<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nthis.maxHeap = nums === undefined ? [] : [...nums];\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\nthis.siftDown(i);\n}\n}\n</code></pre> my_heap.dart<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;int&gt; nums) {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n_maxHeap = nums;\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = _parent(size() - 1); i &gt;= 0; i--) {\nsiftDown(i);\n}\n}\n</code></pre> my_heap.rs<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\nlet mut heap = MaxHeap { max_heap: nums };\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in (0..=Self::parent(heap.size() - 1)).rev() {\nheap.sift_down(i);\n}\nheap\n}\n</code></pre> my_heap.c<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n// \u6240\u6709\u5143\u7d20\u5165\u5806\nMaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\nmaxHeap-&gt;size = size;\nmemcpy(maxHeap-&gt;data, nums, size * sizeof(int));\nfor (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nsiftDown(maxHeap, i);\n}\nreturn maxHeap;\n}\n</code></pre> my_heap.zig<pre><code>// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\nif (self.max_heap != null) return;\nself.max_heap = std.ArrayList(T).init(allocator);\n// \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\ntry self.max_heap.?.appendSlice(nums);\n// \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nvar i: usize = parent(self.size() - 1) + 1;\nwhile (i &gt; 0) : (i -= 1) {\ntry self.siftDown(i - 1);\n}\n}\n</code></pre>"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u63a8\u7b97\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u5047\u8bbe\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1) / 2\\) \u3002</li> <li>\u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(\\log n\\) \u3002</li> </ul> <p>\u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u4f46\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u6027\u8d28\u3002</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u51c6\u786e\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u964d\u4f4e\u8ba1\u7b97\u96be\u5ea6\uff0c\u5047\u8bbe\u7ed9\u5b9a\u4e00\u4e2a\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \u3001\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002</p> <p></p> <p> \u56fe 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf </p> <p>\u5982\u56fe 8-5 \u6240\u793a\uff0c\u8282\u70b9\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>\u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>\u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>\u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>\u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806","text":"<p>\u300c\u5806 heap\u300d\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u4e3b\u8981\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff0c\u5982\u56fe 8-1 \u6240\u793a\u3002</p> <ul> <li>\u300c\u5927\u9876\u5806 max heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> <li>\u300c\u5c0f\u9876\u5806 min heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> </ul> <p></p> <p> \u56fe 8-1 \u00a0 \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806 </p> <p>\u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002</li> <li>\u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u9876\u201d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u5e95\u201d\u3002</li> <li>\u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u6839\u8282\u70b9\uff09\u7684\u503c\u5206\u522b\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002</li> </ul>"},{"location":"chapter_heap/heap/#811","title":"8.1.1 \u00a0 \u5806\u5e38\u7528\u64cd\u4f5c","text":"<p>\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 priority queue\u300d\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u7684\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u201c\u4f18\u5148\u961f\u5217\u201d\u548c\u201c\u5806\u201d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u79f0\u4f5c\u201c\u5806\u201d\u3002</p> <p>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 8-1 \u00a0 \u5806\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\) <p>\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002</p> <p>\u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e2a <code>flag</code> \u6216\u4fee\u6539 <code>Comparator</code> \u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y - x));\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n// Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n// \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n*h = append(*h, x.(int))\n}\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n// \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\nlast := (*h)[len(*h)-1]\n*h = (*h)[:len(*h)-1]\nreturn last\n}\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\nreturn len(*h)\n}\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n// \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\nreturn (*h)[i].(int) &gt; (*h)[j].(int)\n}\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n(*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\nreturn (*h)[0]\n}\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nmaxHeap := &amp;intHeap{}\nheap.Init(maxHeap)\n/* \u5143\u7d20\u5165\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\nheap.Push(maxHeap, 1)\nheap.Push(maxHeap, 3)\nheap.Push(maxHeap, 2)\nheap.Push(maxHeap, 4)\nheap.Push(maxHeap, 5)\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\ntop := maxHeap.Top()\nfmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\nheap.Pop(maxHeap) // 5\nheap.Pop(maxHeap) // 4\nheap.Pop(maxHeap) // 3\nheap.Pop(maxHeap) // 2\nheap.Pop(maxHeap) // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nsize := len(*maxHeap)\nfmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(*maxHeap) == 0\nfmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>// Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.zig<pre><code>\n</code></pre>"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7684\u5b9e\u73b0","text":"<p>\u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p>"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"<p>\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u8bb2\u8fc7\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</p> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002</p> <p>\u5982\u56fe 8-2 \u6240\u793a\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u56fe 8-2 \u00a0 \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8 </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n\"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn 2 * i + 1\ndef right(self, i: int) -&gt; int:\n\"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn 2 * i + 2\ndef parent(self, i: int) -&gt; int:\n\"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\"\"\"\nreturn (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint Parent(int i) {\nreturn (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\nreturn 2*i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\nreturn 2*i + 2\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n// \u5411\u4e0b\u6574\u9664\nreturn (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n2 * i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n(i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\n#left(i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\n#right(i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\n#parent(i) {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nleft(i: number): number {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nright(i: number): number {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nparent(i: number): number {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint _parent(int i) {\nreturn (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n2 * i + 1\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n(i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\nreturn (i - 1) / 2;\n}\n</code></pre> my_heap.zig<pre><code>// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\nreturn 2 * i + 1;\n}\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\nreturn 2 * i + 2;\n}\n// \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n// return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\nreturn @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"<p>\u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def peek(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\nreturn self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\nreturn maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\nreturn h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\nmaxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\nreturn this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\nreturn this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\nreturn _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\nself.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\nreturn maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.zig<pre><code>// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\nreturn self.max_heap.?.items[0];\n}  </code></pre>"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806","text":"<p>\u7ed9\u5b9a\u5143\u7d20 <code>val</code> \uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 heapify\u300d\u3002</p> <p>\u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5982\u56fe 8-3 \u6240\u793a\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u6b65\u9aa4 </p> <p>\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def push(self, val: int):\n\"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n# \u6dfb\u52a0\u8282\u70b9\nself.max_heap.append(val)\n# \u4ece\u5e95\u81f3\u9876\u5806\u5316\nself.sift_up(self.size() - 1)\ndef sift_up(self, i: int):\n\"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\nwhile True:\n# \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\np = self.parent(i)\n# \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, p)\n# \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.push_back(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(maxHeap[i], maxHeap[p]);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.Add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nSiftUp(Size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = Parent(i);\n// \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nSwap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n// \u6dfb\u52a0\u8282\u70b9\nh.data = append(h.data, val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nh.siftUp(len(h.data) - 1)\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\nfor true {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\np := h.parent(i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nh.swap(i, p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap.append(val)\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(i: size() - 1)\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\nvar i = i\nwhile true {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nlet p = parent(i: i)\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i: i, j: p)\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p\n}\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n// \u6dfb\u52a0\u8282\u70b9\nthis.#maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.#siftUp(this.size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nconst p = this.#parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.#swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n// \u6dfb\u52a0\u8282\u70b9\nthis.maxHeap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nthis.siftUp(this.size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nconst p = this.parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n// \u6dfb\u52a0\u8282\u70b9\n_maxHeap.add(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = _parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n_swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfn push(&amp;mut self, val: i32) {\n// \u6dfb\u52a0\u8282\u70b9\nself.max_heap.push(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nself.sift_up(self.size() - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\nloop {\n// \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\nif i == 0 {\nbreak;\n}\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nlet p = Self::parent(i);\n// \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif self.max_heap[i] &lt;= self.max_heap[p] {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n// \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\nif (maxHeap-&gt;size == MAX_SIZE) {\nprintf(\"heap is full!\");\nreturn;\n}\n// \u6dfb\u52a0\u8282\u70b9\nmaxHeap-&gt;data[maxHeap-&gt;size] = val;\nmaxHeap-&gt;size++;\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\nsiftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nint p = parent(maxHeap, i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(maxHeap, i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n// \u6dfb\u52a0\u8282\u70b9\ntry self.max_heap.?.append(val);\n// \u4ece\u5e95\u81f3\u9876\u5806\u5316\ntry self.siftUp(self.size() - 1);\n}  // \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\nvar p = parent(i);\n// \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\nif (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\ntry self.swap(i, p);\n// \u5faa\u73af\u5411\u4e0a\u5806\u5316\ni = p;\n}\n}\n</code></pre>"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806","text":"<p>\u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u8fdb\u884c\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\u3002</p> <ol> <li>\u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\u3002</li> <li>\u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u56e0\u6b64\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\u3002</li> <li>\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\u3002</li> </ol> <p>\u5982\u56fe 8-4 \u6240\u793a\uff0c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u201c\u4ece\u5e95\u81f3\u9876\u5806\u5316\u201d\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\u3002\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-4 \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806\u6b65\u9aa4 </p> <p>\u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def pop(self) -&gt; int:\n\"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n# \u5224\u7a7a\u5904\u7406\nif self.is_empty():\nraise IndexError(\"\u5806\u4e3a\u7a7a\")\n# \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nself.swap(0, self.size() - 1)\n# \u5220\u9664\u8282\u70b9\nval = self.max_heap.pop()\n# \u4ece\u9876\u81f3\u5e95\u5806\u5316\nself.sift_down(0)\n# \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\ndef sift_down(self, i: int):\n\"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\nwhile True:\n# \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl, r, ma = self.left(i), self.right(i), i\nif l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\nma = l\nif r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\nma = r\n# \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, ma)\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty()) {\nthrow out_of_range(\"\u5806\u4e3a\u7a7a\");\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(maxHeap[0], maxHeap[size() - 1]);\n// \u5220\u9664\u8282\u70b9\nmaxHeap.pop_back();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\nma = l;\nif (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\nswap(maxHeap[i], maxHeap[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = maxHeap.remove(size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = left(i), r = right(i), ma = i;\nif (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\nma = l;\nif (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n// \u5224\u7a7a\u5904\u7406\nif (IsEmpty())\nthrow new IndexOutOfRangeException();\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nSwap(0, Size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = maxHeap.Last();\nmaxHeap.RemoveAt(Size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nSiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = Left(i), r = Right(i), ma = i;\nif (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\nma = l;\nif (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\nma = r;\n// \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nSwap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n// \u5224\u7a7a\u5904\u7406\nif h.isEmpty() {\nfmt.Println(\"error\")\nreturn nil\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nh.swap(0, h.size()-1)\n// \u5220\u9664\u8282\u70b9\nval := h.data[len(h.data)-1]\nh.data = h.data[:len(h.data)-1]\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nh.siftDown(0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\nfor true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\nl, r, max := h.left(i), h.right(i), i\nif l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\nmax = l\n}\nif r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\nmax = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif max == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nh.swap(i, max)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max\n}\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -&gt; Int {\n// \u5224\u7a7a\u5904\u7406\nif isEmpty() {\nfatalError(\"\u5806\u4e3a\u7a7a\")\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(i: 0, j: size() - 1)\n// \u5220\u9664\u8282\u70b9\nlet val = maxHeap.remove(at: size() - 1)\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(i: 0)\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = left(i: i)\nlet r = right(i: i)\nvar ma = i\nif l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\nma = l\n}\nif r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(i: i, j: ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop() {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.#swap(0, this.size() - 1);\n// \u5220\u9664\u8282\u70b9\nconst val = this.#maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.#siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.#left(i),\nr = this.#right(i);\nlet ma = i;\nif (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\nif (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.#swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n// \u5224\u7a7a\u5904\u7406\nif (this.isEmpty()) throw new RangeError('Heap is empty.');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nthis.swap(0, this.size() - 1);\n// \u5220\u9664\u8282\u70b9\nconst val = this.maxHeap.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nthis.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nconst l = this.left(i),\nr = this.right(i);\nlet ma = i;\nif (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\nif (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nthis.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n_swap(0, size() - 1);\n// \u5220\u9664\u8282\u70b9\nint val = _maxHeap.removeLast();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = _left(i);\nint r = _right(i);\nint ma = i;\nif (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\nif (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\n_swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&amp;mut self) -&gt; i32 {\n// \u5224\u7a7a\u5904\u7406\nif self.is_empty() {\npanic!(\"index out of bounds\");\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nself.swap(0, self.size() - 1);\n// \u5220\u9664\u8282\u70b9\nlet val = self.max_heap.remove(self.size() - 1);\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nself.sift_down(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nval\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\nloop {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet (l, r, mut ma) = (Self::left(i), Self::right(i), i);\nif l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\nma = l;\n}\nif r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nself.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n// \u5224\u7a7a\u5904\u7406\nif (isEmpty(maxHeap)) {\nprintf(\"heap is empty!\");\nreturn INT_MAX;\n}\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(maxHeap, 0, size(maxHeap) - 1);\n// \u5220\u9664\u8282\u70b9\nint val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\nmaxHeap-&gt;size--;\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\nsiftDown(maxHeap, 0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n}\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\nint l = left(maxHeap, i);\nint r = right(maxHeap, i);\nint max = i;\nif (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\nmax = l;\n}\nif (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\nmax = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (max == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(maxHeap, i, max);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = max;\n}\n}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n// \u5224\u65ad\u5904\u7406\nif (self.isEmpty()) unreachable;\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\ntry self.swap(0, self.size() - 1);\n// \u5220\u9664\u8282\u70b9\nvar val = self.max_heap.?.pop();\n// \u4ece\u9876\u81f3\u5e95\u5806\u5316\ntry self.siftDown(0);\n// \u8fd4\u56de\u5806\u9876\u5143\u7d20\nreturn val;\n} // \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\nvar i = i_;\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nvar l = left(i);\nvar r = right(i);\nvar ma = i;\nif (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\nif (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\ntry self.swap(i, ma);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n</code></pre>"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002</li> <li>\u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u201c\u5806\u6392\u5e8f\u201d\u7ae0\u8282\u3002</li> <li>\u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_heap/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002</li> <li>\u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002</li> <li>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\)\u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002</li> <li>\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</li> <li>\u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002</li> <li>\u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</li> <li>Top-K \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f</p> <p>\u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u5806\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u6f0f\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4f7f\u7528\u4e0d\u5f53\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u6f0f\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-K \u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u65e0\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u524d \\(k\\) \u5927\u7684\u5143\u7d20\u3002</p> <p>\u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"<p>\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u56fe 8-6 \u6240\u793a\u7684 \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002</p> <p>\u6b64\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002</p> <p></p> <p> \u56fe 8-6 \u00a0 \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>Tip</p> <p>\u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u65f6\u7b49\u4ef7\u4e8e\u201c\u9009\u62e9\u6392\u5e8f\u201d\u7b97\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"<p>\u5982\u56fe 8-7 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u5bf9\u6570\u7ec4 <code>nums</code> \u8fdb\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 8-7 \u00a0 \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p>"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"<p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-K \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u56fe 8-8 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\u3002</li> <li>\u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u3002</li> <li>\u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\u3002</li> <li>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-8 \u00a0 \u57fa\u4e8e\u5806\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n\"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n# \u521d\u59cb\u5316\u5c0f\u9876\u5806\nheap = []\n# \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor i in range(k):\nheapq.heappush(heap, nums[i])\n# \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor i in range(k, len(nums)):\n# \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif nums[i] &gt; heap[0]:\nheapq.heappop(heap)\nheapq.heappush(heap, nums[i])\nreturn heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i &lt; k; i++) {\nheap.push(nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i &lt; nums.size(); i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] &gt; heap.top()) {\nheap.pop();\nheap.push(nums[i]);\n}\n}\nreturn heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i &lt; k; i++) {\nheap.offer(nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i &lt; nums.length; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] &gt; heap.peek()) {\nheap.poll();\nheap.offer(nums[i]);\n}\n}\nreturn heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue&lt;int, int&gt; heap = new();\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i &lt; k; i++) {\nheap.Enqueue(nums[i], nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i &lt; nums.Length; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] &gt; heap.Peek()) {\nheap.Dequeue();\nheap.Enqueue(nums[i], nums[i]);\n}\n}\nreturn heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nh := &amp;minHeap{}\nheap.Init(h)\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor i := 0; i &lt; k; i++ {\nheap.Push(h, nums[i])\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor i := k; i &lt; len(nums); i++ {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif nums[i] &gt; h.Top().(int) {\nheap.Pop(h)\nheap.Push(h, nums[i])\n}\n}\nreturn h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\nvar heap = Heap(nums.prefix(k))\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor i in stride(from: k, to: nums.count, by: 1) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif nums[i] &gt; heap.min()! {\n_ = heap.removeMin()\nheap.insert(nums[i])\n}\n}\nreturn heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n// \u5143\u7d20\u53d6\u53cd\nmaxHeap.push(-val);\n}\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n// \u5143\u7d20\u53d6\u53cd\nreturn -maxHeap.pop();\n}\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n// \u5143\u7d20\u53d6\u53cd\nreturn -maxHeap.peek();\n}\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n// \u5143\u7d20\u53d6\u53cd\nreturn maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\n// \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\nconst maxHeap = new MaxHeap([]);\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (let i = 0; i &lt; k; i++) {\npushMinHeap(maxHeap, nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (let i = k; i &lt; nums.length; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] &gt; peekMinHeap(maxHeap)) {\npopMinHeap(maxHeap);\npushMinHeap(maxHeap, nums[i]);\n}\n}\n// \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\nreturn getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n// \u5143\u7d20\u53d6\u53cd\nmaxHeap.push(-val);\n}\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n// \u5143\u7d20\u53d6\u53cd\nreturn -maxHeap.pop();\n}\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n// \u5143\u7d20\u53d6\u53cd\nreturn -maxHeap.peek();\n}\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n// \u5143\u7d20\u53d6\u53cd\nreturn maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\n// \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\nconst maxHeap = new MaxHeap([]);\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (let i = 0; i &lt; k; i++) {\npushMinHeap(maxHeap, nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (let i = k; i &lt; nums.length; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] &gt; peekMinHeap(maxHeap)) {\npopMinHeap(maxHeap);\npushMinHeap(maxHeap, nums[i]);\n}\n}\n// \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\nreturn getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nMinHeap heap = MinHeap(nums.sublist(0, k));\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i &lt; nums.length; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] &gt; heap.peek()) {\nheap.pop();\nheap.push(nums[i]);\n}\n}\nreturn heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n// BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\nlet mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor &amp;num in nums.iter().take(k) {\nheap.push(Reverse(num));\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor &amp;num in nums.iter().skip(k) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif num &gt; heap.peek().unwrap().0 {\nheap.pop();\nheap.push(Reverse(num));\n}\n}\nheap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n// \u5143\u7d20\u53d6\u53cd\npush(maxHeap, -val);\n}\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n// \u5143\u7d20\u53d6\u53cd\nreturn -pop(maxHeap);\n}\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n// \u5143\u7d20\u53d6\u53cd\nreturn -peek(maxHeap);\n}\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n// \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\nint *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\nfor (int i = 0; i &lt; maxHeap-&gt;size; i++) {\nres[i] = -maxHeap-&gt;data[i];\n}\nreturn res;\n}\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n// \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\nint *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\nfor (int i = 0; i &lt; maxHeap-&gt;size; i++) {\nres[i] = -maxHeap-&gt;data[i];\n}\nreturn res;\n}\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\n// \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\nint *empty = (int *)malloc(0);\nMaxHeap *maxHeap = newMaxHeap(empty, 0);\n// \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\nfor (int i = 0; i &lt; k; i++) {\npushMinHeap(maxHeap, nums[i]);\n}\n// \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\nfor (int i = k; i &lt; sizeNums; i++) {\n// \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\nif (nums[i] &gt; peekMinHeap(maxHeap)) {\npopMinHeap(maxHeap);\npushMinHeap(maxHeap, nums[i]);\n}\n}\nint *res = getMinHeap(maxHeap);\n// \u91ca\u653e\u5185\u5b58\ndelMaxHeap(maxHeap);\nreturn res;\n}\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> <p>\u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002</p> <p>\u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002</p>"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":"<p>Abstract</p> <p>\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u4e0e\u6570\u636e\u4ea4\u7ec7\u5728\u4e00\u8d77\uff0c\u88d9\u6446\u4e0a\u98d8\u626c\u7740\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002</p> <p>\u5979\u9080\u8bf7\u4f60\u5171\u821e\uff0c\u8bf7\u7d27\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6ee1\u903b\u8f91\u4e0e\u7f8e\u611f\u7684\u7b97\u6cd5\u4e16\u754c\u3002</p>"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728</li> <li>1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48</li> <li>1.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"<p>\u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002</p> <p>\u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c06\u4e3e\u51e0\u4e2a\u5177\u4f53\u7684\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002</p> <p>\u4f8b\u4e00\uff1a\u67e5\u9605\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u6309\u7167\u56fe 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5b9e\u73b0\u3002</p> <ol> <li>\u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff0c\u5047\u8bbe\u9996\u5b57\u6bcd\u4e3a \\(m\\) \u3002</li> <li>\u7531\u4e8e\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\u3002</li> <li>\u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4 <code>1.</code> \u548c \u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9aa4 </p> <p>\u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u7b97\u6cd5\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u201c\u6570\u7ec4\u201d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u67e5\u627e\u201d\u3002</p> <p>\u4f8b\u4e8c\uff1a\u6574\u7406\u6251\u514b\u3002\u6211\u4eec\u5728\u6253\u724c\u65f6\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u6251\u514b\u724c\uff0c\u4f7f\u5176\u4ece\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5b9e\u73b0\u6d41\u7a0b\u5982\u56fe 1-2 \u6240\u793a\u3002</p> <ol> <li>\u5c06\u6251\u514b\u724c\u5212\u5206\u4e3a\u201c\u6709\u5e8f\u201d\u548c\u201c\u65e0\u5e8f\u201d\u4e24\u90e8\u5206\uff0c\u5e76\u5047\u8bbe\u521d\u59cb\u72b6\u6001\u4e0b\u6700\u5de6 1 \u5f20\u6251\u514b\u724c\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u5728\u65e0\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f20\u6251\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\uff1b\u5b8c\u6210\u540e\u6700\u5de6 2 \u5f20\u6251\u514b\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u6bcf\u4e00\u8f6e\u5c06\u4e00\u5f20\u6251\u514b\u724c\u4ece\u65e0\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u6251\u514b\u724c\u90fd\u6709\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 1-2 \u00a0 \u6251\u514b\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u4e0a\u8ff0\u6574\u7406\u6251\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u5b83\u5728\u5904\u7406\u5c0f\u578b\u6570\u636e\u96c6\u65f6\u975e\u5e38\u9ad8\u6548\u3002\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002</p> <p>\u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u4e86\u6536\u94f6\u5458 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u627e\u6211\u4eec \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u56fe 1-3 \u6240\u793a\u7684\u601d\u8003\u3002</p> <ol> <li>\u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002</li> <li>\u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002</li> <li>\u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002</li> </ol> <p></p> <p> \u56fe 1-3 \u00a0 \u8d27\u5e01\u627e\u96f6\u8fc7\u7a0b </p> <p>\u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u8d2a\u5fc3\u201d\u7b97\u6cd5\u3002</p> <p>\u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8bf7\u7ee7\u7eed\u5f80\u4e0b\u9605\u8bfb\uff0c\u672c\u4e66\u5c06\u5f15\u5bfc\u4f60\u8fc8\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002</li> <li>\u67e5\u9605\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002</li> <li>\u6574\u7406\u6251\u514b\u7684\u8fc7\u7a0b\u4e0e\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u975e\u5e38\u7c7b\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u9002\u5408\u6392\u5e8f\u5c0f\u578b\u6570\u636e\u96c6\u3002</li> <li>\u8d27\u5e01\u627e\u96f6\u7684\u6b65\u9aa4\u672c\u8d28\u4e0a\u662f\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\u3002</li> <li>\u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728\uff0c\u79ef\u6728\u4ee3\u8868\u6570\u636e\uff0c\u79ef\u6728\u7684\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u7b49\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1 \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"<p>\u300c\u7b97\u6cd5 algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002</li> <li>\u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002</li> <li>\u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u5728\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2 \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"<p>\u300c\u6570\u636e\u7ed3\u6784 data structure\u300d\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u8bbe\u8ba1\u76ee\u6807\u3002</p> <ul> <li>\u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u5c11\uff0c\u4ee5\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002</li> <li>\u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002</li> <li>\u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002</li> </ul> <p>\u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4e0b\u9762\u4e3e\u4e24\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>\u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\u3002</li> <li>\u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"<p>\u5982\u56fe 1-4 \u6240\u793a\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\u4ee5\u4e0b\u4e09\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002</li> <li>\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002</li> <li>\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u4f46\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9009\u62e9\u5408\u9002\u7684\u6570\u636e\u7ed3\u6784\u662f\u5173\u952e\u3002</li> </ul> <p></p> <p> \u56fe 1-4 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb </p> <p>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72b9\u5982\u56fe 1-5 \u6240\u793a\u7684\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002</p> <p></p> <p> \u56fe 1-5 \u00a0 \u62fc\u88c5\u79ef\u6728 </p> <p>\u4e24\u8005\u7684\u8be6\u7ec6\u5bf9\u5e94\u5173\u7cfb\u5982\u8868 1-1 \u6240\u793a\u3002</p> <p> \u8868 1-1 \u00a0 \u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728 </p> \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 \u62fc\u88c5\u79ef\u6728 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u662f\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u57fa\u4e8e\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002</p> <p>\u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0</p> <p>\u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u201c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u201d\u7b80\u79f0\u4e3a\u201c\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u67e5\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002</p>"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"<p>Abstract</p> <p>\u7b97\u6cd5\u72b9\u5982\u7f8e\u5999\u7684\u4ea4\u54cd\u4e50\uff0c\u6bcf\u4e00\u884c\u4ee3\u7801\u90fd\u50cf\u97f5\u5f8b\u822c\u6d41\u6dcc\u3002</p> <p>\u613f\u8fd9\u672c\u4e66\u5728\u4f60\u7684\u8111\u6d77\u4e2d\u8f7b\u8f7b\u54cd\u8d77\uff0c\u7559\u4e0b\u72ec\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002</p>"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>0.1 \u00a0 \u5173\u4e8e\u672c\u4e66</li> <li>0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66</li> <li>0.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 \u5173\u4e8e\u672c\u4e66","text":"<p>\u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u3001\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002</p> <ul> <li>\u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u3002</li> <li>\u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Python\u3001C++\u3001Java\u3001C#\u3001Go\u3001Swift\u3001JavaScript\u3001TypeScript\u3001Dart\u3001Rust\u3001C\u3001Zig \u7b49\u8bed\u8a00\u3002</li> <li>\u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\u3002</li> </ul>"},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1 \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"<p>\u82e5\u4f60\u662f\u7b97\u6cd5\u521d\u5b66\u8005\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u672c\u4e66\u6b63\u662f\u4e3a\u4f60\u91cf\u8eab\u5b9a\u5236\u7684\uff01</p> <p>\u5982\u679c\u4f60\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002</p> <p>\u82e5\u4f60\u662f\u7b97\u6cd5\u5927\u795e\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u4f60\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002</p> <p>\u524d\u7f6e\u6761\u4ef6</p> <p>\u4f60\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002</p>"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"<p>\u672c\u4e66\u7684\u4e3b\u8981\u5185\u5bb9\u5982\u56fe 0-1 \u6240\u793a\u3002</p> <ul> <li>\u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u4e0e\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002</li> <li>\u6570\u636e\u7ed3\u6784\uff1a\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002</li> <li>\u7b97\u6cd5\uff1a\u641c\u7d22\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u6548\u7387\u3001\u5e94\u7528\u573a\u666f\u3001\u89e3\u9898\u6b65\u9aa4\u548c\u793a\u4f8b\u95ee\u9898\u7b49\u3002</li> </ul> <p></p> <p> \u56fe 0-1 \u00a0 \u672c\u4e66\u4e3b\u8981\u5185\u5bb9 </p>"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8c22","text":"<p>\u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a</p> <ul> <li>\u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u4f60\u9f13\u52b1\u6211\u201c\u5feb\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\uff1b</li> <li>\u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\uff1b</li> <li>\u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u7f8e\u597d\u56de\u5fc6\uff1b</li> <li>\u611f\u8c22\u6821\u94e8\u5728\u77e5\u8bc6\u4ea7\u6743\u65b9\u9762\u63d0\u4f9b\u7684\u4e13\u4e1a\u5e2e\u52a9\uff0c\u8fd9\u5bf9\u672c\u5f00\u6e90\u4e66\u7684\u53d1\u5c55\u8d77\u5230\u4e86\u91cd\u8981\u7684\u4f5c\u7528\uff1b</li> <li>\u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b</li> <li>\u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4ed6\u5f00\u53d1\u7684\u5f00\u6e90\u6587\u6863\u4e3b\u9898 Material-for-MkDocs \u3002</li> </ul> <p>\u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u4eec\u7684\u6770\u51fa\u8d21\u732e\uff01</p> <p>\u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u7684\u8457\u4f5c\u3002</p> <p>\u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"<p>Tip</p> <p>\u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u4f60\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"<ul> <li>\u6807\u9898\u540e\u6807\u6ce8 <code>*</code> \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002</li> <li>\u91cd\u8981\u4e13\u6709\u540d\u8bcd\u53ca\u5176\u82f1\u6587\u7ffb\u8bd1\u4f1a\u7528 <code>\u300c \u300d</code> \u62ec\u53f7\u6807\u6ce8\uff0c\u4f8b\u5982 <code>\u300c\u6570\u7ec4 array\u300d</code> \u3002\u5efa\u8bae\u8bb0\u4f4f\u5b83\u4eec\uff0c\u4ee5\u4fbf\u9605\u8bfb\u6587\u732e\u3002</li> <li>\u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528 <code>\u201c\u5f15\u53f7\u201d</code> \u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002</li> <li>\u91cd\u8981\u540d\u8bcd\u3001\u91cd\u70b9\u5185\u5bb9\u548c\u603b\u7ed3\u6027\u8bed\u53e5\u4f1a\u88ab \u52a0\u7c97 \uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002</li> <li>\u5f53\u6d89\u53ca\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\u65f6\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002</li> <li>\u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>\"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>// \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n// \u591a\u884c\n// \u6ce8\u91ca\n</code></pre>"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"<p>\u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u548c\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7247\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002</p> <p>\u5982\u679c\u4f60\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u5982\u56fe 0-2 \u6240\u793a\u7684\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u8bf7\u4ee5\u56fe\u4e3a\u4e3b\u3001\u4ee5\u6587\u5b57\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-2 \u00a0 \u52a8\u753b\u56fe\u89e3\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"<p>\u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728 GitHub \u4ed3\u5e93\u3002\u5982\u56fe 0-3 \u6240\u793a\uff0c\u6e90\u4ee3\u7801\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4e00\u952e\u8fd0\u884c\u3002</p> <p>\u5982\u679c\u65f6\u95f4\u5141\u8bb8\uff0c\u5efa\u8bae\u4f60\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u8bf7\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002</p> <p>\u4e0e\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002\u52a8\u624b\u5b66\uff0c\u624d\u662f\u771f\u7684\u5b66\u3002</p> <p></p> <p> \u56fe 0-3 \u00a0 \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b </p> <p>\u8fd0\u884c\u4ee3\u7801\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6240\u793a\u7684\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\uff0c\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u5e93\u3002\u524d\u5f80 GitHub \u4ed3\u5e93\u3002</p> <p>\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\uff1a</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>\u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u56fe 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u70b9\u51fb\u201cDownload ZIP\u201d\u6309\u94ae\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 0-4 \u00a0 \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801 </p> <p>\u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u56fe 0-5 \u6240\u793a\uff0c\u5bf9\u4e8e\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\u7684\u4ee3\u7801\u5757\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684 <code>codes</code> \u6587\u4ef6\u5939\u5185\u627e\u5230\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-5 \u00a0 \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6 </p>"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"<p>\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u8f7b\u6613\u8df3\u8fc7\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u56de\u590d\u3002</p> <p>\u5982\u56fe 0-6 \u6240\u793a\uff0c\u6bcf\u4e2a\u7ae0\u8282\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8bc4\u8bba\u533a\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u5173\u6ce8\u8bc4\u8bba\u533a\u7684\u5185\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\uff0c\u5206\u4eab\u4f60\u7684\u89c1\u89e3\uff0c\u5e2e\u52a9\u4ed6\u4eba\u8fdb\u6b65\u3002</p> <p></p> <p> \u56fe 0-6 \u00a0 \u8bc4\u8bba\u533a\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"<p>\u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9636\u6bb5\u4e00\uff1a\u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5185\u5bb9\u3002</li> <li>\u9636\u6bb5\u4e8c\uff1a\u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5982\u201c\u5251\u6307 Offer\u201d\u548c\u201cLeetCode Hot 100\u201d\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3\uff5e5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002</li> <li>\u9636\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002</li> </ol> <p>\u5982\u56fe 0-7 \u6240\u793a\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u9636\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u9636\u6bb5\u4e8c\u548c\u9636\u6bb5\u4e09\u7684\u5b66\u4e60\u3002</p> <p></p> <p> \u56fe 0-7 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u4e2d\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002</li> <li>\u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002</li> <li>\u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002</li> <li>\u4e66\u4e2d\u7684\u52a8\u753b\u548c\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002</li> <li>\u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u4ee3\u7801\u3002</li> <li>\u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8ba8\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002</li> </ul>"},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.</p> <p>[4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.</p> <p>[5] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.</p> <p>[6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.</p> <p>[7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.</p> <p>[8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition). </p> <p>[9] Aston Zhang, et al. Dive into Deep Learning.</p>"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u7d22","text":"<p>Abstract</p> <p>\u641c\u7d22\u662f\u4e00\u573a\u672a\u77e5\u7684\u5192\u9669\uff0c\u6211\u4eec\u6216\u8bb8\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u95f4\u7684\u6bcf\u4e2a\u89d2\u843d\uff0c\u53c8\u6216\u8bb8\u53ef\u4ee5\u5feb\u901f\u9501\u5b9a\u76ee\u6807\u3002</p> <p>\u5728\u8fd9\u573a\u5bfb\u89c5\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u4e2a\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002</p>"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>10.1 \u00a0 \u4e8c\u5206\u67e5\u627e</li> <li>10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9</li> <li>10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c</li> <li>10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565</li> <li>10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5</li> <li>10.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_searching/binary_search/","title":"10.1 \u00a0 \u4e8c\u5206\u67e5\u627e","text":"<p>\u300c\u4e8c\u5206\u67e5\u627e binary search\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u590d\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 <code>target</code> \u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 10-1 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-1 \u00a0 \u4e8c\u5206\u67e5\u627e\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u56fe 10-2 \u6240\u793a\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u95ed\u533a\u95f4\uff0c\u5176\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002</li> <li>\u5224\u65ad <code>nums[m]</code> \u548c <code>target</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5\u3002<ol> <li>\u5f53 <code>nums[m] &lt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \u3002</li> <li>\u5f53 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \u3002</li> <li>\u5f53 <code>nums[m] = target</code> \u65f6\uff0c\u8bf4\u660e\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002</li> </ol> </li> </ol> <p>\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-2 \u00a0 \u4e8c\u5206\u67e5\u627e\u6d41\u7a0b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f <code>int</code> \u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51fa <code>int</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002</p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n# \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j = 0, len(nums) - 1\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile i &lt;= j:\n# \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u987b\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target:\ni = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] &gt; target:\nj = m - 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\nreturn m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.size() - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint BinarySearch(int[] nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.Length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\ni, j := 0, len(nums)-1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nfor i &lt;= j {\nm := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i = 0\nvar j = nums.count - 1\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile i &lt;= j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0,\nj = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\nconst m = parseInt(i + (j - i) / 2);\nif (nums[m] &lt; target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet i = 0,\nj = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = Math.floor(i + (j - i) / 2);\nif (nums[m] &lt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] &gt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = nums.length - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums[m] &gt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nlet mut i = 0;\nlet mut j = nums.len() as i32 - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile i &lt;= j {\nlet m = i + (j - i) / 2;      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m as usize] &lt; target {         // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if nums[m as usize] &gt; target {  // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}                       }\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nint i = 0, j = len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\nvar i: usize = 0;\nvar j: usize = nums.items.len - 1;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\nvar m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1;\n} else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> <p>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u5faa\u73af\u4e2d\uff0c\u533a\u95f4\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\uff0c\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002</p>"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"<p>\u9664\u4e86\u4e0a\u8ff0\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j]\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\"\"\"\n# \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j = 0, len(nums)\n# \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i &lt; j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target:\ni = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\nelif nums[m] &gt; target:\nj = m  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nelse:\nreturn m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.size();\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt; j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt; j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.Length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt; j) {\nint m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\ni, j := 0, len(nums)\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nfor i &lt; j {\nm := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i = 0\nvar j = nums.count\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i &lt; j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1\n} else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m\n} else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums, target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0,\nj = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt; j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\nconst m = parseInt(i + (j - i) / 2);\nif (nums[m] &lt; target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target)\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nelse return m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet i = 0,\nj = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt; j) {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nconst m = Math.floor(i + (j - i) / 2);\nif (nums[m] &lt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] &gt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\nreturn -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = nums.length;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt; j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums[m] &gt; target) {\n// \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {\n// \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nlet mut i = 0;\nlet mut j = nums.len() as i32;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile i &lt; j {\nlet m = i + (j - i) / 2;      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m as usize] &lt; target {         // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if nums[m as usize] &gt; target {  // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m - 1;\n} else {                      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}                       }\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nint i = 0, j = len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt; j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\nelse if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\nelse // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn m;\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n// \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\nvar i: usize = 0;\nvar j: usize = nums.items.len;\n// \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\nwhile (i &lt;= j) {\nvar m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\ni = m + 1;\n} else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\nj = m;\n} else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\nreturn @intCast(m);\n}\n}\n// \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\nreturn -1;\n}\n</code></pre> <p>\u5982\u56fe 10-3 \u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002</p> <p>\u7531\u4e8e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u901a\u8fc7\u6307\u9488 \\(i\\) \u548c\u6307\u9488 \\(j\\) \u7f29\u5c0f\u533a\u95f4\u7684\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8bae\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002</p> <p></p> <p> \u56fe 10-3 \u00a0 \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49 </p>"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u65e0\u987b\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002</li> </ul> <p>\u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_edge/","title":"10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c","text":""},{"location":"chapter_searching/binary_search_edge/#1031","title":"10.3.1 \u00a0 \u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 <code>target</code> \u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u56de\u5fc6\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u7684\u65b9\u6cd5\uff0c\u641c\u7d22\u5b8c\u6210\u540e \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u67e5\u627e\u63d2\u5165\u70b9\u672c\u8d28\u4e0a\u662f\u5728\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u8003\u8651\u901a\u8fc7\u67e5\u627e\u63d2\u5165\u70b9\u7684\u51fd\u6570\u5b9e\u73b0\u67e5\u627e\u5de6\u8fb9\u754c\u3002\u8bf7\u6ce8\u610f\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b <code>target</code> \uff0c\u8fd9\u79cd\u60c5\u51b5\u53ef\u80fd\u5bfc\u81f4\u4ee5\u4e0b\u4e24\u79cd\u7ed3\u679c\u3002</p> <ul> <li>\u63d2\u5165\u70b9\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002</li> <li>\u5143\u7d20 <code>nums[i]</code> \u4e0e <code>target</code> \u4e0d\u76f8\u7b49\u3002</li> </ul> <p>\u5f53\u9047\u5230\u4ee5\u4e0a\u4e24\u79cd\u60c5\u51b5\u65f6\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target\"\"\"\n# \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\ni = binary_search_insertion(nums, target)\n# \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif i == len(nums) or nums[i] != target:\nreturn -1\n# \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nint i = binarySearchInsertion(nums, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (i == nums.size() || nums[i] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int[] nums, int target) {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nint i = binary_search_insertion.binarySearchInsertion(nums, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (i == nums.length || nums[i] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nint i = binary_search_insertion.BinarySearchInsertion(nums, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (i == nums.Length || nums[i] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\ni := binarySearchInsertion(nums, target)\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif i == len(nums) || nums[i] != target {\nreturn -1\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nlet i = binarySearchInsertion(nums: nums, target: target)\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif i == nums.count || nums[i] != target {\nreturn -1\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums, target) {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nconst i = binarySearchInsertion(nums, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (i === nums.length || nums[i] !== target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nconst i = binarySearchInsertion(nums, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (i === nums.length || nums[i] !== target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nint i = binarySearchInsertion(nums, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (i == nums.length || nums[i] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nlet i = binary_search_insertion(nums, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif i == nums.len() as i32 || nums[i as usize] != target {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\ni\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n// \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\nint i = binarySearchInsertion(nums, numSize, target);\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (i == numSize || nums[i] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\nreturn i;\n}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchLeftEdge}\n</code></pre>"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"<p>\u90a3\u4e48\u5982\u4f55\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u4ee3\u7801\uff0c\u66ff\u6362\u5728 <code>nums[m] == target</code> \u60c5\u51b5\u4e0b\u7684\u6307\u9488\u6536\u7f29\u64cd\u4f5c\u3002\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u4ecb\u7ecd\u4e24\u79cd\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002</p>"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u590d\u7528\u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u67e5\u627e\u6700\u5de6\u5143\u7d20\u7684\u51fd\u6570\u6765\u67e5\u627e\u6700\u53f3\u5143\u7d20\uff0c\u5177\u4f53\u65b9\u6cd5\u4e3a\uff1a\u5c06\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\u3002</p> <p>\u5982\u56fe 10-7 \u6240\u793a\uff0c\u67e5\u627e\u5b8c\u6210\u540e\uff0c\u6307\u9488 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002</p> <p></p> <p> \u56fe 10-7 \u00a0 \u5c06\u67e5\u627e\u53f3\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5de6\u8fb9\u754c </p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u70b9\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c06\u5176\u51cf \\(1\\) \uff0c\u4ece\u800c\u83b7\u5f97 \\(j\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target\"\"\"\n# \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\ni = binary_search_insertion(nums, target + 1)\n# j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nj = i - 1\n# \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif j == -1 or nums[j] != target:\nreturn -1\n# \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nint i = binarySearchInsertion(nums, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nint j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (j == -1 || nums[j] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int[] nums, int target) {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nint i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nint j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (j == -1 || nums[j] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint BinarySearchRightEdge(int[] nums, int target) {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nint i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nint j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (j == -1 || nums[j] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\ni := binarySearchInsertion(nums, target+1)\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nj := i - 1\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif j == -1 || nums[j] != target {\nreturn -1\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nlet i = binarySearchInsertion(nums: nums, target: target + 1)\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nlet j = i - 1\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif j == -1 || nums[j] != target {\nreturn -1\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums, target) {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nconst i = binarySearchInsertion(nums, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nconst j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (j === -1 || nums[j] !== target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nconst i = binarySearchInsertion(nums, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nconst j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (j === -1 || nums[j] !== target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nint i = binarySearchInsertion(nums, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nint j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (j == -1 || nums[j] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nlet i = binary_search_insertion(nums, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nlet j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif j == -1 || nums[j as usize] != target {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nj\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n// \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\nint i = binarySearchInsertion(nums, numSize, target + 1);\n// j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\nint j = i - 1;\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\nif (j == -1 || nums[j] != target) {\nreturn -1;\n}\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\nreturn j;\n}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchRightEdge}\n</code></pre>"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u6700\u7ec8 \\(i\\) \u548c \\(j\\) \u4f1a\u5206\u522b\u6307\u5411\u9996\u4e2a\u5927\u4e8e\u3001\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002</p> <p>\u56e0\u6b64\uff0c\u5982\u56fe 10-8 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u4e8e\u67e5\u627e\u5de6\u53f3\u8fb9\u754c\u3002</p> <ul> <li>\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target - 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(i\\) \u3002</li> <li>\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target + 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(j\\) \u3002</li> </ul> <p></p> <p> \u56fe 10-8 \u00a0 \u5c06\u67e5\u627e\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20 </p> <p>\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u4e24\u70b9\u503c\u5f97\u6ce8\u610f\u3002</p> <ul> <li>\u7ed9\u5b9a\u6570\u7ec4\u4e0d\u5305\u542b\u5c0f\u6570\uff0c\u8fd9\u610f\u5473\u7740\u6211\u4eec\u65e0\u987b\u5173\u5fc3\u5982\u4f55\u5904\u7406\u76f8\u7b49\u7684\u60c5\u51b5\u3002</li> <li>\u56e0\u4e3a\u8be5\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6570\uff0c\u6240\u4ee5\u9700\u8981\u5c06\u51fd\u6570\u4e2d\u7684\u53d8\u91cf <code>target</code> \u6539\u4e3a\u6d6e\u70b9\u6570\u7c7b\u578b\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_insertion/","title":"10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9","text":"<p>\u4e8c\u5206\u67e5\u627e\u4e0d\u4ec5\u53ef\u7528\u4e8e\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff0c\u8fd8\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u53d8\u79cd\u95ee\u9898\uff0c\u6bd4\u5982\u641c\u7d22\u76ee\u6807\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002</p>"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u5143\u7d20 <code>target</code> \uff0c\u6570\u7ec4\u4e0d\u5b58\u5728\u91cd\u590d\u5143\u7d20\u3002\u73b0\u5c06 <code>target</code> \u63d2\u5165\u6570\u7ec4 <code>nums</code> \u4e2d\uff0c\u5e76\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u6570\u7ec4\u4e2d\u5df2\u5b58\u5728\u5143\u7d20 <code>target</code> \uff0c\u5219\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8bf7\u8fd4\u56de\u63d2\u5165\u540e <code>target</code> \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 10-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u679c\u60f3\u590d\u7528\u4e0a\u4e00\u8282\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\uff0c\u5219\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\u3002</p> <p>\u95ee\u9898\u4e00\uff1a\u5f53\u6570\u7ec4\u4e2d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u662f\u5426\u662f\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u9898\u76ee\u8981\u6c42\u5c06 <code>target</code> \u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u8fb9\uff0c\u8fd9\u610f\u5473\u7740\u65b0\u63d2\u5165\u7684 <code>target</code> \u66ff\u6362\u4e86\u539f\u6765 <code>target</code> \u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6570\u7ec4\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u5c31\u662f\u8be5 <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u95ee\u9898\u4e8c\uff1a\u5f53\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728 <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u662f\u54ea\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u67e5\u627e\u8fc7\u7a0b\uff1a\u5f53 <code>nums[m] &lt; target</code> \u65f6 \\(i\\) \u79fb\u52a8\uff0c\u8fd9\u610f\u5473\u7740\u6307\u9488 \\(i\\) \u5728\u5411\u5927\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u9488 \\(j\\) \u59cb\u7ec8\u5728\u5411\u5c0f\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</p> <p>\u56e0\u6b64\u4e8c\u5206\u7ed3\u675f\u65f6\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002\u6613\u5f97\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u7d22\u5f15\u4e3a \\(i\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09\"\"\"\ni, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i &lt;= j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target:\ni = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] &gt; target:\nj = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\nreturn m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n# \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\nint i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\nint i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\ni, j := 0, len(nums)-1\nfor i &lt;= j {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nm := i + (j-i)/2\nif nums[m] &lt; target {\n// target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] &gt; target {\n// target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else {\n// \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\nreturn m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\nvar i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i &lt;= j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target {\ni = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m] &gt; target {\nj = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nconst m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\nnums: Array&lt;number&gt;,\ntarget: number\n): number {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nconst m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\nlet (mut i, mut j) = (0, nums.len() as i32 - 1);    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i &lt;= j {\nlet m = i + (j - i) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m as usize] &lt; target {\ni = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m as usize] &gt; target {\nj = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1;  // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\ni\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\nint i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nreturn m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n}\n}\n// \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertionSimple}\n</code></pre>"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u5728\u4e0a\u4e00\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u89c4\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u5176\u4f59\u4e0d\u53d8\u3002</p> <p>\u5047\u8bbe\u6570\u7ec4\u4e2d\u5b58\u5728\u591a\u4e2a <code>target</code> \uff0c\u5219\u666e\u901a\u4e8c\u5206\u67e5\u627e\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u5143\u7d20\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11 <code>target</code>\u3002</p> <p>\u9898\u76ee\u8981\u6c42\u5c06\u76ee\u6807\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u8fb9\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u8651\u901a\u8fc7\u56fe 10-5 \u6240\u793a\u7684\u6b65\u9aa4\u5b9e\u73b0\u3002</p> <ol> <li>\u6267\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \u3002</li> <li>\u4ece\u7d22\u5f15 \\(k\\) \u5f00\u59cb\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u5f53\u627e\u5230\u6700\u5de6\u8fb9\u7684 <code>target</code> \u65f6\u8fd4\u56de\u3002</li> </ol> <p></p> <p> \u56fe 10-5 \u00a0 \u7ebf\u6027\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9 </p> <p>\u6b64\u65b9\u6cd5\u867d\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5f53\u6570\u7ec4\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684 <code>target</code> \u65f6\uff0c\u8be5\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002</p> <p>\u73b0\u8003\u8651\u62d3\u5c55\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002\u5982\u56fe 10-6 \u6240\u793a\uff0c\u6574\u4f53\u6d41\u7a0b\u4fdd\u6301\u4e0d\u53d8\uff0c\u6bcf\u8f6e\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad <code>target</code> \u548c <code>nums[m]</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5f53 <code>nums[m] &lt; target</code> \u6216 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u91c7\u7528\u666e\u901a\u4e8c\u5206\u67e5\u627e\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411 <code>target</code> \u9760\u8fd1\u3002</li> <li>\u5f53 <code>nums[m] == target</code> \u65f6\uff0c\u8bf4\u660e\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</li> </ul> <p>\u5faa\u73af\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684 <code>target</code> \uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u70b9\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-6 \u00a0 \u4e8c\u5206\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9\u7684\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5224\u65ad\u5206\u652f <code>nums[m] &gt; target</code> \u548c <code>nums[m] == target</code> \u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u4e24\u8005\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5c06\u5224\u65ad\u6761\u4ef6\u4fdd\u6301\u5c55\u5f00\uff0c\u56e0\u4e3a\u5176\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8bfb\u6027\u66f4\u597d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n\"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09\"\"\"\ni, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i &lt;= j:\nm = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target:\ni = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\nelif nums[m] &gt; target:\nj = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nelse:\nj = m - 1  # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n# \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\nint i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\nint i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n// \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\ni, j := 0, len(nums)-1\nfor i &lt;= j {\n// \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nm := i + (j-i)/2\nif nums[m] &lt; target {\n// target \u5728\u533a\u95f4 [m+1, j] \u4e2d\ni = m + 1\n} else if nums[m] &gt; target {\n// target \u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n} else {\n// \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\nj = m - 1\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\nvar i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i &lt;= j {\nlet m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m] &lt; target {\ni = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m] &gt; target {\nj = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nconst m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\nlet i = 0,\nj = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nconst m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\nint i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\nlet (mut i, mut j) = (0, nums.len() as i32 - 1);    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile i &lt;= j {\nlet m = i + (j - i) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif nums[m as usize] &lt; target {\ni = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if nums[m as usize] &gt; target {\nj = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1;  // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\ni\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\nint i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\nwhile (i &lt;= j) {\nint m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\nif (nums[m] &lt; target) {\ni = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n} else if (nums[m] &gt; target) {\nj = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n} else {\nj = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n}\n}\n// \u8fd4\u56de\u63d2\u5165\u70b9 i\nreturn i;\n}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertion}\n</code></pre> <p>Tip</p> <p>\u672c\u8282\u7684\u4ee3\u7801\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002</p> <p>\u603b\u7684\u6765\u770b\uff0c\u4e8c\u5206\u67e5\u627e\u65e0\u975e\u5c31\u662f\u7ed9\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u522b\u8bbe\u5b9a\u641c\u7d22\u76ee\u6807\uff0c\u76ee\u6807\u53ef\u80fd\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u5143\u7d20\uff08\u4f8b\u5982 <code>target</code> \uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u4e2a\u5143\u7d20\u8303\u56f4\uff08\u4f8b\u5982\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff09\u3002</p> <p>\u5728\u4e0d\u65ad\u7684\u5faa\u73af\u4e8c\u5206\u4e2d\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6e10\u903c\u8fd1\u9884\u5148\u8bbe\u5b9a\u7684\u76ee\u6807\u3002\u6700\u7ec8\uff0c\u5b83\u4eec\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u8fc7\u8fb9\u754c\u540e\u505c\u6b62\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"<p>\u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 <code>target</code> \uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3a <code>target</code> \u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"<p>\u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5982\u56fe 10-9 \u6240\u793a\uff0c\u6211\u4eec\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a <code>target</code> \uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 10-9 \u00a0 \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n\"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n# \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor i in range(len(nums) - 1):\nfor j in range(i + 1, len(nums)):\nif nums[i] + nums[j] == target:\nreturn [i, j]\nreturn []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\nint size = nums.size();\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor (int i = 0; i &lt; size - 1; i++) {\nfor (int j = i + 1; j &lt; size; j++) {\nif (nums[i] + nums[j] == target)\nreturn {i, j};\n}\n}\nreturn {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\nint size = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor (int i = 0; i &lt; size - 1; i++) {\nfor (int j = i + 1; j &lt; size; j++) {\nif (nums[i] + nums[j] == target)\nreturn new int[] { i, j };\n}\n}\nreturn new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] TwoSumBruteForce(int[] nums, int target) {\nint size = nums.Length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor (int i = 0; i &lt; size - 1; i++) {\nfor (int j = i + 1; j &lt; size; j++) {\nif (nums[i] + nums[j] == target)\nreturn [i, j];\n}\n}\nreturn [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\nsize := len(nums)\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor i := 0; i &lt; size-1; i++ {\nfor j := i + 1; i &lt; size; j++ {\nif nums[i]+nums[j] == target {\nreturn []int{i, j}\n}\n}\n}\nreturn nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor i in nums.indices.dropLast() {\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[i] + nums[j] == target {\nreturn [i, j]\n}\n}\n}\nreturn [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor (let i = 0; i &lt; n; i++) {\nfor (let j = i + 1; j &lt; n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\nconst n = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor (let i = 0; i &lt; n; i++) {\nfor (let j = i + 1; j &lt; n; j++) {\nif (nums[i] + nums[j] === target) {\nreturn [i, j];\n}\n}\n}\nreturn [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\nint size = nums.length;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor (var i = 0; i &lt; size - 1; i++) {\nfor (var j = i + 1; j &lt; size; j++) {\nif (nums[i] + nums[j] == target) return [i, j];\n}\n}\nreturn [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\nlet size = nums.len();\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nfor i in 0..size - 1 {\nfor j in i + 1..size {\nif nums[i] + nums[j] == target {\nreturn Some(vec![i as i32, j as i32]);\n}\n}\n}\nNone\n}\n</code></pre> two_sum.c<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\nfor (int i = 0; i &lt; numsSize; ++i) {\nfor (int j = i + 1; j &lt; numsSize; ++j) {\nif (nums[i] + nums[j] == target) {\nint *res = malloc(sizeof(int) * 2);\nres[0] = i, res[1] = j;\n*returnSize = 2;\nreturn res;\n}\n}\n}\n*returnSize = 0;\nreturn NULL;\n}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\nvar size: usize = nums.len;\nvar i: usize = 0;\n// \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\nwhile (i &lt; size - 1) : (i += 1) {\nvar j = i + 1;\nwhile (j &lt; size) : (j += 1) {\nif (nums[i] + nums[j] == target) {\nreturn [_]i32{@intCast(i), @intCast(j)};\n}\n}\n}\nreturn null;\n}\n</code></pre> <p>\u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"<p>\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\u56fe 10-10 \u6240\u793a\u7684\u6b65\u9aa4\u3002</p> <ol> <li>\u5224\u65ad\u6570\u5b57 <code>target - nums[i]</code> \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u3002</li> <li>\u5c06\u952e\u503c\u5bf9 <code>nums[i]</code> \u548c\u7d22\u5f15 <code>i</code> \u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> \u56fe 10-10 \u00a0 \u8f85\u52a9\u54c8\u5e0c\u8868\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n\"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n# \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndic = {}\n# \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor i in range(len(nums)):\nif target - nums[i] in dic:\nreturn [dic[target - nums[i]], i]\ndic[nums[i]] = i\nreturn []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\nint size = nums.size();\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nunordered_map&lt;int, int&gt; dic;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor (int i = 0; i &lt; size; i++) {\nif (dic.find(target - nums[i]) != dic.end()) {\nreturn {dic[target - nums[i]], i};\n}\ndic.emplace(nums[i], i);\n}\nreturn {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\nint size = nums.length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nMap&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor (int i = 0; i &lt; size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn new int[] { dic.get(target - nums[i]), i };\n}\ndic.put(nums[i], i);\n}\nreturn new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\nint size = nums.Length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nDictionary&lt;int, int&gt; dic = [];\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor (int i = 0; i &lt; size; i++) {\nif (dic.ContainsKey(target - nums[i])) {\nreturn [dic[target - nums[i]], i];\n}\ndic.Add(nums[i], i);\n}\nreturn [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nhashTable := map[int]int{}\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor idx, val := range nums {\nif preIdx, ok := hashTable[target-val]; ok {\nreturn []int{preIdx, idx}\n}\nhashTable[val] = idx\n}\nreturn nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nvar dic: [Int: Int] = [:]\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor i in nums.indices {\nif let j = dic[target - nums[i]] {\nreturn [j, i]\n}\ndic[nums[i]] = i\n}\nreturn [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nlet m = {};\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor (let i = 0; i &lt; nums.length; i++) {\nif (m[target - nums[i]] !== undefined) {\nreturn [m[target - nums[i]], i];\n} else {\nm[nums[i]] = i;\n}\n}\nreturn [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nlet m: Map&lt;number, number&gt; = new Map();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor (let i = 0; i &lt; nums.length; i++) {\nlet index = m.get(target - nums[i]);\nif (index !== undefined) {\nreturn [index, i];\n} else {\nm.set(nums[i], i);\n}\n}\nreturn [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\nint size = nums.length;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nMap&lt;int, int&gt; dic = HashMap();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor (var i = 0; i &lt; size; i++) {\nif (dic.containsKey(target - nums[i])) {\nreturn [dic[target - nums[i]]!, i];\n}\ndic.putIfAbsent(nums[i], () =&gt; i);\n}\nreturn [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nlet mut dic = HashMap::new();\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nfor (i, num) in nums.iter().enumerate() {\nmatch dic.get(&amp;(target - num)) {\nSome(v) =&gt; return Some(vec![*v as i32, i as i32]),\nNone =&gt; dic.insert(num, i as i32)\n};\n}\nNone\n}\n</code></pre> two_sum.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\nint key;\nint val;\nUT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nHashTable *find(HashTable *h, int key) {\nHashTable *tmp;\nHASH_FIND_INT(h, &amp;key, tmp);\nreturn tmp;\n}\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable *h, int key, int val) {\nHashTable *t = find(h, key);\nif (t == NULL) {\nHashTable *tmp = malloc(sizeof(HashTable));\ntmp-&gt;key = key, tmp-&gt;val = val;\nHASH_ADD_INT(h, key, tmp);\n} else {\nt-&gt;val = val;\n}\n}\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\nHashTable *hashtable = NULL;\nfor (int i = 0; i &lt; numsSize; i++) {\nHashTable *t = find(hashtable, target - nums[i]);\nif (t != NULL) {\nint *res = malloc(sizeof(int) * 2);\nres[0] = t-&gt;val, res[1] = i;\n*returnSize = 2;\nreturn res;\n}\ninsert(hashtable, nums[i], i);\n}\n*returnSize = 0;\nreturn NULL;\n}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\nvar size: usize = nums.len;\n// \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nvar dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\ndefer dic.deinit();\nvar i: usize = 0;\n// \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nwhile (i &lt; size) : (i += 1) {\nif (dic.contains(target - nums[i])) {\nreturn [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n}\ntry dic.put(nums[i], @intCast(i));\n}\nreturn null;\n}\n</code></pre> <p>\u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002</p> <p>\u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"<p>\u300c\u641c\u7d22\u7b97\u6cd5 searching algorithm\u300d\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002</p> <p>\u641c\u7d22\u7b97\u6cd5\u53ef\u6839\u636e\u5b9e\u73b0\u601d\u8def\u5206\u4e3a\u4ee5\u4e0b\u4e24\u7c7b\u3002</p> <ul> <li>\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002</li> <li>\u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002</li> </ul> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u7d22","text":"<p>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u7ebf\u6027\u641c\u7d22\u201d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u201c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u548c\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002</li> </ul> <p>\u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"<p>\u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u4e8c\u5206\u67e5\u627e\u201d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002</li> <li>\u201c\u54c8\u5e0c\u67e5\u627e\u201d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002</li> <li>\u201c\u6811\u67e5\u627e\u201d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</li> </ul> <p>\u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</p> <p>\u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u9500\u3002</p> <p>Tip</p> <p>\u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"<p>\u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u4ece\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u56fe 10-11 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-11 \u00a0 \u591a\u79cd\u641c\u7d22\u7b56\u7565 </p> <p>\u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002</p> <p> \u8868 10-1 \u00a0 \u67e5\u627e\u7b97\u6cd5\u6548\u7387\u5bf9\u6bd4 </p> \u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f <p>\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002</p> <p>\u7ebf\u6027\u641c\u7d22</p> <ul> <li>\u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u987b\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002</li> <li>\u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002</li> <li>\u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002</li> <li>\u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002</li> </ul> <p>\u54c8\u5e0c\u67e5\u627e</p> <ul> <li>\u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> <li>\u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002</li> <li>\u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002</li> <li>\u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002</li> </ul> <p>\u6811\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u5206\u6563\u5b58\u50a8\u7684\u3002</li> <li>\u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002</li> <li>\u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_searching/summary/","title":"10.6 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u8fdb\u884c\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002</li> <li>\u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002</li> <li>\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u7528\u4e8e\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u987b\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002</li> <li>\u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</li> </ul>"},{"location":"chapter_sorting/","title":"\u7b2c 11 \u7ae0 \u00a0 \u6392\u5e8f","text":"<p>Abstract</p> <p>\u6392\u5e8f\u72b9\u5982\u4e00\u628a\u5c06\u6df7\u4e71\u53d8\u4e3a\u79e9\u5e8f\u7684\u9b54\u6cd5\u94a5\u5319\uff0c\u4f7f\u6211\u4eec\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u4e0e\u5904\u7406\u6570\u636e\u3002</p> <p>\u65e0\u8bba\u662f\u7b80\u5355\u7684\u5347\u5e8f\uff0c\u8fd8\u662f\u590d\u6742\u7684\u5206\u7c7b\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u4eec\u5c55\u793a\u4e86\u6570\u636e\u7684\u548c\u8c10\u7f8e\u611f\u3002</p>"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5</li> <li>11.2 \u00a0 \u9009\u62e9\u6392\u5e8f</li> <li>11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f</li> <li>11.4 \u00a0 \u63d2\u5165\u6392\u5e8f</li> <li>11.5 \u00a0 \u5feb\u901f\u6392\u5e8f</li> <li>11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f</li> <li>11.7 \u00a0 \u5806\u6392\u5e8f</li> <li>11.8 \u00a0 \u6876\u6392\u5e8f</li> <li>11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f</li> <li>11.10 \u00a0 \u57fa\u6570\u6392\u5e8f</li> <li>11.11 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_sorting/bubble_sort/","title":"11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"<p>\u300c\u5192\u6ce1\u6392\u5e8f bubble sort\u300d\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002</p> <p>\u5982\u56fe 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u8fc7\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6765\u6a21\u62df\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 &gt; \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u4e8c\u8005\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u5192\u6ce1 </p>"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u56fe 11-5 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c</li> <li>\u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> <p></p> <p> \u56fe 11-5 \u00a0 \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n\"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(n - 1, 0, -1):\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j in range(i):\nif nums[j] &gt; nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\n}\n}\n}\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n(nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n}\n}\n}\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i &gt; 0; i-- {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j &lt; i; j++ {\nif nums[j] &gt; nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\n}\n}\n}\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] &gt; nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\n}\n}\n}\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (let j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (let j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List&lt;int&gt; nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i &gt; 0; i--) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfn bubble_sort(nums: &amp;mut [i32]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in (1..nums.len()).rev() {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in 0..i {\nif nums[j] &gt; nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = 0; i &lt; size - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; size - 1 - i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\nint temp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = temp;\n}\n}\n}\n}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: usize = nums.len - 1;\nwhile (i &gt; 0) : (i -= 1) {\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j &lt; i) : (j += 1) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\n}\n}\n}\n}\n</code></pre>"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u4f18\u5316","text":"<p>\u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d <code>flag</code> \u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002</p> <p>\u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n\"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in range(n - 1, 0, -1):\nflag = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n# \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j in range(i):\nif nums[j] &gt; nums[j + 1]:\n# \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j + 1] = nums[j + 1], nums[j]\nflag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\nif not flag:\nbreak  # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.size() - 1; i &gt; 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n// \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\nswap(nums[j], nums[j + 1]);\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag)\nbreak; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i &gt; 0; i--) {\nboolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag)\nbreak; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.Length - 1; i &gt; 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n(nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := len(nums) - 1; i &gt; 0; i-- {\nflag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor j := 0; j &lt; i; j++ {\nif nums[j] &gt; nums[j+1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nnums[j], nums[j+1] = nums[j+1], nums[j]\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\nvar flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nfor j in stride(from: 0, to: i, by: 1) {\nif nums[j] &gt; nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j]\nnums[j] = nums[j + 1]\nnums[j + 1] = tmp\nflag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\nbreak\n}\n}\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i &gt; 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (let j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (let i = nums.length - 1; i &gt; 0; i--) {\nlet flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (let j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = nums.length - 1; i &gt; 0; i--) {\nbool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nint tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i in (1..nums.len()).rev() {\nlet mut flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nfor j in 0..i {\nif nums[j] &gt; nums[j + 1] {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nlet tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n}\n}\nif !flag {break};  // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor (int i = 0; i &lt; size - 1; i++) {\nbool flag = false;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\nfor (int j = 0; j &lt; size - 1 - i; j++) {\nif (nums[j] &gt; nums[j + 1]) {\nint temp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = temp;\nflag = true;\n}\n}\nif (!flag)\nbreak;\n}\n}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nvar i: usize = nums.len - 1;\nwhile (i &gt; 0) : (i -= 1) {\nvar flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\nvar j: usize = 0;\n// \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \nwhile (j &lt; i) : (j += 1) {\nif (nums[j] &gt; nums[j + 1]) {\n// \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\nvar tmp = nums[j];\nnums[j] = nums[j + 1];\nnums[j + 1] = tmp;\nflag = true;\n}\n}\nif (!flag) break;   // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n}\n}\n</code></pre>"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u603b\u548c\u4e3a \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165 <code>flag</code> \u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/","title":"11.8 \u00a0 \u6876\u6392\u5e8f","text":"<p>\u524d\u8ff0\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u9636\u3002</p> <p>\u300c\u6876\u6392\u5e8f bucket sort\u300d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002</p>"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5176\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u5185\u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-13 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\u3002</li> <li>\u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u8fd9\u91cc\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\u3002</li> <li>\u6309\u7167\u6876\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u5408\u5e76\u7ed3\u679c\u3002</li> </ol> <p></p> <p> \u56fe 11-13 \u00a0 \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n\"\"\"\u6876\u6392\u5e8f\"\"\"\n# \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nk = len(nums) // 2\nbuckets = [[] for _ in range(k)]\n# 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor num in nums:\n# \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\ni = int(num * k)\n# \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].append(num)\n# 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor bucket in buckets:\n# \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort()\n# 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\ni = 0\nfor bucket in buckets:\nfor num in bucket:\nnums[i] = num\ni += 1\n</code></pre> bucket_sort.cpp<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.size() / 2;\nvector&lt;vector&lt;float&gt;&gt; buckets(k);\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (float num : nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = num * k;\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\nbuckets[i].push_back(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (vector&lt;float&gt; &amp;bucket : buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nsort(bucket.begin(), bucket.end());\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (vector&lt;float&gt; &amp;bucket : buckets) {\nfor (float num : bucket) {\nnums[i++] = num;\n}\n}\n}\n</code></pre> bucket_sort.java<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.length / 2;\nList&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\nfor (int i = 0; i &lt; k; i++) {\nbuckets.add(new ArrayList&lt;&gt;());\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (float num : nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (int) (num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets.get(i).add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (List&lt;Float&gt; bucket : buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nCollections.sort(bucket);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (List&lt;Float&gt; bucket : buckets) {\nfor (float num : bucket) {\nnums[i++] = num;\n}\n}\n}\n</code></pre> bucket_sort.cs<pre><code>/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.Length / 2;\nList&lt;List&lt;float&gt;&gt; buckets = [];\nfor (int i = 0; i &lt; k; i++) {\nbuckets.Add([]);\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nforeach (float num in nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (int)(num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].Add(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nforeach (List&lt;float&gt; bucket in buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.Sort();\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint j = 0;\nforeach (List&lt;float&gt; bucket in buckets) {\nforeach (float num in bucket) {\nnums[j++] = num;\n}\n}\n}\n</code></pre> bucket_sort.go<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nk := len(nums) / 2\nbuckets := make([][]float64, k)\nfor i := 0; i &lt; k; i++ {\nbuckets[i] = make([]float64, 0)\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor _, num := range nums {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\ni := int(num * float64(k))\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i] = append(buckets[i], num)\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor i := 0; i &lt; k; i++ {\n// \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nsort.Float64s(buckets[i])\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\ni := 0\nfor _, bucket := range buckets {\nfor _, num := range bucket {\nnums[i] = num\ni++\n}\n}\n}\n</code></pre> bucket_sort.swift<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nlet k = nums.count / 2\nvar buckets = (0 ..&lt; k).map { _ in [Double]() }\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor num in nums {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nlet i = Int(num * Double(k))\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].append(num)\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor i in buckets.indices {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbuckets[i].sort()\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nvar i = nums.startIndex\nfor bucket in buckets {\nfor num in bucket {\nnums[i] = num\nnums.formIndex(after: &amp;i)\n}\n}\n}\n</code></pre> bucket_sort.js<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nconst k = nums.length / 2;\nconst buckets = [];\nfor (let i = 0; i &lt; k; i++) {\nbuckets.push([]);\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (const num of nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nconst i = Math.floor(num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].push(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (const bucket of buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort((a, b) =&gt; a - b);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nlet i = 0;\nfor (const bucket of buckets) {\nfor (const num of bucket) {\nnums[i++] = num;\n}\n}\n}\n</code></pre> bucket_sort.ts<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nconst k = nums.length / 2;\nconst buckets: number[][] = [];\nfor (let i = 0; i &lt; k; i++) {\nbuckets.push([]);\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (const num of nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nconst i = Math.floor(num * k);\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].push(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (const bucket of buckets) {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort((a, b) =&gt; a - b);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nlet i = 0;\nfor (const bucket of buckets) {\nfor (const num of bucket) {\nnums[i++] = num;\n}\n}\n}\n</code></pre> bucket_sort.dart<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(List&lt;double&gt; nums) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = nums.length ~/ 2;\nList&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (double _num in nums) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 _num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint i = (_num * k).toInt();\n// \u5c06 _num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\nbuckets[i].add(_num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (List&lt;double&gt; bucket in buckets) {\nbucket.sort();\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nint i = 0;\nfor (List&lt;double&gt; bucket in buckets) {\nfor (double _num in bucket) {\nnums[i++] = _num;\n}\n}\n}\n</code></pre> bucket_sort.rs<pre><code>/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &amp;mut [f64]) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nlet k = nums.len() / 2;\nlet mut buckets = vec![vec![]; k];\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor &amp;mut num in &amp;mut *nums {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nlet i = (num * k as f64) as usize;\n// \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\nbuckets[i].push(num);\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor bucket in &amp;mut buckets {\n// \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\nbucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nlet mut i = 0;\nfor bucket in &amp;mut buckets {\nfor &amp;mut num in bucket {\nnums[i] = num;\ni += 1;\n}\n}\n}\n</code></pre> bucket_sort.c<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n// \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\nint k = size / 2;\nfloat **buckets = calloc(k, sizeof(float *));\nfor (int i = 0; i &lt; k; i++) {\n// \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d k \u4e2a\u5143\u7d20\nbuckets[i] = calloc(ARRAY_SIZE, sizeof(float));\n}\n// 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\nfor (int i = 0; i &lt; size; i++) {\n// \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\nint bucket_idx = nums[i] * k;\nint j = 0;\n// \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\nwhile (buckets[bucket_idx][j] &gt; 0 &amp;&amp; buckets[bucket_idx][j] &lt; nums[i]) {\nj++;\n}\nfloat temp = nums[i];\nwhile (j &lt; ARRAY_SIZE &amp;&amp; buckets[bucket_idx][j] &gt; 0) {\nswap(&amp;temp, &amp;buckets[bucket_idx][j]);\nj++;\n}\nbuckets[bucket_idx][j] = temp;\n}\n// 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\nfor (int i = 0; i &lt; k; i++) {\nqsort(buckets[i], ARRAY_SIZE, sizeof(float), compare_float);\n}\n// 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\nfor (int i = 0, j = 0; j &lt; k; j++) {\nfor (int l = 0; l &lt; ARRAY_SIZE; l++) {\nif (buckets[j][l] &gt; 0) {\nnums[i++] = buckets[j][l];\n}\n}\n}\n// \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\nfor (int i = 0; i &lt; k; i++) {\nfree(buckets[i]);\n}\nfree(buckets);\n}\n</code></pre> bucket_sort.zig<pre><code>[class]{}-[func]{bucketSort}\n</code></pre>"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8d39 \\(O(n + k)\\) \u65f6\u95f4\u3002</li> <li>\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3 \u00a0 \u5982\u4f55\u5b9e\u73b0\u5e73\u5747\u5206\u914d","text":"<p>\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4e2a\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002</p> <p>\u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u6761\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002</p> <p>\u5982\u56fe 11-14 \u6240\u793a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u76ee\u6807\u662f\u8ba9\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002</p> <p></p> <p> \u56fe 11-14 \u00a0 \u9012\u5f52\u5212\u5206\u6876 </p> <p>\u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002</p> <p>\u5982\u56fe 11-15 \u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 11-15 \u00a0 \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876 </p>"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"<p>\u300c\u8ba1\u6570\u6392\u5e8f counting sort\u300d\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"<p>\u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\uff0c\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-16 \u6240\u793a\u3002</p> <ol> <li>\u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 <code>counter</code> \u3002</li> <li>\u501f\u52a9 <code>counter</code> \u7edf\u8ba1 <code>nums</code> \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d <code>counter[num]</code> \u5bf9\u5e94\u6570\u5b57 <code>num</code> \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 <code>nums</code>\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a <code>num</code>\uff09\uff0c\u6bcf\u8f6e\u5c06 <code>counter[num]</code> \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002</li> <li>\u7531\u4e8e <code>counter</code> \u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 <code>counter</code> \uff0c\u6839\u636e\u5404\u6570\u5b57\u51fa\u73b0\u6b21\u6570\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 <code>nums</code> \u5373\u53ef\u3002</li> </ol> <p></p> <p> \u56fe 11-16 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n\"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n# \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = 0\nfor num in nums:\nm = max(m, num)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\ni = 0\nfor num in range(m + 1):\nfor _ in range(counter[num]):\nnums[i] = num\ni += 1\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector&lt;int&gt; counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num &lt; m + 1; num++) {\nfor (int j = 0; j &lt; counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num &lt; m + 1; num++) {\nfor (int j = 0; j &lt; counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid CountingSortNaive(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nforeach (int num in nums) {\nm = Math.Max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nforeach (int num in nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num &lt; m + 1; num++) {\nfor (int j = 0; j &lt; counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor _, num := range nums {\nif num &gt; m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nfor i, num := 0, 0; num &lt; m+1; num++ {\nfor j := 0; j &lt; counter[num]; j++ {\nnums[i] = num\ni++\n}\n}\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nvar i = 0\nfor num in stride(from: 0, to: m + 1, by: 1) {\nfor _ in stride(from: 0, to: counter[num], by: 1) {\nnums[i] = num\ni += 1\n}\n}\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter = new Array(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nlet i = 0;\nfor (let num = 0; num &lt; m + 1; num++) {\nfor (let j = 0; j &lt; counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nlet i = 0;\nfor (let num = 0; num &lt; m + 1; num++) {\nfor (let j = 0; j &lt; counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List&lt;int&gt; nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int _num in nums) {\nm = max(m, _num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\nList&lt;int&gt; counter = List.filled(m + 1, 0);\nfor (int _num in nums) {\ncounter[_num]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int _num = 0; _num &lt; m + 1; _num++) {\nfor (int j = 0; j &lt; counter[_num]; j++, i++) {\nnums[i] = _num;\n}\n}\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = *nums.into_iter().max().unwrap();\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nlet mut counter = vec![0; m as usize + 1];\nfor &amp;num in &amp;*nums {\ncounter[num as usize] += 1;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nlet mut i = 0;\nfor num in 0..m + 1 {\nfor _ in 0..counter[num as usize] {\nnums[i] = num;\ni += 1;\n}\n}\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int i = 0; i &lt; size; i++) {\nif (nums[i] &gt; m) {\nm = nums[i];\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint *counter = calloc(m, sizeof(int));\nfor (int i = 0; i &lt; size; i++) {\ncounter[nums[i]]++;\n}\n// 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\nint i = 0;\nfor (int num = 0; num &lt; m + 1; num++) {\nfor (int j = 0; j &lt; counter[num]; j++, i++) {\nnums[i] = num;\n}\n}\n// 4. \u91ca\u653e\u5185\u5b58\nfree(counter);\n}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSortNaive}\n</code></pre> <p>\u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb</p> <p>\u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4 <code>counter</code> \u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 <code>3.</code> \u5c31\u5931\u6548\u4e86\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p> <p>\u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 <code>counter</code> \u7684\u201c\u524d\u7f00\u548c\u201d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 <code>i</code> \u5904\u7684\u524d\u7f00\u548c <code>prefix[i]</code> \u7b49\u4e8e\u6570\u7ec4\u524d <code>i</code> \u4e2a\u5143\u7d20\u4e4b\u548c\uff1a</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>\u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0c<code>prefix[num] - 1</code> \u4ee3\u8868\u5143\u7d20 <code>num</code> \u5728\u7ed3\u679c\u6570\u7ec4 <code>res</code> \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 <code>nums</code> \u7684\u6bcf\u4e2a\u5143\u7d20 <code>num</code> \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u5c06 <code>num</code> \u586b\u5165\u6570\u7ec4 <code>res</code> \u7684\u7d22\u5f15 <code>prefix[num] - 1</code> \u5904\u3002</li> <li>\u4ee4\u524d\u7f00\u548c <code>prefix[num]</code> \u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e <code>num</code> \u7684\u7d22\u5f15\u3002</li> </ol> <p>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4 <code>res</code> \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 <code>res</code> \u8986\u76d6\u539f\u6570\u7ec4 <code>nums</code> \u5373\u53ef\u3002\u56fe 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-17 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n\"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n# \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n# 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm = max(nums)\n# 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n# counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter = [0] * (m + 1)\nfor num in nums:\ncounter[num] += 1\n# 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n# \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in range(m):\ncounter[i + 1] += counter[i]\n# 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n# \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn = len(nums)\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nnum = nums[i]\nres[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n# \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvector&lt;int&gt; counter(m + 1, 0);\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i &lt; m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.size();\nvector&lt;int&gt; res(n);\nfor (int i = n - 1; i &gt;= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int num : nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nfor (int num : nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i &lt; m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nint[] res = new int[n];\nfor (int i = n - 1; i &gt;= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i &lt; n; i++) {\nnums[i] = res[i];\n}\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nforeach (int num in nums) {\nm = Math.Max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint[] counter = new int[m + 1];\nforeach (int num in nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i &lt; m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.Length;\nint[] res = new int[n];\nfor (int i = n - 1; i &gt;= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i &lt; n; i++) {\nnums[i] = res[i];\n}\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nm := 0\nfor _, num := range nums {\nif num &gt; m {\nm = num\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\ncounter := make([]int, m+1)\nfor _, num := range nums {\ncounter[num]++\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i := 0; i &lt; m; i++ {\ncounter[i+1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nn := len(nums)\nres := make([]int, n)\nfor i := n - 1; i &gt;= 0; i-- {\nnum := nums[i]\n// \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\nres[counter[num]-1] = num\n// \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\ncounter[num]--\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\ncopy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = nums.max()!\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nvar counter = Array(repeating: 0, count: m + 1)\nfor num in nums {\ncounter[num] += 1\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in stride(from: 0, to: m, by: 1) {\ncounter[i + 1] += counter[i]\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nvar res = Array(repeating: 0, count: nums.count)\nfor i in stride(from: nums.count - 1, through: 0, by: -1) {\nlet num = nums[i]\nres[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in stride(from: 0, to: nums.count, by: 1) {\nnums[i] = res[i]\n}\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter = new Array(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (let i = 0; i &lt; m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nconst n = nums.length;\nconst res = new Array(n);\nfor (let i = n - 1; i &gt;= 0; i--) {\nconst num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i &lt; n; i++) {\nnums[i] = res[i];\n}\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = 0;\nfor (const num of nums) {\nm = Math.max(m, num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nconst counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\nfor (const num of nums) {\ncounter[num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (let i = 0; i &lt; m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nconst n = nums.length;\nconst res: number[] = new Array&lt;number&gt;(n);\nfor (let i = n - 1; i &gt;= 0; i--) {\nconst num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i &lt; n; i++) {\nnums[i] = res[i];\n}\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List&lt;int&gt; nums) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int _num in nums) {\nm = max(m, _num);\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\nList&lt;int&gt; counter = List.filled(m + 1, 0);\nfor (int _num in nums) {\ncounter[_num]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i &lt; m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint n = nums.length;\nList&lt;int&gt; res = List.filled(n, 0);\nfor (int i = n - 1; i &gt;= 0; i--) {\nint _num = nums[i];\nres[counter[_num] - 1] = _num; // \u5c06 _num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[_num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nnums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &amp;mut [i32]) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nlet m = *nums.into_iter().max().unwrap();\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nlet mut counter = vec![0; m as usize + 1];\nfor &amp;num in &amp;*nums {\ncounter[num as usize] += 1;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor i in 0..m as usize {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nlet n = nums.len();\nlet mut res = vec![0; n];\nfor i in (0..n).rev() {\nlet num = nums[i];\nres[counter[num as usize] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num as usize] -= 1; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in 0..n {\nnums[i] = res[i];\n}\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n// 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\nint m = 0;\nfor (int i = 0; i &lt; size; i++) {\nif (nums[i] &gt; m) {\nm = nums[i];\n}\n}\n// 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n// counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\nint *counter = calloc(m, sizeof(int));\nfor (int i = 0; i &lt; size; i++) {\ncounter[nums[i]]++;\n}\n// 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n// \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\nfor (int i = 0; i &lt; m; i++) {\ncounter[i + 1] += counter[i];\n}\n// 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n// \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\nint *res = malloc(sizeof(int) * size);\nfor (int i = size - 1; i &gt;= 0; i--) {\nint num = nums[i];\nres[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\ncounter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n}\n// \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\nmemcpy(nums, res, size * sizeof(int));\n// 5. \u91ca\u653e\u5185\u5b58\nfree(counter);\n}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSort}\n</code></pre>"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 <code>nums</code> \u548c\u904d\u5386 <code>counter</code> \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411 <code>res</code> \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 <code>nums</code> \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386 <code>nums</code> \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002</li> </ul>"},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4 \u00a0 \u5c40\u9650\u6027","text":"<p>\u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002</p>"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u6392\u5e8f","text":"<p>Tip</p> <p>\u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u201c\u5806\u201c\u7ae0\u8282\u3002</p> <p>\u300c\u5806\u6392\u5e8f heap sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002</li> </ol> <p>\u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002</p>"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-12 \u6240\u793a\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002</li> <li>\u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08Sift Down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002</li> </ol> <p>Tip</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-12 \u00a0 \u5806\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u201c\u5806\u201d\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316 <code>sift_down()</code> \u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9 <code>sift_down()</code> \u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n\"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\nwhile True:\n# \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl = 2 * i + 1\nr = 2 * i + 2\nma = i\nif l &lt; n and nums[l] &gt; nums[ma]:\nma = l\nif r &lt; n and nums[r] &gt; nums[ma]:\nma = r\n# \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i:\nbreak\n# \u4ea4\u6362\u4e24\u8282\u70b9\nnums[i], nums[ma] = nums[ma], nums[i]\n# \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\ndef heap_sort(nums: list[int]):\n\"\"\"\u5806\u6392\u5e8f\"\"\"\n# \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in range(len(nums) // 2 - 1, -1, -1):\nsift_down(nums, len(nums), i)\n# \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i in range(len(nums) - 1, 0, -1):\n# \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nnums[0], nums[i] = nums[i], nums[0]\n# \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\nma = l;\nif (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nswap(nums[i], nums[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\nsiftDown(nums, nums.size(), i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.size() - 1; i &gt; 0; --i) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nswap(nums[0], nums[i]);\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n</code></pre> heap_sort.java<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\nma = l;\nif (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.length - 1; i &gt; 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n</code></pre> heap_sort.cs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\nma = l;\nif (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i)\nbreak;\n// \u4ea4\u6362\u4e24\u8282\u70b9\n(nums[ma], nums[i]) = (nums[i], nums[ma]);\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\nSiftDown(nums, nums.Length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.Length - 1; i &gt; 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n(nums[i], nums[0]) = (nums[0], nums[i]);\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nSiftDown(nums, i, 0);\n}\n}\n</code></pre> heap_sort.go<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\nfor true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nl := 2*i + 1\nr := 2*i + 2\nma := i\nif l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\nma = l\n}\nif r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n(*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i := len(*nums)/2 - 1; i &gt;= 0; i-- {\nsiftDown(nums, len(*nums), i)\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i := len(*nums) - 1; i &gt; 0; i-- {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n(*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0)\n}\n}\n</code></pre> heap_sort.swift<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\nvar i = i\nwhile true {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1\nlet r = 2 * i + 2\nvar ma = i\nif l &lt; n, nums[l] &gt; nums[ma] {\nma = l\n}\nif r &lt; n, nums[r] &gt; nums[ma] {\nma = r\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nnums.swapAt(i, ma)\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma\n}\n}\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\nsiftDown(nums: &amp;nums, n: nums.count, i: i)\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i in stride(from: nums.count - 1, to: 0, by: -1) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nnums.swapAt(0, i)\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums: &amp;nums, n: i, i: 0)\n}\n}\n</code></pre> heap_sort.js<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1;\nlet r = 2 * i + 2;\nlet ma = i;\nif (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\nma = l;\n}\nif (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n[nums[i], nums[ma]] = [nums[ma], nums[i]];\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (let i = nums.length - 1; i &gt; 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n[nums[0], nums[i]] = [nums[i], nums[0]];\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n</code></pre> heap_sort.ts<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1;\nlet r = 2 * i + 2;\nlet ma = i;\nif (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\nma = l;\n}\nif (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma === i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\n[nums[i], nums[ma]] = [nums[ma], nums[i]];\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (let i = nums.length - 1; i &gt; 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n[nums[0], nums[i]] = [nums[i], nums[0]];\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n</code></pre> heap_sort.dart<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\nwhile (true) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\nif (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) break;\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List&lt;int&gt; nums) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\nsiftDown(nums, nums.length, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = nums.length - 1; i &gt; 0; i--) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n</code></pre> heap_sort.rs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\nloop {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nlet l = 2 * i + 1;\nlet r = 2 * i + 2;\nlet mut ma = i;\nif l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\nma = l;\n}\nif r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\nma = r;\n}\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif ma == i {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nlet temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nfn heap_sort(nums: &amp;mut [i32]) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor i in (0..=nums.len() / 2 - 1).rev() {\nsift_down(nums, nums.len(), i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor i in (1..=nums.len() - 1).rev() {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nlet tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsift_down(nums, i, 0);\n}\n}\n</code></pre> heap_sort.c<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\nwhile (1) {\n// \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\nint l = 2 * i + 1;\nint r = 2 * i + 2;\nint ma = i;\nif (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\nma = l;\nif (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\nma = r;\n// \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\nif (ma == i) {\nbreak;\n}\n// \u4ea4\u6362\u4e24\u8282\u70b9\nint temp = nums[i];\nnums[i] = nums[ma];\nnums[ma] = temp;\n// \u5faa\u73af\u5411\u4e0b\u5806\u5316\ni = ma;\n}\n}\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n// \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\nfor (int i = n / 2 - 1; i &gt;= 0; --i) {\nsiftDown(nums, n, i);\n}\n// \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\nfor (int i = n - 1; i &gt; 0; --i) {\n// \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\nint tmp = nums[0];\nnums[0] = nums[i];\nnums[i] = tmp;\n// \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\nsiftDown(nums, i, 0);\n}\n}\n</code></pre> heap_sort.zig<pre><code>[class]{}-[func]{siftDown}\n[class]{}-[func]{heapSort}\n</code></pre>"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/","title":"11.4 \u00a0 \u63d2\u5165\u6392\u5e8f","text":"<p>\u300c\u63d2\u5165\u6392\u5e8f insertion sort\u300d\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002</p> <p>\u56fe 11-6 \u5c55\u793a\u4e86\u6570\u7ec4\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a <code>base</code> \uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 <code>base</code> \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u5c06 <code>base</code> \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 11-6 \u00a0 \u5355\u6b21\u63d2\u5165\u64cd\u4f5c </p>"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-7 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n\"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n# \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\nfor i in range(1, len(nums)):\nbase = nums[i]\nj = i - 1\n# \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j &gt;= 0 and nums[j] &gt; base:\nnums[j + 1] = nums[j]  # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\nnums[j + 1] = base  # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n</code></pre> insertion_sort.cpp<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i &lt; nums.size(); i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.java<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i &lt; nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base;        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.cs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i &lt; nums.Length; i++) {\nint bas = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = bas;         // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.go<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\nfor i := 1; i &lt; len(nums); i++ {\nbase := nums[i]\nj := i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nfor j &gt;= 0 &amp;&amp; nums[j] &gt; base {\nnums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--\n}\nnums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.swift<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor i in stride(from: 1, to: nums.count, by: 1) {\nlet base = nums[i]\nvar j = i - 1\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j &gt;= 0, nums[j] &gt; base {\nnums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1\n}\nnums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.js<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (let i = 1; i &lt; nums.length; i++) {\nlet base = nums[i],\nj = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.ts<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (let i = 1; i &lt; nums.length; i++) {\nconst base = nums[i];\nlet j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.dart<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List&lt;int&gt; nums) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i &lt; nums.length; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\nnums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj--;\n}\nnums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.rs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &amp;mut [i32]) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor i in 1..nums.len() {\nlet (base, mut j) = (nums[i],  (i - 1) as i32);\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\nnums[(j + 1) as usize] = nums[j as usize]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nj -= 1;\n}\nnums[(j + 1) as usize] = base;  // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre> insertion_sort.c<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nfor (int i = 1; i &lt; size; i++) {\nint base = nums[i], j = i - 1;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n// \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\nnums[j + 1] = nums[j];\nj--;\n}\n// \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\nnums[j + 1] = base;\n}\n}\n</code></pre> insertion_sort.zig<pre><code>// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n// \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\nvar i: usize = 1;\nwhile (i &lt; nums.len) : (i += 1) {\nvar base = nums[i];\nvar j: usize = i;\n// \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\nwhile (j &gt;= 1 and nums[j - 1] &gt; base) : (j -= 1) {\nnums[j] = nums[j - 1];  // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n}\nnums[j] = base;             // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n}\n}\n</code></pre>"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3 \u00a0 \u63d2\u5165\u6392\u5e8f\u4f18\u52bf","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002</p> <p>\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u5730\u4f4d\uff1b\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u4f5c\u7528\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002</p> <p>\u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/","title":"11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"<p>\u300c\u5f52\u5e76\u6392\u5e8f merge sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5305\u542b\u56fe 11-10 \u6240\u793a\u7684\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u9636\u6bb5\u3002</p> <ol> <li>\u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-10 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5 </p>"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5982\u56fe 11-11 \u6240\u793a\uff0c\u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002</p> <ol> <li>\u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 <code>mid</code> \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[left, mid]</code> \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[mid + 1, right]</code> \uff09\u3002</li> <li>\u9012\u5f52\u6267\u884c\u6b65\u9aa4 <code>1.</code> \uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u3002</li> </ol> <p>\u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-11 \u00a0 \u5f52\u5e76\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u4e0e\u4e8c\u53c9\u6811\u540e\u5e8f\u904d\u5386\u7684\u9012\u5f52\u987a\u5e8f\u662f\u4e00\u81f4\u7684\u3002</p> <ul> <li>\u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002</li> </ul> <p>\u5f52\u5e76\u6392\u5e8f\u7684\u5b9e\u73b0\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\u3002\u8bf7\u6ce8\u610f\uff0c<code>nums</code> \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a <code>[left, right]</code> \uff0c\u800c <code>tmp</code> \u7684\u5bf9\u5e94\u533a\u95f4\u4e3a <code>[0, right - left]</code> \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n\"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n# \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n# \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\ntmp = [0] * (right - left + 1)\n# \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\ni, j, k = left, mid + 1, 0\n# \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile i &lt;= mid and j &lt;= right:\nif nums[i] &lt;= nums[j]:\ntmp[k] = nums[i]\ni += 1\nelse:\ntmp[k] = nums[j]\nj += 1\nk += 1\n# \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile i &lt;= mid:\ntmp[k] = nums[i]\ni += 1\nk += 1\nwhile j &lt;= right:\ntmp[k] = nums[j]\nj += 1\nk += 1\n# \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor k in range(0, len(tmp)):\nnums[left + k] = tmp[k]\ndef merge_sort(nums: list[int], left: int, right: int):\n\"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n# \u7ec8\u6b62\u6761\u4ef6\nif left &gt;= right:\nreturn  # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n# \u5212\u5206\u9636\u6bb5\nmid = (left + right) // 2  # \u8ba1\u7b97\u4e2d\u70b9\nmerge_sort(nums, left, mid)  # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmerge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n# \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nvector&lt;int&gt; tmp(right - left + 1);\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nint i = left, j = mid + 1, k = 0;\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid &amp;&amp; j &lt;= right) {\nif (nums[i] &lt;= nums[j])\ntmp[k++] = nums[i++];\nelse\ntmp[k++] = nums[j++];\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid) {\ntmp[k++] = nums[i++];\n}\nwhile (j &lt;= right) {\ntmp[k++] = nums[j++];\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor (k = 0; k &lt; tmp.size(); k++) {\nnums[left + k] = tmp[k];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right)\nreturn; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int[] nums, int left, int mid, int right) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nint[] tmp = new int[right - left + 1];\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nint i = left, j = mid + 1, k = 0;\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid &amp;&amp; j &lt;= right) {\nif (nums[i] &lt;= nums[j])\ntmp[k++] = nums[i++];\nelse\ntmp[k++] = nums[j++];\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid) {\ntmp[k++] = nums[i++];\n}\nwhile (j &lt;= right) {\ntmp[k++] = nums[j++];\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor (k = 0; k &lt; tmp.length; k++) {\nnums[left + k] = tmp[k];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right)\nreturn; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nint[] tmp = new int[right - left + 1];\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nint i = left, j = mid + 1, k = 0;\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid &amp;&amp; j &lt;= right) {\nif (nums[i] &lt;= nums[j])\ntmp[k++] = nums[i++];\nelse\ntmp[k++] = nums[j++];\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid) {\ntmp[k++] = nums[i++];\n}\nwhile (j &lt;= right) {\ntmp[k++] = nums[j++];\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor (k = 0; k &lt; tmp.Length; ++k) {\nnums[left + k] = tmp[k];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nMergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nMergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nMerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums []int, left, mid, right int) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\ntmp := make([]int, right-left+1)\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\ni, j, k := left, mid+1, 0\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nfor i &lt;= mid &amp;&amp; j &lt;= right {\nif nums[i] &lt;= nums[j] {\ntmp[k] = nums[i]\ni++\n} else {\ntmp[k] = nums[j]\nj++\n}\nk++\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nfor i &lt;= mid {\ntmp[k] = nums[i]\ni++\nk++\n}\nfor j &lt;= right {\ntmp[k] = nums[j]\nj++\nk++\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor k := 0; k &lt; len(tmp); k++ {\nnums[left+k] = tmp[k]\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left &gt;= right {\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nmid := (left + right) / 2\nmergeSort(nums, left, mid)\nmergeSort(nums, mid+1, right)\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nvar tmp = Array(repeating: 0, count: right - left + 1)\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nvar i = left, j = mid + 1, k = 0\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile i &lt;= mid, j &lt;= right {\nif nums[i] &lt;= nums[j] {\ntmp[k] = nums[i]\ni += 1\nk += 1\n} else {\ntmp[k] = nums[j]\nj += 1\nk += 1\n}\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile i &lt;= mid {\ntmp[k] = nums[i]\ni += 1\nk += 1\n}\nwhile j &lt;= right {\ntmp[k] = nums[j]\nj += 1\nk += 1\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor k in tmp.indices {\nnums[left + k] = tmp[k]\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left &gt;= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nreturn\n}\n// \u5212\u5206\u9636\u6bb5\nlet mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums: &amp;nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums: &amp;nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums, left, mid, right) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nconst tmp = new Array(right - left + 1);\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nlet i = left,\nj = mid + 1,\nk = 0;\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid &amp;&amp; j &lt;= right) {\nif (nums[i] &lt;= nums[j]) {\ntmp[k++] = nums[i++];\n} else {\ntmp[k++] = nums[j++];\n}\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid) {\ntmp[k++] = nums[i++];\n}\nwhile (j &lt;= right) {\ntmp[k++] = nums[j++];\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor (k = 0; k &lt; tmp.length; k++) {\nnums[left + k] = tmp[k];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nconst tmp = new Array(right - left + 1);\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nlet i = left,\nj = mid + 1,\nk = 0;\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid &amp;&amp; j &lt;= right) {\nif (nums[i] &lt;= nums[j]) {\ntmp[k++] = nums[i++];\n} else {\ntmp[k++] = nums[j++];\n}\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid) {\ntmp[k++] = nums[i++];\n}\nwhile (j &lt;= right) {\ntmp[k++] = nums[j++];\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor (k = 0; k &lt; tmp.length; k++) {\nnums[left + k] = tmp[k];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nList&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nint i = left, j = mid + 1, k = 0;\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid &amp;&amp; j &lt;= right) {\nif (nums[i] &lt;= nums[j])\ntmp[k++] = nums[i++];\nelse\ntmp[k++] = nums[j++];\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid) {\ntmp[k++] = nums[i++];\n}\nwhile (j &lt;= right) {\ntmp[k++] = nums[j++];\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor (k = 0; k &lt; tmp.length; k++) {\nnums[left + k] = tmp[k];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nlet tmp_size = right - left + 1;\nlet mut tmp = vec![0; tmp_size];\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nlet (mut i, mut j, mut k) = (left, mid + 1, 0);\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile i &lt;= mid &amp;&amp; j &lt;= right {\nif nums[i] &lt;= nums[j] {\ntmp[k] = nums[j];\ni += 1;\n} else {\ntmp[k] = nums[j];\nj += 1;\n}\nk += 1;\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile i &lt;= mid {\ntmp[k] = nums[i];\nk += 1;\ni += 1;\n}\nwhile j &lt;= right {\ntmp[k] = nums[j];\nk += 1;\nj += 1;\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor k in 0..tmp_size {\nnums[left + k] = tmp[k];\n}\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n// \u7ec8\u6b62\u6761\u4ef6\nif left &gt;= right { return; }             // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nlet mid = (left + right) / 2;     // \u8ba1\u7b97\u4e2d\u70b9\nmerge_sort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmerge_sort(nums, mid + 1, right);  // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int *nums, int left, int mid, int right) {\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n// \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\nint tmpSize = right - left + 1;\nint *tmp = (int *)malloc(tmpSize * sizeof(int));\n// \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\nint i = left, j = mid + 1, k = 0;\n// \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid &amp;&amp; j &lt;= right) {\nif (nums[i] &lt;= nums[j]) {\ntmp[k++] = nums[i++];\n} else {\ntmp[k++] = nums[j++];\n}\n}\n// \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\nwhile (i &lt;= mid) {\ntmp[k++] = nums[i++];\n}\nwhile (j &lt;= right) {\ntmp[k++] = nums[j++];\n}\n// \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\nfor (k = 0; k &lt; tmpSize; ++k) {\nnums[left + k] = tmp[k];\n}\n// \u91ca\u653e\u5185\u5b58\nfree(tmp);\n}\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right)\nreturn; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nint mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\nmergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\nmergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\nmerge(nums, left, mid, right);\n}\n</code></pre> merge_sort.zig<pre><code>// \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n// \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\ndefer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar tmp = try mem_allocator.alloc(i32, right + 1 - left);\nstd.mem.copy(i32, tmp, nums[left..right+1]);\n// \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \nvar leftStart = left - left;\nvar leftEnd = mid - left;\n// \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \nvar rightStart = mid + 1 - left;\nvar rightEnd = right - left;\n// i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\nvar i = leftStart;\nvar j = rightStart;\n// \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\nvar k = left;\nwhile (k &lt;= right) : (k += 1) {\n// \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\nif (i &gt; leftEnd) {\nnums[k] = tmp[j];\nj += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &lt;= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n} else if  (j &gt; rightEnd or tmp[i] &lt;= tmp[j]) {\nnums[k] = tmp[i];\ni += 1;\n// \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &gt; \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n} else {\nnums[k] = tmp[j];\nj += 1;\n}\n}\n}\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n// \u7ec8\u6b62\u6761\u4ef6\nif (left &gt;= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n// \u5212\u5206\u9636\u6bb5\nvar mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\ntry mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\ntry mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n// \u5408\u5e76\u9636\u6bb5\ntry merge(nums, left, mid, right);\n}\n</code></pre>"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3 \u00a0 \u94fe\u8868\u6392\u5e8f","text":"<p>\u5bf9\u4e8e\u94fe\u8868\uff0c\u5f52\u5e76\u6392\u5e8f\u76f8\u8f83\u4e8e\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u6392\u5e8f\u4efb\u52a1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</p> <ul> <li>\u5212\u5206\u9636\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u201d\u6765\u5b9e\u73b0\u94fe\u8868\u5212\u5206\u5de5\u4f5c\uff0c\u4ece\u800c\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5728\u94fe\u8868\u4e2d\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4ec5\u9700\u6539\u53d8\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u5b9e\u73b0\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u987b\u521b\u5efa\u989d\u5916\u94fe\u8868\u3002</li> </ul> <p>\u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002</p>"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"<p>\u300c\u5feb\u901f\u6392\u5e8f quick sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u5982\u56fe 11-8 \u6240\u793a\u3002</p> <ol> <li>\u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 <code>i</code> \u548c <code>j</code> \u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\u3002</li> <li>\u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528 <code>i</code>\uff08<code>j</code>\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\u3002</li> <li>\u5faa\u73af\u6267\u884c\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230 <code>i</code> \u548c <code>j</code> \u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-8 \u00a0 \u54e8\u5175\u5212\u5206\u6b65\u9aa4 </p> <p>\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565</p> <p>\u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n\"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n# \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i &lt; j:\nwhile i &lt; j and nums[j] &gt;= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i &lt; j and nums[i] &lt;= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector&lt;int&gt; &amp;nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid Swap(int[] nums, int i, int j) {\n(nums[j], nums[i]) = (nums[i], nums[j]);\n}\n/* \u54e8\u5175\u5212\u5206 */\nint Partition(int[] nums, int left, int right) {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nSwap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nSwap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i &lt; j {\nfor i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\nj-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\ni++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\nlet tmp = nums[i]\nnums[i] = nums[j]\nnums[j] = tmp\n}\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nvar i = left\nvar j = right\nwhile i &lt; j {\nwhile i &lt; j, nums[j] &gt;= nums[left] {\nj -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile i &lt; j, nums[i] &lt;= nums[left] {\ni += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nswap(nums: &amp;nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums: &amp;nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.js<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\nlet tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\nj -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\ni += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n// \u5143\u7d20\u4ea4\u6362\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n_swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\n_swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nlet (mut i, mut j) = (left, right);\nwhile i &lt; j {\nwhile i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\nj -= 1;      // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\ni += 1;      // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nnums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nnums.swap(i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\ni                    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\nint tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n// \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nj--;\n}\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n// \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\ni++;\n}\n// \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\nswap(nums, i, j);\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nswap(nums, i, left);\n// \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\nreturn i;\n}\n</code></pre> quick_sort.zig<pre><code>// \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\nvar tmp = nums[i];\nnums[i] = nums[j];\nnums[j] = tmp;\n}\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre>"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5feb\u901f\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-9 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u3002</li> <li>\u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u201c\u54e8\u5175\u5212\u5206\u201d\u3002</li> <li>\u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n\"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left &gt;= right:\nreturn\n# \u54e8\u5175\u5212\u5206\npivot = self.partition(nums, left, right)\n# \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nself.quick_sort(nums, left, pivot - 1)\nself.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right)\nreturn;\n// \u54e8\u5175\u5212\u5206\nint pivot = Partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nQuickSort(nums, left, pivot - 1);\nQuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left &gt;= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\npivot := q.partition(nums, left, right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nq.quickSort(nums, left, pivot-1)\nq.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left &gt;= right {\nreturn\n}\n// \u54e8\u5175\u5212\u5206\nlet pivot = partition(nums: &amp;nums, left: left, right: right)\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums: &amp;nums, left: left, right: pivot - 1)\nquickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right) return;\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nconst pivot = this.partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nthis.quickSort(nums, left, pivot - 1);\nthis.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right) return;\n// \u54e8\u5175\u5212\u5206\nint pivot = _partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif left &gt;= right {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nlet pivot = Self::partition(nums, left as usize, right as usize) as i32;\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nSelf::quick_sort(left, pivot - 1, nums);\nSelf::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n// \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nj--;\n}\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n// \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\ni++;\n}\n// \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\nswap(nums, i, j);\n}\n// \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nswap(nums, i, left);\n// \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\nreturn i;\n}\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right) {\nreturn;\n}\n// \u54e8\u5175\u5212\u5206\nint pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nif (left &gt;= right) return;\n// \u54e8\u5175\u5212\u5206\nvar pivot = partition(nums, left, right);\n// \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\nquickSort(nums, left, pivot - 1);\nquickSort(nums, pivot + 1, right);\n}\n</code></pre>"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5728\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\uff0c\u672a\u501f\u52a9\u989d\u5916\u6570\u7ec4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5212\u5206\u7684\u6700\u540e\u4e00\u6b65\uff0c\u57fa\u51c6\u6570\u53ef\u80fd\u4f1a\u88ab\u4ea4\u6362\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3 \u00a0 \u5feb\u901f\u6392\u5e8f\u4e3a\u4ec0\u4e48\u5feb","text":"<p>\u4ece\u540d\u79f0\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u5e94\u8be5\u5177\u6709\u4e00\u5b9a\u7684\u4f18\u52bf\u3002\u5c3d\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u548c\u201c\u5806\u6392\u5e8f\u201d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u6ca1\u6709\u5f52\u5e76\u6392\u5e8f\u7a33\u5b9a\uff0c\u4f46\u5728\u7edd\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8fd0\u884c\u3002</li> <li>\u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u6267\u884c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u7cfb\u7edf\u53ef\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5230\u7f13\u5b58\uff0c\u56e0\u6b64\u8bbf\u95ee\u5143\u7d20\u7684\u6548\u7387\u8f83\u9ad8\u3002\u800c\u50cf\u201c\u5806\u6392\u5e8f\u201d\u8fd9\u7c7b\u7b97\u6cd5\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u4ece\u800c\u7f3a\u4e4f\u8fd9\u4e00\u7279\u6027\u3002</li> <li>\u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u5c0f\uff1a\u5728\u4e0a\u8ff0\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362\u7b49\u64cd\u4f5c\u7684\u603b\u6570\u91cf\u6700\u5c11\u3002\u8fd9\u4e0e\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6bd4\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u66f4\u5feb\u7684\u539f\u56e0\u7c7b\u4f3c\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4 \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"<p>\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u4e3e\u4e00\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u62e9\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u5bfc\u81f4\u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\)\u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u5982\u6b64\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u7684\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u90fd\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u3002</p> <p>\u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u54e8\u5175\u5212\u5206\u4e2d\u7684\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u7136\u800c\uff0c\u5982\u679c\u8fd0\u6c14\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9009\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u51c6\u6570\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u5c3d\u5982\u4eba\u610f\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002</p> <p>\u4e3a\u4e86\u8fdb\u4e00\u6b65\u6539\u8fdb\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\uff08\u901a\u5e38\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u8fd9\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6982\u7387\u5c06\u5927\u5e45\u63d0\u5347\u3002\u5f53\u7136\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u7528\u8fd9\u79cd\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u5927\u5927\u964d\u4f4e\u3002</p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n\"\"\"\u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n# \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n# \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]):\nreturn left\nelif (nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]):\nreturn mid\nreturn right\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n\"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n# \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nmed = self.median_three(nums, left, (left + right) // 2, right)\n# \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n# \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\ni, j = left, right\nwhile i &lt; j:\nwhile i &lt; j and nums[j] &gt;= nums[left]:\nj -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile i &lt; j and nums[i] &lt;= nums[left]:\ni += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n# \u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n# \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\nreturn left;\nelse if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\nreturn left;\nelse if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\nreturn left;\nelse if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = MedianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nSwap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\nj--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nSwap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nSwap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] &lt; nums[mid]) != (nums[left] &lt; nums[right]) {\nreturn left\n} else if (nums[mid] &lt; nums[left]) != (nums[mid] &lt; nums[right]) {\nreturn mid\n}\nreturn right\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nmed := q.medianThree(nums, left, (left+right)/2, right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums[left], nums[med] = nums[med], nums[left]\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\ni, j := left, right\nfor i &lt; j {\nfor i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\nj-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nfor i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\ni++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\n//\u5143\u7d20\u4ea4\u6362\nnums[i], nums[j] = nums[j], nums[i]\n}\n//\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nnums[i], nums[left] = nums[left], nums[i]\nreturn i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\nif (nums[left] &lt; nums[mid]) != (nums[left] &lt; nums[right]) {\nreturn left\n} else if (nums[mid] &lt; nums[left]) != (nums[mid] &lt; nums[right]) {\nreturn mid\n} else {\nreturn right\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums: &amp;nums, i: left, j: med)\nreturn partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right])) return left;\nelse if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\nreturn mid;\nelse return right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(\nnums,\nleft,\nMath.floor((left + right) / 2),\nright\n);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\nnums: number[],\nleft: number,\nmid: number,\nright: number\n): number {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (Number(nums[left] &lt; nums[mid]) ^ Number(nums[left] &lt; nums[right])) {\nreturn left;\n} else if (\nNumber(nums[mid] &lt; nums[left]) ^ Number(nums[mid] &lt; nums[right])\n) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = this.medianThree(\nnums,\nleft,\nMath.floor((left + right) / 2),\nright\n);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nthis.swap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nlet i = left,\nj = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\ni++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nthis.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nthis.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\nreturn left;\nelse if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = _medianThree(nums, left, (left + right) ~/ 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n_swap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n_swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\n_swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif (nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]) {\nreturn left;\n} else if (nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]) {\nreturn mid;\n} right\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nlet med = Self::median_three(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nnums.swap(left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nlet (mut i, mut j) = (left, right);\nwhile i &lt; j {\nwhile i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\nj -= 1;      // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nwhile i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\ni += 1;      // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n}\nnums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nnums.swap(i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\ni                    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\nreturn left;\nelse if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\nreturn mid;\nelse\nreturn right;\n}\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */ int partitionMedian(int nums[], int left, int right) {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nint i = left, j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\nj--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\ni++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.zig<pre><code>// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n// \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n// \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\nif ((nums[left] &lt; nums[mid]) != (nums[left] &lt; nums[right])) {\nreturn left;\n} else if ((nums[mid] &lt; nums[left]) != (nums[mid] &lt; nums[right])) {\nreturn mid;\n} else {\nreturn right;\n}\n}\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n// \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nvar med = medianThree(nums, left, (left + right) / 2, right);\n// \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\nswap(nums, left, med);\n// \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\nvar i = left;\nvar j = right;\nwhile (i &lt; j) {\nwhile (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nwhile (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\nswap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n}\nswap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\nreturn i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre>"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"<p>\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u8bbe\u9012\u5f52\u4e2d\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(m\\) \uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u4ea7\u751f\u957f\u5ea6\u4e3a \\(0\\) \u7684\u5de6\u5b50\u6570\u7ec4\u548c\u957f\u5ea6\u4e3a \\(m - 1\\) \u7684\u53f3\u5b50\u6570\u7ec4\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u4e00\u5c42\u9012\u5f52\u8c03\u7528\u51cf\u5c11\u7684\u95ee\u9898\u89c4\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(n / 2\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n\"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n# \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left &lt; right:\n# \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot = self.partition(nums, left, right)\n# \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif pivot - left &lt; right - pivot:\nself.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\nelse:\nself.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = Partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nQuickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nQuickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nfor left &lt; right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\npivot := q.partition(nums, left, right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif pivot-left &lt; right-pivot {\nq.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nq.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\nvar left = left\nvar right = right\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left &lt; right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = partition(nums: &amp;nums, left: left, right: right)\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left) &lt; (right - pivot) {\nquickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = this.partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nthis.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nthis.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = _partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nquickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile left &lt; right {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nlet pivot = Self::partition(nums, left as usize, right as usize) as i32;\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif  pivot - left &lt; right - pivot {\nSelf::quick_sort(left, pivot - 1, nums);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nSelf::quick_sort(pivot + 1, right, nums); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nint pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nquickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\nvar left = left_;\nvar right = right_;\n// \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\nwhile (left &lt; right) {\n// \u54e8\u5175\u5212\u5206\u64cd\u4f5c\nvar pivot = partition(nums, left, right);\n// \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\nif (pivot - left &lt; right - pivot) {\nquickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\nleft = pivot + 1;                   // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n} else {\nquickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\nright = pivot - 1;                  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n}\n}\n}\n</code></pre>"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6570\u6392\u5e8f","text":"<p>\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002</p> <p>\u300c\u57fa\u6570\u6392\u5e8f radix sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-18 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \u3002</li> <li>\u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u201c\u8ba1\u6570\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002</li> <li>\u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4 <code>2.</code> \u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-18 \u00a0 \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4e0b\u9762\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u4f59\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002</p> <p>\u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n\"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n# \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num // exp) % 10\ndef counting_sort_digit(nums: list[int], exp: int):\n\"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n# \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\ncounter = [0] * 10\nn = len(nums)\n# \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i in range(n):\nd = digit(nums[i], exp)  # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1  # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n# \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i in range(1, 10):\ncounter[i] += counter[i - 1]\n# \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nres = [0] * n\nfor i in range(n - 1, -1, -1):\nd = digit(nums[i], exp)\nj = counter[d] - 1  # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]  # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1  # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n# \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in range(n):\nnums[i] = res[i]\ndef radix_sort(nums: list[int]):\n\"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n# \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nm = max(nums)\n# \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nexp = 1\nwhile exp &lt;= m:\n# \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n# k = 1 -&gt; exp = 1\n# k = 2 -&gt; exp = 10\n# \u5373 exp = 10^(k-1)\ncounting_sort_digit(nums, exp)\nexp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nvector&lt;int&gt; counter(10, 0);\nint n = nums.size();\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i &lt; n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i &lt; 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvector&lt;int&gt; res(n, 0);\nfor (int i = n - 1; i &gt;= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i &lt; n; i++)\nnums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = *max_element(nums.begin(), nums.end());\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp &lt;= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nint[] counter = new int[10];\nint n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i &lt; n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i &lt; 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint[] res = new int[n];\nfor (int i = n - 1; i &gt;= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i &lt; n; i++)\nnums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = Integer.MIN_VALUE;\nfor (int num : nums)\nif (num &gt; m)\nm = num;\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp &lt;= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.cs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nint[] counter = new int[10];\nint n = nums.Length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i &lt; n; i++) {\nint d = Digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i &lt; 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint[] res = new int[n];\nfor (int i = n - 1; i &gt;= 0; i--) {\nint d = Digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i &lt; n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint m = int.MinValue;\nforeach (int num in nums) {\nif (num &gt; m) m = num;\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp &lt;= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\nCountingSortDigit(nums, exp);\n}\n}\n</code></pre> radix_sort.go<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\ncounter := make([]int, 10)\nn := len(nums)\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i := 0; i &lt; n; i++ {\nd := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++             // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i := 1; i &lt; 10; i++ {\ncounter[i] += counter[i-1]\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nres := make([]int, n)\nfor i := n - 1; i &gt;= 0; i-- {\nd := digit(nums[i], exp)\nj := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]    // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i := 0; i &lt; n; i++ {\nnums[i] = res[i]\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nmax := math.MinInt\nfor _, num := range nums {\nif num &gt; max {\nmax = num\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor exp := 1; max &gt;= exp; exp *= 10 {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp)\n}\n}\n</code></pre> radix_sort.swift<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n(num / exp) % 10\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nvar counter = Array(repeating: 0, count: 10)\nlet n = nums.count\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i in nums.indices {\nlet d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i in 1 ..&lt; 10 {\ncounter[i] += counter[i - 1]\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvar res = Array(repeating: 0, count: n)\nfor i in stride(from: n - 1, through: 0, by: -1) {\nlet d = digit(num: nums[i], exp: exp)\nlet j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in nums.indices {\nnums[i] = res[i]\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nvar m = Int.min\nfor num in nums {\nif num &gt; m {\nm = num\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums: &amp;nums, exp: exp)\n}\n}\n</code></pre> radix_sort.js<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn Math.floor(num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nconst counter = new Array(10).fill(0);\nconst n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (let i = 0; i &lt; n; i++) {\nconst d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (let i = 1; i &lt; 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nconst res = new Array(n).fill(0);\nfor (let i = n - 1; i &gt;= 0; i--) {\nconst d = digit(nums[i], exp);\nconst j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i &lt; n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nlet m = Number.MIN_VALUE;\nfor (const num of nums) {\nif (num &gt; m) {\nm = num;\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (let exp = 1; exp &lt;= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n</code></pre> radix_sort.ts<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn Math.floor(num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nconst counter = new Array(10).fill(0);\nconst n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (let i = 0; i &lt; n; i++) {\nconst d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (let i = 1; i &lt; 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nconst res = new Array(n).fill(0);\nfor (let i = n - 1; i &gt;= 0; i--) {\nconst d = digit(nums[i], exp);\nconst j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (let i = 0; i &lt; n; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nlet m = Number.MIN_VALUE;\nfor (const num of nums) {\nif (num &gt; m) {\nm = num;\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (let exp = 1; exp &lt;= m; exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n}\n</code></pre> radix_sort.dart<pre><code>/* \u83b7\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (_num ~/ exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nList&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\nint n = nums.length;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i &lt; n; i++) {\nint d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i &lt; 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nList&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\nfor (int i = n - 1; i &gt;= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List&lt;int&gt; nums) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n// dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\nint m = -1 &lt;&lt; 63;\nfor (int _num in nums) if (_num &gt; m) m = _num;\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; exp &lt;= m; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn ((num / exp) % 10) as usize;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nlet mut counter = [0; 10];\nlet n = nums.len();\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor i in 0..n {\nlet d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor i in 1..10 {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nlet mut res = vec![0; n];\nfor i in (0..n).rev() {\nlet d = digit(nums[i], exp);\nlet j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor i in 0..n {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nfn radix_sort(nums: &amp;mut [i32]) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nlet m = *nums.into_iter().max().unwrap();\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nlet mut exp = 1;\nwhile exp &lt;= m {\ncounting_sort_digit(nums, exp);\nexp *= 10;\n}\n}\n</code></pre> radix_sort.c<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn (num / exp) % 10;\n}\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nint *counter = (int *)malloc((sizeof(int) * 10));\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (int i = 0; i &lt; size; i++) {\n// \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\nint d = digit(nums[i], exp);\n// \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\ncounter[d]++;\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nfor (int i = 1; i &lt; 10; i++) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nint *res = (int *)malloc(sizeof(int) * size);\nfor (int i = size - 1; i &gt;= 0; i--) {\nint d = digit(nums[i], exp);\nint j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\nfor (int i = 0; i &lt; size; i++) {\nnums[i] = res[i];\n}\n}\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nint max = INT32_MIN;\nfor (size_t i = 0; i &lt; size - 1; i++) {\nif (nums[i] &gt; max) {\nmax = nums[i];\n}\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nfor (int exp = 1; max &gt;= exp; exp *= 10)\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ncountingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.zig<pre><code>// \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n// \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\nreturn @mod(@divFloor(num, exp), 10);\n}\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n// \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\nvar mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n// defer mem_arena.deinit();\nconst mem_allocator = mem_arena.allocator();\nvar counter = try mem_allocator.alloc(usize, 10);\n@memset(counter, 0);\nvar n = nums.len;\n// \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\nfor (nums) |num| {\nvar d: u32 = @bitCast(digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\ncounter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n}\n// \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\nvar i: usize = 1;\nwhile (i &lt; 10) : (i += 1) {\ncounter[i] += counter[i - 1];\n}\n// \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\nvar res = try mem_allocator.alloc(i32, n);\ni = n - 1;\nwhile (i &gt;= 0) : (i -= 1) {\nvar d: u32 = @bitCast(digit(nums[i], exp));\nvar j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\nres[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\ncounter[d] -= 1;        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\nif (i == 0) break;\n}\n// \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\ni = 0;\nwhile (i &lt; n) : (i += 1) {\nnums[i] = res[i];\n}\n}\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n// \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\nvar m: i32 = std.math.minInt(i32);\nfor (nums) |num| {\nif (num &gt; m) m = num;\n}\n// \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\nvar exp: i32 = 1;\nwhile (exp &lt;= m) : (exp *= 10) {\n// \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n// k = 1 -&gt; exp = 1\n// k = 2 -&gt; exp = 10\n// \u5373 exp = 10^(k-1)\ntry countingSortDigit(nums, exp);    }\n} </code></pre> <p>\u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f</p> <p>\u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &lt; b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &gt; b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u56e0\u6b64\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk)\\)\uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\)\u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_sorting/selection_sort/","title":"11.2 \u00a0 \u9009\u62e9\u6392\u5e8f","text":"<p>\u300c\u9009\u62e9\u6392\u5e8f selection sort\u300d\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7b80\u5355\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002</p> <p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 11-2 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-2 \u00a0 \u9009\u62e9\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n\"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\nn = len(nums)\n# \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i in range(n - 1):\n# \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nk = i\nfor j in range(i + 1, n):\nif nums[j] &lt; nums[k]:\nk = j  # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n# \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\nint n = nums.size();\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i &lt; n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j &lt; n; j++) {\nif (nums[j] &lt; nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nswap(nums[i], nums[k]);\n}\n}\n</code></pre> selection_sort.java<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\nint n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i &lt; n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j &lt; n; j++) {\nif (nums[j] &lt; nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n</code></pre> selection_sort.cs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\nint n = nums.Length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i &lt; n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j &lt; n; j++) {\nif (nums[j] &lt; nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n(nums[k], nums[i]) = (nums[i], nums[k]);\n}\n}\n</code></pre> selection_sort.go<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\nn := len(nums)\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i := 0; i &lt; n-1; i++ {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nk := i\nfor j := i + 1; j &lt; n; j++ {\nif nums[j] &lt; nums[k] {\n// \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\nk = j\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums[i], nums[k] = nums[k], nums[i]\n}\n}\n</code></pre> selection_sort.swift<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i in nums.indices.dropLast() {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nvar k = i\nfor j in nums.indices.dropFirst(i + 1) {\nif nums[j] &lt; nums[k] {\nk = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums.swapAt(i, k)\n}\n}\n</code></pre> selection_sort.js<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\nlet n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (let i = 0; i &lt; n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nlet k = i;\nfor (let j = i + 1; j &lt; n; j++) {\nif (nums[j] &lt; nums[k]) {\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n[nums[i], nums[k]] = [nums[k], nums[i]];\n}\n}\n</code></pre> selection_sort.ts<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\nlet n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (let i = 0; i &lt; n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nlet k = i;\nfor (let j = i + 1; j &lt; n; j++) {\nif (nums[j] &lt; nums[k]) {\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n[nums[i], nums[k]] = [nums[k], nums[i]];\n}\n}\n</code></pre> selection_sort.dart<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List&lt;int&gt; nums) {\nint n = nums.length;\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i &lt; n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j &lt; n; j++) {\nif (nums[j] &lt; nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n</code></pre> selection_sort.rs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfn selection_sort(nums: &amp;mut [i32]) {\nlet n = nums.len();\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor i in 0..n-1 {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nlet mut k = i;\nfor j in i+1..n {\nif nums[j] &lt; nums[k] {\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nnums.swap(i, k);\n}\n}\n</code></pre> selection_sort.c<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n// \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\nfor (int i = 0; i &lt; n - 1; i++) {\n// \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\nint k = i;\nfor (int j = i + 1; j &lt; n; j++) {\nif (nums[j] &lt; nums[k])\nk = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n}\n// \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\nint temp = nums[i];\nnums[i] = nums[k];\nnums[k] = temp;\n}\n}\n</code></pre> selection_sort.zig<pre><code>[class]{}-[func]{selectionSort}\n</code></pre>"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5982\u56fe 11-3 \u6240\u793a\uff0c\u5143\u7d20 <code>nums[i]</code> \u6709\u53ef\u80fd\u88ab\u4ea4\u6362\u81f3\u4e0e\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002</li> </ul> <p> \u56fe 11-3 \u00a0 \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"<p>\u300c\u6392\u5e8f\u7b97\u6cd5 sorting algorithm\u300d\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002</p> <p>\u5982\u56fe 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002</p> <p></p> <p> \u56fe 11-1 \u00a0 \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"<p>\u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002</p> <p>\u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u300c\u539f\u5730\u6392\u5e8f\u300d\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u987b\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002</p> <p>\u7a33\u5b9a\u6027\uff1a\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002</p> <p>\u7a33\u5b9a\u6392\u5e8f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\uff1a</p> <pre><code># \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n('A', 19)\n('B', 18)\n('C', 21)\n('D', 19)\n('E', 23)\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n('B', 18)\n('D', 19)\n('A', 19)\n('C', 21)\n('E', 23)\n</code></pre> <p>\u81ea\u9002\u5e94\u6027\uff1a\u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002</p> <p>\u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002</p> <p>\u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u300c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u300d\u4f9d\u8d56\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(&lt;\\)\u3001\\(=\\)\u3001\\(&gt;\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u300c\u975e\u6bd4\u8f83\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002</p>"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"<p>\u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002</li> <li>\u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</li> <li>\u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002</li> <li>\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002</li> <li>\u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002</li> <li>\u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002</li> <li>\u56fe 11-19 \u5bf9\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a33\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9002\u5e94\u6027\u7b49\u3002</li> </ul> <p> \u56fe 11-19 \u00a0 \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4 </p>"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u9700\u7684\uff1f</p> <p>\u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u662f\u57fa\u4e8e\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f\uff1a</p> <p>\u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> \uff1b\u518d\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> \u3002</p> <p>\u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002</p> <p>\u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f</p> <p>\u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002</p> <p>\u54e8\u5175\u5212\u5206 <code>partition()</code> \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 <code>nums[left]</code> \u548c <code>nums[i]</code> \u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <code>&lt;=</code> \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d <code>nums[left] &gt;= nums[i]</code> \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 <code>i == j</code> \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd <code>nums[j] == nums[i] &gt; nums[left]</code>\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 <code>[0, 0, 0, 0, 1]</code> \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a <code>[1, 0, 0, 0, 0]</code> \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002</p> <p>\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 <code>nums[right]</code> \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002</p> <p>\u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff1f</p> <p>\u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002</p> <p>\u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002</p> <p>\u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f</p> <p>\u662f\u7684\u3002\u5bf9\u4e8e\u8fd9\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002</p> <p>\u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f</p> <p>\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u6808\u4e0e\u961f\u5217","text":"<p>Abstract</p> <p>\u6808\u5982\u540c\u53e0\u732b\u732b\uff0c\u800c\u961f\u5217\u5c31\u50cf\u732b\u732b\u6392\u961f\u3002</p> <p>\u4e24\u8005\u5206\u522b\u4ee3\u8868\u5148\u5165\u540e\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u903b\u8f91\u5173\u7cfb\u3002</p>"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>5.1 \u00a0 \u6808</li> <li>5.2 \u00a0 \u961f\u5217</li> <li>5.3 \u00a0 \u53cc\u5411\u961f\u5217</li> <li>5.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 \u53cc\u5411\u961f\u5217","text":"<p>\u5728\u961f\u5217\u4e2d\uff0c\u6211\u4eec\u4ec5\u80fd\u5220\u9664\u5934\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u5982\u56fe 5-7 \u6240\u793a\uff0c\u300c\u53cc\u5411\u961f\u5217 double-ended queue\u300d\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 5-7 \u00a0 \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c </p>"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 5-3 \u00a0 \u53cc\u5411\u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) <p>\u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig deque.py<pre><code>from collections import deque\n# \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: deque[int] = deque()\n# \u5143\u7d20\u5165\u961f\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n</code></pre> deque.cpp<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque&lt;int&gt; deque;\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList&lt;int&gt; deque = new();\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n</code></pre> deque.ts<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n</code></pre> deque.dart<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u961f\u5c3e\u5143\u7d20\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;W\n</code></pre> deque.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n/* \u8bbf\u95ee\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u961f\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() {   // \u961f\u5c3e\u5143\u7d20\n}\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop_front) = deque.pop_front() { // \u961f\u9996\u5143\u7d20\u51fa\u961f\n}\nif let Some(pop_rear) = deque.pop_back() {   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n</code></pre> deque.zig<pre><code>\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p>"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002</p> <p>\u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u201c\u53cc\u5411\u94fe\u8868\u201d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-8 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002</p> LinkedListDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-8 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_deque.py<pre><code>class ListNode:\n\"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\ndef __init__(self, val: int):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself.val: int = val\nself.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\nself.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\nclass LinkedListDeque:\n\"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._front: ListNode | None = None  # \u5934\u8282\u70b9 front\nself._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\nself._size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self._size\ndef is_empty(self) -&gt; bool:\n\"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self.size() == 0\ndef push(self, num: int, is_front: bool):\n\"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\nnode = ListNode(num)\n# \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif self.is_empty():\nself._front = self._rear = node\n# \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelif is_front:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself._front.prev = node\nnode.next = self._front\nself._front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n# \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse:\n# \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself._rear.next = node\nnode.prev = self._rear\nself._rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\nself._size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\ndef push_first(self, num: int):\n\"\"\"\u961f\u9996\u5165\u961f\"\"\"\nself.push(num, True)\ndef push_last(self, num: int):\n\"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\nself.push(num, False)\ndef pop(self, is_front: bool) -&gt; int:\n\"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n# \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif is_front:\nval: int = self._front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n# \u5220\u9664\u5934\u8282\u70b9\nfnext: ListNode | None = self._front.next\nif fnext != None:\nfnext.prev = None\nself._front.next = None\nself._front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n# \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse:\nval: int = self._rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n# \u5220\u9664\u5c3e\u8282\u70b9\nrprev: ListNode | None = self._rear.prev\nif rprev != None:\nrprev.next = None\nself._rear.prev = None\nself._rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\nself._size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\ndef pop_first(self) -&gt; int:\n\"\"\"\u961f\u9996\u51fa\u961f\"\"\"\nreturn self.pop(True)\ndef pop_last(self) -&gt; int:\n\"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\nreturn self.pop(False)\ndef peek_first(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\nreturn self._front.val\ndef peek_last(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\nreturn self._rear.val\ndef to_array(self) -&gt; list[int]:\n\"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\nnode = self._front\nres = [0] * self.size()\nfor i in range(self.size()):\nres[i] = node.val\nnode = node.next\nreturn res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\nint val;              // \u8282\u70b9\u503c\nDoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\nDoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\nDoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n}\n};\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate:\nDoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nLinkedListDeque() : front(nullptr), rear(nullptr) {\n}\n/* \u6790\u6784\u65b9\u6cd5 */\n~LinkedListDeque() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nDoublyListNode *pre, *cur = front;\nwhile (cur != nullptr) {\npre = cur;\ncur = cur-&gt;next;\ndelete pre;\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int num, bool isFront) {\nDoublyListNode *node = new DoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront-&gt;prev = node;\nnode-&gt;next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear-&gt;next = node;\nnode-&gt;prev = rear;\nrear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint pop(bool isFront) {\nif (isEmpty())\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front-&gt;val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nDoublyListNode *fNext = front-&gt;next;\nif (fNext != nullptr) {\nfNext-&gt;prev = nullptr;\nfront-&gt;next = nullptr;\ndelete front;\n}\nfront = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear-&gt;val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nDoublyListNode *rPrev = rear-&gt;prev;\nif (rPrev != nullptr) {\nrPrev-&gt;next = nullptr;\nrear-&gt;prev = nullptr;\ndelete rear;\n}\nrear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn front-&gt;val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn rear-&gt;val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector&lt;int&gt; toVector() {\nDoublyListNode *node = front;\nvector&lt;int&gt; res(size());\nfor (int i = 0; i &lt; res.size(); i++) {\nres[i] = node-&gt;val;\nnode = node-&gt;next;\n}\nreturn res;\n}\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\nListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\nListNode(int val) {\nthis.val = val;\nprev = next = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nprivate int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate void push(int num, boolean isFront) {\nListNode node = new ListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (isEmpty())\nfront = rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear.next = node;\nnode.prev = rear;\nrear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\npush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\npush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate int pop(boolean isFront) {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode fNext = front.next;\nif (fNext != null) {\nfNext.prev = null;\nfront.next = null;\n}\nfront = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode rPrev = rear.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear.prev = null;\n}\nrear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn front.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn rear.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i &lt; res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\npublic int val = val;       // \u8282\u70b9\u503c\npublic ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\npublic ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\nint queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npublic LinkedListDeque() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int Size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool IsEmpty() {\nreturn Size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid Push(int num, bool isFront) {\nListNode node = new(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (IsEmpty()) {\nfront = node;\nrear = node;\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront!.prev = node;\nnode.next = front;\nfront = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear!.next = node;\nnode.prev = rear;\nrear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npublic void PushFirst(int num) {\nPush(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void PushLast(int num) {\nPush(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint? Pop(bool isFront) {\nif (IsEmpty())\nthrow new Exception();\nint? val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode? fNext = front?.next;\nif (fNext != null) {\nfNext.prev = null;\nfront!.next = null;\n}\nfront = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear?.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode? rPrev = rear?.prev;\nif (rPrev != null) {\nrPrev.next = null;\nrear!.prev = null;\n}\nrear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int? PopFirst() {\nreturn Pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int? PopLast() {\nreturn Pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int? PeekFirst() {\nif (IsEmpty())\nthrow new Exception();\nreturn front?.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int? PeekLast() {\nif (IsEmpty())\nthrow new Exception();\nreturn rear?.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int?[] ToArray() {\nListNode? node = front;\nint?[] res = new int?[Size()];\nfor (int i = 0; i &lt; res.Length; i++) {\nres[i] = node?.val;\nnode = node?.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\nreturn &amp;linkedListDeque{\ndata: list.New(),\n}\n}\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\ns.data.PushFront(value)\n}\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\ns.data.PushBack(value)\n}\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\nreturn s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nvar val: Int // \u8282\u70b9\u503c\nvar next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\nweak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\ninit(val: Int) {\nself.val = val\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate var front: ListNode? // \u5934\u8282\u70b9 front\nprivate var rear: ListNode? // \u5c3e\u8282\u70b9 rear\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\ninit() {\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -&gt; Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -&gt; Bool {\nsize() == 0\n}\n/* \u5165\u961f\u64cd\u4f5c */\nprivate func push(num: Int, isFront: Bool) {\nlet node = ListNode(val: num)\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif isEmpty() {\nfront = node\nrear = node\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if isFront {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nfront?.prev = node\nnode.next = front\nfront = node // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nrear?.next = node\nnode.prev = rear\nrear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\npush(num: num, isFront: true)\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\npush(num: num, isFront: false)\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nprivate func pop(isFront: Bool) -&gt; Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nlet val: Int\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif isFront {\nval = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet fNext = front?.next\nif fNext != nil {\nfNext?.prev = nil\nfront?.next = nil\n}\nfront = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet rPrev = rear?.prev\nif rPrev != nil {\nrPrev?.next = nil\nrear?.prev = nil\n}\nrear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nqueSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -&gt; Int {\npop(isFront: true)\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -&gt; Int {\npop(isFront: false)\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -&gt; Int? {\nisEmpty() ? nil : front?.val\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -&gt; Int? {\nisEmpty() ? nil : rear?.val\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -&gt; [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nprev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nnext; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nval; // \u8282\u70b9\u503c\nconstructor(val) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n#front; // \u5934\u8282\u70b9 front\n#rear; // \u5c3e\u8282\u70b9 rear\n#queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\nthis.#queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.#rear.next = node;\nnode.prev = this.#rear;\nthis.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val) {\nconst node = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (this.#queSize === 0) {\nthis.#front = node;\nthis.#rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.#front.prev = node;\nnode.next = this.#front;\nthis.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet temp = this.#rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.#rear.prev = null;\n}\nthis.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst() {\nif (this.#queSize === 0) {\nreturn null;\n}\nconst value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet temp = this.#front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.#front.next = null;\n}\nthis.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\nthis.#queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nreturn this.#queSize === 0 ? null : this.#rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nreturn this.#queSize === 0 ? null : this.#front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint() {\nconst arr = [];\nlet temp = this.#front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log('[' + arr.join(', ') + ']');\n}\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nprev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nnext: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\nval: number; // \u8282\u70b9\u503c\nconstructor(val: number) {\nthis.val = val;\nthis.next = null;\nthis.prev = null;\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\nprivate front: ListNode; // \u5934\u8282\u70b9 front\nprivate rear: ListNode; // \u5c3e\u8282\u70b9 rear\nprivate queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nconstructor() {\nthis.front = null;\nthis.rear = null;\nthis.queSize = 0;\n}\n/* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\npushLast(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nthis.rear.next = node;\nnode.prev = this.rear;\nthis.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\npushFirst(val: number): void {\nconst node: ListNode = new ListNode(val);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (this.queSize === 0) {\nthis.front = node;\nthis.rear = node;\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nthis.front.prev = node;\nnode.next = this.front;\nthis.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\nthis.queSize++;\n}\n/* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\npopLast(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nlet temp: ListNode = this.rear.prev;\nif (temp !== null) {\ntemp.next = null;\nthis.rear.prev = null;\n}\nthis.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\npopFirst(): number {\nif (this.queSize === 0) {\nreturn null;\n}\nconst value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nlet temp: ListNode = this.front.next;\nif (temp !== null) {\ntemp.prev = null;\nthis.front.next = null;\n}\nthis.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\nthis.queSize--;\nreturn value;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nreturn this.queSize === 0 ? null : this.rear.val;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nreturn this.queSize === 0 ? null : this.front.val;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u6253\u5370\u53cc\u5411\u961f\u5217 */\nprint(): void {\nconst arr: number[] = [];\nlet temp: ListNode = this.front;\nwhile (temp !== null) {\narr.push(temp.val);\ntemp = temp.next;\n}\nconsole.log('[' + arr.join(', ') + ']');\n}\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\nint val; // \u8282\u70b9\u503c\nListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\nListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\nListNode(this.val, {this.next, this.prev});\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\nlate ListNode? _front; // \u5934\u8282\u70b9 _front\nlate ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\nint _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nLinkedListDeque() {\nthis._front = null;\nthis._rear = null;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\nint size() {\nreturn this._queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\nvoid push(int _num, bool isFront) {\nfinal ListNode node = ListNode(_num);\nif (isEmpty()) {\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n_front = _rear = node;\n} else if (isFront) {\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n_front!.prev = node;\nnode.next = _front;\n_front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n} else {\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n_rear!.next = node;\nnode.prev = _rear;\n_rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\n_queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int _num) {\npush(_num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int _num) {\npush(_num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\nint? pop(bool isFront) {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\nif (isEmpty()) {\nreturn null;\n}\nfinal int val;\nif (isFront) {\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nval = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nListNode? fNext = _front!.next;\nif (fNext != null) {\nfNext.prev = null;\n_front!.next = null;\n}\n_front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n} else {\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nval = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nListNode? rPrev = _rear!.prev;\nif (rPrev != null) {\nrPrev.next = null;\n_rear!.prev = null;\n}\n_rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\n_queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint? popFirst() {\nreturn pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint? popLast() {\nreturn pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint? peekFirst() {\nreturn _front?.val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint? peekLast() {\nreturn _rear?.val;\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nList&lt;int&gt; toArray() {\nListNode? node = _front;\nfinal List&lt;int&gt; res = [];\nfor (int i = 0; i &lt; _queSize; i++) {\nres.add(node!.val);\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode&lt;T&gt; {\npub val: T,                                 // \u8282\u70b9\u503c\npub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\npub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\nimpl&lt;T&gt; ListNode&lt;T&gt; {\npub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\nRc::new(RefCell::new(ListNode {\nval,\nnext: None,\nprev: None,\n}))\n}\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\nfront: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,    // \u5934\u8282\u70b9 front\nrear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,     // \u5c3e\u8282\u70b9 rear \nque_size: usize,                            // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\npub fn new() -&gt; Self {\nSelf {\nfront: None,\nrear: None,\nque_size: 0, }\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npub fn size(&amp;self) -&gt; usize {\nreturn self.que_size;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npub fn is_empty(&amp;self) -&gt; bool {\nreturn self.size() == 0;\n}\n/* \u5165\u961f\u64cd\u4f5c */\npub fn push(&amp;mut self, num: T, is_front: bool) {\nlet node = ListNode::new(num);\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nif is_front {\nmatch self.front.take() {\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nNone =&gt; {\nself.rear = Some(node.clone());\nself.front = Some(node);\n}\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nSome(old_front) =&gt; {\nold_front.borrow_mut().prev = Some(node.clone());\nnode.borrow_mut().next = Some(old_front);\nself.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n}\n}\n} // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\nmatch self.rear.take() {\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nNone =&gt; {\nself.front = Some(node.clone());\nself.rear = Some(node);\n}\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nSome(old_rear) =&gt; {\nold_rear.borrow_mut().next = Some(node.clone());\nnode.borrow_mut().prev = Some(old_rear);\nself.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\n}\n}\nself.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\npub fn push_first(&amp;mut self, num: T) {\nself.push(num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\npub fn push_last(&amp;mut self, num: T) {\nself.push(num, false);\n}\n/* \u51fa\u961f\u64cd\u4f5c */\npub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n// \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\nif self.is_empty() { return None };\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif is_front {\nself.front.take().map(|old_front| {\nmatch old_front.borrow_mut().next.take() {\nSome(new_front) =&gt; {\nnew_front.borrow_mut().prev.take();\nself.front = Some(new_front);   // \u66f4\u65b0\u5934\u8282\u70b9\n}\nNone =&gt; {\nself.rear.take();\n}\n}\nself.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nRc::try_unwrap(old_front).ok().unwrap().into_inner().val\n})\n} // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nself.rear.take().map(|old_rear| {\nmatch old_rear.borrow_mut().prev.take() {\nSome(new_rear) =&gt; {\nnew_rear.borrow_mut().next.take();\nself.rear = Some(new_rear);     // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nNone =&gt; {\nself.front.take();\n}\n}\nself.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nRc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n})\n}\n}\n/* \u961f\u9996\u51fa\u961f */\npub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\nreturn self.pop(true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\npub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\nreturn self.pop(false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\nself.front.as_ref()\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\nself.rear.as_ref()\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\nif let Some(node) = head {\nlet mut nums = self.to_array(node.borrow().next.as_ref());\nnums.insert(0, node.borrow().val);\nreturn nums;\n}\nreturn Vec::new();\n}\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\nint val;                     // \u8282\u70b9\u503c\nstruct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\nstruct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\nDoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\nnew-&gt;val = num;\nnew-&gt;next = NULL;\nnew-&gt;prev = NULL;\nreturn new;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\nfree(node);\n}\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\nDoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\nLinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\ndeque-&gt;front = NULL;\ndeque-&gt;rear = NULL;\ndeque-&gt;queSize = 0;\nreturn deque;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n// \u91ca\u653e\u6240\u6709\u8282\u70b9\nfor (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\nDoublyListNode *tmp = deque-&gt;front;\ndeque-&gt;front = deque-&gt;front-&gt;next;\nfree(tmp);\n}\n// \u91ca\u653e deque \u7ed3\u6784\u4f53\nfree(deque);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\nreturn deque-&gt;queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\nreturn (size(deque) == 0);\n}\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\nDoublyListNode *node = newDoublyListNode(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\nif (empty(deque)) {\ndeque-&gt;front = deque-&gt;rear = node;\n}\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\nelse if (isFront) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\ndeque-&gt;front-&gt;prev = node;\nnode-&gt;next = deque-&gt;front;\ndeque-&gt;front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\nelse {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\ndeque-&gt;rear-&gt;next = node;\nnode-&gt;prev = deque-&gt;rear;\ndeque-&gt;rear = node;\n}\ndeque-&gt;queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\npush(deque, num, true);\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\npush(deque, num, false);\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\nassert(size(deque) &amp;&amp; deque-&gt;front);\nreturn deque-&gt;front-&gt;val;\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\nassert(size(deque) &amp;&amp; deque-&gt;rear);\nreturn deque-&gt;rear-&gt;val;\n}\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\nif (empty(deque))\nreturn -1;\nint val;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (isFront) {\nval = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\nDoublyListNode *fNext = deque-&gt;front-&gt;next;\nif (fNext) {\nfNext-&gt;prev = NULL;\ndeque-&gt;front-&gt;next = NULL;\ndelDoublyListNode(deque-&gt;front);\n}\ndeque-&gt;front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n}\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\nelse {\nval = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\nDoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\nif (rPrev) {\nrPrev-&gt;next = NULL;\ndeque-&gt;rear-&gt;prev = NULL;\ndelDoublyListNode(deque-&gt;rear);\n}\ndeque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\ndeque-&gt;queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\nreturn pop(deque, true);\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\nreturn pop(deque, false);\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\nint *arr = malloc(sizeof(int) * deque-&gt;queSize);\n// \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\nint i;\nDoublyListNode *node;\nfor (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\narr[i] = node-&gt;val;\nnode = node-&gt;next;\n}\nprintArray(arr, deque-&gt;queSize);\nfree(arr);\n}\n</code></pre> linkedlist_deque.zig<pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nval: T = undefined,     // \u8282\u70b9\u503c\nnext: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u6307\u9488\nprev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u6307\u9488\n// Initialize a list node with specific value\npub fn init(self: *Self, x: i32) void {\nself.val = x;\nself.next = null;\nself.prev = null;\n}\n};\n}\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\nrear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\nque_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u5165\u961f\u64cd\u4f5c\npub fn push(self: *Self, num: T, is_front: bool) !void {\nvar node = try self.mem_allocator.create(ListNode(T));\nnode.init(num);\n// \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\nif (self.isEmpty()) {\nself.front = node;\nself.rear = node;\n// \u961f\u9996\u5165\u961f\u64cd\u4f5c\n} else if (is_front) {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\nself.front.?.prev = node;\nnode.next = self.front;\nself.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n} else {\n// \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\nself.rear.?.next = node;\nnode.prev = self.rear;\nself.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nself.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n} // \u961f\u9996\u5165\u961f\npub fn pushFirst(self: *Self, num: T) !void {\ntry self.push(num, true);\n} // \u961f\u5c3e\u5165\u961f\npub fn pushLast(self: *Self, num: T) !void {\ntry self.push(num, false);\n} // \u51fa\u961f\u64cd\u4f5c\npub fn pop(self: *Self, is_front: bool) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nvar val: T = undefined;\n// \u961f\u9996\u51fa\u961f\u64cd\u4f5c\nif (is_front) {\nval = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n// \u5220\u9664\u5934\u8282\u70b9\nvar fNext = self.front.?.next;\nif (fNext != null) {\nfNext.?.prev = null;\nself.front.?.next = null;\n}\nself.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n// \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n} else {\nval = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n// \u5220\u9664\u5c3e\u8282\u70b9\nvar rPrev = self.rear.?.prev;\nif (rPrev != null) {\nrPrev.?.next = null;\nself.rear.?.prev = null;\n}\nself.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n}\nself.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\nreturn val;\n} // \u961f\u9996\u51fa\u961f\npub fn popFirst(self: *Self) T {\nreturn self.pop(true);\n} // \u961f\u5c3e\u51fa\u961f\npub fn popLast(self: *Self) T {\nreturn self.pop(false);\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peekFirst(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\npub fn peekLast(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn self.rear.?.val;\n}\n// \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i &lt; res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-9 \u6240\u793a\uff0c\u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002</p> ArrayDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-9 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_deque.py<pre><code>class ArrayDeque:\n\"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\ndef __init__(self, capacity: int):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._nums: list[int] = [0] * capacity\nself._front: int = 0\nself._size: int = 0\ndef capacity(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\nreturn len(self._nums)\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self._size\ndef is_empty(self) -&gt; bool:\n\"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self._size == 0\ndef index(self, i: int) -&gt; int:\n\"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n# \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + self.capacity()) % self.capacity()\ndef push_first(self, num: int):\n\"\"\"\u961f\u9996\u5165\u961f\"\"\"\nif self._size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nself._front = self.index(self._front - 1)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nself._nums[self._front] = num\nself._size += 1\ndef push_last(self, num: int):\n\"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\nif self._size == self.capacity():\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear = self.index(self._front + self._size)\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself._nums[rear] = num\nself._size += 1\ndef pop_first(self) -&gt; int:\n\"\"\"\u961f\u9996\u51fa\u961f\"\"\"\nnum = self.peek_first()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nself._front = self.index(self._front + 1)\nself._size -= 1\nreturn num\ndef pop_last(self) -&gt; int:\n\"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\nnum = self.peek_last()\nself._size -= 1\nreturn num\ndef peek_first(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\nreturn self._nums[self._front]\ndef peek_last(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n# \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast = self.index(self._front + self._size - 1)\nreturn self._nums[last]\ndef to_array(self) -&gt; list[int]:\n\"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n# \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres = []\nfor i in range(self._size):\nres.append(self._nums[self.index(self._front + i)])\nreturn res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate:\nvector&lt;int&gt; nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nnums.resize(capacity);\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn nums.size();\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int num) {\nif (queSize == capacity()) {\ncout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int num) {\nif (queSize == capacity()) {\ncout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty())\nthrow out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nvector&lt;int&gt; toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector&lt;int&gt; res(queSize);\nfor (int i = 0, j = front; i &lt; queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nthis.nums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate int index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void pushFirst(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void pushLast(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int popFirst() {\nint num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int popLast() {\nint num = peekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peekFirst() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int peekLast() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i &lt; queSize; i++, j++) {\nres[i] = nums[index(j)];\n}\nreturn res;\n}\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nint[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayDeque(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint Capacity() {\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npublic int Size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool IsEmpty() {\nreturn queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint Index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + Capacity()) % Capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npublic void PushFirst(int num) {\nif (queSize == Capacity()) {\nConsole.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = Index(front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num;\nqueSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npublic void PushLast(int num) {\nif (queSize == Capacity()) {\nConsole.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = Index(front + queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npublic int PopFirst() {\nint num = PeekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = Index(front + 1);\nqueSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npublic int PopLast() {\nint num = PeekLast();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int PeekFirst() {\nif (IsEmpty()) {\nthrow new InvalidOperationException();\n}\nreturn nums[front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npublic int PeekLast() {\nif (IsEmpty()) {\nthrow new InvalidOperationException();\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = Index(front + queSize - 1);\nreturn nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\npublic int[] ToArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i &lt; queSize; i++, j++) {\nres[i] = nums[Index(j)];\n}\nreturn res;\n}\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\nreturn &amp;arrayDeque{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + q.queCapacity) % q.queCapacity\n}\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nq.front = q.index(q.front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[q.front] = num\nq.queSize++\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\nif q.queSize == q.queCapacity {\nfmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nrear := q.index(q.front + q.queSize)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nq.nums[rear] = num\nq.queSize++\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\nnum := q.peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nq.front = q.index(q.front + 1)\nq.queSize--\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\nnum := q.peekLast()\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\nif q.isEmpty() {\nreturn nil\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlast := q.index(q.front + q.queSize - 1)\nreturn q.nums[last]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nres := make([]int, q.queSize)\nfor i, j := 0, q.front; i &lt; q.queSize; i++ {\nres[i] = q.nums[q.index(j)]\nj++\n}\nreturn res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(capacity: Int) {\nnums = Array(repeating: 0, count: capacity)\nfront = 0\nqueSize = 0\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -&gt; Int {\nnums.count\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -&gt; Int {\nqueSize\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -&gt; Bool {\nsize() == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nprivate func index(i: Int) -&gt; Int {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n(i + capacity()) % capacity()\n}\n/* \u961f\u9996\u5165\u961f */\nfunc pushFirst(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nfront = index(i: front - 1)\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nnums[front] = num\nqueSize += 1\n}\n/* \u961f\u5c3e\u5165\u961f */\nfunc pushLast(num: Int) {\nif size() == capacity() {\nprint(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nlet rear = index(i: front + size())\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u961f\u9996\u51fa\u961f */\nfunc popFirst() -&gt; Int {\nlet num = peekFirst()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nfront = index(i: front + 1)\nqueSize -= 1\nreturn num\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfunc popLast() -&gt; Int {\nlet num = peekLast()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peekFirst() -&gt; Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc peekLast() -&gt; Int {\nif isEmpty() {\nfatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlet last = index(i: front + size() - 1)\nreturn nums[last]\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfunc toArray() -&gt; [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: size())\nfor (i, j) in sequence(first: (0, front), next: { $0 &lt; self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[index(i: j)]\n}\nreturn res\n}\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n#nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\nthis.#front = 0;\nthis.#queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.#front = this.index(this.#front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.#nums[this.#front] = num;\nthis.#queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num) {\nif (this.#queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear = this.index(this.#front + this.#queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst() {\nconst num = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.#front = this.index(this.#front + 1);\nthis.#queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast() {\nconst num = this.peekLast();\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst() {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\nreturn this.#nums[this.#front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast() {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.#front + this.#queSize - 1);\nreturn this.#nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res = [];\nfor (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\nres[i] = this.#nums[this.index(j)];\n}\nreturn res;\n}\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nprivate nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = 0;\nthis.queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\ncapacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nindex(i: number): number {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + this.capacity()) % this.capacity();\n}\n/* \u961f\u9996\u5165\u961f */\npushFirst(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nthis.front = this.index(this.front - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nthis.nums[this.front] = num;\nthis.queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\npushLast(num: number): void {\nif (this.queSize === this.capacity()) {\nconsole.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nconst rear: number = this.index(this.front + this.queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\npopFirst(): number {\nconst num: number = this.peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nthis.front = this.index(this.front + 1);\nthis.queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\npopLast(): number {\nconst num: number = this.peekLast();\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeekFirst(): number {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\nreturn this.nums[this.front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\npeekLast(): number {\nif (this.isEmpty()) throw new Error('The Deque Is Empty.');\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nconst last = this.index(this.front + this.queSize - 1);\nreturn this.nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst res: number[] = [];\nfor (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\nres[i] = this.nums[this.index(j)];\n}\nreturn res;\n}\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\nlate List&lt;int&gt; _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nlate int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nlate int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayDeque(int capacity) {\nthis._nums = List.filled(capacity, 0);\nthis._front = this._queSize = 0;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn _nums.length;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint index(int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn (i + capacity()) % capacity();\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(int _num) {\nif (_queSize == capacity()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n_front = index(_front - 1);\n// \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n_nums[_front] = _num;\n_queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(int _num) {\nif (_queSize == capacity()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = index(_front + _queSize);\n// \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n_nums[rear] = _num;\n_queSize++;\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst() {\nint _num = peekFirst();\n// \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n_front = index(_front + 1);\n_queSize--;\nreturn _num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast() {\nint _num = peekLast();\n_queSize--;\nreturn _num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst() {\nif (isEmpty()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n}\nreturn _nums[_front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast() {\nif (isEmpty()) {\nthrow Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n}\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nint last = index(_front + _queSize - 1);\nreturn _nums[last];\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nList&lt;int&gt; toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nList&lt;int&gt; res = List.filled(_queSize, 0);\nfor (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\nres[i] = _nums[index(j)];\n}\nreturn res;\n}\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\nnums: Vec&lt;i32&gt;,     // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront: usize,       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nque_size: usize,    // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\nimpl ArrayDeque {\n/* \u6784\u9020\u65b9\u6cd5 */\npub fn new(capacity: usize) -&gt; Self {\nSelf {\nnums: vec![0; capacity],\nfront: 0,\nque_size: 0,\n}\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\npub fn capacity(&amp;self) -&gt; usize {\nself.nums.len()\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\npub fn size(&amp;self) -&gt; usize {\nself.que_size\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npub fn is_empty(&amp;self) -&gt; bool {\nself.que_size == 0\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfn index(&amp;self, i: i32) -&gt; usize {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n}\n/* \u961f\u9996\u5165\u961f */\npub fn push_first(&amp;mut self, num: i32) {\nif self.que_size == self.capacity() {\nprintln!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\nself.front = self.index(self.front as i32 - 1);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\nself.nums[self.front] = num;\nself.que_size += 1;\n}\n/* \u961f\u5c3e\u5165\u961f */\npub fn push_last(&amp;mut self, num: i32) {\nif self.que_size == self.capacity() {\nprintln!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nlet rear = self.index(self.front as i32 + self.que_size as i32);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.nums[rear] = num;\nself.que_size += 1;\n}\n/* \u961f\u9996\u51fa\u961f */\nfn pop_first(&amp;mut self) -&gt; i32 {\nlet num = self.peek_first();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\nself.front = self.index(self.front as i32 + 1);\nself.que_size -= 1;\nnum\n}\n/* \u961f\u5c3e\u51fa\u961f */\nfn pop_last(&amp;mut self) -&gt; i32 {\nlet num = self.peek_last();\nself.que_size -= 1;\nnum\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfn peek_first(&amp;self) -&gt; i32 {\nif self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\nself.nums[self.front]\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfn peek_last(&amp;self) -&gt; i32 {\nif self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\n// \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\nlet last = self.index(self.front as i32 + self.que_size as i32 - 1);\nself.nums[last]\n}\n/* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\nfn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nlet mut res = vec![0; self.que_size];\nlet mut j = self.front;\nfor i in 0..self.que_size {\nres[i] = self.nums[self.index(j as i32)];\nj += 1;\n}\nres\n}\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\nArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n// \u521d\u59cb\u5316\u6570\u7ec4\ndeque-&gt;queCapacity = capacity;\ndeque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\ndeque-&gt;front = deque-&gt;queSize = 0;\nreturn deque;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\nfree(deque-&gt;nums);\nfree(deque);\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\nreturn deque-&gt;queCapacity;\n}\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\nreturn deque-&gt;queSize;\n}\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\nreturn deque-&gt;queSize == 0;\n}\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n// \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\nreturn ((i + capacity(deque)) % capacity(deque));\n}\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\nif (deque-&gt;queSize == capacity(deque)) {\nprintf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\ndeque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n// \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\ndeque-&gt;nums[deque-&gt;front] = num;\ndeque-&gt;queSize++;\n}\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\nif (deque-&gt;queSize == capacity(deque)) {\nprintf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\nint rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque-&gt;nums[rear] = num;\ndeque-&gt;queSize++;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n// \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\nassert(empty(deque) == 0);\nreturn deque-&gt;nums[deque-&gt;front];\n}\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n// \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\nassert(empty(deque) == 0);\nint last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\nreturn deque-&gt;nums[last];\n}\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\nint num = peekFirst(deque);\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\ndeque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\ndeque-&gt;queSize--;\nreturn num;\n}\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\nint num = peekLast(deque);\ndeque-&gt;queSize--;\nreturn num;\n}\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"<p>\u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002</p> <p>\u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c <code>push</code> \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 <code>pop</code> \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u961f\u5217","text":"<p>\u300c\u961f\u5217 queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002</p> <p>\u5982\u56fe 5-4 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u201c\u961f\u9996\u201d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u201c\u961f\u5c3e\u201d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u5165\u961f\u201d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u51fa\u961f\u201d\u3002</p> <p></p> <p> \u56fe 5-4 \u00a0 \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002</p> <p> \u8868 5-2 \u00a0 \u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) <p>\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig queue.py<pre><code>from collections import deque\n# \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u5f53\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u8350\nque: deque[int] = deque()\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0];\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nqueue&lt;int&gt; queue;\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;int&gt; queue = new();\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue&lt;int&gt; queue = Queue();\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u53cc\u5411\u961f\u5217\u4f5c\u4e3a\u666e\u901a\u961f\u5217\u6765\u4f7f\u7528\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop) = deque.pop_front() {\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n</code></pre> queue.zig<pre><code>\n</code></pre>"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u961f\u5217\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\u3002\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u7b26\u5408\u8981\u6c42\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u201c\u5934\u8282\u70b9\u201d\u548c\u201c\u5c3e\u8282\u70b9\u201d\u5206\u522b\u89c6\u4e3a\u201c\u961f\u9996\u201d\u548c\u201c\u961f\u5c3e\u201d\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-5 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n\"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._front: ListNode | None = None  # \u5934\u8282\u70b9 front\nself._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\nself._size: int = 0\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self._size\ndef is_empty(self) -&gt; bool:\n\"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn not self._front\ndef push(self, num: int):\n\"\"\"\u5165\u961f\"\"\"\n# \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nnode = ListNode(num)\n# \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif self._front is None:\nself._front = node\nself._rear = node\n# \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse:\nself._rear.next = node\nself._rear = node\nself._size += 1\ndef pop(self) -&gt; int:\n\"\"\"\u51fa\u961f\"\"\"\nnum = self.peek()\n# \u5220\u9664\u5934\u8282\u70b9\nself._front = self._front.next\nself._size -= 1\nreturn num\ndef peek(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn self._front.val\ndef to_list(self) -&gt; list[int]:\n\"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nqueue = []\ntemp = self._front\nwhile temp:\nqueue.append(temp.val)\ntemp = temp.next\nreturn queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate:\nListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nint queSize;\npublic:\nLinkedListQueue() {\nfront = nullptr;\nrear = nullptr;\nqueSize = 0;\n}\n~LinkedListQueue() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(front);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode *node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == nullptr) {\nfront = node;\nrear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nrear-&gt;next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nListNode *tmp = front;\nfront = front-&gt;next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (size() == 0)\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn front-&gt;val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector&lt;int&gt; toVector() {\nListNode *node = front;\nvector&lt;int&gt; res(size());\nfor (int i = 0; i &lt; res.size(); i++) {\nres[i] = node-&gt;val;\nnode = node-&gt;next;\n}\nreturn res;\n}\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\nprivate int queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = front;\nint[] res = new int[size()];\nfor (int i = 0; i &lt; res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \nint queSize = 0;\npublic LinkedListQueue() {\nfront = null;\nrear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int Size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool IsEmpty() {\nreturn Size() == 0;\n}\n/* \u5165\u961f */\npublic void Push(int num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nListNode node = new(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (front == null) {\nfront = node;\nrear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else if (rear != null) {\nrear.next = node;\nrear = node;\n}\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int Pop() {\nint num = Peek();\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front?.next;\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int Peek() {\nif (IsEmpty())\nthrow new Exception();\nreturn front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] ToArray() {\nif (front == null)\nreturn [];\nListNode? node = front;\nint[] res = new int[Size()];\nfor (int i = 0; i &lt; res.Length; i++) {\nres[i] = node!.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\nreturn &amp;linkedListQueue{\ndata: list.New(),\n}\n}\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\ns.data.PushBack(value)\n}\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Front()\nreturn e.Value\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\nreturn s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate var front: ListNode? // \u5934\u8282\u70b9\nprivate var rear: ListNode? // \u5c3e\u8282\u70b9\nprivate var _size = 0\ninit() {}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -&gt; Int {\n_size\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -&gt; Bool {\nsize() == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nlet node = ListNode(x: num)\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif front == nil {\nfront = node\nrear = node\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nrear?.next = node\nrear = node\n}\n_size += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -&gt; Int {\nlet num = peek()\n// \u5220\u9664\u5934\u8282\u70b9\nfront = front?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -&gt; Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn front!.val\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -&gt; [Int] {\nvar node = front\nvar res = Array(repeating: 0, count: size())\nfor i in res.indices {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n#front; // \u5934\u8282\u70b9 #front\n#rear; // \u5c3e\u8282\u70b9 #rear\n#queSize = 0;\nconstructor() {\nthis.#front = null;\nthis.#rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (!this.#front) {\nthis.#front = node;\nthis.#rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nthis.#rear.next = node;\nthis.#rear = node;\n}\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u5220\u9664\u5934\u8282\u70b9\nthis.#front = this.#front.next;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.#front.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#front;\nconst res = new Array(this.size);\nfor (let i = 0; i &lt; res.length; i++) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nprivate front: ListNode | null; // \u5934\u8282\u70b9 front\nprivate rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\nprivate queSize: number = 0;\nconstructor() {\nthis.front = null;\nthis.rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size === 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nconst node = new ListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (!this.front) {\nthis.front = node;\nthis.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nthis.rear!.next = node;\nthis.rear = node;\n}\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\nif (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n// \u5220\u9664\u5934\u8282\u70b9\nthis.front = this.front.next;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.front;\nconst res = new Array&lt;number&gt;(this.size);\nfor (let i = 0; i &lt; res.length; i++) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\nListNode? _front; // \u5934\u8282\u70b9 _front\nListNode? _rear; // \u5c3e\u8282\u70b9 _rear\nint _queSize = 0; // \u961f\u5217\u957f\u5ea6\nLinkedListQueue() {\n_front = null;\n_rear = null;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int _num) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\nfinal node = ListNode(_num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (_front == null) {\n_front = node;\n_rear = node;\n} else {\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n_rear!.next = node;\n_rear = node;\n}\n_queSize++;\n}\n/* \u51fa\u961f */\nint pop() {\nfinal int _num = peek();\n// \u5220\u9664\u5934\u8282\u70b9\n_front = _front!.next;\n_queSize--;\nreturn _num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (_queSize == 0) {\nthrow Exception('\u961f\u5217\u4e3a\u7a7a');\n}\nreturn _front!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nList&lt;int&gt; toArray() {\nListNode? node = _front;\nfinal List&lt;int&gt; queue = [];\nwhile (node != null) {\nqueue.add(node.val);\nnode = node.next;\n}\nreturn queue;\n}\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\nfront: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,    // \u5934\u8282\u70b9 front\nrear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,     // \u5c3e\u8282\u70b9 rear \nque_size: usize,                            // \u961f\u5217\u7684\u957f\u5ea6\n}\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\npub fn new() -&gt; Self {\nSelf {\nfront: None,\nrear: None,\nque_size: 0, }\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npub fn size(&amp;self) -&gt; usize {\nreturn self.que_size;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npub fn is_empty(&amp;self) -&gt; bool {\nreturn self.size() == 0;\n}\n/* \u5165\u961f */\npub fn push(&amp;mut self, num: T) {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nlet new_rear = ListNode::new(num);\nmatch self.rear.take() {\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nSome(old_rear) =&gt; {\nold_rear.borrow_mut().next = Some(new_rear.clone());\nself.rear = Some(new_rear);\n}\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nNone =&gt; {\nself.front = Some(new_rear.clone());\nself.rear = Some(new_rear);\n}\n}\nself.que_size += 1;\n}\n/* \u51fa\u961f */\npub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\nself.front.take().map(|old_front| {\nmatch old_front.borrow_mut().next.take() {\nSome(new_front) =&gt; {\nself.front = Some(new_front);\n}\nNone =&gt; {\nself.rear.take();\n}\n}\nself.que_size -= 1;\nRc::try_unwrap(old_front).ok().unwrap().into_inner().val\n})\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\nself.front.as_ref()\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\nif let Some(node) = head {\nlet mut nums = self.to_array(node.borrow().next.as_ref());\nnums.insert(0, node.borrow().val);\nreturn nums;\n}\nreturn Vec::new();\n}\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\nListNode *front, *rear;\nint queSize;\n} LinkedListQueue;\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\nLinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\nqueue-&gt;front = NULL;\nqueue-&gt;rear = NULL;\nqueue-&gt;queSize = 0;\nreturn queue;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n// \u91ca\u653e\u6240\u6709\u8282\u70b9\nfor (int i = 0; i &lt; queue-&gt;queSize &amp;&amp; queue-&gt;front != NULL; i++) {\nListNode *tmp = queue-&gt;front;\nqueue-&gt;front = queue-&gt;front-&gt;next;\nfree(tmp);\n}\n// \u91ca\u653e queue \u7ed3\u6784\u4f53\nfree(queue);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\nreturn queue-&gt;queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\nreturn (size(queue) == 0);\n}\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n// \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\nListNode *node = newListNode(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (queue-&gt;front == NULL) {\nqueue-&gt;front = node;\nqueue-&gt;rear = node;\n}\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\nelse {\nqueue-&gt;rear-&gt;next = node;\nqueue-&gt;rear = node;\n}\nqueue-&gt;queSize++;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\nassert(size(queue) &amp;&amp; queue-&gt;front);\nreturn queue-&gt;front-&gt;val;\n}\n/* \u51fa\u961f */\nvoid pop(LinkedListQueue *queue) {\nint num = peek(queue);\nListNode *tmp = queue-&gt;front;\nqueue-&gt;front = queue-&gt;front-&gt;next;\nfree(tmp);\nqueue-&gt;queSize--;\n}\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\nint *arr = malloc(sizeof(int) * queue-&gt;queSize);\n// \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\nint i;\nListNode *node;\nfor (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\narr[i] = node-&gt;val;\nnode = node-&gt;next;\n}\nprintArray(arr, queue-&gt;queSize);\nfree(arr);\n}\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nfront: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\nrear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\nque_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.front = null;\nself.rear = null;\nself.que_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.que_size;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.front.?.val;\n}  // \u5165\u961f\npub fn push(self: *Self, num: T) !void {\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\n// \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\nif (self.front == null) {\nself.front = node;\nself.rear = node;\n// \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n} else {\nself.rear.?.next = node;\nself.rear = node;\n}\nself.que_size += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u5220\u9664\u5934\u8282\u70b9\nself.front = self.front.?.next;\nself.que_size -= 1;\nreturn num;\n} // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.front;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i &lt; res.len) : (i += 1) {\nres[i] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>front</code> \u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cf <code>size</code> \u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49 <code>rear = front + size</code> \uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684 <code>rear</code> \u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002</p> <p>\u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a <code>[front, rear - 1]</code>\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u6cd5\u5982\u56fe 5-6 \u6240\u793a\u3002</p> <ul> <li>\u5165\u961f\u64cd\u4f5c\uff1a\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 <code>rear</code> \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 <code>size</code> \u589e\u52a0 1 \u3002</li> <li>\u51fa\u961f\u64cd\u4f5c\uff1a\u53ea\u9700\u5c06 <code>front</code> \u589e\u52a0 1 \uff0c\u5e76\u5c06 <code>size</code> \u51cf\u5c11 1 \u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-6 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0c<code>front</code> \u548c <code>rear</code> \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u73af\u5f62\u6570\u7ec4\u201d\u3002</p> <p>\u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9 <code>front</code> \u6216 <code>rear</code> \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_queue.py<pre><code>class ArrayQueue:\n\"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\ndef __init__(self, size: int):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nself._front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nself._size: int = 0  # \u961f\u5217\u957f\u5ea6\ndef capacity(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\nreturn len(self._nums)\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\nreturn self._size\ndef is_empty(self) -&gt; bool:\n\"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self._size == 0\ndef push(self, num: int):\n\"\"\"\u5165\u961f\"\"\"\nif self._size == self.capacity():\nraise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n# \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n# \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear: int = (self._front + self._size) % self.capacity()\n# \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself._nums[rear] = num\nself._size += 1\ndef pop(self) -&gt; int:\n\"\"\"\u51fa\u961f\"\"\"\nnum: int = self.peek()\n# \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself._front = (self._front + 1) % self.capacity()\nself._size -= 1\nreturn num\ndef peek(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\nreturn self._nums[self._front]\ndef to_list(self) -&gt; list[int]:\n\"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nres = [0] * self.size()\nj: int = self._front\nfor i in range(self.size()):\nres[i] = self._nums[(j % self.capacity())]\nj += 1\nreturn res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate:\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u961f\u5217\u957f\u5ea6\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\npublic:\nArrayQueue(int capacity) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = new int[capacity];\nqueCapacity = capacity;\nfront = queSize = 0;\n}\n~ArrayQueue() {\ndelete[] nums;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity() {\nreturn queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u961f */\nvoid push(int num) {\nif (queSize == queCapacity) {\ncout &lt;&lt; \"\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\nvoid pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % queCapacity;\nqueSize--;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (isEmpty())\nthrow out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn nums[front];\n}\n/* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\nvector&lt;int&gt; toVector() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvector&lt;int&gt; arr(queSize);\nfor (int i = 0, j = front; i &lt; queSize; i++, j++) {\narr[i] = nums[j % queCapacity];\n}\nreturn arr;\n}\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate int queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\npublic int capacity() {\nreturn nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void push(int num) {\nif (queSize == capacity()) {\nSystem.out.println(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int pop() {\nint num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i &lt; queSize; i++, j++) {\nres[i] = nums[j % capacity()];\n}\nreturn res;\n}\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nint[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize; // \u961f\u5217\u957f\u5ea6\npublic ArrayQueue(int capacity) {\nnums = new int[capacity];\nfront = queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint Capacity() {\nreturn nums.Length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\npublic int Size() {\nreturn queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\npublic bool IsEmpty() {\nreturn queSize == 0;\n}\n/* \u5165\u961f */\npublic void Push(int num) {\nif (queSize == Capacity()) {\nConsole.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (front + queSize) % Capacity();\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num;\nqueSize++;\n}\n/* \u51fa\u961f */\npublic int Pop() {\nint num = Peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % Capacity();\nqueSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npublic int Peek() {\nif (IsEmpty())\nthrow new Exception();\nreturn nums[front];\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\npublic int[] ToArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nint[] res = new int[queSize];\nfor (int i = 0, j = front; i &lt; queSize; i++, j++) {\nres[i] = nums[j % this.Capacity()];\n}\nreturn res;\n}\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\nnums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize     int   // \u961f\u5217\u957f\u5ea6\nqueCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\nreturn &amp;arrayQueue{\nnums:        make([]int, queCapacity),\nqueCapacity: queCapacity,\nfront:       0,\nqueSize:     0,\n}\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\nreturn q.queSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\nreturn q.queSize == 0\n}\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n// \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\nif q.queSize == q.queCapacity {\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nrear := (q.front + q.queSize) % q.queCapacity\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nq.nums[rear] = num\nq.queSize++\n}\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\nnum := q.peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nq.front = (q.front + 1) % q.queCapacity\nq.queSize--\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\nif q.isEmpty() {\nreturn nil\n}\nreturn q.nums[q.front]\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\nrear := (q.front + q.queSize)\nif rear &gt;= q.queCapacity {\nrear %= q.queCapacity\nreturn append(q.nums[q.front:], q.nums[:rear]...)\n}\nreturn q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate var queSize = 0 // \u961f\u5217\u957f\u5ea6\ninit(capacity: Int) {\n// \u521d\u59cb\u5316\u6570\u7ec4\nnums = Array(repeating: 0, count: capacity)\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nfunc capacity() -&gt; Int {\nnums.count\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc size() -&gt; Int {\nqueSize\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -&gt; Bool {\nqueSize == 0\n}\n/* \u5165\u961f */\nfunc push(num: Int) {\nif size() == capacity() {\nprint(\"\u961f\u5217\u5df2\u6ee1\")\nreturn\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nlet rear = (front + queSize) % capacity()\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nnums[rear] = num\nqueSize += 1\n}\n/* \u51fa\u961f */\n@discardableResult\nfunc pop() -&gt; Int {\nlet num = peek()\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nfront = (front + 1) % capacity()\nqueSize -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc peek() -&gt; Int {\nif isEmpty() {\nfatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n}\nreturn nums[front]\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\nfunc toArray() -&gt; [Int] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = Array(repeating: 0, count: queSize)\nfor (i, j) in sequence(first: (0, front), next: { $0 &lt; self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\nres[i] = nums[j % capacity()]\n}\nreturn res\n}\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n#nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n#front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n#queSize = 0; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity) {\nthis.#nums = new Array(capacity);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity() {\nreturn this.#nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#queSize === 0;\n}\n/* \u5165\u961f */\npush(num) {\nif (this.size === this.capacity) {\nconsole.log('\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.#front + this.size) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.#nums[rear] = num;\nthis.#queSize++;\n}\n/* \u51fa\u961f */\npop() {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.#front = (this.#front + 1) % this.capacity;\nthis.#queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek() {\nif (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.#nums[this.#front];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\narr[i] = this.#nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nprivate nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nprivate front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nprivate queSize: number; // \u961f\u5217\u957f\u5ea6\nconstructor(capacity: number) {\nthis.nums = new Array(capacity);\nthis.front = this.queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nget capacity(): number {\nreturn this.nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.queSize === 0;\n}\n/* \u5165\u961f */\npush(num: number): void {\nif (this.size === this.capacity) {\nconsole.log('\u961f\u5217\u5df2\u6ee1');\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nconst rear = (this.front + this.queSize) % this.capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nthis.nums[rear] = num;\nthis.queSize++;\n}\n/* \u51fa\u961f */\npop(): number {\nconst num = this.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nthis.front = (this.front + 1) % this.capacity;\nthis.queSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek(): number {\nif (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\nreturn this.nums[this.front];\n}\n/* \u8fd4\u56de Array */\ntoArray(): number[] {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nconst arr = new Array(this.size);\nfor (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\narr[i] = this.nums[j % this.capacity];\n}\nreturn arr;\n}\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\nlate List&lt;int&gt; _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nlate int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nlate int _queSize; // \u961f\u5217\u957f\u5ea6\nArrayQueue(int capacity) {\n_nums = List.filled(capacity, 0);\n_front = _queSize = 0;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capaCity() {\nreturn _nums.length;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size() {\nreturn _queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _queSize == 0;\n}\n/* \u5165\u961f */\nvoid push(int _num) {\nif (_queSize == capaCity()) {\nthrow Exception(\"\u961f\u5217\u5df2\u6ee1\");\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (_front + _queSize) % capaCity();\n// \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n_nums[rear] = _num;\n_queSize++;\n}\n/* \u51fa\u961f */\nint pop() {\nint _num = peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n_front = (_front + 1) % capaCity();\n_queSize--;\nreturn _num;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek() {\nif (isEmpty()) {\nthrow Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n}\nreturn _nums[_front];\n}\n/* \u8fd4\u56de Array */\nList&lt;int&gt; toArray() {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nfinal List&lt;int&gt; res = List.filled(_queSize, 0);\nfor (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\nres[i] = _nums[j % capaCity()];\n}\nreturn res;\n}\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\nnums: Vec&lt;i32&gt;,     // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nfront: i32,         // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nque_size: i32,      // \u961f\u5217\u957f\u5ea6\nque_capacity: i32,  // \u961f\u5217\u5bb9\u91cf\n}\nimpl ArrayQueue {\n/* \u6784\u9020\u65b9\u6cd5 */\nfn new(capacity: i32) -&gt; ArrayQueue {\nArrayQueue {\nnums: vec![0; capacity as usize],\nfront: 0,\nque_size: 0,\nque_capacity: capacity,\n}\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nfn capacity(&amp;self) -&gt; i32 {\nself.que_capacity\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfn size(&amp;self) -&gt; i32 {\nself.que_size\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfn is_empty(&amp;self) -&gt; bool {\nself.que_size == 0\n}\n/* \u5165\u961f */\nfn push(&amp;mut self, num: i32) {\nif self.que_size == self.capacity() {\nprintln!(\"\u961f\u5217\u5df2\u6ee1\");\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nlet rear = (self.front + self.que_size) % self.que_capacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nself.nums[rear as usize] = num;\nself.que_size += 1;\n}\n/* \u51fa\u961f */\nfn pop(&amp;mut self) -&gt; i32 {\nlet num = self.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.front = (self.front + 1) % self.que_capacity;\nself.que_size -= 1;\nnum\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfn peek(&amp;self) -&gt; i32 {\nif self.is_empty() {\npanic!(\"index out of bounds\");\n}\nself.nums[self.front as usize]\n}\n/* \u8fd4\u56de\u6570\u7ec4 */\nfn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\nlet cap = self.que_capacity;\nlet mut j = self.front;\nlet mut arr = vec![0; self.que_size as usize];\nfor i in 0..self.que_size {\narr[i as usize] = self.nums[(j % cap) as usize];\nj += 1;\n}\narr\n}\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\nint *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\nint front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nint queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nint queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\nArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n// \u521d\u59cb\u5316\u6570\u7ec4\nqueue-&gt;queCapacity = capacity;\nqueue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\nqueue-&gt;front = queue-&gt;queSize = 0;\nreturn queue;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\nfree(queue-&gt;nums);\nfree(queue);\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\nreturn queue-&gt;queCapacity;\n}\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\nreturn queue-&gt;queSize;\n}\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\nreturn queue-&gt;queSize == 0;\n}\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\nassert(size(queue) != 0);\nreturn queue-&gt;nums[queue-&gt;front];\n}\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\nif (size(queue) == capacity(queue)) {\nprintf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\nreturn;\n}\n// \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nint rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n// \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\nqueue-&gt;nums[rear] = num;\nqueue-&gt;queSize++;\n}\n/* \u51fa\u961f */\nvoid pop(ArrayQueue *queue) {\nint num = peek(queue);\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nqueue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\nqueue-&gt;queSize--;\n}\n</code></pre> array_queue.zig<pre><code>// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nnums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \ncap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\nfront: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\nqueSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.cap = cap;\nself.nums = try self.mem_allocator.alloc(T, self.cap);\n@memset(self.nums, @as(T, 0));\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\npub fn capacity(self: *Self) usize {\nreturn self.cap;\n}\n// \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.queSize;\n}\n// \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.queSize == 0;\n}\n// \u5165\u961f\npub fn push(self: *Self, num: T) !void {\nif (self.size() == self.capacity()) {\nstd.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\nreturn;\n}\n// \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n// \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\nvar rear = (self.front + self.queSize) % self.capacity();\n// \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\nself.nums[rear] = num;\nself.queSize += 1;\n} // \u51fa\u961f\npub fn pop(self: *Self) T {\nvar num = self.peek();\n// \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\nself.front = (self.front + 1) % self.capacity();\nself.queSize -= 1;\nreturn num;\n} // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\nreturn self.nums[self.front];\n} // \u8fd4\u56de\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\n// \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nvar j: usize = self.front;\nwhile (i &lt; self.size()) : ({ i += 1; j += 1; }) {\nres[i] = self.nums[j % self.capacity()];\n}\nreturn res;\n}\n};\n}\n</code></pre> <p>\u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff1a\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002</li> <li>\u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\uff0c\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 \u6808","text":"<p>\u300c\u6808 stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u903b\u8f91\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u8fd9\u79cd\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-1 \u6240\u793a\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u201c\u6808\u9876\u201d\uff0c\u5e95\u90e8\u79f0\u4e3a\u201c\u6808\u5e95\u201d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u6808\u201d\uff0c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u6808\u201d\u3002</p> <p></p> <p> \u56fe 5-1 \u00a0 \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"<p>\u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 <code>push()</code>\u3001<code>pop()</code>\u3001<code>peek()</code> \u547d\u540d\u4e3a\u4f8b\u3002</p> <p> \u8868 5-1 \u00a0 \u6808\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\) <p>\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u201c\u6570\u7ec4\u201d\u6216\u201c\u94fe\u8868\u201d\u5f53\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u5728\u7a0b\u5e8f\u903b\u8f91\u4e0a\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig stack.py<pre><code># \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: list[int] = []\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* \u521d\u59cb\u5316\u6808 */\nstack&lt;int&gt; stack;\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;int&gt; stack = new();\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList&lt;int&gt; stack = [];\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u628a Vec \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet top = stack.last().unwrap();\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.pop().unwrap();\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.len();\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n</code></pre> stack.zig<pre><code>\n</code></pre>"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u6808\u7684\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002</p> <p>\u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002</p> <p>\u5982\u56fe 5-2 \u6240\u793a\uff0c\u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-2 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_stack.py<pre><code>class LinkedListStack:\n\"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._peek: ListNode | None = None\nself._size: int = 0\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\nreturn self._size\ndef is_empty(self) -&gt; bool:\n\"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn not self._peek\ndef push(self, val: int):\n\"\"\"\u5165\u6808\"\"\"\nnode = ListNode(val)\nnode.next = self._peek\nself._peek = node\nself._size += 1\ndef pop(self) -&gt; int:\n\"\"\"\u51fa\u6808\"\"\"\nnum = self.peek()\nself._peek = self._peek.next\nself._size -= 1\nreturn num\ndef peek(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u6808\u4e3a\u7a7a\")\nreturn self._peek.val\ndef to_list(self) -&gt; list[int]:\n\"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\narr = []\nnode = self._peek\nwhile node:\narr.append(node.val)\nnode = node.next\narr.reverse()\nreturn arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate:\nListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint stkSize;        // \u6808\u7684\u957f\u5ea6\npublic:\nLinkedListStack() {\nstackTop = nullptr;\nstkSize = 0;\n}\n~LinkedListStack() {\n// \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\nfreeMemoryLinkedList(stackTop);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nListNode *node = new ListNode(num);\nnode-&gt;next = stackTop;\nstackTop = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint num = top();\nListNode *tmp = stackTop;\nstackTop = stackTop-&gt;next;\n// \u91ca\u653e\u5185\u5b58\ndelete tmp;\nstkSize--;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (isEmpty())\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stackTop-&gt;val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nvector&lt;int&gt; toVector() {\nListNode *node = stackTop;\nvector&lt;int&gt; res(size());\nfor (int i = res.size() - 1; i &gt;= 0; i--) {\nres[i] = node-&gt;val;\nnode = node-&gt;next;\n}\nreturn res;\n}\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate int stkSize = 0; // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nListNode node = new ListNode(num);\nnode.next = stackPeek;\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int pop() {\nint num = peek();\nstackPeek = stackPeek.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn stackPeek.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] toArray() {\nListNode node = stackPeek;\nint[] res = new int[size()];\nfor (int i = res.length - 1; i &gt;= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint stkSize = 0;   // \u6808\u7684\u957f\u5ea6\npublic LinkedListStack() {\nstackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int Size() {\nreturn stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool IsEmpty() {\nreturn Size() == 0;\n}\n/* \u5165\u6808 */\npublic void Push(int num) {\nListNode node = new(num) {\nnext = stackPeek\n};\nstackPeek = node;\nstkSize++;\n}\n/* \u51fa\u6808 */\npublic int Pop() {\nint num = Peek();\nstackPeek = stackPeek!.next;\nstkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int Peek() {\nif (IsEmpty())\nthrow new Exception();\nreturn stackPeek!.val;\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] ToArray() {\nif (stackPeek == null)\nreturn [];\nListNode? node = stackPeek;\nint[] res = new int[Size()];\nfor (int i = res.Length - 1; i &gt;= 0; i--) {\nres[i] = node!.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n// \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\ndata *list.List\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\nreturn &amp;linkedListStack{\ndata: list.New(),\n}\n}\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\ns.data.PushBack(value)\n}\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\ns.data.Remove(e)\nreturn e.Value\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\ne := s.data.Back()\nreturn e.Value\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\nreturn s.data.Len()\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\nreturn s.data.Len() == 0\n}\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\nreturn s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate var _size = 0 // \u6808\u7684\u957f\u5ea6\ninit() {}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -&gt; Int {\n_size\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -&gt; Bool {\nsize() == 0\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nlet node = ListNode(x: num)\nnode.next = _peek\n_peek = node\n_size += 1\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -&gt; Int {\nlet num = peek()\n_peek = _peek?.next\n_size -= 1\nreturn num\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn _peek!.val\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -&gt; [Int] {\nvar node = _peek\nvar res = Array(repeating: 0, count: _size)\nfor i in sequence(first: res.count - 1, next: { $0 &gt;= 0 + 1 ? $0 - 1 : nil }) {\nres[i] = node!.val\nnode = node?.next\n}\nreturn res\n}\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n#stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n#stkSize = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.#stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.size === 0;\n}\n/* \u5165\u6808 */\npush(num) {\nconst node = new ListNode(num);\nnode.next = this.#stackPeek;\nthis.#stackPeek = node;\nthis.#stkSize++;\n}\n/* \u51fa\u6808 */\npop() {\nconst num = this.peek();\nthis.#stackPeek = this.#stackPeek.next;\nthis.#stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek() {\nif (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray() {\nlet node = this.#stackPeek;\nconst res = new Array(this.size);\nfor (let i = res.length - 1; i &gt;= 0; i--) {\nres[i] = node.val;\nnode = node.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nprivate stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nprivate stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\nconstructor() {\nthis.stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.size === 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nconst node = new ListNode(num);\nnode.next = this.stackPeek;\nthis.stackPeek = node;\nthis.stkSize++;\n}\n/* \u51fa\u6808 */\npop(): number {\nconst num = this.peek();\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nthis.stackPeek = this.stackPeek.next;\nthis.stkSize--;\nreturn num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek(): number {\nif (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stackPeek.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\ntoArray(): number[] {\nlet node = this.stackPeek;\nconst res = new Array&lt;number&gt;(this.size);\nfor (let i = res.length - 1; i &gt;= 0; i--) {\nres[i] = node!.val;\nnode = node!.next;\n}\nreturn res;\n}\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\nListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint _stkSize = 0; // \u6808\u7684\u957f\u5ea6\nLinkedListStack() {\n_stackPeek = null;\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn _stkSize;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _stkSize == 0;\n}\n/* \u5165\u6808 */\nvoid push(int _num) {\nfinal ListNode node = ListNode(_num);\nnode.next = _stackPeek;\n_stackPeek = node;\n_stkSize++;\n}\n/* \u51fa\u6808 */\nint pop() {\nfinal int _num = peek();\n_stackPeek = _stackPeek!.next;\n_stkSize--;\nreturn _num;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek() {\nif (_stackPeek == null) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stackPeek!.val;\n}\n/* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\nList&lt;int&gt; toList() {\nListNode? node = _stackPeek;\nList&lt;int&gt; list = [];\nwhile (node != null) {\nlist.add(node.val);\nnode = node.next;\n}\nlist = list.reversed.toList();\nreturn list;\n}\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\nstack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,   // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nstk_size: usize,                                // \u6808\u7684\u957f\u5ea6\n}\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\npub fn new() -&gt; Self {\nSelf {\nstack_peek: None,\nstk_size: 0,\n}\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npub fn size(&amp;self) -&gt; usize {\nreturn self.stk_size;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npub fn is_empty(&amp;self) -&gt; bool {\nreturn self.size() == 0;\n}\n/* \u5165\u6808 */\npub fn push(&amp;mut self, num: T) {\nlet node = ListNode::new(num);\nnode.borrow_mut().next = self.stack_peek.take();\nself.stack_peek = Some(node);\nself.stk_size += 1;\n}\n/* \u51fa\u6808 */\npub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\nself.stack_peek.take().map(|old_head| {\nmatch old_head.borrow_mut().next.take() {\nSome(new_head) =&gt; {\nself.stack_peek = Some(new_head);\n}\nNone =&gt; {\nself.stack_peek = None;\n}\n}\nself.stk_size -= 1;\nRc::try_unwrap(old_head).ok().unwrap().into_inner().val\n})\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\nself.stack_peek.as_ref()\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\nif let Some(node) = head {\nlet mut nums = self.to_array(node.borrow().next.as_ref());\nnums.push(node.borrow().val);\nreturn nums;\n}\nreturn Vec::new();\n}\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\nListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nint size;      // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\nLinkedListStack *s = malloc(sizeof(LinkedListStack));\ns-&gt;top = NULL;\ns-&gt;size = 0;\nreturn s;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\nwhile (s-&gt;top) {\nListNode *n = s-&gt;top-&gt;next;\nfree(s-&gt;top);\ns-&gt;top = n;\n}\nfree(s);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\nassert(s);\nreturn s-&gt;size;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\nassert(s);\nreturn size(s) == 0;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\nassert(s);\nassert(size(s) != 0);\nreturn s-&gt;top-&gt;val;\n}\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\nassert(s);\nListNode *node = (ListNode *)malloc(sizeof(ListNode));\nnode-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\nnode-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\ns-&gt;top = node;       // \u66f4\u65b0\u6808\u9876\ns-&gt;size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\nif (s-&gt;size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nassert(s);\nint val = peek(s);\nListNode *tmp = s-&gt;top;\ns-&gt;top = s-&gt;top-&gt;next;\n// \u91ca\u653e\u5185\u5b58\nfree(tmp);\ns-&gt;size--;\nreturn val;\n}\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\nstk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\nmem_arena: ?std.heap.ArenaAllocator = null,\nmem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n// \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) !void {\nif (self.mem_arena == null) {\nself.mem_arena = std.heap.ArenaAllocator.init(allocator);\nself.mem_allocator = self.mem_arena.?.allocator();\n}\nself.stack_top = null;\nself.stk_size = 0;\n}\n// \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.mem_arena == null) return;\nself.mem_arena.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stk_size;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack_top.?.val;\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\nvar node = try self.mem_allocator.create(inc.ListNode(T));\nnode.init(num);\nnode.next = self.stack_top;\nself.stack_top = node;\nself.stk_size += 1;\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.peek();\nself.stack_top = self.stack_top.?.next;\nself.stk_size -= 1;\nreturn num;\n} // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\npub fn toArray(self: *Self) ![]T {\nvar node = self.stack_top;\nvar res = try self.mem_allocator.alloc(T, self.size());\n@memset(res, @as(T, 0));\nvar i: usize = 0;\nwhile (i &lt; res.len) : (i += 1) {\nres[res.len - i - 1] = node.?.val;\nnode = node.?.next;\n}\nreturn res;\n}\n};\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5982\u56fe 5-3 \u6240\u793a\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-3 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u987b\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_stack.py<pre><code>class ArrayStack:\n\"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\ndef __init__(self):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._stack: list[int] = []\ndef size(self) -&gt; int:\n\"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\nreturn len(self._stack)\ndef is_empty(self) -&gt; bool:\n\"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\nreturn self._stack == []\ndef push(self, item: int):\n\"\"\"\u5165\u6808\"\"\"\nself._stack.append(item)\ndef pop(self) -&gt; int:\n\"\"\"\u51fa\u6808\"\"\"\nif self.is_empty():\nraise IndexError(\"\u6808\u4e3a\u7a7a\")\nreturn self._stack.pop()\ndef peek(self) -&gt; int:\n\"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\nif self.is_empty():\nraise IndexError(\"\u6808\u4e3a\u7a7a\")\nreturn self._stack[-1]\ndef to_list(self) -&gt; list[int]:\n\"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\nreturn self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate:\nvector&lt;int&gt; stack;\npublic:\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn stack.size() == 0;\n}\n/* \u5165\u6808 */\nvoid push(int num) {\nstack.push_back(num);\n}\n/* \u51fa\u6808 */\nvoid pop() {\nint oldTop = top();\nstack.pop_back();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top() {\nif (isEmpty())\nthrow out_of_range(\"\u6808\u4e3a\u7a7a\");\nreturn stack.back();\n}\n/* \u8fd4\u56de Vector */\nvector&lt;int&gt; toVector() {\nreturn stack;\n}\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate ArrayList&lt;Integer&gt; stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = new ArrayList&lt;&gt;();\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int size() {\nreturn stack.size();\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic boolean isEmpty() {\nreturn size() == 0;\n}\n/* \u5165\u6808 */\npublic void push(int num) {\nstack.add(num);\n}\n/* \u51fa\u6808 */\npublic int pop() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn stack.remove(size() - 1);\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int peek() {\nif (isEmpty())\nthrow new IndexOutOfBoundsException();\nreturn stack.get(size() - 1);\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic Object[] toArray() {\nreturn stack.toArray();\n}\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nList&lt;int&gt; stack;\npublic ArrayStack() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\npublic int Size() {\nreturn stack.Count;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\npublic bool IsEmpty() {\nreturn Size() == 0;\n}\n/* \u5165\u6808 */\npublic void Push(int num) {\nstack.Add(num);\n}\n/* \u51fa\u6808 */\npublic int Pop() {\nif (IsEmpty())\nthrow new Exception();\nvar val = Peek();\nstack.RemoveAt(Size() - 1);\nreturn val;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npublic int Peek() {\nif (IsEmpty())\nthrow new Exception();\nreturn stack[Size() - 1];\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\npublic int[] ToArray() {\nreturn [.. stack];\n}\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\ndata []int // \u6570\u636e\n}\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\nreturn &amp;arrayStack{\n// \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\ndata: make([]int, 0, 16),\n}\n}\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\nreturn len(s.data)\n}\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\nreturn s.size() == 0\n}\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n// \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\ns.data = append(s.data, v)\n}\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\nval := s.peek()\ns.data = s.data[:len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\nif s.isEmpty() {\nreturn nil\n}\nval := s.data[len(s.data)-1]\nreturn val\n}\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\nreturn s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate var stack: [Int]\ninit() {\n// \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\nstack = []\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc size() -&gt; Int {\nstack.count\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc isEmpty() -&gt; Bool {\nstack.isEmpty\n}\n/* \u5165\u6808 */\nfunc push(num: Int) {\nstack.append(num)\n}\n/* \u51fa\u6808 */\n@discardableResult\nfunc pop() -&gt; Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.removeLast()\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\nif isEmpty() {\nfatalError(\"\u6808\u4e3a\u7a7a\")\n}\nreturn stack.last!\n}\n/* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nfunc toArray() -&gt; [Int] {\nstack\n}\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n#stack;\nconstructor() {\nthis.#stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size() {\nreturn this.#stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty() {\nreturn this.#stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num) {\nthis.#stack.push(num);\n}\n/* \u51fa\u6808 */\npop() {\nif (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop() {\nif (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.#stack[this.#stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.#stack;\n}\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nprivate stack: number[];\nconstructor() {\nthis.stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nget size(): number {\nreturn this.stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nisEmpty(): boolean {\nreturn this.stack.length === 0;\n}\n/* \u5165\u6808 */\npush(num: number): void {\nthis.stack.push(num);\n}\n/* \u51fa\u6808 */\npop(): number | undefined {\nif (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack.pop();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\ntop(): number | undefined {\nif (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\nreturn this.stack[this.stack.length - 1];\n}\n/* \u8fd4\u56de Array */\ntoArray() {\nreturn this.stack;\n}\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\nlate List&lt;int&gt; _stack;\nArrayStack() {\n_stack = [];\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size() {\nreturn _stack.length;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty() {\nreturn _stack.isEmpty;\n}\n/* \u5165\u6808 */\nvoid push(int _num) {\n_stack.add(_num);\n}\n/* \u51fa\u6808 */\nint pop() {\nif (isEmpty()) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stack.removeLast();\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek() {\nif (isEmpty()) {\nthrow Exception(\"\u6808\u4e3a\u7a7a\");\n}\nreturn _stack.last;\n}\n/* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\nList&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack&lt;T&gt; {\nstack: Vec&lt;T&gt;,\n}\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n/* \u521d\u59cb\u5316\u6808 */\nfn new() -&gt; ArrayStack&lt;T&gt; {\nArrayStack::&lt;T&gt; { stack: Vec::&lt;T&gt;::new() }\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfn size(&amp;self) -&gt; usize {\nself.stack.len()\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfn is_empty(&amp;self) -&gt; bool {\nself.size() == 0\n}\n/* \u5165\u6808 */\nfn push(&amp;mut self, num: T) {\nself.stack.push(num);\n}\n/* \u51fa\u6808 */\nfn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\nmatch self.stack.pop() {\nSome(num) =&gt; Some(num),\nNone =&gt; None,\n}\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\nif self.is_empty() { panic!(\"\u6808\u4e3a\u7a7a\") };\nself.stack.last()\n}\n/* \u8fd4\u56de &amp;Vec */\nfn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n&amp;self.stack\n}\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\nint *data;\nint size;\n} ArrayStack;\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\nArrayStack *stack = malloc(sizeof(ArrayStack));\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\nstack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\nstack-&gt;size = 0;\nreturn stack;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\nfree(stack-&gt;data);\nfree(stack);\n}\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\nreturn stack-&gt;size;\n}\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\nreturn stack-&gt;size == 0;\n}\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\nif (stack-&gt;size == MAX_SIZE) {\nprintf(\"stack is full.\\n\");\nreturn;\n}\nstack-&gt;data[stack-&gt;size] = num;\nstack-&gt;size++;\n}\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\nif (stack-&gt;size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nreturn stack-&gt;data[stack-&gt;size - 1];\n}\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\nif (stack-&gt;size == 0) {\nprintf(\"stack is empty.\\n\");\nreturn INT_MAX;\n}\nint val = peek(stack);\nstack-&gt;size--;\nreturn val;\n}\n</code></pre> array_stack.zig<pre><code>// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\nreturn struct {\nconst Self = @This();\nstack: ?std.ArrayList(T) = null,     // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\npub fn init(self: *Self, allocator: std.mem.Allocator) void {\nif (self.stack == null) {\nself.stack = std.ArrayList(T).init(allocator);\n}\n}\n// \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\npub fn deinit(self: *Self) void {\nif (self.stack == null) return;\nself.stack.?.deinit();\n}\n// \u83b7\u53d6\u6808\u7684\u957f\u5ea6\npub fn size(self: *Self) usize {\nreturn self.stack.?.items.len;\n}\n// \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\npub fn isEmpty(self: *Self) bool {\nreturn self.size() == 0;\n}\n// \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npub fn peek(self: *Self) T {\nif (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\nreturn self.stack.?.items[self.size() - 1];\n}  // \u5165\u6808\npub fn push(self: *Self, num: T) !void {\ntry self.stack.?.append(num);\n} // \u51fa\u6808\npub fn pop(self: *Self) T {\nvar num = self.stack.?.pop();\nreturn num;\n} // \u8fd4\u56de ArrayList\npub fn toList(self: *Self) std.ArrayList(T) {\nreturn self.stack.?;\n}\n};\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"<p>\u652f\u6301\u64cd\u4f5c</p> <p>\u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002</p> <p>\u65f6\u95f4\u6548\u7387</p> <p>\u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002</p> <p>\u5728\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\u65f6\uff0c\u4f8b\u5982 <code>int</code> \u6216 <code>double</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> </ul> <p>\u7a7a\u95f4\u6548\u7387</p> <p>\u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\uff1b\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\u7684\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002</p> <p>\u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002</p> <p>\u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5bf9\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u540e\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u4e2a\u7f51\u9875\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002</li> <li>\u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u4e0d\u65ad\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_stack_and_queue/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> <li>\u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002</li> <li>\u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002</li> <li>\u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f</p> <p>\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u201c\u53cc\u5411\u961f\u5217\u201d\u7ae0\u8282\u6709\u63d0\u5230\u3002</p> <p>\u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f</p> <p>\u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0c<code>Java</code> \u548c <code>Python</code> \u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728 <code>C</code> \u548c <code>C++</code> \u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002</p> <p>\u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f</p> <p>\u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\uff0c\u6216\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002</p> <p>\u64a4\u9500\uff08undo\uff09\u548c\u53cd\u64a4\u9500\uff08redo\uff09\u5177\u4f53\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff1f</p> <p>\u4f7f\u7528\u4e24\u4e2a\u5806\u6808\uff0c\u6808 <code>A</code> \u7528\u4e8e\u64a4\u9500\uff0c\u6808 <code>B</code> \u7528\u4e8e\u53cd\u64a4\u9500\u3002</p> <ol> <li>\u6bcf\u5f53\u7528\u6237\u6267\u884c\u4e00\u4e2a\u64cd\u4f5c\uff0c\u5c06\u8fd9\u4e2a\u64cd\u4f5c\u538b\u5165\u6808 <code>A</code> \uff0c\u5e76\u6e05\u7a7a\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>A</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u53cd\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>B</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>A</code> \u3002</li> </ol>"},{"location":"chapter_tree/","title":"\u7b2c 7 \u7ae0 \u00a0 \u6811","text":"<p>Abstract</p> <p>\u53c2\u5929\u5927\u6811\u5145\u6ee1\u751f\u547d\u529b\uff0c\u6839\u6df1\u53f6\u8302\uff0c\u5206\u679d\u6276\u758f\u3002</p> <p>\u5b83\u4e3a\u6211\u4eec\u5c55\u73b0\u4e86\u6570\u636e\u5206\u6cbb\u7684\u751f\u52a8\u5f62\u6001\u3002</p>"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>7.1 \u00a0 \u4e8c\u53c9\u6811</li> <li>7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386</li> <li>7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a</li> <li>7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811</li> <li>7.5 \u00a0 AVL \u6811 *</li> <li>7.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a","text":"<p>\u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9 <code>TreeNode</code> \uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u80fd\u5426\u7528\u6570\u7ec4\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\u3002\u7ed9\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u6240\u6709\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u5219\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\u3002</p> <p>\u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002\u56fe 7-12 \u5c55\u793a\u4e86\u5404\u4e2a\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u6620\u5c04\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"<p>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u7279\u4f8b\uff0c\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\u901a\u5e38\u5b58\u5728\u8bb8\u591a \\(\\text{None}\\) \u3002\u7531\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b \\(\\text{None}\\) \uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b \\(\\text{None}\\) \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\u3002\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</p> <p>\u5982\u56fe 7-13 \u6240\u793a\uff0c\u7ed9\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u4e0a\u8ff0\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002</p> <p></p> <p> \u56fe 7-13 \u00a0 \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 \\(\\text{None}\\) \u3002\u5982\u56fe 7-14 \u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6807\u8bb0\u7a7a\u4f4d\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 None \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> \u56fe 7-14 \u00a0 \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c\\(\\text{None}\\) \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 \\(\\text{None}\\) \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 \\(\\text{None}\\) \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u56fe 7-15 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 7-15 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u68f5\u57fa\u4e8e\u6570\u7ec4\u8868\u793a\u7684\u4e8c\u53c9\u6811\uff0c\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u7ed9\u5b9a\u67d0\u8282\u70b9\uff0c\u83b7\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3001\u7236\u8282\u70b9\u3002</li> <li>\u83b7\u53d6\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3001\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n\"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\ndef __init__(self, arr: list[int | None]):\n\"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\nself._tree = list(arr)\ndef size(self):\n\"\"\"\u8282\u70b9\u6570\u91cf\"\"\"\nreturn len(self._tree)\ndef val(self, i: int) -&gt; int:\n\"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n# \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif i &lt; 0 or i &gt;= self.size():\nreturn None\nreturn self._tree[i]\ndef left(self, i: int) -&gt; int | None:\n\"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\nreturn 2 * i + 1\ndef right(self, i: int) -&gt; int | None:\n\"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\nreturn 2 * i + 2\ndef parent(self, i: int) -&gt; int | None:\n\"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\nreturn (i - 1) // 2\ndef level_order(self) -&gt; list[int]:\n\"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\nself.res = []\n# \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor i in range(self.size()):\nif self.val(i) is not None:\nself.res.append(self.val(i))\nreturn self.res\ndef dfs(self, i: int, order: str):\n\"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\nif self.val(i) is None:\nreturn\n# \u524d\u5e8f\u904d\u5386\nif order == \"pre\":\nself.res.append(self.val(i))\nself.dfs(self.left(i), order)\n# \u4e2d\u5e8f\u904d\u5386\nif order == \"in\":\nself.res.append(self.val(i))\nself.dfs(self.right(i), order)\n# \u540e\u5e8f\u904d\u5386\nif order == \"post\":\nself.res.append(self.val(i))\ndef pre_order(self) -&gt; list[int]:\n\"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\nself.res = []\nself.dfs(0, order=\"pre\")\nreturn self.res\ndef in_order(self) -&gt; list[int]:\n\"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\nself.res = []\nself.dfs(0, order=\"in\")\nreturn self.res\ndef post_order(self) -&gt; list[int]:\n\"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\nself.res = []\nself.dfs(0, order=\"post\")\nreturn self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\npublic:\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayBinaryTree(vector&lt;int&gt; arr) {\ntree = arr;\n}\n/* \u8282\u70b9\u6570\u91cf */\nint size() {\nreturn tree.size();\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(int i) {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif (i &lt; 0 || i &gt;= size())\nreturn INT_MAX;\nreturn tree[i];\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint parent(int i) {\nreturn (i - 1) / 2;\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nvector&lt;int&gt; levelOrder() {\nvector&lt;int&gt; res;\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; size(); i++) {\nif (val(i) != INT_MAX)\nres.push_back(val(i));\n}\nreturn res;\n}\n/* \u524d\u5e8f\u904d\u5386 */\nvector&lt;int&gt; preOrder() {\nvector&lt;int&gt; res;\ndfs(0, \"pre\", res);\nreturn res;\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvector&lt;int&gt; inOrder() {\nvector&lt;int&gt; res;\ndfs(0, \"in\", res);\nreturn res;\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvector&lt;int&gt; postOrder() {\nvector&lt;int&gt; res;\ndfs(0, \"post\", res);\nreturn res;\n}\nprivate:\nvector&lt;int&gt; tree;\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif (val(i) == INT_MAX)\nreturn;\n// \u524d\u5e8f\u904d\u5386\nif (order == \"pre\")\nres.push_back(val(i));\ndfs(left(i), order, res);\n// \u4e2d\u5e8f\u904d\u5386\nif (order == \"in\")\nres.push_back(val(i));\ndfs(right(i), order, res);\n// \u540e\u5e8f\u904d\u5386\nif (order == \"post\")\nres.push_back(val(i));\n}\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\nprivate List&lt;Integer&gt; tree;\n/* \u6784\u9020\u65b9\u6cd5 */\npublic ArrayBinaryTree(List&lt;Integer&gt; arr) {\ntree = new ArrayList&lt;&gt;(arr);\n}\n/* \u8282\u70b9\u6570\u91cf */\npublic int size() {\nreturn tree.size();\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\npublic Integer val(int i) {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif (i &lt; 0 || i &gt;= size())\nreturn null;\nreturn tree.get(i);\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\npublic Integer left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\npublic Integer right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\npublic Integer parent(int i) {\nreturn (i - 1) / 2;\n}\n/* \u5c42\u5e8f\u904d\u5386 */\npublic List&lt;Integer&gt; levelOrder() {\nList&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; size(); i++) {\nif (val(i) != null)\nres.add(val(i));\n}\nreturn res;\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nprivate void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif (val(i) == null)\nreturn;\n// \u524d\u5e8f\u904d\u5386\nif (\"pre\".equals(order))\nres.add(val(i));\ndfs(left(i), order, res);\n// \u4e2d\u5e8f\u904d\u5386\nif (\"in\".equals(order))\nres.add(val(i));\ndfs(right(i), order, res);\n// \u540e\u5e8f\u904d\u5386\nif (\"post\".equals(order))\nres.add(val(i));\n}\n/* \u524d\u5e8f\u904d\u5386 */\npublic List&lt;Integer&gt; preOrder() {\nList&lt;Integer&gt; res = new ArrayList&lt;&gt;();\ndfs(0, \"pre\", res);\nreturn res;\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\npublic List&lt;Integer&gt; inOrder() {\nList&lt;Integer&gt; res = new ArrayList&lt;&gt;();\ndfs(0, \"in\", res);\nreturn res;\n}\n/* \u540e\u5e8f\u904d\u5386 */\npublic List&lt;Integer&gt; postOrder() {\nList&lt;Integer&gt; res = new ArrayList&lt;&gt;();\ndfs(0, \"post\", res);\nreturn res;\n}\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\nList&lt;int?&gt; tree = new(arr);\n/* \u8282\u70b9\u6570\u91cf */\npublic int Size() {\nreturn tree.Count;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\npublic int? Val(int i) {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif (i &lt; 0 || i &gt;= Size())\nreturn null;\nreturn tree[i];\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\npublic int Left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\npublic int Right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\npublic int Parent(int i) {\nreturn (i - 1) / 2;\n}\n/* \u5c42\u5e8f\u904d\u5386 */\npublic List&lt;int&gt; LevelOrder() {\nList&lt;int&gt; res = [];\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; Size(); i++) {\nif (Val(i).HasValue)\nres.Add(Val(i)!.Value);\n}\nreturn res;\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid DFS(int i, string order, List&lt;int&gt; res) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif (!Val(i).HasValue)\nreturn;\n// \u524d\u5e8f\u904d\u5386\nif (order == \"pre\")\nres.Add(Val(i)!.Value);\nDFS(Left(i), order, res);\n// \u4e2d\u5e8f\u904d\u5386\nif (order == \"in\")\nres.Add(Val(i)!.Value);\nDFS(Right(i), order, res);\n// \u540e\u5e8f\u904d\u5386\nif (order == \"post\")\nres.Add(Val(i)!.Value);\n}\n/* \u524d\u5e8f\u904d\u5386 */\npublic List&lt;int&gt; PreOrder() {\nList&lt;int&gt; res = [];\nDFS(0, \"pre\", res);\nreturn res;\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\npublic List&lt;int&gt; InOrder() {\nList&lt;int&gt; res = [];\nDFS(0, \"in\", res);\nreturn res;\n}\n/* \u540e\u5e8f\u904d\u5386 */\npublic List&lt;int&gt; PostOrder() {\nList&lt;int&gt; res = [];\nDFS(0, \"post\", res);\nreturn res;\n}\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\ntree []any\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\nreturn &amp;arrayBinaryTree{\ntree: arr,\n}\n}\n/* \u8282\u70b9\u6570\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\nreturn len(abt.tree)\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif i &lt; 0 || i &gt;= abt.size() {\nreturn nil\n}\nreturn abt.tree[i]\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\nreturn 2*i + 1\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\nreturn 2*i + 2\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\nreturn (i - 1) / 2\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\nvar res []any\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor i := 0; i &lt; abt.size(); i++ {\nif abt.val(i) != nil {\nres = append(res, abt.val(i))\n}\n}\nreturn res\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif abt.val(i) == nil {\nreturn\n}\n// \u524d\u5e8f\u904d\u5386\nif order == \"pre\" {\n*res = append(*res, abt.val(i))\n}\nabt.dfs(abt.left(i), order, res)\n// \u4e2d\u5e8f\u904d\u5386\nif order == \"in\" {\n*res = append(*res, abt.val(i))\n}\nabt.dfs(abt.right(i), order, res)\n// \u540e\u5e8f\u904d\u5386\nif order == \"post\" {\n*res = append(*res, abt.val(i))\n}\n}\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\nvar res []any\nabt.dfs(0, \"pre\", &amp;res)\nreturn res\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\nvar res []any\nabt.dfs(0, \"in\", &amp;res)\nreturn res\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\nvar res []any\nabt.dfs(0, \"post\", &amp;res)\nreturn res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\nprivate var tree: [Int?]\n/* \u6784\u9020\u65b9\u6cd5 */\ninit(arr: [Int?]) {\ntree = arr\n}\n/* \u8282\u70b9\u6570\u91cf */\nfunc size() -&gt; Int {\ntree.count\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc val(i: Int) -&gt; Int? {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif i &lt; 0 || i &gt;= size() {\nreturn nil\n}\nreturn tree[i]\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n2 * i + 1\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n(i - 1) / 2\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder() -&gt; [Int] {\nvar res: [Int] = []\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor i in stride(from: 0, to: size(), by: 1) {\nif let val = val(i: i) {\nres.append(val)\n}\n}\nreturn res\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nprivate func dfs(i: Int, order: String, res: inout [Int]) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nguard let val = val(i: i) else {\nreturn\n}\n// \u524d\u5e8f\u904d\u5386\nif order == \"pre\" {\nres.append(val)\n}\ndfs(i: left(i: i), order: order, res: &amp;res)\n// \u4e2d\u5e8f\u904d\u5386\nif order == \"in\" {\nres.append(val)\n}\ndfs(i: right(i: i), order: order, res: &amp;res)\n// \u540e\u5e8f\u904d\u5386\nif order == \"post\" {\nres.append(val)\n}\n}\n/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder() -&gt; [Int] {\nvar res: [Int] = []\ndfs(i: 0, order: \"pre\", res: &amp;res)\nreturn res\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder() -&gt; [Int] {\nvar res: [Int] = []\ndfs(i: 0, order: \"in\", res: &amp;res)\nreturn res\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder() -&gt; [Int] {\nvar res: [Int] = []\ndfs(i: 0, order: \"post\", res: &amp;res)\nreturn res\n}\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n#tree;\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(arr) {\nthis.#tree = arr;\n}\n/* \u8282\u70b9\u6570\u91cf */\nsize() {\nreturn this.#tree.length;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nval(i) {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif (i &lt; 0 || i &gt;= this.size()) return null;\nreturn this.#tree[i];\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nleft(i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nright(i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nparent(i) {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nlevelOrder() {\nlet res = [];\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i &lt; this.size(); i++) {\nif (this.val(i) !== null) res.push(this.val(i));\n}\nreturn res;\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n#dfs(i, order, res) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif (this.val(i) === null) return;\n// \u524d\u5e8f\u904d\u5386\nif (order === 'pre') res.push(this.val(i));\nthis.#dfs(this.left(i), order, res);\n// \u4e2d\u5e8f\u904d\u5386\nif (order === 'in') res.push(this.val(i));\nthis.#dfs(this.right(i), order, res);\n// \u540e\u5e8f\u904d\u5386\nif (order === 'post') res.push(this.val(i));\n}\n/* \u524d\u5e8f\u904d\u5386 */\npreOrder() {\nconst res = [];\nthis.#dfs(0, 'pre', res);\nreturn res;\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\ninOrder() {\nconst res = [];\nthis.#dfs(0, 'in', res);\nreturn res;\n}\n/* \u540e\u5e8f\u904d\u5386 */\npostOrder() {\nconst res = [];\nthis.#dfs(0, 'post', res);\nreturn res;\n}\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n#tree: (number | null)[];\n/* \u6784\u9020\u65b9\u6cd5 */\nconstructor(arr: (number | null)[]) {\nthis.#tree = arr;\n}\n/* \u8282\u70b9\u6570\u91cf */\nsize(): number {\nreturn this.#tree.length;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nval(i: number): number | null {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif (i &lt; 0 || i &gt;= this.size()) return null;\nreturn this.#tree[i];\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nleft(i: number): number {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nright(i: number): number {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nparent(i: number): number {\nreturn Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nlevelOrder(): number[] {\nlet res = [];\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (let i = 0; i &lt; this.size(); i++) {\nif (this.val(i) !== null) res.push(this.val(i));\n}\nreturn res;\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n#dfs(i: number, order: Order, res: (number | null)[]): void {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif (this.val(i) === null) return;\n// \u524d\u5e8f\u904d\u5386\nif (order === 'pre') res.push(this.val(i));\nthis.#dfs(this.left(i), order, res);\n// \u4e2d\u5e8f\u904d\u5386\nif (order === 'in') res.push(this.val(i));\nthis.#dfs(this.right(i), order, res);\n// \u540e\u5e8f\u904d\u5386\nif (order === 'post') res.push(this.val(i));\n}\n/* \u524d\u5e8f\u904d\u5386 */\npreOrder(): (number | null)[] {\nconst res = [];\nthis.#dfs(0, 'pre', res);\nreturn res;\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\ninOrder(): (number | null)[] {\nconst res = [];\nthis.#dfs(0, 'in', res);\nreturn res;\n}\n/* \u540e\u5e8f\u904d\u5386 */\npostOrder(): (number | null)[] {\nconst res = [];\nthis.#dfs(0, 'post', res);\nreturn res;\n}\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\nlate List&lt;int?&gt; _tree;\n/* \u6784\u9020\u65b9\u6cd5 */\nArrayBinaryTree(this._tree);\n/* \u8282\u70b9\u6570\u91cf */\nint size() {\nreturn _tree.length;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint? val(int i) {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif (i &lt; 0 || i &gt;= size()) {\nreturn null;\n}\nreturn _tree[i];\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint? left(int i) {\nreturn 2 * i + 1;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nint? right(int i) {\nreturn 2 * i + 2;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nint? parent(int i) {\nreturn (i - 1) ~/ 2;\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; levelOrder() {\nList&lt;int&gt; res = [];\nfor (int i = 0; i &lt; size(); i++) {\nif (val(i) != null) {\nres.add(val(i)!);\n}\n}\nreturn res;\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(int i, String order, List&lt;int?&gt; res) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif (val(i) == null) {\nreturn;\n}\n// \u524d\u5e8f\u904d\u5386\nif (order == 'pre') {\nres.add(val(i));\n}\ndfs(left(i)!, order, res);\n// \u4e2d\u5e8f\u904d\u5386\nif (order == 'in') {\nres.add(val(i));\n}\ndfs(right(i)!, order, res);\n// \u540e\u5e8f\u904d\u5386\nif (order == 'post') {\nres.add(val(i));\n}\n}\n/* \u524d\u5e8f\u904d\u5386 */\nList&lt;int?&gt; preOrder() {\nList&lt;int?&gt; res = [];\ndfs(0, 'pre', res);\nreturn res;\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nList&lt;int?&gt; inOrder() {\nList&lt;int?&gt; res = [];\ndfs(0, 'in', res);\nreturn res;\n}\n/* \u540e\u5e8f\u904d\u5386 */\nList&lt;int?&gt; postOrder() {\nList&lt;int?&gt; res = [];\ndfs(0, 'post', res);\nreturn res;\n}\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\ntree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\nimpl ArrayBinaryTree {\n/* \u6784\u9020\u65b9\u6cd5 */\nfn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\nSelf { tree: arr }\n}\n/* \u8282\u70b9\u6570\u91cf */\nfn size(&amp;self) -&gt; i32 {\nself.tree.len() as i32\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif i &lt; 0 || i &gt;= self.size() {\nNone\n} else {\nself.tree[i as usize]\n}\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn left(&amp;self, i: i32) -&gt; i32 {\n2 * i + 1\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfn right(&amp;self, i: i32) -&gt; i32 {\n2 * i + 2\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfn parent(&amp;self, i: i32) -&gt; i32 {\n(i - 1) / 2\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\nlet mut res = vec![];\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor i in 0..self.size() {\nif let Some(val) = self.val(i) {\nres.push(val)\n}\n}\nres\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfn dfs(&amp;self, i: i32, order: &amp;str, res: &amp;mut Vec&lt;i32&gt;) {\nif self.val(i).is_none() {\nreturn;\n}\nlet val = self.val(i).unwrap();\n// \u524d\u5e8f\u904d\u5386\nif order == \"pre\" {\nres.push(val);\n}\nself.dfs(self.left(i), order, res);\n// \u4e2d\u5e8f\u904d\u5386\nif order == \"in\" {\nres.push(val);\n}\nself.dfs(self.right(i), order, res);\n// \u540e\u5e8f\u904d\u5386\nif order == \"post\" {\nres.push(val);\n}\n}\n/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\nlet mut res = vec![];\nself.dfs(0, \"pre\", &amp;mut res);\nres\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\nlet mut res = vec![];\nself.dfs(0, \"in\", &amp;mut res);\nres\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\nlet mut res = vec![];\nself.dfs(0, \"post\", &amp;mut res);\nres\n}\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\nint *tree;\nint size;\n} ArrayBinaryTree;\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\nArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\nabt-&gt;tree = malloc(sizeof(int) * arrSize);\nmemcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\nabt-&gt;size = arrSize;\nreturn abt;\n}\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\nfree(abt-&gt;tree);\nfree(abt);\n}\n/* \u8282\u70b9\u6570\u91cf */\nint size(ArrayBinaryTree *abt) {\nreturn abt-&gt;size;\n}\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n// \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\nif (i &lt; 0 || i &gt;= size(abt))\nreturn INT_MAX;\nreturn abt-&gt;tree[i];\n}\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\nint *res = (int *)malloc(sizeof(int) * size(abt));\nint index = 0;\n// \u76f4\u63a5\u904d\u5386\u6570\u7ec4\nfor (int i = 0; i &lt; size(abt); i++) {\nif (val(abt, i) != INT_MAX)\nres[index++] = val(abt, i);\n}\n*returnSize = index;\nreturn res;\n}\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n// \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\nif (val(abt, i) == INT_MAX)\nreturn;\n// \u524d\u5e8f\u904d\u5386\nif (strcmp(order, \"pre\") == 0)\nres[(*index)++] = val(abt, i);\ndfs(abt, left(i), order, res, index);\n// \u4e2d\u5e8f\u904d\u5386\nif (strcmp(order, \"in\") == 0)\nres[(*index)++] = val(abt, i);\ndfs(abt, right(i), order, res, index);\n// \u540e\u5e8f\u904d\u5386\nif (strcmp(order, \"post\") == 0)\nres[(*index)++] = val(abt, i);\n}\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\nint *res = (int *)malloc(sizeof(int) * size(abt));\nint index = 0;\ndfs(abt, 0, \"pre\", res, &amp;index);\n*returnSize = index;\nreturn res;\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\nint *res = (int *)malloc(sizeof(int) * size(abt));\nint index = 0;\ndfs(abt, 0, \"in\", res, &amp;index);\n*returnSize = index;\nreturn res;\n}\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\nint *res = (int *)malloc(sizeof(int) * size(abt));\nint index = 0;\ndfs(abt, 0, \"post\", res, &amp;index);\n*returnSize = index;\nreturn res;\n}\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre>"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\u3002</li> <li>\u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\u3002</li> <li>\u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\u3002</li> </ul> <p>\u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002</li> <li>\u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf \\(\\text{None}\\) \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL \u6811 *","text":"<p>\u5728\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u6076\u5316\u4e3a \\(O(n)\\) \u3002</p> <p>\u5982\u56fe 7-24 \u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002</p> <p></p> <p> \u56fe 7-24 \u00a0 AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>\u518d\u4f8b\u5982\uff0c\u5728\u56fe 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u6076\u5316\u3002</p> <p></p> <p> \u56fe 7-25 \u00a0 AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002</p>"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"<p>AVL \u6811\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u4e5f\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree\u300d\u3002</p>"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u8282\u70b9\u9ad8\u5ea6","text":"<p>\u7531\u4e8e AVL \u6811\u7684\u76f8\u5173\u64cd\u4f5c\u9700\u8981\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e3a\u8282\u70b9\u7c7b\u6dfb\u52a0 <code>height</code> \u53d8\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class TreeNode:\n\"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                 # \u8282\u70b9\u503c\nself.height: int = 0                # \u8282\u70b9\u9ad8\u5ea6\nself.left: TreeNode | None = None   # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nself.right: TreeNode | None = None  # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\nint val{};          // \u8282\u70b9\u503c\nint height = 0;     // \u8282\u70b9\u9ad8\u5ea6\nTreeNode *left{};   // \u5de6\u5b50\u8282\u70b9\nTreeNode *right{};  // \u53f3\u5b50\u8282\u70b9\nTreeNode() = default;\nexplicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\npublic int val;        // \u8282\u70b9\u503c\npublic int height;     // \u8282\u70b9\u9ad8\u5ea6\npublic TreeNode left;  // \u5de6\u5b50\u8282\u70b9\npublic TreeNode right; // \u53f3\u5b50\u8282\u70b9\npublic TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\npublic int? val = x;    // \u8282\u70b9\u503c\npublic int height;      // \u8282\u70b9\u9ad8\u5ea6\npublic TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\npublic TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal    int       // \u8282\u70b9\u503c\nHeight int       // \u8282\u70b9\u9ad8\u5ea6\nLeft   *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nRight  *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u8282\u70b9\u503c\nvar height: Int // \u8282\u70b9\u9ad8\u5ea6\nvar left: TreeNode? // \u5de6\u5b50\u8282\u70b9\nvar right: TreeNode? // \u53f3\u5b50\u8282\u70b9\ninit(x: Int) {\nval = x\nheight = 0\n}\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval; // \u8282\u70b9\u503c\nheight; //\u8282\u70b9\u9ad8\u5ea6\nleft; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nright; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nconstructor(val, left, right, height) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height;\nthis.left = left === undefined ? null : left;\nthis.right = right === undefined ? null : right;\n}\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval: number;            // \u8282\u70b9\u503c\nheight: number;         // \u8282\u70b9\u9ad8\u5ea6\nleft: TreeNode | null;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nright: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nconstructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val;\nthis.height = height === undefined ? 0 : height; this.left = left === undefined ? null : left; this.right = right === undefined ? null : right; }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nint height;      // \u8282\u70b9\u9ad8\u5ea6\nTreeNode? left;  // \u5de6\u5b50\u8282\u70b9\nTreeNode? right; // \u53f3\u5b50\u8282\u70b9\nTreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nval: i32,                               // \u8282\u70b9\u503c\nheight: i32,                            // \u8282\u70b9\u9ad8\u5ea6\nleft: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\nright: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\n}\nimpl TreeNode {\n/* \u6784\u9020\u65b9\u6cd5 */\nfn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\nRc::new(RefCell::new(Self {\nval,\nheight: 0,\nleft: None,\nright: None\n}))\n}\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nTreeNode struct TreeNode {\nint val;\nint height;\nstruct TreeNode *left;\nstruct TreeNode *right;\n} TreeNode;\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\nTreeNode *node;\nnode = (TreeNode *)malloc(sizeof(TreeNode));\nnode-&gt;val = val;\nnode-&gt;height = 0;\nnode-&gt;left = NULL;\nnode-&gt;right = NULL;\nreturn node;\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u5176\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(0\\) \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(-1\\) \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n\"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n# \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif node is not None:\nreturn node.height\nreturn -1\ndef update_height(self, node: TreeNode | None):\n\"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n# \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == nullptr ? -1 : node-&gt;height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif node != nil {\nreturn node.Height\n}\nreturn -1\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\nlh := t.height(node.Left)\nrh := t.height(node.Right)\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif lh &gt; rh {\nnode.Height = lh + 1\n} else {\nnode.Height = rh + 1\n}\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nnode == nil ? -1 : node!.height\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height =\nMath.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node === null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.height =\nMath.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn node == null ? -1 : node.height;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nmatch node {\nSome(node) =&gt; node.borrow().height,\nNone =&gt; -1,\n}\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\nif let Some(node) = node {\nlet left = node.borrow().left.clone();\nlet right = node.borrow().right.clone();\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n}\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nif (node != NULL) {\nreturn node-&gt;height;\n}\nreturn -1;\n}\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\nint lh = height(node-&gt;left);\nint rh = height(node-&gt;right);\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nif (lh &gt; rh) {\nnode-&gt;height = lh + 1;\n} else {\nnode-&gt;height = rh + 1;\n}\n}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n_ = self;\n// \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\nreturn if (node == null) -1 else node.?.height;\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n// \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\nnode.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"<p>\u8282\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 balance factor\u300d\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a \\(0\\) \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n\"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n# \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node is None:\nreturn 0\n# \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == nullptr)\nreturn 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null)\nreturn 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif node == nil {\nreturn 0\n}\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nguard let node = node else { return 0 }\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node === null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\nmatch node {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nNone =&gt; 0,\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nSome(node) =&gt; {\nSelf::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n}\n}\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == NULL) {\nreturn 0;\n}\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n// \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\nif (node == null) return 0;\n// \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\nreturn self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Note</p> <p>\u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002</p>"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6811\u65cb\u8f6c","text":"<p>AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u201c\u65cb\u8f6c\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7684\u6027\u8d28\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u201c\u5e73\u8861\u4e8c\u53c9\u6811\u201d\u3002</p> <p>\u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(&gt; 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5931\u8861\u8282\u70b9\u201d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"<p>\u5982\u56fe 7-26 \u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a <code>node</code> \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3a <code>child</code> \uff0c\u6267\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9aa4 </p> <p>\u5982\u56fe 7-27 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grandChild</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grandChild</code> \u4f5c\u4e3a <code>node</code> \u7684\u5de6\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-27 \u00a0 \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c </p> <p>\u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n\"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\nchild = node.left\ngrand_child = child.right\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node\nnode.left = grand_child\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.update_height(node)\nself.update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\nTreeNode *child = node-&gt;left;\nTreeNode *grandChild = child-&gt;right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild-&gt;right = node;\nnode-&gt;left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\nTreeNode child = node.left;\nTreeNode grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\nTreeNode? child = node?.left;\nTreeNode? grandChild = child?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nUpdateHeight(node);\nUpdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\nchild := node.Left\ngrandChild := child.Right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.Right = node\nnode.Left = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\nlet child = node?.left\nlet grandChild = child?.right\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild?.right = node\nnode?.left = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\nconst child = node.left;\nconst grandChild = child.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\nTreeNode? child = node!.left;\nTreeNode? grandChild = child!.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.right = node;\nnode.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\nmatch node {\nSome(node) =&gt; {\nlet child = node.borrow().left.clone().unwrap();\nlet grand_child = child.borrow().right.clone();\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.borrow_mut().right = Some(node.clone());\nnode.borrow_mut().left = grand_child;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nSelf::update_height(Some(node));\nSelf::update_height(Some(child.clone()));\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nSome(child)\n}\nNone =&gt; None,\n}\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\nTreeNode *child, *grandChild;\nchild = node-&gt;left;\ngrandChild = child-&gt;right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild-&gt;right = node;\nnode-&gt;left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.left;\nvar grandChild = child.?.right;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\nchild.?.right = node;\nnode.?.left = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"<p>\u76f8\u5e94\u5730\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u56fe 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c </p> <p>\u540c\u7406\uff0c\u5982\u56fe 7-29 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grandChild</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grandChild</code> \u4f5c\u4e3a <code>node</code> \u7684\u53f3\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-29 \u00a0 \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c </p> <p>\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u53f3\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\u4e2d\u7684\u6240\u6709\u7684 <code>left</code> \u66ff\u6362\u4e3a <code>right</code> \uff0c\u5c06\u6240\u6709\u7684 <code>right</code> \u66ff\u6362\u4e3a <code>left</code> \uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n\"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\nchild = node.right\ngrand_child = child.left\n# \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node\nnode.right = grand_child\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.update_height(node)\nself.update_height(child)\n# \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\nTreeNode *child = node-&gt;right;\nTreeNode *grandChild = child-&gt;left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild-&gt;left = node;\nnode-&gt;right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\nTreeNode child = node.right;\nTreeNode grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\nTreeNode? child = node?.right;\nTreeNode? grandChild = child?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nUpdateHeight(node);\nUpdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\nchild := node.Right\ngrandChild := child.Left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.Left = node\nnode.Right = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\nt.updateHeight(child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\nlet child = node?.right\nlet grandChild = child?.left\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild?.left = node\nnode?.right = grandChild\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node: node)\nupdateHeight(node: child)\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.#updateHeight(node);\nthis.#updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\nconst child = node.right;\nconst grandChild = child.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nthis.updateHeight(node);\nthis.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\nTreeNode? child = node!.right;\nTreeNode? grandChild = child!.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.left = node;\nnode.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\nmatch node {\nSome(node) =&gt; {\nlet child = node.borrow().right.clone().unwrap();\nlet grand_child = child.borrow().left.clone();\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.borrow_mut().left = Some(node.clone());\nnode.borrow_mut().right = grand_child;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nSelf::update_height(Some(node));\nSelf::update_height(Some(child.clone()));\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nSome(child)\n}\nNone =&gt; None,\n}\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\nTreeNode *child, *grandChild;\nchild = node-&gt;right;\ngrandChild = child-&gt;left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild-&gt;left = node;\nnode-&gt;right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\nupdateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\nvar child = node.?.right;\nvar grandChild = child.?.left;\n// \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\nchild.?.left = node;\nnode.?.right = grandChild;\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.updateHeight(node);\nself.updateHeight(child);\n// \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"<p>\u5bf9\u4e8e\u56fe 7-30 \u4e2d\u7684\u5931\u8861\u8282\u70b9 3 \uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-30 \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb </p>"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"<p>\u5982\u56fe 7-31 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-31 \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb </p>"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f6c\u7684\u9009\u62e9","text":"<p>\u56fe 7-32 \u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-32 \u00a0 AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5 </p> <p>\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u56fe 7-32 \u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002</p> <p> \u8868 7-3 \u00a0 \u56db\u79cd\u65cb\u8f6c\u60c5\u51b5\u7684\u9009\u62e9\u6761\u4ef6 </p> \u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(&lt;0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(&gt;0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb <p>\u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n\"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n# \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nbalance_factor = self.balance_factor(node)\n# \u5de6\u504f\u6811\nif balance_factor &gt; 1:\nif self.balance_factor(node.left) &gt;= 0:\n# \u53f3\u65cb\nreturn self.right_rotate(node)\nelse:\n# \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = self.left_rotate(node.left)\nreturn self.right_rotate(node)\n# \u53f3\u504f\u6811\nelif balance_factor &lt; -1:\nif self.balance_factor(node.right) &lt;= 0:\n# \u5de6\u65cb\nreturn self.left_rotate(node)\nelse:\n# \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = self.right_rotate(node.right)\nreturn self.left_rotate(node)\n# \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n</code></pre> avl_tree.cpp<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint _balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (_balanceFactor &gt; 1) {\nif (balanceFactor(node-&gt;left) &gt;= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode-&gt;left = leftRotate(node-&gt;left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (_balanceFactor &lt; -1) {\nif (balanceFactor(node-&gt;right) &lt;= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode-&gt;right = rightRotate(node-&gt;right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor &gt; 1) {\nif (balanceFactor(node.left) &gt;= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor &lt; -1) {\nif (balanceFactor(node.right) &lt;= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint balanceFactorInt = BalanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactorInt &gt; 1) {\nif (BalanceFactor(node?.left) &gt;= 0) {\n// \u53f3\u65cb\nreturn RightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode!.left = LeftRotate(node!.left);\nreturn RightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactorInt &lt; -1) {\nif (BalanceFactor(node?.right) &lt;= 0) {\n// \u5de6\u65cb\nreturn LeftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode!.right = RightRotate(node!.right);\nreturn LeftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n// Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\nbf := t.balanceFactor(node)\n// \u5de6\u504f\u6811\nif bf &gt; 1 {\nif t.balanceFactor(node.Left) &gt;= 0 {\n// \u53f3\u65cb\nreturn t.rightRotate(node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.Left = t.leftRotate(node.Left)\nreturn t.rightRotate(node)\n}\n}\n// \u53f3\u504f\u6811\nif bf &lt; -1 {\nif t.balanceFactor(node.Right) &lt;= 0 {\n// \u5de6\u65cb\nreturn t.leftRotate(node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.Right = t.rightRotate(node.Right)\nreturn t.leftRotate(node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nlet balanceFactor = balanceFactor(node: node)\n// \u5de6\u504f\u6811\nif balanceFactor &gt; 1 {\nif self.balanceFactor(node: node?.left) &gt;= 0 {\n// \u53f3\u65cb\nreturn rightRotate(node: node)\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode?.left = leftRotate(node: node?.left)\nreturn rightRotate(node: node)\n}\n}\n// \u53f3\u504f\u6811\nif balanceFactor &lt; -1 {\nif self.balanceFactor(node: node?.right) &lt;= 0 {\n// \u5de6\u65cb\nreturn leftRotate(node: node)\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode?.right = rightRotate(node: node?.right)\nreturn leftRotate(node: node)\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n</code></pre> avl_tree.js<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor &gt; 1) {\nif (this.balanceFactor(node.left) &gt;= 0) {\n// \u53f3\u65cb\nreturn this.#rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.#leftRotate(node.left);\nreturn this.#rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor &lt; -1) {\nif (this.balanceFactor(node.right) &lt;= 0) {\n// \u5de6\u65cb\nreturn this.#leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.#rightRotate(node.right);\nreturn this.#leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nconst balanceFactor = this.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balanceFactor &gt; 1) {\nif (this.balanceFactor(node.left) &gt;= 0) {\n// \u53f3\u65cb\nreturn this.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = this.leftRotate(node.left);\nreturn this.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balanceFactor &lt; -1) {\nif (this.balanceFactor(node.right) &lt;= 0) {\n// \u5de6\u65cb\nreturn this.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = this.rightRotate(node.right);\nreturn this.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint factor = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (factor &gt; 1) {\nif (balanceFactor(node!.left) &gt;= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.left = leftRotate(node.left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (factor &lt; -1) {\nif (balanceFactor(node!.right) &lt;= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.right = rightRotate(node.right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nlet balance_factor = Self::balance_factor(node.clone());\n// \u5de6\u504f\u6811\nif balance_factor &gt; 1 {\nlet node = node.unwrap();\nif Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n// \u53f3\u65cb\nSelf::right_rotate(Some(node))\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nlet left = node.borrow().left.clone();\nnode.borrow_mut().left = Self::left_rotate(left);\nSelf::right_rotate(Some(node))\n}\n}\n// \u53f3\u504f\u6811\nelse if balance_factor &lt; -1 {\nlet node = node.unwrap();\nif Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n// \u5de6\u65cb\nSelf::left_rotate(Some(node))\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nlet right = node.borrow().right.clone();\nnode.borrow_mut().right = Self::right_rotate(right);\nSelf::left_rotate(Some(node))\n}\n} else {\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nnode\n}\n}\n</code></pre> avl_tree.c<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nint bf = balanceFactor(node);\n// \u5de6\u504f\u6811\nif (bf &gt; 1) {\nif (balanceFactor(node-&gt;left) &gt;= 0) {\n// \u53f3\u65cb\nreturn rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode-&gt;left = leftRotate(node-&gt;left);\nreturn rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (bf &lt; -1) {\nif (balanceFactor(node-&gt;right) &lt;= 0) {\n// \u5de6\u65cb\nreturn leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode-&gt;right = rightRotate(node-&gt;right);\nreturn leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre> avl_tree.zig<pre><code>// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n// \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\nvar balance_factor = self.balanceFactor(node);\n// \u5de6\u504f\u6811\nif (balance_factor &gt; 1) {\nif (self.balanceFactor(node.?.left) &gt;= 0) {\n// \u53f3\u65cb\nreturn self.rightRotate(node);\n} else {\n// \u5148\u5de6\u65cb\u540e\u53f3\u65cb\nnode.?.left = self.leftRotate(node.?.left);\nreturn self.rightRotate(node);\n}\n}\n// \u53f3\u504f\u6811\nif (balance_factor &lt; -1) {\nif (self.balanceFactor(node.?.right) &lt;= 0) {\n// \u5de6\u65cb\nreturn self.leftRotate(node);\n} else {\n// \u5148\u53f3\u65cb\u540e\u5de6\u65cb\nnode.?.right = self.rightRotate(node.?.right);\nreturn self.leftRotate(node);\n}\n}\n// \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def insert(self, val):\n\"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\nself._root = self.insert_helper(self._root, val)\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n\"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn TreeNode(val)\n# 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\nif val &lt; node.val:\nnode.left = self.insert_helper(node.left, val)\nelif val &gt; node.val:\nnode.right = self.insert_helper(node.right, val)\nelse:\n# \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\nif (node == nullptr)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val &lt; node-&gt;val)\nnode-&gt;left = insertHelper(node-&gt;left, val);\nelse if (val &gt; node-&gt;val)\nnode-&gt;right = insertHelper(node-&gt;right, val);\nelse\nreturn node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\nif (node == null)\nreturn new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val &lt; node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\nroot = InsertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\nif (node == null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val &lt; node.val)\nnode.left = InsertHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = InsertHelper(node.right, val);\nelse\nreturn node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nUpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = Rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\nt.root = t.insertHelper(t.root, val)\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn NewTreeNode(val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif val &lt; node.Val.(int) {\nnode.Left = t.insertHelper(node.Left, val)\n} else if val &gt; node.Val.(int) {\nnode.Right = t.insertHelper(node.Right, val)\n} else {\n// \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\nroot = insertHelper(node: root, val: val)\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\nvar node = node\nif node == nil {\nreturn TreeNode(x: val)\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif val &lt; node!.val {\nnode?.left = insertHelper(node: node?.left, val: val)\n} else if val &gt; node!.val {\nnode?.right = insertHelper(node: node?.right, val: val)\n} else {\nreturn node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nupdateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\nthis.root = this.#insertHelper(this.root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = this.#insertHelper(node.right, val);\nelse return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nthis.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\nthis.root = this.insertHelper(this.root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return new TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val &lt; node.val) {\nnode.left = this.insertHelper(node.left, val);\n} else if (val &gt; node.val) {\nnode.right = this.insertHelper(node.right, val);\n} else {\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nthis.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\nroot = insertHelper(root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\nif (node == null) return TreeNode(val);\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val &lt; node.val)\nnode.left = insertHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = insertHelper(node.right, val);\nelse\nreturn node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&amp;mut self, val: i32) {\nself.root = Self::insert_helper(self.root.clone(), val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\nmatch node {\nSome(mut node) =&gt; {\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nmatch {\nlet node_val = node.borrow().val;\nnode_val\n}\n.cmp(&amp;val)\n{\nOrdering::Greater =&gt; {\nlet left = node.borrow().left.clone();\nnode.borrow_mut().left = Self::insert_helper(left, val);\n}\nOrdering::Less =&gt; {\nlet right = node.borrow().right.clone();\nnode.borrow_mut().right = Self::insert_helper(right, val);\n}\nOrdering::Equal =&gt; {\nreturn Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\n}\nSelf::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = Self::rotate(Some(node)).unwrap();\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nSome(node)\n}\nNone =&gt; Some(TreeNode::new(val)),\n}\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\ntree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\nif (node == NULL) {\nreturn newTreeNode(val);\n}\n/* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\nif (val &lt; node-&gt;val) {\nnode-&gt;left = insertHelper(node-&gt;left, val);\n} else if (val &gt; node-&gt;val) {\nnode-&gt;right = insertHelper(node-&gt;right, val);\n} else {\n// \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\nreturn node;\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\nself.root = (try self.insertHelper(self.root, val)).?;\n}\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) {\nvar tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\ntmp_node.init(val);\nreturn tmp_node;\n}\n// 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\nif (val &lt; node.?.val) {\nnode.?.left = try self.insertHelper(node.?.left, val);\n} else if (val &gt; node.?.val) {\nnode.?.right = try self.insertHelper(node.?.right, val);\n} else {\nreturn node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n}\nself.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def remove(self, val: int):\n\"\"\"\u5220\u9664\u8282\u70b9\"\"\"\nself._root = self.remove_helper(self._root, val)\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n\"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\nif node is None:\nreturn None\n# 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif val &lt; node.val:\nnode.left = self.remove_helper(node.left, val)\nelif val &gt; node.val:\nnode.right = self.remove_helper(node.right, val)\nelse:\nif node.left is None or node.right is None:\nchild = node.left or node.right\n# \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child is None:\nreturn None\n# \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse:\nnode = child\nelse:\n# \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\ntemp = node.right\nwhile temp.left is not None:\ntemp = temp.left\nnode.right = self.remove_helper(node.right, temp.val)\nnode.val = temp.val\n# \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nself.update_height(node)\n# 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nreturn self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\nif (node == nullptr)\nreturn nullptr;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val &lt; node-&gt;val)\nnode-&gt;left = removeHelper(node-&gt;left, val);\nelse if (val &gt; node-&gt;val)\nnode-&gt;right = removeHelper(node-&gt;right, val);\nelse {\nif (node-&gt;left == nullptr || node-&gt;right == nullptr) {\nTreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == nullptr) {\ndelete node;\nreturn nullptr;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\ndelete node;\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode *temp = node-&gt;right;\nwhile (temp-&gt;left != nullptr) {\ntemp = temp-&gt;left;\n}\nint tempVal = temp-&gt;val;\nnode-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\nnode-&gt;val = tempVal;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\nif (node == null)\nreturn null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val &lt; node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode child = node.left != null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode temp = node.right;\nwhile (temp.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\nroot = RemoveHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val &lt; node.val)\nnode.left = RemoveHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = RemoveHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode? child = node.left ?? node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode? temp = node.right;\nwhile (temp.left != null) {\ntemp = temp.left;\n}\nnode.right = RemoveHelper(node.right, temp.val!.Value);\nnode.val = temp.val;\n}\n}\nUpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = Rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\nt.root = t.removeHelper(t.root, val)\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val &lt; node.Val.(int) {\nnode.Left = t.removeHelper(node.Left, val)\n} else if val &gt; node.Val.(int) {\nnode.Right = t.removeHelper(node.Right, val)\n} else {\nif node.Left == nil || node.Right == nil {\nchild := node.Left\nif node.Right != nil {\nchild = node.Right\n}\nif child == nil {\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nreturn nil\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\ntemp := node.Right\nfor temp.Left != nil {\ntemp = temp.Left\n}\nnode.Right = t.removeHelper(node.Right, temp.Val.(int))\nnode.Val = temp.Val\n}\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nt.updateHeight(node)\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = t.rotate(node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\nroot = removeHelper(node: root, val: val)\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\nvar node = node\nif node == nil {\nreturn nil\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val &lt; node!.val {\nnode?.left = removeHelper(node: node?.left, val: val)\n} else if val &gt; node!.val {\nnode?.right = removeHelper(node: node?.right, val: val)\n} else {\nif node?.left == nil || node?.right == nil {\nlet child = node?.left != nil ? node?.left : node?.right\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif child == nil {\nreturn nil\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse {\nnode = child\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nvar temp = node?.right\nwhile temp?.left != nil {\ntemp = temp?.left\n}\nnode?.right = removeHelper(node: node?.right, val: temp!.val)\nnode?.val = temp!.val\n}\n}\nupdateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node: node)\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node\n}\n</code></pre> avl_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\nthis.root = this.#removeHelper(this.root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = this.#removeHelper(node.right, val);\nelse {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) return null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse node = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nlet temp = node.right;\nwhile (temp.left !== null) {\ntemp = temp.left;\n}\nnode.right = this.#removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.#rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\nthis.root = this.removeHelper(this.root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\nif (node === null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val &lt; node.val) {\nnode.left = this.removeHelper(node.left, val);\n} else if (val &gt; node.val) {\nnode.right = this.removeHelper(node.right, val);\n} else {\nif (node.left === null || node.right === null) {\nconst child = node.left !== null ? node.left : node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child === null) {\nreturn null;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nlet temp = node.right;\nwhile (temp.left !== null) {\ntemp = temp.left;\n}\nnode.right = this.removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nthis.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = this.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\nroot = removeHelper(root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\nif (node == null) return null;\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val &lt; node.val)\nnode.left = removeHelper(node.left, val);\nelse if (val &gt; node.val)\nnode.right = removeHelper(node.right, val);\nelse {\nif (node.left == null || node.right == null) {\nTreeNode? child = node.left ?? node.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null)\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nelse\nnode = child;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode? temp = node.right;\nwhile (temp!.left != null) {\ntemp = temp.left;\n}\nnode.right = removeHelper(node.right, temp.val);\nnode.val = temp.val;\n}\n}\nupdateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfn remove(&amp;self, val: i32) {\nSelf::remove_helper(self.root.clone(), val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\nmatch node {\nSome(mut node) =&gt; {\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif val &lt; node.borrow().val {\nlet left = node.borrow().left.clone();\nnode.borrow_mut().left = Self::remove_helper(left, val);\n} else if val &gt; node.borrow().val {\nlet right = node.borrow().right.clone();\nnode.borrow_mut().right = Self::remove_helper(right, val);\n} else if node.borrow().left.is_none() || node.borrow().right.is_none() {\nlet child = if node.borrow().left.is_some() {\nnode.borrow().left.clone()\n} else {\nnode.borrow().right.clone()\n};\nmatch child {\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nNone =&gt; {\nreturn None;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nSome(child) =&gt; node = child,\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nlet mut temp = node.borrow().right.clone().unwrap();\nloop {\nlet temp_left = temp.borrow().left.clone();\nif temp_left.is_none() {\nbreak;\n}\ntemp = temp_left.unwrap();\n}\nlet right = node.borrow().right.clone();\nnode.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\nnode.borrow_mut().val = temp.borrow().val;\n}\nSelf::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = Self::rotate(Some(node)).unwrap();\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nSome(node)\n}\nNone =&gt; None,\n}\n}\n</code></pre> avl_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\nTreeNode *root = removeHelper(tree-&gt;root, val);\n}\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\nTreeNode *child, *grandChild;\nif (node == NULL) {\nreturn NULL;\n}\n/* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\nif (val &lt; node-&gt;val) {\nnode-&gt;left = removeHelper(node-&gt;left, val);\n} else if (val &gt; node-&gt;val) {\nnode-&gt;right = removeHelper(node-&gt;right, val);\n} else {\nif (node-&gt;left == NULL || node-&gt;right == NULL) {\nchild = node-&gt;left;\nif (node-&gt;right != NULL) {\nchild = node-&gt;right;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == NULL) {\nreturn NULL;\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nTreeNode *temp = node-&gt;right;\nwhile (temp-&gt;left != NULL) {\ntemp = temp-&gt;left;\n}\nint tempVal = temp-&gt;val;\nnode-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\nnode-&gt;val = tempVal;\n}\n}\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nupdateHeight(node);\n/* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nnode = rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre> avl_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\nself.root = self.removeHelper(self.root, val).?;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\nvar node = node_;\nif (node == null) return null;\n// 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\nif (val &lt; node.?.val) {\nnode.?.left = self.removeHelper(node.?.left, val);\n} else if (val &gt; node.?.val) {\nnode.?.right = self.removeHelper(node.?.right, val);\n} else {\nif (node.?.left == null or node.?.right == null) {\nvar child = if (node.?.left != null) node.?.left else node.?.right;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\nif (child == null) {\nreturn null;\n// \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n} else {\nnode = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\nvar temp = node.?.right;\nwhile (temp.?.left != null) {\ntemp = temp.?.left;\n}\nnode.?.right = self.removeHelper(node.?.right, temp.?.val);\nnode.?.val = temp.?.val;\n}\n}\nself.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n// 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nnode = self.rotate(node);\n// \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\nreturn node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\u3002</li> <li>\u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\u3002</li> <li>\u7ea2\u9ed1\u6811\u5728\u8bb8\u591a\u5e94\u7528\u4e2d\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\u3002\u8fd9\u662f\u56e0\u4e3a\u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5176\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"<p>\u5982\u56fe 7-16 \u6240\u793a\uff0c\u300c\u4e8c\u53c9\u641c\u7d22\u6811 binary search tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u3002</p> <ol> <li>\u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(&lt;\\) \u6839\u8282\u70b9\u7684\u503c \\(&lt;\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\u3002</li> <li>\u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6 <code>1.</code> \u3002</li> </ol> <p></p> <p> \u56fe 7-16 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811 </p>"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":"<p>\u6211\u4eec\u5c06\u4e8c\u53c9\u641c\u7d22\u6811\u5c01\u88c5\u4e3a\u4e00\u4e2a\u7c7b <code>BinarySearchTree</code> \uff0c\u5e76\u58f0\u660e\u4e00\u4e2a\u6210\u5458\u53d8\u91cf <code>root</code> \uff0c\u6307\u5411\u6811\u7684\u6839\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c <code>num</code> \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u5982\u56fe 7-17 \u6240\u793a\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9 <code>cur</code> \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9 <code>root</code> \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503c <code>cur.val</code> \u548c <code>num</code> \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <ul> <li>\u82e5 <code>cur.val &lt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.right</code> \u3002</li> <li>\u82e5 <code>cur.val &gt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.left</code> \u3002</li> <li>\u82e5 <code>cur.val = num</code> \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\u3002</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-17 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u8282\u70b9\u793a\u4f8b </p> <p>\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n\"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\ncur = self._root\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur is not None:\n# \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val &lt; num:\ncur = cur.right\n# \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelif cur.val &gt; num:\ncur = cur.left\n# \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse:\nbreak\nreturn cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\nTreeNode *cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur-&gt;val &lt; num)\ncur = cur-&gt;right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur-&gt;val &gt; num)\ncur = cur-&gt;left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\nTreeNode cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num)\ncur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val &gt; num)\ncur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\nTreeNode? cur = root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num) cur =\ncur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val &gt; num)\ncur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\nnode := bst.root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor node != nil {\nif node.Val.(int) &lt; num {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nnode = node.Right\n} else if node.Val.(int) &gt; num {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nnode = node.Left\n} else {\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -&gt; TreeNode? {\nvar cur = root\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val &lt; num {\ncur = cur?.right\n}\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if cur!.val &gt; num {\ncur = cur?.left\n}\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse {\nbreak\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\nlet cur = this.root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num) cur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val &gt; num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\nlet cur = this.root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num) cur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val &gt; num) cur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse break;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\nTreeNode? cur = _root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; _num)\ncur = cur.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if (cur.val &gt; _num)\ncur = cur.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse\nbreak;\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&amp;self, num: i32) -&gt; Option&lt;TreeNodeRc&gt; {\nlet mut cur = self.root.clone();\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile let Some(node) = cur.clone() {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif node.borrow().val &lt; num {\ncur = node.borrow().right.clone();\n}\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse if node.borrow().val &gt; num {\ncur = node.borrow().left.clone();\n}\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nelse {\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\ncur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\nTreeNode *cur = bst-&gt;root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\nif (cur-&gt;val &lt; num) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur-&gt;right;\n} else if (cur-&gt;val &gt; num) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur-&gt;left;\n} else {\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\nvar cur = self.root;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val &lt; num) {\ncur = cur.?.right;\n// \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else if (cur.?.val &gt; num) {\ncur = cur.?.left;\n// \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n} else {\nbreak;\n}\n}\n// \u8fd4\u56de\u76ee\u6807\u8282\u70b9\nreturn cur;\n}\n</code></pre>"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 <code>num</code> \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-18 \u6240\u793a\u3002</p> <ol> <li>\u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c <code>num</code> \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 \\(\\text{None}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\u3002</li> <li>\u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9 <code>num</code> \uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e \\(\\text{None}\\) \u7684\u4f4d\u7f6e\u3002</li> </ol> <p></p> <p> \u56fe 7-18 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002</li> <li>\u4e3a\u4e86\u5b9e\u73b0\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u8282\u70b9 <code>pre</code> \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\u3002\u8fd9\u6837\u5728\u904d\u5386\u81f3 \\(\\text{None}\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py<pre><code>def insert(self, num: int):\n\"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif self._root is None:\nself._root = TreeNode(num)\nreturn\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\ncur, pre = self._root, None\nwhile cur is not None:\n# \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur.val == num:\nreturn\npre = cur\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val &lt; num:\ncur = cur.right\n# \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u63d2\u5165\u8282\u70b9\nnode = TreeNode(num)\nif pre.val &lt; num:\npre.right = node\nelse:\npre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (root == nullptr) {\nroot = new TreeNode(num);\nreturn;\n}\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur-&gt;val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur-&gt;val &lt; num)\ncur = cur-&gt;right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur-&gt;left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode *node = new TreeNode(num);\nif (pre-&gt;val &lt; num)\npre-&gt;right = node;\nelse\npre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (root == null) {\nroot = new TreeNode(num);\nreturn;\n}\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num)\ncur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode node = new TreeNode(num);\nif (pre.val &lt; num)\npre.right = node;\nelse\npre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (root == null) {\nroot = new TreeNode(num);\nreturn;\n}\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == num)\nreturn;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num)\ncur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode node = new(num);\nif (pre != null) {\nif (pre.val &lt; num)\npre.right = node;\nelse\npre.left = node;\n}\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif cur == nil {\nbst.root = NewTreeNode(num)\nreturn\n}\n// \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nreturn\n}\npre = cur\nif cur.Val.(int) &lt; num {\ncur = cur.Right\n} else {\ncur = cur.Left\n}\n}\n// \u63d2\u5165\u8282\u70b9\nnode := NewTreeNode(num)\nif pre.Val.(int) &lt; num {\npre.Right = node\n} else {\npre.Left = node\n}\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif root == nil {\nroot = TreeNode(x: num)\nreturn\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif cur!.val == num {\nreturn\n}\npre = cur\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val &lt; num {\ncur = cur?.right\n}\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = TreeNode(x: num)\nif pre!.val &lt; num {\npre?.right = node\n} else {\npre?.left = node\n}\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (this.root === null) {\nthis.root = new TreeNode(num);\nreturn;\n}\nlet cur = this.root,\npre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val === num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nconst node = new TreeNode(num);\nif (pre.val &lt; num) pre.right = node;\nelse pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (this.root === null) {\nthis.root = new TreeNode(num);\nreturn;\n}\nlet cur: TreeNode | null = this.root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val === num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num) cur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nconst node = new TreeNode(num);\nif (pre!.val &lt; num) pre!.right = node;\nelse pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (_root == null) {\n_root = TreeNode(_num);\nreturn;\n}\nTreeNode? cur = _root;\nTreeNode? pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.val == _num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; _num)\ncur = cur.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode? node = TreeNode(_num);\nif (pre!.val &lt; _num)\npre.right = node;\nelse\npre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&amp;mut self, num: i32) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif self.root.is_none() {\nself.root = Some(TreeNode::new(num));\nreturn;\n}\nlet mut cur = self.root.clone();\nlet mut pre = None;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile let Some(node) = cur.clone() {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif node.borrow().val == num {\nreturn;\n}\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\npre = cur.clone();\nif node.borrow().val &lt; num {\ncur = node.borrow().right.clone();\n}\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = node.borrow().left.clone();\n}\n}\n// \u63d2\u5165\u8282\u70b9\nlet node = TreeNode::new(num);\nlet pre = pre.unwrap();\nif pre.borrow().val &lt; num {\npre.borrow_mut().right = Some(Rc::clone(&amp;node));\n} else {\npre.borrow_mut().left = Some(Rc::clone(&amp;node));\n}\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (bst-&gt;root == NULL) {\nbst-&gt;root = newTreeNode(num);\nreturn;\n}\nTreeNode *cur = bst-&gt;root, *pre = NULL;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur-&gt;val == num) {\nreturn;\n}\npre = cur;\nif (cur-&gt;val &lt; num) {\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur-&gt;right;\n} else {\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur-&gt;left;\n}\n}\n// \u63d2\u5165\u8282\u70b9\nTreeNode *node = newTreeNode(num);\nif (pre-&gt;val &lt; num) {\npre-&gt;right = node;\n} else {\npre-&gt;left = node;\n}\n}\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\nif (self.root == null) {\nself.root = try self.mem_allocator.create(inc.TreeNode(T));\nreturn;\n}\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur.?.val == num) return;\npre = cur;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val &lt; num) {\ncur = cur.?.right;\n// \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u63d2\u5165\u8282\u70b9\nvar node = try self.mem_allocator.create(inc.TreeNode(T));\nnode.init(num);\nif (pre.?.val &lt; num) {\npre.?.right = node;\n} else {\npre.?.left = node;\n}\n}\n</code></pre> <p>\u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5148\u5728\u4e8c\u53c9\u6811\u4e2d\u67e5\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u518d\u5c06\u5176\u5220\u9664\u3002</p> <p>\u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u4ecd\u7136\u6ee1\u8db3\u3002</p> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u6839\u636e\u76ee\u6807\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u79cd\u60c5\u51b5\uff0c\u6267\u884c\u5bf9\u5e94\u7684\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>\u5982\u56fe 7-19 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u8be5\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002</p> <p></p> <p> \u56fe 7-19 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0 \uff09 </p> <p>\u5982\u56fe 7-20 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 7-20 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1 \uff09 </p> <p>\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002</p> <p>\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-21 \u6240\u793a\u3002</p> <ol> <li>\u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a <code>tmp</code> \u3002</li> <li>\u7528 <code>tmp</code> \u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9 <code>tmp</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-21 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 2 \uff09 </p> <p>\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py<pre><code>def remove(self, num: int):\n\"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n# \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self._root is None:\nreturn\n# \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\ncur, pre = self._root, None\nwhile cur is not None:\n# \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur.val == num:\nbreak\npre = cur\n# \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur.val &lt; num:\ncur = cur.right\n# \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse:\ncur = cur.left\n# \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur is None:\nreturn\n# \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif cur.left is None or cur.right is None:\n# \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nchild = cur.left or cur.right\n# \u5220\u9664\u8282\u70b9 cur\nif cur != self._root:\nif pre.left == cur:\npre.left = child\nelse:\npre.right = child\nelse:\n# \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nself._root = child\n# \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse:\n# \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\ntmp: TreeNode = cur.right\nwhile tmp.left is not None:\ntmp = tmp.left\n# \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nself.remove(tmp.val)\n# \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == nullptr)\nreturn;\nTreeNode *cur = root, *pre = nullptr;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != nullptr) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur-&gt;val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur-&gt;val &lt; num)\ncur = cur-&gt;right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur-&gt;left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == nullptr)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\nTreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre-&gt;left == cur)\npre-&gt;left = child;\nelse\npre-&gt;right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n// \u91ca\u653e\u5185\u5b58\ndelete cur;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode *tmp = cur-&gt;right;\nwhile (tmp-&gt;left != nullptr) {\ntmp = tmp-&gt;left;\n}\nint tmpVal = tmp-&gt;val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp-&gt;val);\n// \u7528 tmp \u8986\u76d6 cur\ncur-&gt;val = tmpVal;\n}\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num)\ncur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nTreeNode child = cur.left != null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre.left == cur)\npre.left = child;\nelse\npre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode tmp = cur.right;\nwhile (tmp.left != null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (root == null)\nreturn;\nTreeNode? cur = root, pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == num)\nbreak;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num)\ncur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null)\nreturn;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nTreeNode? child = cur.left ?? cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != root) {\nif (pre!.left == cur)\npre.left = child;\nelse\npre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode? tmp = cur.right;\nwhile (tmp.left != null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nRemove(tmp.val!.Value);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\ncur := bst.root\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\nvar pre *TreeNode = nil\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nfor cur != nil {\nif cur.Val == num {\nbreak\n}\npre = cur\nif cur.Val.(int) &lt; num {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\ncur = cur.Right\n} else {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\ncur = cur.Left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\nif cur.Left == nil || cur.Right == nil {\nvar child *TreeNode = nil\n// \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\nif cur.Left != nil {\nchild = cur.Left\n} else {\nchild = cur.Right\n}\n// \u5220\u9664\u8282\u70b9 cur\nif cur != bst.root {\nif pre.Left == cur {\npre.Left = child\n} else {\npre.Right = child\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nbst.root = child\n}\n// \u5b50\u8282\u70b9\u6570\u4e3a 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\ntmp := cur.Right\nfor tmp.Left != nil {\ntmp = tmp.Left\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nbst.remove(tmp.Val.(int))\n// \u7528 tmp \u8986\u76d6 cur\ncur.Val = tmp.Val\n}\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif root == nil {\nreturn\n}\nvar cur = root\nvar pre: TreeNode?\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile cur != nil {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif cur!.val == num {\nbreak\n}\npre = cur\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif cur!.val &lt; num {\ncur = cur?.right\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = cur?.left\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur == nil {\nreturn\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif cur?.left == nil || cur?.right == nil {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nlet child = cur?.left != nil ? cur?.left : cur?.right\n// \u5220\u9664\u8282\u70b9 cur\nif cur !== root {\nif pre?.left === cur {\npre?.left = child\n} else {\npre?.right = child\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nroot = child\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nvar tmp = cur?.right\nwhile tmp?.left != nil {\ntmp = tmp?.left\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(num: tmp!.val)\n// \u7528 tmp \u8986\u76d6 cur\ncur?.val = tmp!.val\n}\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (this.root === null) return;\nlet cur = this.root,\npre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num) cur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nconst child = cur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur !== this.root) {\nif (pre.left === cur) pre.left = child;\nelse pre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nthis.root = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nlet tmp = cur.right;\nwhile (tmp.left !== null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nthis.remove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (this.root === null) return;\nlet cur: TreeNode | null = this.root,\npre: TreeNode | null = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur !== null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val === num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; num) cur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse cur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur === null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left === null || cur.right === null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nconst child: TreeNode | null =\ncur.left !== null ? cur.left : cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur !== this.root) {\nif (pre!.left === cur) pre!.left = child;\nelse pre!.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nthis.root = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nlet tmp: TreeNode | null = cur.right;\nwhile (tmp!.left !== null) {\ntmp = tmp!.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nthis.remove(tmp!.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp!.val;\n}\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (_root == null) return;\nTreeNode? cur = _root;\nTreeNode? pre = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.val == _num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.val &lt; _num)\ncur = cur.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse\ncur = cur.left;\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.left == null || cur.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nTreeNode? child = cur.left ?? cur.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (cur != _root) {\nif (pre!.left == cur)\npre.left = child;\nelse\npre.right = child;\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n_root = child;\n}\n} else {\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode? tmp = cur.right;\nwhile (tmp!.left != null) {\ntmp = tmp.left;\n}\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremove(tmp.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.val = tmp.val;\n}\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&amp;mut self, num: i32) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif self.root.is_none() { return; }\nlet mut cur = self.root.clone();\nlet mut pre = None;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile let Some(node) = cur.clone() {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif node.borrow().val == num {\nbreak;\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\npre = cur.clone();\nif node.borrow().val &lt; num {\ncur = node.borrow().right.clone();\n}\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\nelse {\ncur = node.borrow().left.clone();\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif cur.is_none() {\nreturn;\n}\nlet cur = cur.unwrap();\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif cur.borrow().left.is_none() || cur.borrow().right.is_none() {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\nlet child = cur.borrow().left.clone().or_else(|| cur.borrow().right.clone());\nlet pre = pre.unwrap();\nlet left = pre.borrow().left.clone().unwrap();\n// \u5220\u9664\u8282\u70b9 cur\nif !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\nif Rc::ptr_eq(&amp;left, &amp;cur) {\npre.borrow_mut().left = child;\n} else {\npre.borrow_mut().right = child;\n}\n} else {\n// \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\nself.root = child;\n}\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\nelse {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nlet mut tmp = cur.borrow().right.clone();\nwhile let Some(node) = tmp.clone() {\nif node.borrow().left.is_some() {\ntmp = node.borrow().left.clone();\n} else {\nbreak;\n}\n}\nlet tmpval = tmp.unwrap().borrow().val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nself.remove(tmpval);\n// \u7528 tmp \u8986\u76d6 cur\ncur.borrow_mut().val = tmpval;\n}\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (bst-&gt;root == NULL)\nreturn;\nTreeNode *cur = bst-&gt;root, *pre = NULL;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != NULL) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur-&gt;val == num)\nbreak;\npre = cur;\nif (cur-&gt;val &lt; num) {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\ncur = cur-&gt;right;\n} else {\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\ncur = cur-&gt;left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == NULL)\nreturn;\n// \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\nif (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n/* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\nTreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n// \u5220\u9664\u8282\u70b9 cur\nif (pre-&gt;left == cur) {\npre-&gt;left = child;\n} else {\npre-&gt;right = child;\n}\n} else {\n/* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nTreeNode *tmp = cur-&gt;right;\nwhile (tmp-&gt;left != NULL) {\ntmp = tmp-&gt;left;\n}\nint tmpVal = tmp-&gt;val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nremoveItem(bst, tmp-&gt;val);\n// \u7528 tmp \u8986\u76d6 cur\ncur-&gt;val = tmpVal;\n}\n}\n</code></pre> binary_search_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n// \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\nif (self.root == null) return;\nvar cur = self.root;\nvar pre: ?*inc.TreeNode(T) = null;\n// \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\nwhile (cur != null) {\n// \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\nif (cur.?.val == num) break;\npre = cur;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\nif (cur.?.val &lt; num) {\ncur = cur.?.right;\n// \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n} else {\ncur = cur.?.left;\n}\n}\n// \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\nif (cur == null) return;\n// \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\nif (cur.?.left == null or cur.?.right == null) {\n// \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\nvar child = if (cur.?.left != null) cur.?.left else cur.?.right;\n// \u5220\u9664\u8282\u70b9 cur\nif (pre.?.left == cur) {\npre.?.left = child;\n} else {\npre.?.right = child;\n}\n// \u5b50\u8282\u70b9\u6570\u91cf = 2\n} else {\n// \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\nvar tmp = cur.?.right;\nwhile (tmp.?.left != null) {\ntmp = tmp.?.left;\n}\nvar tmp_val = tmp.?.val;\n// \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\nself.remove(tmp.?.val);\n// \u7528 tmp \u8986\u76d6 cur\ncur.?.val = tmp_val;\n}\n}\n</code></pre>"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u904d\u5386\u6709\u5e8f","text":"<p>\u5982\u56fe 7-22 \u6240\u793a\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002</p> <p>\u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u987b\u8fdb\u884c\u989d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002</p> <p></p> <p> \u56fe 7-22 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 </p>"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"<p>\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002\u89c2\u5bdf\u8868 7-2 \uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u6570\u636e\u7684\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002</p> <p> \u8868 7-2 \u00a0 \u6570\u7ec4\u4e0e\u641c\u7d22\u6811\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) <p>\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u56fe 7-23 \u6240\u793a\u7684\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 7-23 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316 </p>"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002</li> <li>\u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 \u4e8c\u53c9\u6811","text":"<p>\u300c\u4e8c\u53c9\u6811 binary tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u503c\u3001\u5de6\u5b50\u8282\u70b9\u5f15\u7528\u548c\u53f3\u5b50\u8282\u70b9\u5f15\u7528\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class TreeNode:\n\"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\ndef __init__(self, val: int):\nself.val: int = val                # \u8282\u70b9\u503c\nself.left: TreeNode | None = None  # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nself.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nint val;          // \u8282\u70b9\u503c\nTreeNode *left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nTreeNode *right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nTreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nTreeNode left;   // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nTreeNode right;  // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\nTreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\npublic int? val = x;    // \u8282\u70b9\u503c\npublic TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\npublic TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\nVal   int\nLeft  *TreeNode\nRight *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\nreturn &amp;TreeNode{\nLeft:  nil, // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nRight: nil, // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nVal:   v,   // \u8282\u70b9\u503c\n}\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nvar val: Int // \u8282\u70b9\u503c\nvar left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nvar right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\ninit(x: Int) {\nval = x\n}\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval; // \u8282\u70b9\u503c\nleft; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nright; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\nconstructor(val, left, right) {\nthis.val = val === undefined ? 0 : val;\nthis.left = left === undefined ? null : left;\nthis.right = right === undefined ? null : right;\n}\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nval: number;\nleft: TreeNode | null;\nright: TreeNode | null;\nconstructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\nthis.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\nthis.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nthis.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\nint val;         // \u8282\u70b9\u503c\nTreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nTreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\nTreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\nval: i32,                               // \u8282\u70b9\u503c\nleft: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\nright: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\nimpl TreeNode {\n/* \u6784\u9020\u65b9\u6cd5 */\nfn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\nRc::new(RefCell::new(Self {\nval,\nleft: None,\nright: None\n}))\n}\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct TreeNode {\nint val;                // \u8282\u70b9\u503c\nint height;             // \u8282\u70b9\u9ad8\u5ea6\nstruct TreeNode *left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\nstruct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n} TreeNode;\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\nTreeNode *node;\nnode = (TreeNode *)malloc(sizeof(TreeNode));\nnode-&gt;val = val;\nnode-&gt;height = 0;\nnode-&gt;left = NULL;\nnode-&gt;right = NULL;\nreturn node;\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u8282\u70b9 left-child node\u300d\u548c\u300c\u53f3\u5b50\u8282\u70b9 right-child node\u300d\uff0c\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u300c\u7236\u8282\u70b9 parent node\u300d\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u300c\u5de6\u5b50\u6811 left subtree\u300d\uff0c\u540c\u7406\u53ef\u5f97\u300c\u53f3\u5b50\u6811 right subtree\u300d\u3002</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u5982\u56fe 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002</p> <p></p> <p> \u56fe 7-1 \u00a0 \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811 </p>"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"<p>\u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed\u5982\u56fe 7-2 \u6240\u793a\u3002</p> <ul> <li>\u300c\u6839\u8282\u70b9 root node\u300d\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\u3002</li> <li>\u300c\u53f6\u8282\u70b9 leaf node\u300d\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 \\(\\text{None}\\) \u3002</li> <li>\u300c\u8fb9 edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</li> <li>\u8282\u70b9\u6240\u5728\u7684\u300c\u5c42 level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \u3002</li> <li>\u8282\u70b9\u7684\u300c\u5ea6 degree\u300d\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u8303\u56f4\u662f 0\u30011\u30012 \u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u300c\u6df1\u5ea6 depth\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u8ddd\u79bb\u8be5\u8282\u70b9\u6700\u8fdc\u7684\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> </ul> <p></p> <p> \u56fe 7-2 \u00a0 \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002</p>"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u53c9\u6811","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py<pre><code># \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\nn2 = new TreeNode(2),\nn3 = new TreeNode(3),\nn4 = new TreeNode(4),\nn5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.zig<pre><code>\n</code></pre>"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u3002\u56fe 7-3 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 7-3 \u00a0 \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py<pre><code># \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u5220\u9664\u8282\u70b9 p\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> <p>Note</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-4 \u6240\u793a\uff0c\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree\u300d\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u3002</p> <p></p> <p> \u56fe 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-5 \u6240\u793a\uff0c\u300c\u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002</p> <p></p> <p> \u56fe 7-5 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-6 \u6240\u793a\uff0c\u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree\u300d\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-6 \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-7 \u6240\u793a\uff0c\u300c\u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree\u300d\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002</p> <p></p> <p> \u56fe 7-7 \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"<p>\u56fe 7-8 \u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u7684\u7406\u60f3\u7ed3\u6784\u4e0e\u9000\u5316\u7ed3\u6784\u3002\u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u201c\u94fe\u8868\u201d\u3002</p> <ul> <li>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\u3002</li> <li>\u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</li> </ul> <p></p> <p> \u56fe 7-8 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> <p>\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7ed3\u6784\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u3002</p> <p> \u8868 7-1 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u7684\u6811\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"<p>\u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002</p> <p>\u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002</p>"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c42\u5e8f\u904d\u5386","text":"<p>\u5982\u56fe 7-9 \u6240\u793a\uff0c\u300c\u5c42\u5e8f\u904d\u5386 level-order traversal\u300d\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002</p> <p>\u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal, BFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\u3002</p> <p></p> <p> \u56fe 7-9 \u00a0 \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u201c\u961f\u5217\u201d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n\"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n# \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue: deque[TreeNode] = deque()\nqueue.append(root)\n# \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nres = []\nwhile queue:\nnode: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\nres.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\nif node.left is not None:\nqueue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif node.right is not None:\nqueue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\nreturn res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue&lt;TreeNode *&gt; queue;\nqueue.push(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvector&lt;int&gt; vec;\nwhile (!queue.empty()) {\nTreeNode *node = queue.front();\nqueue.pop();              // \u961f\u5217\u51fa\u961f\nvec.push_back(node-&gt;val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node-&gt;left != nullptr)\nqueue.push(node-&gt;left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node-&gt;right != nullptr)\nqueue.push(node-&gt;right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\nqueue.add(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList&lt;Integer&gt; list = new ArrayList&lt;&gt;();\nwhile (!queue.isEmpty()) {\nTreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\nlist.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null)\nqueue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null)\nqueue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue&lt;TreeNode&gt; queue = new();\nqueue.Enqueue(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList&lt;int&gt; list = [];\nwhile (queue.Count != 0) {\nTreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\nlist.Add(node.val!.Value);       // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null)\nqueue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null)\nqueue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nqueue := list.New()\nqueue.PushBack(root)\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nnums := make([]any, 0)\nfor queue.Len() &gt; 0 {\n// \u961f\u5217\u51fa\u961f\nnode := queue.Remove(queue.Front()).(*TreeNode)\n// \u4fdd\u5b58\u8282\u70b9\u503c\nnums = append(nums, node.Val)\nif node.Left != nil {\n// \u5de6\u5b50\u8282\u70b9\u5165\u961f\nqueue.PushBack(node.Left)\n}\nif node.Right != nil {\n// \u53f3\u5b50\u8282\u70b9\u5165\u961f\nqueue.PushBack(node.Right)\n}\n}\nreturn nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nvar queue: [TreeNode] = [root]\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list: [Int] = []\nwhile !queue.isEmpty {\nlet node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\nlist.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\nif let left = node.left {\nqueue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif let right = node.right {\nqueue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\n}\nreturn list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list = [];\nwhile (queue.length) {\nlet node = queue.shift(); // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst queue = [root];\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nconst list: number[] = [];\nwhile (queue.length) {\nlet node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\nlist.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left) {\nqueue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif (node.right) {\nqueue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\n}\nreturn list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nQueue&lt;TreeNode?&gt; queue = Queue();\nqueue.add(root);\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nList&lt;int&gt; res = [];\nwhile (queue.isNotEmpty) {\nTreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\nres.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\nif (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}\nreturn res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nlet mut que = VecDeque::new();\nque.push_back(Rc::clone(&amp;root));\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nlet mut vec = Vec::new();\nwhile let Some(node) = que.pop_front() {                 // \u961f\u5217\u51fa\u961f\nvec.push(node.borrow().val);                         // \u4fdd\u5b58\u8282\u70b9\u503c\nif let Some(left) = node.borrow().left.as_ref() {\nque.push_back(Rc::clone(left));                  // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif let Some(right) = node.borrow().right.as_ref() {\nque.push_back(Rc::clone(right));                 // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n};\n}\nvec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n/* \u8f85\u52a9\u961f\u5217 */\nint front, rear;\nint index, *arr;\nTreeNode *node;\nTreeNode **queue;\n/* \u8f85\u52a9\u961f\u5217 */\nqueue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n// \u961f\u5217\u6307\u9488\nfront = 0, rear = 0;\n// \u52a0\u5165\u6839\u8282\u70b9\nqueue[rear++] = root;\n// \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n/* \u8f85\u52a9\u6570\u7ec4 */\narr = (int *)malloc(sizeof(int) * MAX_SIZE);\n// \u6570\u7ec4\u6307\u9488\nindex = 0;\nwhile (front &lt; rear) {\n// \u961f\u5217\u51fa\u961f\nnode = queue[front++];\n// \u4fdd\u5b58\u8282\u70b9\u503c\narr[index++] = node-&gt;val;\nif (node-&gt;left != NULL) {\n// \u5de6\u5b50\u8282\u70b9\u5165\u961f\nqueue[rear++] = node-&gt;left;\n}\nif (node-&gt;right != NULL) {\n// \u53f3\u5b50\u8282\u70b9\u5165\u961f\nqueue[rear++] = node-&gt;right;\n}\n}\n// \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n*size = index;\narr = realloc(arr, sizeof(int) * (*size));\n// \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\nfree(queue);\nreturn arr;\n}\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n// \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\nconst L = std.TailQueue(*inc.TreeNode(T));\nvar queue = L{};\nvar root_node = try mem_allocator.create(L.Node);\nroot_node.data = root;\nqueue.append(root_node); // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\nvar list = std.ArrayList(T).init(std.heap.page_allocator);\nwhile (queue.len &gt; 0) {\nvar queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\nvar node = queue_node.data;\ntry list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\nif (node.left != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.left.?;\nqueue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n}\nif (node.right != null) {\nvar tmp_node = try mem_allocator.create(L.Node);\ntmp_node.data = node.right.?;\nqueue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n}        }\nreturn list;\n}\n</code></pre>"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\((n + 1) / 2\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2 \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"<p>\u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal, DFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002</p> <p>\u56fe 7-10 \u5c55\u793a\u4e86\u5bf9\u4e8c\u53c9\u6811\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u68f5\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002</p> <p></p> <p> \u56fe 7-10 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n\"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nres.append(root.val)\npre_order(root=root.left)\npre_order(root=root.right)\ndef in_order(root: TreeNode | None):\n\"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\nin_order(root=root.left)\nres.append(root.val)\nin_order(root=root.right)\ndef post_order(root: TreeNode | None):\n\"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\nif root is None:\nreturn\n# \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npost_order(root=root.left)\npost_order(root=root.right)\nres.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nvec.push_back(root-&gt;val);\npreOrder(root-&gt;left);\npreOrder(root-&gt;right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(root-&gt;left);\nvec.push_back(root-&gt;val);\ninOrder(root-&gt;right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\nif (root == nullptr)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(root-&gt;left);\npostOrder(root-&gt;right);\nvec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nlist.add(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.add(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\nif (root == null)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nlist.Add(root.val!.Value);\nPreOrder(root.left);\nPreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\nInOrder(root.left);\nlist.Add(root.val!.Value);\nInOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\nPostOrder(root.left);\nPostOrder(root.right);\nlist.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nnums = append(nums, node.Val)\npreOrder(node.Left)\npreOrder(node.Right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(node.Left)\nnums = append(nums, node.Val)\ninOrder(node.Right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\nif node == nil {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(node.Left)\npostOrder(node.Right)\nnums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nlist.append(root.val)\npreOrder(root: root.left)\npreOrder(root: root.right)\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(root: root.left)\nlist.append(root.val)\ninOrder(root: root.right)\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\nguard let root = root else {\nreturn\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(root: root.left)\npostOrder(root: root.right)\nlist.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\nif (root === null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nlist.push(root.val);\npreOrder(root.left);\npreOrder(root.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(root.left);\nlist.push(root.val);\ninOrder(root.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\nif (root === null) {\nreturn;\n}\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(root.left);\npostOrder(root.right);\nlist.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nlist.add(node.val);\npreOrder(node.left);\npreOrder(node.right);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(node.left);\nlist.add(node.val);\ninOrder(node.right);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\nif (node == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(node.left);\npostOrder(node.right);\nlist.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\nlet mut result = vec![];\nif let Some(node) = root {\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\nresult.push(node.borrow().val);\nresult.append(&amp;mut pre_order(node.borrow().left.as_ref()));\nresult.append(&amp;mut pre_order(node.borrow().right.as_ref()));\n}\nresult\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\nlet mut result = vec![];\nif let Some(node) = root {\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\nresult.append(&amp;mut in_order(node.borrow().left.as_ref()));\nresult.push(node.borrow().val);\nresult.append(&amp;mut in_order(node.borrow().right.as_ref()));\n}\nresult\n}\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\nlet mut result = vec![];\nif let Some(node) = root {\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\nresult.append(&amp;mut post_order(node.borrow().left.as_ref()));\nresult.append(&amp;mut post_order(node.borrow().right.as_ref()));\nresult.push(node.borrow().val);\n}\nresult\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\narr[(*size)++] = root-&gt;val;\npreOrder(root-&gt;left, size);\npreOrder(root-&gt;right, size);\n}\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ninOrder(root-&gt;left, size);\narr[(*size)++] = root-&gt;val;\ninOrder(root-&gt;right, size);\n}\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\nif (root == NULL)\nreturn;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\npostOrder(root-&gt;left, size);\npostOrder(root-&gt;right, size);\narr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\ntry list.append(root.?.val);\ntry preOrder(T, root.?.left);\ntry preOrder(T, root.?.right);\n}\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\ntry inOrder(T, root.?.left);\ntry list.append(root.?.val);\ntry inOrder(T, root.?.right);\n}\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\nif (root == null) return;\n// \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\ntry postOrder(T, root.?.left);\ntry postOrder(T, root.?.right);\ntry list.append(root.?.val);\n}\n</code></pre> <p>Tip</p> <p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4e5f\u53ef\u4ee5\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u7814\u7a76\u3002</p> <p>\u56fe 7-11 \u5c55\u793a\u4e86\u524d\u5e8f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5176\u53ef\u5206\u4e3a\u201c\u9012\u201d\u548c\u201c\u5f52\u201d\u4e24\u4e2a\u9006\u5411\u7684\u90e8\u5206\u3002</p> <ol> <li>\u201c\u9012\u201d\u8868\u793a\u5f00\u542f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5e8f\u5728\u6b64\u8fc7\u7a0b\u4e2d\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</li> <li>\u201c\u5f52\u201d\u8868\u793a\u51fd\u6570\u8fd4\u56de\uff0c\u4ee3\u8868\u5f53\u524d\u8282\u70b9\u5df2\u7ecf\u8bbf\u95ee\u5b8c\u6bd5\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-11 \u00a0 \u524d\u5e8f\u904d\u5386\u7684\u9012\u5f52\u8fc7\u7a0b </p>"},{"location":"chapter_tree/binary_tree_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u5360\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_tree/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u6bcf\u4e2a\u4e8c\u53c9\u6811\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u503c\u4ee5\u53ca\u4e24\u4e2a\u6307\u9488\uff0c\u5206\u522b\u6307\u5411\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u3002</li> <li>\u5bf9\u4e8e\u4e8c\u53c9\u6811\u4e2d\u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u88ab\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u76f8\u5173\u672f\u8bed\u5305\u62ec\u6839\u8282\u70b9\u3001\u53f6\u8282\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u8282\u70b9\u63d2\u5165\u548c\u8282\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002</li> <li>\u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u6709\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u5b8c\u6ee1\u4e8c\u53c9\u6811\u548c\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u6700\u7406\u60f3\u7684\u72b6\u6001\uff0c\u800c\u94fe\u8868\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002</li> <li>\u4e8c\u53c9\u6811\u53ef\u4ee5\u7528\u6570\u7ec4\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c06\u8282\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u5c42\u5e8f\u904d\u5386\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u6839\u636e\u7236\u8282\u70b9\u4e0e\u5b50\u8282\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u5173\u7cfb\u6765\u5b9e\u73b0\u6307\u9488\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u65b9\u6cd5\uff0c\u5b83\u4f53\u73b0\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u901a\u8fc7\u961f\u5217\u6765\u5b9e\u73b0\u3002</li> <li>\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u5b83\u4eec\u4f53\u73b0\u4e86\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>AVL \u6811\uff0c\u4e5f\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002</li> <li>AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\u3002</p> <p>\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f</p> <p>\u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>\u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f</p> <p>\u4e0e\u987a\u5e8f\u548c\u9006\u5e8f\u904d\u5386\u6570\u7ec4\u7c7b\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u4e09\u79cd\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5b83\u4eec\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u8282\u70b9\u5927\u5c0f\u6ee1\u8db3 <code>\u5de6\u5b50\u8282\u70b9\u503c &lt; \u6839\u8282\u70b9\u503c &lt; \u53f3\u5b50\u8282\u70b9\u503c</code> \uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167 <code>\u5de6 $\\rightarrow$ \u6839 $\\rightarrow$ \u53f3</code> \u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002</p> <p>\u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9 <code>node</code>\u3001<code>child</code>\u3001<code>grand_child</code> \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3 <code>node</code> \u7684\u7236\u8282\u70b9\u548c <code>node</code> \u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f</p> <p>\u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c <code>right_rotate(root)</code> \u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8 <code>return child</code> \u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002</p> <p>\u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230 <code>private</code> \u548c <code>public</code> \u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06 <code>height()</code> \u51fd\u6570\u548c <code>updateHeight()</code> \u51fd\u6570\u5206\u522b\u653e\u5728 <code>public</code> \u548c <code>private</code> \u4e2d\u5462\uff1f</p> <p>\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a <code>private</code> \u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528 <code>updateHeight()</code> \u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c <code>height()</code> \u662f\u8bbf\u95ee\u8282\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8e <code>vector.size()</code> \uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210 <code>public</code> \u4ee5\u4fbf\u4f7f\u7528\u3002</p> <p>\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f</p> <p>\u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4ee3\u7801\u4e2d\u7684 <code>build_tree()</code> \u65b9\u6cd5\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u5c06\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002</p> <p>\u5728 Java \u4e2d\uff0c\u5b57\u7b26\u4e32\u5bf9\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528 <code>equals()</code> \u65b9\u6cd5\uff1f</p> <p>\u5728 Java \u4e2d\uff0c\u5bf9\u4e8e\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c<code>==</code> \u7528\u4e8e\u5bf9\u6bd4\u4e24\u4e2a\u53d8\u91cf\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5bf9\u4e8e\u5f15\u7528\u7c7b\u578b\uff0c\u4e24\u79cd\u7b26\u53f7\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li><code>==</code> \uff1a\u7528\u6765\u6bd4\u8f83\u4e24\u4e2a\u53d8\u91cf\u662f\u5426\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5373\u5b83\u4eec\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002</li> <li><code>equals()</code>\uff1a\u7528\u6765\u5bf9\u6bd4\u4e24\u4e2a\u5bf9\u8c61\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002</li> </ul> <p>\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5bf9\u6bd4\u503c\uff0c\u6211\u4eec\u5e94\u8be5\u4f7f\u7528 <code>equals()</code> \u3002\u7136\u800c\uff0c\u901a\u8fc7 <code>String a = \"hi\"; String b = \"hi\";</code> \u521d\u59cb\u5316\u7684\u5b57\u7b26\u4e32\u90fd\u5b58\u50a8\u5728\u5b57\u7b26\u4e32\u5e38\u91cf\u6c60\u4e2d\uff0c\u5b83\u4eec\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528 <code>a == b</code> \u6765\u6bd4\u8f83\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5185\u5bb9\u3002</p> <p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u7684\u8282\u70b9\u6570\u91cf\u662f \\(2^h\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6ee1\u4e8c\u53c9\u6811\uff0c\u5176\u8282\u70b9\u603b\u6570 \\(n = 7\\) \uff0c\u5219\u5e95\u5c42\u8282\u70b9\u6570\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002</p>"}]}
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+{"config":{"lang":["en"],"separator":"[\\s\\u200b\\u3000\\-\u3001\u3002\uff0c\uff0e\uff1f\uff01\uff1b]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Home","text":"\u300a Hello \u7b97\u6cd5 \u300b <p>\u52a8\u753b\u56fe\u89e3\u3001\u4e00\u952e\u8fd0\u884c\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6559\u7a0b</p> <p> </p> <p>        \u5f00\u59cb\u9605\u8bfb           \u4e0b\u8f7d\u4ee3\u7801           \u4e0b\u8f7d PDF    </p> <p> </p>  \u63a8\u8350\u8bed  <p>Quote</p> <p>\u201c\u4e00\u672c\u901a\u4fd7\u6613\u61c2\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u4e66\uff0c\u5f15\u5bfc\u8bfb\u8005\u624b\u8111\u5e76\u7528\u5730\u5b66\u4e60\uff0c\u5f3a\u70c8\u63a8\u8350\u7b97\u6cd5\u521d\u5b66\u8005\u9605\u8bfb\u3002\u201d</p> <p>\u2014\u2014 \u9093\u4fca\u8f89\uff0c\u6e05\u534e\u5927\u5b66\u8ba1\u7b97\u673a\u7cfb\u6559\u6388</p> <p>Quote</p> <p>\u201c\u5982\u679c\u6211\u5f53\u5e74\u5b66\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u65f6\u5019\u6709\u300aHello \u7b97\u6cd5\u300b\uff0c\u5b66\u8d77\u6765\u5e94\u8be5\u4f1a\u7b80\u5355 10 \u500d\uff01\u201d</p> <p>\u2014\u2014 \u674e\u6c90\uff0c\u4e9a\u9a6c\u900a\u8d44\u6df1\u9996\u5e2d\u79d1\u5b66\u5bb6</p> \u52a8\u753b\u56fe\u89e3 <p>\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1</p> <p>\"A picture is worth a thousand words.\"</p> <p>\u201c\u4e00\u56fe\u80dc\u5343\u8a00\u201d</p> \u4e00\u952e\u8fd0\u884c <p>\u5341\u4f59\u79cd\u7f16\u7a0b\u8bed\u8a00\u4ee3\u7801\u53ef\u76f4\u63a5\u8fd0\u884c</p> <p>\"Talk is cheap. Show me the code.\"</p> <p>\u201c\u5c11\u5439\u725b\uff0c\u770b\u4ee3\u7801\u201d</p> \u4e92\u52a9\u5b66\u4e60 <p>\u6b22\u8fce\u8ba8\u8bba\u4e0e\u63d0\u95ee\u8bfb\u8005\u95f4\u643a\u624b\u5171\u8fdb</p> <p>\"\u8ffd\u98ce\u8d76\u6708\u83ab\u505c\u7559\"</p> <p>\u201c\u5e73\u829c\u5c3d\u5904\u662f\u6625\u5c71\u201d</p>  \u5e8f  <p>\u4e24\u5e74\u524d\uff0c\u6211\u5728\u529b\u6263\u4e0a\u5206\u4eab\u4e86\u201c\u5251\u6307 Offer\u201d\u7cfb\u5217\u9898\u89e3\uff0c\u53d7\u5230\u4e86\u8bb8\u591a\u540c\u5b66\u7684\u559c\u7231\u548c\u652f\u6301\u3002\u5728\u4e0e\u8bfb\u8005\u4ea4\u6d41\u671f\u95f4\uff0c\u6211\u6700\u5e38\u6536\u5230\u7684\u4e00\u4e2a\u95ee\u9898\u662f\u201c\u5982\u4f55\u5165\u95e8\u7b97\u6cd5\u201d\u3002\u9010\u6e10\u5730\uff0c\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u4ea7\u751f\u4e86\u6d53\u539a\u7684\u5174\u8da3\u3002</p> <p>\u4e24\u773c\u4e00\u62b9\u9ed1\u5730\u5237\u9898\u4f3c\u4e4e\u662f\u6700\u53d7\u6b22\u8fce\u7684\u65b9\u6cd5\uff0c\u7b80\u5355\u76f4\u63a5\u4e14\u6709\u6548\u3002\u7136\u800c\u5237\u9898\u5c31\u5982\u540c\u73a9\u201c\u626b\u96f7\u201d\u6e38\u620f\uff0c\u81ea\u5b66\u80fd\u529b\u5f3a\u7684\u540c\u5b66\u80fd\u591f\u987a\u5229\u5c06\u5730\u96f7\u9010\u4e2a\u6392\u6389\uff0c\u800c\u57fa\u7840\u4e0d\u8db3\u7684\u540c\u5b66\u5f88\u53ef\u80fd\u88ab\u70b8\u7684\u6ee1\u5934\u662f\u5305\uff0c\u5e76\u5728\u632b\u6298\u4e2d\u6b65\u6b65\u9000\u7f29\u3002\u901a\u8bfb\u6559\u6750\u4e5f\u662f\u4e00\u79cd\u5e38\u89c1\u505a\u6cd5\uff0c\u4f46\u5bf9\u4e8e\u9762\u5411\u6c42\u804c\u7684\u540c\u5b66\u6765\u8bf4\uff0c\u6bd5\u4e1a\u5b63\u3001\u6295\u9012\u7b80\u5386\u3001\u51c6\u5907\u7b14\u8bd5\u9762\u8bd5\u5df2\u7ecf\u6d88\u8017\u4e86\u5927\u90e8\u5206\u7cbe\u529b\uff0c\u5543\u539a\u91cd\u7684\u4e66\u5f80\u5f80\u53d8\u6210\u4e86\u4e00\u9879\u8270\u5de8\u7684\u6311\u6218\u3002</p> <p>\u5982\u679c\u4f60\u4e5f\u9762\u4e34\u7c7b\u4f3c\u7684\u56f0\u6270\uff0c\u90a3\u4e48\u5f88\u5e78\u8fd0\u8fd9\u672c\u4e66\u627e\u5230\u4e86\u4f60\u3002\u672c\u4e66\u662f\u6211\u5bf9\u8fd9\u4e2a\u95ee\u9898\u7ed9\u51fa\u7684\u7b54\u6848\uff0c\u5373\u4f7f\u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u4e5f\u81f3\u5c11\u662f\u4e00\u6b21\u79ef\u6781\u7684\u5c1d\u8bd5\u3002\u672c\u4e66\u867d\u7136\u4e0d\u8db3\u4ee5\u8ba9\u4f60\u76f4\u63a5\u62ff\u5230 Offer \uff0c\u4f46\u4f1a\u5f15\u5bfc\u4f60\u63a2\u7d22\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u201c\u77e5\u8bc6\u5730\u56fe\u201d\uff0c\u5e26\u4f60\u4e86\u89e3\u4e0d\u540c\u201c\u5730\u96f7\u201d\u7684\u5f62\u72b6\u3001\u5927\u5c0f\u548c\u5206\u5e03\u4f4d\u7f6e\uff0c\u8ba9\u4f60\u638c\u63e1\u5404\u79cd\u201c\u6392\u96f7\u65b9\u6cd5\u201d\u3002\u6709\u4e86\u8fd9\u4e9b\u672c\u9886\uff0c\u76f8\u4fe1\u4f60\u53ef\u4ee5\u66f4\u52a0\u81ea\u5982\u5730\u5237\u9898\u548c\u9605\u8bfb\u6587\u732e\uff0c\u9010\u6b65\u6784\u5efa\u8d77\u5b8c\u6574\u7684\u77e5\u8bc6\u4f53\u7cfb\u3002</p> <p>\u6211\u6df1\u6df1\u8d5e\u540c\u8d39\u66fc\u6559\u6388\u6240\u8a00\uff1a\u201cKnowledge isn't free. You have to pay attention.\u201d\u4ece\u8fd9\u4e2a\u610f\u4e49\u4e0a\u770b\uff0c\u8fd9\u672c\u4e66\u5e76\u975e\u5b8c\u5168\u201c\u514d\u8d39\u201d\u3002\u4e3a\u4e86\u4e0d\u8f9c\u8d1f\u4f60\u4e3a\u672c\u4e66\u6240\u4ed8\u51fa\u7684\u5b9d\u8d35\u201c\u6ce8\u610f\u529b\u201d\uff0c\u6211\u4f1a\u5c3d\u6211\u6240\u80fd\uff0c\u6295\u5165\u6700\u5927\u7684\u201c\u6ce8\u610f\u529b\u201d\u6765\u5b8c\u6210\u8fd9\u672c\u4e66\u7684\u521b\u4f5c\u3002</p>  \u4f5c\u8005\u7b80\u4ecb  <p>\u9773\u5b87\u680b (Krahets)\uff0c\u5927\u5382\u9ad8\u7ea7\u7b97\u6cd5\u5de5\u7a0b\u5e08\uff0c\u4e0a\u6d77\u4ea4\u901a\u5927\u5b66\u7855\u58eb\u3002\u529b\u6263\uff08LeetCode\uff09\u5168\u7f51\u9605\u8bfb\u91cf\u6700\u9ad8\u535a\u4e3b\uff0c\u53d1\u8868\u7684\u300a\u56fe\u89e3\u7b97\u6cd5\u6570\u636e\u7ed3\u6784\u300b\u5df2\u88ab\u8ba2\u9605 30 \u4e07\u672c\u3002</p>  \u81f4\u8c22  <p>\u672c\u4e66\u5728\u5f00\u6e90\u793e\u533a\u4f17\u591a\u8d21\u732e\u8005\u7684\u5171\u540c\u52aa\u529b\u4e0b\u4e0d\u65ad\u5b8c\u5584\u3002\u611f\u8c22\u6bcf\u4e00\u4f4d\u6295\u5165\u65f6\u95f4\u4e0e\u7cbe\u529b\u7684\u64b0\u7a3f\u4eba\uff0c\u4ed6\u4eec\u662f\uff08\u6309\u7167 GitHub \u81ea\u52a8\u751f\u6210\u7684\u987a\u5e8f\u6392\u5217\uff09\uff1a</p> <p> </p> <p>\u672c\u4e66\u7684\u4ee3\u7801\u5ba1\u9605\u5de5\u4f5c\u7531 Gonglja\u3001gvenusleo\u3001hpstory\u3001justin\u2010tse\u3001krahets\u3001night-cruise\u3001nuomi1\u3001Reanon \u548c sjinzh \u5b8c\u6210\uff08\u6309\u7167\u9996\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\uff09\u3002\u611f\u8c22\u4ed6\u4eec\u4ed8\u51fa\u7684\u65f6\u95f4\u4e0e\u7cbe\u529b\uff0c\u6b63\u662f\u4ed6\u4eec\u786e\u4fdd\u4e86\u5404\u8bed\u8a00\u4ee3\u7801\u7684\u89c4\u8303\u4e0e\u7edf\u4e00\u3002</p> <sub>Gonglja</sub><sub>C, C++</sub> <sub>gvenusleo</sub><sub>Dart</sub> <sub>hpstory</sub><sub>C#</sub> <sub>justin-tse</sub><sub>JS, TS</sub> <sub>krahets</sub><sub>Java, Python</sub> <sub>night-cruise</sub><sub>Rust</sub> <sub>nuomi1</sub><sub>Swift</sub> <sub>Reanon</sub><sub>Go, C</sub> <sub>sjinzh</sub><sub>Rust, Zig</sub>"},{"location":"chapter_appendix/","title":"\u7b2c 16 \u7ae0 \u00a0 \u9644\u5f55","text":""},{"location":"chapter_appendix/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5</li> <li>16.2 \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c</li> <li>16.3 \u00a0 \u672f\u8bed\u8868</li> </ul>"},{"location":"chapter_appendix/contribution/","title":"16.2 \u00a0 \u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c","text":"<p>\u7531\u4e8e\u7b14\u8005\u80fd\u529b\u6709\u9650\uff0c\u4e66\u4e2d\u96be\u514d\u5b58\u5728\u4e00\u4e9b\u9057\u6f0f\u548c\u9519\u8bef\uff0c\u8bf7\u60a8\u8c05\u89e3\u3002\u5982\u679c\u60a8\u53d1\u73b0\u4e86\u7b14\u8bef\u3001\u94fe\u63a5\u5931\u6548\u3001\u5185\u5bb9\u7f3a\u5931\u3001\u6587\u5b57\u6b67\u4e49\u3001\u89e3\u91ca\u4e0d\u6e05\u6670\u6216\u884c\u6587\u7ed3\u6784\u4e0d\u5408\u7406\u7b49\u95ee\u9898\uff0c\u8bf7\u534f\u52a9\u6211\u4eec\u8fdb\u884c\u4fee\u6b63\uff0c\u4ee5\u7ed9\u8bfb\u8005\u63d0\u4f9b\u66f4\u4f18\u8d28\u7684\u5b66\u4e60\u8d44\u6e90\u3002</p> <p>\u6240\u6709\u64b0\u7a3f\u4eba\u7684 GitHub ID \u5c06\u5728\u672c\u4e66\u4ed3\u5e93\u3001\u7f51\u9875\u7248\u548c PDF \u7248\u7684\u4e3b\u9875\u4e0a\u8fdb\u884c\u5c55\u793a\uff0c\u4ee5\u611f\u8c22\u4ed6\u4eec\u5bf9\u5f00\u6e90\u793e\u533a\u7684\u65e0\u79c1\u5949\u732e\u3002</p> <p>\u5f00\u6e90\u7684\u9b45\u529b</p> <p>\u7eb8\u8d28\u56fe\u4e66\u7684\u4e24\u6b21\u5370\u5237\u7684\u95f4\u9694\u65f6\u95f4\u5f80\u5f80\u8f83\u4e45\uff0c\u5185\u5bb9\u66f4\u65b0\u975e\u5e38\u4e0d\u65b9\u4fbf\u3002</p> <p>\u800c\u5728\u672c\u5f00\u6e90\u4e66\u4e2d\uff0c\u5185\u5bb9\u66f4\u8fed\u7684\u65f6\u95f4\u88ab\u7f29\u77ed\u81f3\u6570\u65e5\u751a\u81f3\u51e0\u4e2a\u5c0f\u65f6\u3002</p>"},{"location":"chapter_appendix/contribution/#1","title":"1. \u00a0 \u5185\u5bb9\u5fae\u8c03","text":"<p>\u5982\u56fe 16-3 \u6240\u793a\uff0c\u6bcf\u4e2a\u9875\u9762\u7684\u53f3\u4e0a\u89d2\u90fd\u6709\u201c\u7f16\u8f91\u56fe\u6807\u201d\u3002\u60a8\u53ef\u4ee5\u6309\u7167\u4ee5\u4e0b\u6b65\u9aa4\u4fee\u6539\u6587\u672c\u6216\u4ee3\u7801\u3002</p> <ol> <li>\u70b9\u51fb\u201c\u7f16\u8f91\u56fe\u6807\u201d\uff0c\u5982\u679c\u9047\u5230\u201c\u9700\u8981 Fork \u6b64\u4ed3\u5e93\u201d\u7684\u63d0\u793a\uff0c\u8bf7\u540c\u610f\u8be5\u64cd\u4f5c\u3002</li> <li>\u4fee\u6539 Markdown \u6e90\u6587\u4ef6\u5185\u5bb9\uff0c\u68c0\u67e5\u5185\u5bb9\u7684\u6b63\u786e\u6027\uff0c\u5e76\u5c3d\u91cf\u4fdd\u6301\u6392\u7248\u683c\u5f0f\u7684\u7edf\u4e00\u3002</li> <li>\u5728\u9875\u9762\u5e95\u90e8\u586b\u5199\u4fee\u6539\u8bf4\u660e\uff0c\u7136\u540e\u70b9\u51fb\u201cPropose file change\u201d\u6309\u94ae\u3002\u9875\u9762\u8df3\u8f6c\u540e\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol> <p></p> <p> \u56fe 16-3 \u00a0 \u9875\u9762\u7f16\u8f91\u6309\u952e </p> <p>\u56fe\u7247\u65e0\u6cd5\u76f4\u63a5\u4fee\u6539\uff0c\u9700\u8981\u901a\u8fc7\u65b0\u5efa Issue \u6216\u8bc4\u8bba\u7559\u8a00\u6765\u63cf\u8ff0\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5c3d\u5feb\u91cd\u65b0\u7ed8\u5236\u5e76\u66ff\u6362\u56fe\u7247\u3002</p>"},{"location":"chapter_appendix/contribution/#2","title":"2. \u00a0 \u5185\u5bb9\u521b\u4f5c","text":"<p>\u5982\u679c\u60a8\u6709\u5174\u8da3\u53c2\u4e0e\u6b64\u5f00\u6e90\u9879\u76ee\uff0c\u5305\u62ec\u5c06\u4ee3\u7801\u7ffb\u8bd1\u6210\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u3001\u6269\u5c55\u6587\u7ae0\u5185\u5bb9\u7b49\uff0c\u90a3\u4e48\u9700\u8981\u5b9e\u65bd\u4ee5\u4e0b Pull Request \u5de5\u4f5c\u6d41\u7a0b\u3002</p> <ol> <li>\u767b\u5f55 GitHub \uff0c\u5c06\u672c\u4e66\u7684\u4ee3\u7801\u4ed3\u5e93 Fork \u5230\u4e2a\u4eba\u8d26\u53f7\u4e0b\u3002</li> <li>\u8fdb\u5165\u60a8\u7684 Fork \u4ed3\u5e93\u7f51\u9875\uff0c\u4f7f\u7528 <code>git clone</code> \u547d\u4ee4\u5c06\u4ed3\u5e93\u514b\u9686\u81f3\u672c\u5730\u3002</li> <li>\u5728\u672c\u5730\u8fdb\u884c\u5185\u5bb9\u521b\u4f5c\uff0c\u5e76\u8fdb\u884c\u5b8c\u6574\u6d4b\u8bd5\uff0c\u9a8c\u8bc1\u4ee3\u7801\u7684\u6b63\u786e\u6027\u3002</li> <li>\u5c06\u672c\u5730\u6240\u505a\u66f4\u6539 Commit \uff0c\u7136\u540e Push \u81f3\u8fdc\u7a0b\u4ed3\u5e93\u3002</li> <li>\u5237\u65b0\u4ed3\u5e93\u7f51\u9875\uff0c\u70b9\u51fb\u201cCreate pull request\u201d\u6309\u94ae\u5373\u53ef\u53d1\u8d77\u62c9\u53d6\u8bf7\u6c42\u3002</li> </ol>"},{"location":"chapter_appendix/contribution/#3-docker","title":"3. \u00a0 Docker \u90e8\u7f72","text":"<p>\u5728 <code>hello-algo</code> \u6839\u76ee\u5f55\u4e0b\uff0c\u6267\u884c\u4ee5\u4e0b Docker \u811a\u672c\uff0c\u5373\u53ef\u5728 <code>http://localhost:8000</code> \u8bbf\u95ee\u672c\u9879\u76ee\uff1a</p> <pre><code>docker-compose up -d\n</code></pre> <p>\u4f7f\u7528\u4ee5\u4e0b\u547d\u4ee4\u5373\u53ef\u5220\u9664\u90e8\u7f72\uff1a</p> <pre><code>docker-compose down\n</code></pre>"},{"location":"chapter_appendix/installation/","title":"16.1 \u00a0 \u7f16\u7a0b\u73af\u5883\u5b89\u88c5","text":""},{"location":"chapter_appendix/installation/#1611-ide","title":"16.1.1 \u00a0 \u5b89\u88c5 IDE","text":"<p>\u63a8\u8350\u4f7f\u7528\u5f00\u6e90\u3001\u8f7b\u91cf\u7684 VSCode \u4f5c\u4e3a\u672c\u5730\u96c6\u6210\u5f00\u53d1\u73af\u5883\uff08IDE\uff09\u3002\u8bbf\u95ee VSCode \u5b98\u7f51\uff0c\u6839\u636e\u64cd\u4f5c\u7cfb\u7edf\u9009\u62e9\u76f8\u5e94\u7248\u672c\u7684 VSCode \u8fdb\u884c\u4e0b\u8f7d\u548c\u5b89\u88c5\u3002</p> <p></p> <p> \u56fe 16-1 \u00a0 \u4ece\u5b98\u7f51\u4e0b\u8f7d VSCode </p> <p>VSCode \u62e5\u6709\u5f3a\u5927\u7684\u6269\u5c55\u5305\u751f\u6001\u7cfb\u7edf\uff0c\u652f\u6301\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u7684\u8fd0\u884c\u548c\u8c03\u8bd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5b89\u88c5\u201cPython Extension Pack\u201d\u6269\u5c55\u5305\u4e4b\u540e\uff0c\u5373\u53ef\u8fdb\u884c Python \u4ee3\u7801\u8c03\u8bd5\u3002\u5b89\u88c5\u6b65\u9aa4\u5982\u56fe 16-2 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 16-2 \u00a0 \u5b89\u88c5 VSCode \u6269\u5c55\u5305 </p>"},{"location":"chapter_appendix/installation/#1612","title":"16.1.2 \u00a0 \u5b89\u88c5\u8bed\u8a00\u73af\u5883","text":""},{"location":"chapter_appendix/installation/#1-python","title":"1. \u00a0 Python \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Miniconda3 \uff0c\u9700\u8981 Python 3.10 \u6216\u66f4\u65b0\u7248\u672c\u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>python</code> \uff0c\u5b89\u88c5 Python Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728\u547d\u4ee4\u884c\u8f93\u5165 <code>pip install black</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#2-cc","title":"2. \u00a0 C/C++ \u73af\u5883","text":"<ol> <li>Windows \u7cfb\u7edf\u9700\u8981\u5b89\u88c5 MinGW\uff08\u914d\u7f6e\u6559\u7a0b\uff09\uff1bMacOS \u81ea\u5e26 Clang \uff0c\u65e0\u987b\u5b89\u88c5\u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>c++</code> \uff0c\u5b89\u88c5 C/C++ Extension Pack \u3002</li> <li>\uff08\u53ef\u9009\uff09\u6253\u5f00 Settings \u9875\u9762\uff0c\u641c\u7d22 <code>Clang_format_fallback Style</code> \u4ee3\u7801\u683c\u5f0f\u5316\u9009\u9879\uff0c\u8bbe\u7f6e\u4e3a <code>{ BasedOnStyle: Microsoft, BreakBeforeBraces: Attach }</code> \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#3-java","title":"3. \u00a0 Java \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 OpenJDK\uff08\u7248\u672c\u9700\u6ee1\u8db3 &gt; JDK 9\uff09\u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>java</code> \uff0c\u5b89\u88c5 Extension Pack for Java \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#4-c","title":"4. \u00a0 C# \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 .Net 8.0 \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>C# Dev Kit</code> \uff0c\u5b89\u88c5 C# Dev Kit \uff08\u914d\u7f6e\u6559\u7a0b\uff09\u3002</li> <li>\u4e5f\u53ef\u4f7f\u7528 Visual Studio\uff08\u5b89\u88c5\u6559\u7a0b\uff09\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#5-go","title":"5. \u00a0 Go \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 go \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>go</code> \uff0c\u5b89\u88c5 Go \u3002</li> <li>\u6309\u5feb\u6377\u952e <code>Ctrl + Shift + P</code> \u547c\u51fa\u547d\u4ee4\u680f\uff0c\u8f93\u5165 go \uff0c\u9009\u62e9 <code>Go: Install/Update Tools</code> \uff0c\u5168\u90e8\u52fe\u9009\u5e76\u5b89\u88c5\u5373\u53ef\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#6-swift","title":"6. \u00a0 Swift \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Swift \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>swift</code> \uff0c\u5b89\u88c5 Swift for Visual Studio Code \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#7-javascript","title":"7. \u00a0 JavaScript \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 node.js \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>javascript</code> \uff0c\u5b89\u88c5 JavaScript (ES6) code snippets \u3002</li> <li>\uff08\u53ef\u9009\uff09\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>Prettier</code> \uff0c\u5b89\u88c5\u4ee3\u7801\u683c\u5f0f\u5316\u5de5\u5177\u3002</li> </ol>"},{"location":"chapter_appendix/installation/#8-dart","title":"8. \u00a0 Dart \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Dart \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>dart</code> \uff0c\u5b89\u88c5 Dart \u3002</li> </ol>"},{"location":"chapter_appendix/installation/#9-rust","title":"9. \u00a0 Rust \u73af\u5883","text":"<ol> <li>\u4e0b\u8f7d\u5e76\u5b89\u88c5 Rust \u3002</li> <li>\u5728 VSCode \u7684\u63d2\u4ef6\u5e02\u573a\u4e2d\u641c\u7d22 <code>rust</code> \uff0c\u5b89\u88c5 rust-analyzer \u3002</li> </ol>"},{"location":"chapter_appendix/terminology/","title":"16.3 \u00a0 \u672f\u8bed\u8868","text":"<p>\u8868 16-1 \u5217\u51fa\u4e86\u4e66\u4e2d\u51fa\u73b0\u7684\u91cd\u8981\u672f\u8bed\u3002\u5efa\u8bae\u4f60\u540c\u65f6\u8bb0\u4f4f\u5b83\u4eec\u7684\u4e2d\u82f1\u6587\u53eb\u6cd5\uff0c\u4ee5\u4fbf\u9605\u8bfb\u82f1\u6587\u6587\u732e\u3002</p> <p> \u8868 16-1 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u91cd\u8981\u540d\u8bcd </p> \u4e2d\u6587 English \u4e2d\u6587 English \u7b97\u6cd5 algorithm \u5c42\u5e8f\u904d\u5386 level-order traversal \u6570\u636e\u7ed3\u6784 data structure \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal \u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis \u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal \u65f6\u95f4\u590d\u6742\u5ea6 time complexity \u4e8c\u53c9\u641c\u7d22\u6811 binary search tree \u7a7a\u95f4\u590d\u6742\u5ea6 space complexity \u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree \u8fed\u4ee3 iteration \u5e73\u8861\u56e0\u5b50 balance factor \u9012\u5f52 recursion \u5806 heap \u5c3e\u9012\u5f52 tail recursion \u5927\u9876\u5806 max heap \u9012\u5f52\u6811 recursion tree \u5c0f\u9876\u5806 min heap \u5927 big- \u4f18\u5148\u961f\u5217 priority queue \u8bb0\u53f7 notation \u6e10\u8fd1\u4e0a\u754c asymptotic upper bound \u5806\u5316 heapify \u539f\u7801 sign\u2013magnitude \u56fe graph \u53cd\u7801 1\u2019s complement \u9876\u70b9 vertex \u8865\u7801 2\u2019s complement \u65e0\u5411\u56fe undirected graph \u6570\u7ec4 array \u6709\u5411\u56fe directed graph \u7d22\u5f15 index \u8fde\u901a\u56fe connected graph \u94fe\u8868 linked list \u975e\u8fde\u901a\u56fe disconnected graph \u94fe\u8868\u8282\u70b9 linked list node, list node \u6709\u6743\u56fe weighted graph \u5217\u8868 list \u90bb\u63a5 adjacency \u52a8\u6001\u6570\u7ec4 dynamic array \u8def\u5f84 path \u786c\u76d8 hard disk \u5165\u5ea6 in-degree \u5185\u5b58 random-access memory (RAM) \u51fa\u5ea6 out-degree \u7f13\u5b58 cache memory \u90bb\u63a5\u77e9\u9635 adjacency matrix \u7f13\u5b58\u672a\u547d\u4e2d cache miss \u90bb\u63a5\u8868 adjacency list \u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate \u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search \u6808 stack \u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search \u961f\u5217 queue \u4e8c\u5206\u67e5\u627e binary search \u53cc\u5411\u961f\u5217 double-ended queue \u641c\u7d22\u7b97\u6cd5 searching algorithm \u54c8\u5e0c\u8868 hash table \u6392\u5e8f\u7b97\u6cd5 sorting algorithm \u6876 bucket \u9009\u62e9\u6392\u5e8f selection sort \u54c8\u5e0c\u51fd\u6570 hash function \u5192\u6ce1\u6392\u5e8f bubble sort \u54c8\u5e0c\u51b2\u7a81 hash collision \u63d2\u5165\u6392\u5e8f insertion sort \u8d1f\u8f7d\u56e0\u5b50 load factor \u5feb\u901f\u6392\u5e8f quick sort \u94fe\u5f0f\u5730\u5740 separate chaining \u5f52\u5e76\u6392\u5e8f merge sort \u5f00\u653e\u5bfb\u5740 open addressing \u5806\u6392\u5e8f heap sort \u7ebf\u6027\u63a2\u6d4b linear probing \u6876\u6392\u5e8f bucket sort \u61d2\u5220\u9664 lazy deletion \u8ba1\u6570\u6392\u5e8f counting sort \u4e8c\u53c9\u6811 binary tree \u57fa\u6570\u6392\u5e8f radix sort \u6811\u8282\u70b9 tree node \u5206\u6cbb divide and conquer \u5de6\u5b50\u8282\u70b9 left-child node \u6c49\u8bfa\u5854\u95ee\u9898 hanota problem \u53f3\u5b50\u8282\u70b9 right-child node \u56de\u6eaf\u7b97\u6cd5 backtracking algorithm \u7236\u8282\u70b9 parent node \u7ea6\u675f constraint \u5de6\u5b50\u6811 left subtree \u89e3 solution \u53f3\u5b50\u6811 right subtree \u72b6\u6001 state \u6839\u8282\u70b9 root node \u526a\u679d pruning \u53f6\u8282\u70b9 leaf node \u5168\u6392\u5217\u95ee\u9898 permutations problem \u8fb9 edge \u5b50\u96c6\u548c\u95ee\u9898 subset-sum problem \u5c42 level N \u7687\u540e\u95ee\u9898 N-queens problem \u5ea6 degree \u52a8\u6001\u89c4\u5212 dynamic programming \u9ad8\u5ea6 height \u521d\u59cb\u72b6\u6001 initial state \u6df1\u5ea6 depth \u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b state-trasition equation \u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree \u80cc\u5305\u95ee\u9898 knapsack problem \u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 edit distance problem \u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree \u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm \u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree AVL \u6811 AVL tree \u7ea2\u9ed1\u6811 red-black tree"},{"location":"chapter_array_and_linkedlist/","title":"\u7b2c 4 \u7ae0 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u7684\u4e16\u754c\u5982\u540c\u4e00\u5835\u539a\u5b9e\u7684\u7816\u5899\u3002</p> <p>\u6570\u7ec4\u7684\u7816\u5757\u6574\u9f50\u6392\u5217\uff0c\u9010\u4e2a\u7d27\u8d34\u3002\u94fe\u8868\u7684\u7816\u5757\u5206\u6563\u5404\u5904\uff0c\u8fde\u63a5\u7684\u85e4\u8513\u81ea\u7531\u5730\u7a7f\u68ad\u4e8e\u7816\u7f1d\u4e4b\u95f4\u3002</p>"},{"location":"chapter_array_and_linkedlist/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>4.1 \u00a0 \u6570\u7ec4</li> <li>4.2 \u00a0 \u94fe\u8868</li> <li>4.3 \u00a0 \u5217\u8868</li> <li>4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *</li> <li>4.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/","title":"4.1 \u00a0 \u6570\u7ec4","text":"<p>\u300c\u6570\u7ec4 array\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5c06\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\u3002\u6211\u4eec\u5c06\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u79f0\u4e3a\u8be5\u5143\u7d20\u7684\u300c\u7d22\u5f15 index\u300d\u3002\u56fe 4-1 \u5c55\u793a\u4e86\u6570\u7ec4\u7684\u4e3b\u8981\u672f\u8bed\u548c\u6982\u5ff5\u3002</p> <p></p> <p> \u56fe 4-1 \u00a0 \u6570\u7ec4\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p>"},{"location":"chapter_array_and_linkedlist/array/#411","title":"4.1.1 \u00a0 \u6570\u7ec4\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/array/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u6570\u7ec4","text":"<p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9700\u6c42\u9009\u7528\u6570\u7ec4\u7684\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u5f0f\uff1a\u65e0\u521d\u59cb\u503c\u3001\u7ed9\u5b9a\u521d\u59cb\u503c\u3002\u5728\u672a\u6307\u5b9a\u521d\u59cb\u503c\u7684\u60c5\u51b5\u4e0b\uff0c\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4f1a\u5c06\u6570\u7ec4\u5143\u7d20\u521d\u59cb\u5316\u4e3a \\(0\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code># \u521d\u59cb\u5316\u6570\u7ec4\narr: list[int] = [0] * 5  # [ 0, 0, 0, 0, 0 ]\nnums: list[int] = [1, 3, 2, 5, 4]  \n</code></pre> array.cpp<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\n// \u5b58\u50a8\u5728\u6808\u4e0a\nint arr[5];\nint nums[5] = { 1, 3, 2, 5, 4 };\n// \u5b58\u50a8\u5728\u5806\u4e0a\uff08\u9700\u8981\u624b\u52a8\u91ca\u653e\u7a7a\u95f4\uff09\nint* arr1 = new int[5];\nint* nums1 = new int[5] { 1, 3, 2, 5, 4 };\n</code></pre> array.java<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = { 1, 3, 2, 5, 4 };\n</code></pre> array.cs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint[] arr = new int[5]; // { 0, 0, 0, 0, 0 }\nint[] nums = [1, 3, 2, 5, 4];\n</code></pre> array.go<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr [5]int\n// \u5728 Go \u4e2d\uff0c\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[5]int\uff09\u4e3a\u6570\u7ec4\uff0c\u4e0d\u6307\u5b9a\u957f\u5ea6\u65f6\uff08[]int\uff09\u4e3a\u5207\u7247\n// \u7531\u4e8e Go \u7684\u6570\u7ec4\u88ab\u8bbe\u8ba1\u4e3a\u5728\u7f16\u8bd1\u671f\u786e\u5b9a\u957f\u5ea6\uff0c\u56e0\u6b64\u53ea\u80fd\u4f7f\u7528\u5e38\u91cf\u6765\u6307\u5b9a\u957f\u5ea6\n// \u4e3a\u4e86\u65b9\u4fbf\u5b9e\u73b0\u6269\u5bb9 extend() \u65b9\u6cd5\uff0c\u4ee5\u4e0b\u5c06\u5207\u7247\uff08Slice\uff09\u770b\u4f5c\u6570\u7ec4\uff08Array\uff09\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> array.swift<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr = Array(repeating: 0, count: 5) // [0, 0, 0, 0, 0]\nlet nums = [1, 3, 2, 5, 4]\n</code></pre> array.js<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nvar arr = new Array(5).fill(0);\nvar nums = [1, 3, 2, 5, 4];\n</code></pre> array.ts<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: number[] = new Array(5).fill(0);\nlet nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> array.dart<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nList&lt;int&gt; arr = List.filled(5, 0); // [0, 0, 0, 0, 0]\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> array.rs<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nlet arr: Vec&lt;i32&gt; = vec![0; 5]; // [0, 0, 0, 0, 0]\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> array.c<pre><code>/* \u521d\u59cb\u5316\u6570\u7ec4 */\nint arr[5] = { 0 }; // { 0, 0, 0, 0, 0 }\nint nums[5] = { 1, 3, 2, 5, 4 };\n</code></pre> array.zig<pre><code>// \u521d\u59cb\u5316\u6570\u7ec4\nvar arr = [_]i32{0} ** 5; // { 0, 0, 0, 0, 0 }\nvar nums = [_]i32{ 1, 3, 2, 5, 4 };\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u8ba1\u7b97\u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u975e\u5e38\u5bb9\u6613\u3002\u7ed9\u5b9a\u6570\u7ec4\u5185\u5b58\u5730\u5740\uff08\u9996\u5143\u7d20\u5185\u5b58\u5730\u5740\uff09\u548c\u67d0\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u56fe 4-2 \u6240\u793a\u7684\u516c\u5f0f\u8ba1\u7b97\u5f97\u5230\u8be5\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u76f4\u63a5\u8bbf\u95ee\u8be5\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 4-2 \u00a0 \u6570\u7ec4\u5143\u7d20\u7684\u5185\u5b58\u5730\u5740\u8ba1\u7b97 </p> <p>\u89c2\u5bdf\u56fe 4-2 \uff0c\u6211\u4eec\u53d1\u73b0\u6570\u7ec4\u9996\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u4e3a \\(0\\) \uff0c\u8fd9\u4f3c\u4e4e\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\u4f1a\u66f4\u81ea\u7136\u3002\u4f46\u4ece\u5730\u5740\u8ba1\u7b97\u516c\u5f0f\u7684\u89d2\u5ea6\u770b\uff0c\u7d22\u5f15\u672c\u8d28\u4e0a\u662f\u5185\u5b58\u5730\u5740\u7684\u504f\u79fb\u91cf\u3002\u9996\u4e2a\u5143\u7d20\u7684\u5730\u5740\u504f\u79fb\u91cf\u662f \\(0\\) \uff0c\u56e0\u6b64\u5b83\u7684\u7d22\u5f15\u4e3a \\(0\\) \u662f\u5408\u7406\u7684\u3002</p> <p>\u5728\u6570\u7ec4\u4e2d\u8bbf\u95ee\u5143\u7d20\u975e\u5e38\u9ad8\u6548\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u968f\u673a\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def random_access(nums: list[int]) -&gt; int:\n    \"\"\"\u968f\u673a\u8bbf\u95ee\u5143\u7d20\"\"\"\n    # \u5728\u533a\u95f4 [0, len(nums)-1] \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    random_index = random.randint(0, len(nums) - 1)\n    # \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    random_num = nums[random_index]\n    return random_num\n</code></pre> array.cpp<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.java<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int[] nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = ThreadLocalRandom.current().nextInt(0, nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.cs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint RandomAccess(int[] nums) {\n    Random random = new();\n    // \u5728\u533a\u95f4 [0, nums.Length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = random.Next(nums.Length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.go<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums []int) (randomNum int) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    randomIndex := rand.Intn(len(nums))\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    randomNum = nums[randomIndex]\n    return\n}\n</code></pre> array.swift<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunc randomAccess(nums: [Int]) -&gt; Int {\n    // \u5728\u533a\u95f4 [0, nums.count) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let randomIndex = nums.indices.randomElement()!\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let randomNum = nums[randomIndex]\n    return randomNum\n}\n</code></pre> array.js<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums) {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.ts<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfunction randomAccess(nums: number[]): number {\n    // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    const random_index = Math.floor(Math.random() * nums.length);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    const random_num = nums[random_index];\n    return random_num;\n}\n</code></pre> array.dart<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(List&lt;int&gt; nums) {\n  // \u5728\u533a\u95f4 [0, nums.length) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n  int randomIndex = Random().nextInt(nums.length);\n  // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n  int randomNum = nums[randomIndex];\n  return randomNum;\n}\n</code></pre> array.rs<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nfn random_access(nums: &amp;[i32]) -&gt; i32 {\n    // \u5728\u533a\u95f4 [0, nums.len()) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    let random_index = rand::thread_rng().gen_range(0..nums.len());\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    let random_num = nums[random_index];\n    random_num\n}\n</code></pre> array.c<pre><code>/* \u968f\u673a\u8bbf\u95ee\u5143\u7d20 */\nint randomAccess(int *nums, int size) {\n    // \u5728\u533a\u95f4 [0, size) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6570\u5b57\n    int randomIndex = rand() % size;\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    int randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre> array.zig<pre><code>// \u968f\u673a\u8bbf\u95ee\u5143\u7d20\nfn randomAccess(nums: []i32) i32 {\n    // \u5728\u533a\u95f4 [0, nums.len) \u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u6574\u6570\n    var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len);\n    // \u83b7\u53d6\u5e76\u8fd4\u56de\u968f\u673a\u5143\u7d20\n    var randomNum = nums[randomIndex];\n    return randomNum;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#3","title":"3. \u00a0 \u63d2\u5165\u5143\u7d20","text":"<p>\u6570\u7ec4\u5143\u7d20\u5728\u5185\u5b58\u4e2d\u662f\u201c\u7d27\u6328\u7740\u7684\u201d\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u7a7a\u95f4\u518d\u5b58\u653e\u4efb\u4f55\u6570\u636e\u3002\u5982\u56fe 4-3 \u6240\u793a\uff0c\u5982\u679c\u60f3\u5728\u6570\u7ec4\u4e2d\u95f4\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\uff0c\u5219\u9700\u8981\u5c06\u8be5\u5143\u7d20\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u4e4b\u540e\u518d\u628a\u5143\u7d20\u8d4b\u503c\u7ed9\u8be5\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 4-3 \u00a0 \u6570\u7ec4\u63d2\u5165\u5143\u7d20\u793a\u4f8b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u56fa\u5b9a\u7684\uff0c\u56e0\u6b64\u63d2\u5165\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u4f1a\u5bfc\u81f4\u6570\u7ec4\u5c3e\u90e8\u5143\u7d20\u201c\u4e22\u5931\u201d\u3002\u6211\u4eec\u5c06\u8fd9\u4e2a\u95ee\u9898\u7684\u89e3\u51b3\u65b9\u6848\u7559\u5728\u201c\u5217\u8868\u201d\u7ae0\u8282\u4e2d\u8ba8\u8bba\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def insert(nums: list[int], num: int, index: int):\n    \"\"\"\u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\"\"\"\n    # \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in range(len(nums) - 1, index, -1):\n        nums[i] = nums[i - 1]\n    # \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid Insert(int[] nums, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = nums.Length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums []int, num int, index int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i := len(nums) - 1; i &gt; index; i-- {\n        nums[i] = nums[i-1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunc insert(nums: inout [Int], num: Int, index: Int) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).reversed() {\n        nums[i] = nums[i - 1]\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums, num, index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfunction insert(nums: number[], num: number, index: number): void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (let i = nums.length - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 _num */\nvoid insert(List&lt;int&gt; nums, int _num, int index) {\n  // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n  for (var i = nums.length - 1; i &gt; index; i--) {\n    nums[i] = nums[i - 1];\n  }\n  // \u5c06 _num \u8d4b\u7ed9 index \u5904\u5143\u7d20\n  nums[index] = _num;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nfn insert(nums: &amp;mut Vec&lt;i32&gt;, num: i32, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for i in (index + 1..nums.len()).rev() {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num */\nvoid insert(int *nums, int size, int num, int index) {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (int i = size - 1; i &gt; index; i--) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u7684\u7d22\u5f15 index \u5904\u63d2\u5165\u5143\u7d20 num\nfn insert(nums: []i32, num: i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    var i = nums.len - 1;\n    while (i &gt; index) : (i -= 1) {\n        nums[i] = nums[i - 1];\n    }\n    // \u5c06 num \u8d4b\u7ed9 index \u5904\u7684\u5143\u7d20\n    nums[index] = num;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#4","title":"4. \u00a0 \u5220\u9664\u5143\u7d20","text":"<p>\u540c\u7406\uff0c\u5982\u56fe 4-4 \u6240\u793a\uff0c\u82e5\u60f3\u5220\u9664\u7d22\u5f15 \\(i\\) \u5904\u7684\u5143\u7d20\uff0c\u5219\u9700\u8981\u628a\u7d22\u5f15 \\(i\\) \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\u3002</p> <p></p> <p> \u56fe 4-4 \u00a0 \u6570\u7ec4\u5220\u9664\u5143\u7d20\u793a\u4f8b </p> <p>\u8bf7\u6ce8\u610f\uff0c\u5220\u9664\u5143\u7d20\u5b8c\u6210\u540e\uff0c\u539f\u5148\u672b\u5c3e\u7684\u5143\u7d20\u53d8\u5f97\u201c\u65e0\u610f\u4e49\u201d\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u987b\u7279\u610f\u53bb\u4fee\u6539\u5b83\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def remove(nums: list[int], index: int):\n    \"\"\"\u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\"\"\"\n    # \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in range(index, len(nums) - 1):\n        nums[i] = nums[i + 1]\n</code></pre> array.cpp<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.java<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.cs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid Remove(int[] nums, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; nums.Length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.go<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums []int, index int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i := index; i &lt; len(nums)-1; i++ {\n        nums[i] = nums[i+1]\n    }\n}\n</code></pre> array.swift<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunc remove(nums: inout [Int], index: Int) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in nums.indices.dropFirst(index).dropLast() {\n        nums[i] = nums[i + 1]\n    }\n}\n</code></pre> array.js<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums, index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.ts<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfunction remove(nums: number[], index: number): void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (let i = index; i &lt; nums.length - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.dart<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nvoid remove(List&lt;int&gt; nums, int index) {\n  // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n  for (var i = index; i &lt; nums.length - 1; i++) {\n    nums[i] = nums[i + 1];\n  }\n}\n</code></pre> array.rs<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\nfn remove(nums: &amp;mut Vec&lt;i32&gt;, index: usize) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for i in index..nums.len() - 1 {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.c<pre><code>/* \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(int *nums, int size, int index) {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (int i = index; i &lt; size - 1; i++) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> array.zig<pre><code>// \u5220\u9664\u7d22\u5f15 index \u5904\u7684\u5143\u7d20\nfn remove(nums: []i32, index: usize) void {\n    // \u628a\u7d22\u5f15 index \u4e4b\u540e\u7684\u6240\u6709\u5143\u7d20\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    var i = index;\n    while (i &lt; nums.len - 1) : (i += 1) {\n        nums[i] = nums[i + 1];\n    }\n}\n</code></pre> <p>\u603b\u7684\u6765\u770b\uff0c\u6570\u7ec4\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u6709\u4ee5\u4e0b\u7f3a\u70b9\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u9ad8\uff1a\u6570\u7ec4\u7684\u63d2\u5165\u548c\u5220\u9664\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u957f\u5ea6\u3002</li> <li>\u4e22\u5931\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u5728\u63d2\u5165\u5143\u7d20\u540e\uff0c\u8d85\u51fa\u6570\u7ec4\u957f\u5ea6\u8303\u56f4\u7684\u5143\u7d20\u4f1a\u4e22\u5931\u3002</li> <li>\u5185\u5b58\u6d6a\u8d39\uff1a\u6211\u4eec\u53ef\u4ee5\u521d\u59cb\u5316\u4e00\u4e2a\u6bd4\u8f83\u957f\u7684\u6570\u7ec4\uff0c\u53ea\u7528\u524d\u9762\u4e00\u90e8\u5206\uff0c\u8fd9\u6837\u5728\u63d2\u5165\u6570\u636e\u65f6\uff0c\u4e22\u5931\u7684\u672b\u5c3e\u5143\u7d20\u90fd\u662f\u201c\u65e0\u610f\u4e49\u201d\u7684\uff0c\u4f46\u8fd9\u6837\u505a\u4f1a\u9020\u6210\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#5","title":"5. \u00a0 \u904d\u5386\u6570\u7ec4","text":"<p>\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e2\u53ef\u4ee5\u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u83b7\u53d6\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def traverse(nums: list[int]):\n    \"\"\"\u904d\u5386\u6570\u7ec4\"\"\"\n    count = 0\n    # \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in range(len(nums)):\n        count += nums[i]\n    # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums:\n        count += num\n    # \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num in enumerate(nums):\n        count += nums[i]\n        count += num\n</code></pre> array.cpp<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.java<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (int num : nums) {\n        count += num;\n    }\n}\n</code></pre> array.cs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid Traverse(int[] nums) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    foreach (int num in nums) {\n        count += num;\n    }\n}\n</code></pre> array.go<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums []int) {\n    count := 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; len(nums); i++ {\n        count += nums[i]\n    }\n    count = 0\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for _, num := range nums {\n        count += num\n    }\n    // \u540c\u65f6\u904d\u5386\u6570\u636e\u7d22\u5f15\u548c\u5143\u7d20\n    for i, num := range nums {\n        count += nums[i]\n        count += num\n    }\n}\n</code></pre> array.swift<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunc traverse(nums: [Int]) {\n    var count = 0\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in nums.indices {\n        count += nums[i]\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        count += num\n    }\n}\n</code></pre> array.js<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums) {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.ts<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfunction traverse(nums: number[]): void {\n    let count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (const num of nums) {\n        count += num;\n    }\n}\n</code></pre> array.dart<pre><code>/* \u904d\u5386\u6570\u7ec4\u5143\u7d20 */\nvoid traverse(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n  }\n  // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n  for (int _num in nums) {\n    count += _num;\n  }\n  // \u901a\u8fc7 forEach \u65b9\u6cd5\u904d\u5386\u6570\u7ec4\n  nums.forEach((_num) {\n    count += _num;\n  });\n}\n</code></pre> array.rs<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nfn traverse(nums: &amp;[i32]) {\n    let mut _count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for i in 0..nums.len() {\n        _count += nums[i];\n    }\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for num in nums {\n        _count += num;\n    }\n}\n</code></pre> array.c<pre><code>/* \u904d\u5386\u6570\u7ec4 */\nvoid traverse(int *nums, int size) {\n    int count = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        count += nums[i];\n    }\n}\n</code></pre> array.zig<pre><code>// \u904d\u5386\u6570\u7ec4\nfn traverse(nums: []i32) void {\n    var count: i32 = 0;\n    // \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u6570\u7ec4\n    var i: i32 = 0;\n    while (i &lt; nums.len) : (i += 1) {\n        count += nums[i];\n    }\n    count = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\u5143\u7d20\n    for (nums) |num| {\n        count += num;\n    }\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#6","title":"6. \u00a0 \u67e5\u627e\u5143\u7d20","text":"<p>\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\u9700\u8981\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u5224\u65ad\u5143\u7d20\u503c\u662f\u5426\u5339\u914d\uff0c\u82e5\u5339\u914d\u5219\u8f93\u51fa\u5bf9\u5e94\u7d22\u5f15\u3002</p> <p>\u56e0\u4e3a\u6570\u7ec4\u662f\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u6240\u4ee5\u4e0a\u8ff0\u67e5\u627e\u64cd\u4f5c\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u67e5\u627e\u201d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def find(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\"\"\"\n    for i in range(len(nums)):\n        if nums[i] == target:\n            return i\n    return -1\n</code></pre> array.cpp<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.java<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.cs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint Find(int[] nums, int target) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.go<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums []int, target int) (index int) {\n    index = -1\n    for i := 0; i &lt; len(nums); i++ {\n        if nums[i] == target {\n            index = i\n            break\n        }\n    }\n    return\n}\n</code></pre> array.swift<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunc find(nums: [Int], target: Int) -&gt; Int {\n    for i in nums.indices {\n        if nums[i] == target {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> array.js<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums, target) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) return i;\n    }\n    return -1;\n}\n</code></pre> array.ts<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfunction find(nums: number[], target: number): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (nums[i] === target) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> array.dart<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(List&lt;int&gt; nums, int target) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    if (nums[i] == target) return i;\n  }\n  return -1;\n}\n</code></pre> array.rs<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nfn find(nums: &amp;[i32], target: i32) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        if nums[i] == target {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> array.c<pre><code>/* \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20 */\nint find(int *nums, int size, int target) {\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] == target)\n            return i;\n    }\n    return -1;\n}\n</code></pre> array.zig<pre><code>// \u5728\u6570\u7ec4\u4e2d\u67e5\u627e\u6307\u5b9a\u5143\u7d20\nfn find(nums: []i32, target: i32) i32 {\n    for (nums, 0..) |num, i| {\n        if (num == target) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#7","title":"7. \u00a0 \u6269\u5bb9\u6570\u7ec4","text":"<p>\u5728\u590d\u6742\u7684\u7cfb\u7edf\u73af\u5883\u4e2d\uff0c\u7a0b\u5e8f\u96be\u4ee5\u4fdd\u8bc1\u6570\u7ec4\u4e4b\u540e\u7684\u5185\u5b58\u7a7a\u95f4\u662f\u53ef\u7528\u7684\uff0c\u4ece\u800c\u65e0\u6cd5\u5b89\u5168\u5730\u6269\u5c55\u6570\u7ec4\u5bb9\u91cf\u3002\u56e0\u6b64\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6570\u7ec4\u7684\u957f\u5ea6\u662f\u4e0d\u53ef\u53d8\u7684\u3002</p> <p>\u5982\u679c\u6211\u4eec\u5e0c\u671b\u6269\u5bb9\u6570\u7ec4\uff0c\u5219\u9700\u91cd\u65b0\u5efa\u7acb\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u7136\u540e\u628a\u539f\u6570\u7ec4\u5143\u7d20\u4f9d\u6b21\u590d\u5236\u5230\u65b0\u6570\u7ec4\u3002\u8fd9\u662f\u4e00\u4e2a \\(O(n)\\) \u7684\u64cd\u4f5c\uff0c\u5728\u6570\u7ec4\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\u975e\u5e38\u8017\u65f6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array.py<pre><code>def extend(nums: list[int], enlarge: int) -&gt; list[int]:\n    \"\"\"\u6269\u5c55\u6570\u7ec4\u957f\u5ea6\"\"\"\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res = [0] * (len(nums) + enlarge)\n    # \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in range(len(nums)):\n        res[i] = nums[i]\n    # \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n</code></pre> array.cpp<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = new int[size + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    delete[] nums;\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.java<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.cs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint[] Extend(int[] nums, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int[] res = new int[nums.Length + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; nums.Length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.go<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums []int, enlarge int) []int {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    res := make([]int, len(nums)+enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i, num := range nums {\n        res[i] = num\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.swift<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfunc extend(nums: [Int], enlarge: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = Array(repeating: 0, count: nums.count + enlarge)\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for i in nums.indices {\n        res[i] = nums[i]\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res\n}\n</code></pre> array.js<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cJavaScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums, enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.ts<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\n// \u8bf7\u6ce8\u610f\uff0cTypeScript \u7684 Array \u662f\u52a8\u6001\u6570\u7ec4\uff0c\u53ef\u4ee5\u76f4\u63a5\u6269\u5c55\n// \u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60\uff0c\u672c\u51fd\u6570\u5c06 Array \u770b\u4f5c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6570\u7ec4\nfunction extend(nums: number[], enlarge: number): number[] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    const res = new Array(nums.length + enlarge).fill(0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (let i = 0; i &lt; nums.length; i++) {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.dart<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nList&lt;int&gt; extend(List&lt;int&gt; nums, int enlarge) {\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n  List&lt;int&gt; res = List.filled(nums.length + enlarge, 0);\n  // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n  for (var i = 0; i &lt; nums.length; i++) {\n    res[i] = nums[i];\n  }\n  // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n  return res;\n}\n</code></pre> array.rs<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nfn extend(nums: Vec&lt;i32&gt;, enlarge: usize) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    let mut res: Vec&lt;i32&gt; = vec![0; nums.len() + enlarge];\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\n    for i in 0..nums.len() {\n        res[i] = nums[i];\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    res\n}\n</code></pre> array.c<pre><code>/* \u6269\u5c55\u6570\u7ec4\u957f\u5ea6 */\nint *extend(int *nums, int size, int enlarge) {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    int *res = (int *)malloc(sizeof(int) * (size + enlarge));\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    for (int i = 0; i &lt; size; i++) {\n        res[i] = nums[i];\n    }\n    // \u521d\u59cb\u5316\u6269\u5c55\u540e\u7684\u7a7a\u95f4\n    for (int i = size; i &lt; size + enlarge; i++) {\n        res[i] = 0;\n    }\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre> array.zig<pre><code>// \u6269\u5c55\u6570\u7ec4\u957f\u5ea6\nfn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u6269\u5c55\u957f\u5ea6\u540e\u7684\u6570\u7ec4\n    var res = try mem_allocator.alloc(i32, nums.len + enlarge);\n    @memset(res, 0);\n    // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n    std.mem.copy(i32, res, nums);\n    // \u8fd4\u56de\u6269\u5c55\u540e\u7684\u65b0\u6570\u7ec4\n    return res;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/array/#412","title":"4.1.2 \u00a0 \u6570\u7ec4\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u4e14\u5143\u7d20\u7c7b\u578b\u76f8\u540c\u3002\u8fd9\u79cd\u505a\u6cd5\u5305\u542b\u4e30\u5bcc\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u7cfb\u7edf\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u4fe1\u606f\u6765\u4f18\u5316\u6570\u636e\u7ed3\u6784\u7684\u64cd\u4f5c\u6548\u7387\u3002</p> <ul> <li>\u7a7a\u95f4\u6548\u7387\u9ad8\uff1a\u6570\u7ec4\u4e3a\u6570\u636e\u5206\u914d\u4e86\u8fde\u7eed\u7684\u5185\u5b58\u5757\uff0c\u65e0\u987b\u989d\u5916\u7684\u7ed3\u6784\u5f00\u9500\u3002</li> <li>\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff1a\u6570\u7ec4\u5141\u8bb8\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u4efb\u4f55\u5143\u7d20\u3002</li> <li>\u7f13\u5b58\u5c40\u90e8\u6027\uff1a\u5f53\u8bbf\u95ee\u6570\u7ec4\u5143\u7d20\u65f6\uff0c\u8ba1\u7b97\u673a\u4e0d\u4ec5\u4f1a\u52a0\u8f7d\u5b83\uff0c\u8fd8\u4f1a\u7f13\u5b58\u5176\u5468\u56f4\u7684\u5176\u4ed6\u6570\u636e\uff0c\u4ece\u800c\u501f\u52a9\u9ad8\u901f\u7f13\u5b58\u6765\u63d0\u5347\u540e\u7eed\u64cd\u4f5c\u7684\u6267\u884c\u901f\u5ea6\u3002</li> </ul> <p>\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u662f\u4e00\u628a\u53cc\u5203\u5251\uff0c\u5176\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u63d2\u5165\u4e0e\u5220\u9664\u6548\u7387\u4f4e\uff1a\u5f53\u6570\u7ec4\u4e2d\u5143\u7d20\u8f83\u591a\u65f6\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20\u3002</li> <li>\u957f\u5ea6\u4e0d\u53ef\u53d8\uff1a\u6570\u7ec4\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u5c31\u56fa\u5b9a\u4e86\uff0c\u6269\u5bb9\u6570\u7ec4\u9700\u8981\u5c06\u6240\u6709\u6570\u636e\u590d\u5236\u5230\u65b0\u6570\u7ec4\uff0c\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u7a7a\u95f4\u6d6a\u8d39\uff1a\u5982\u679c\u6570\u7ec4\u5206\u914d\u7684\u5927\u5c0f\u8d85\u8fc7\u5b9e\u9645\u6240\u9700\uff0c\u90a3\u4e48\u591a\u4f59\u7684\u7a7a\u95f4\u5c31\u88ab\u6d6a\u8d39\u4e86\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/array/#413","title":"4.1.3 \u00a0 \u6570\u7ec4\u5178\u578b\u5e94\u7528","text":"<p>\u6570\u7ec4\u662f\u4e00\u79cd\u57fa\u7840\u4e14\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\uff0c\u65e2\u9891\u7e41\u5e94\u7528\u5728\u5404\u7c7b\u7b97\u6cd5\u4e4b\u4e2d\uff0c\u4e5f\u53ef\u7528\u4e8e\u5b9e\u73b0\u5404\u79cd\u590d\u6742\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1a\u5982\u679c\u6211\u4eec\u60f3\u968f\u673a\u62bd\u53d6\u4e00\u4e9b\u6837\u672c\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u6570\u7ec4\u5b58\u50a8\uff0c\u5e76\u751f\u6210\u4e00\u4e2a\u968f\u673a\u5e8f\u5217\uff0c\u6839\u636e\u7d22\u5f15\u5b9e\u73b0\u968f\u673a\u62bd\u6837\u3002</li> <li>\u6392\u5e8f\u548c\u641c\u7d22\uff1a\u6570\u7ec4\u662f\u6392\u5e8f\u548c\u641c\u7d22\u7b97\u6cd5\u6700\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u4e8c\u5206\u67e5\u627e\u7b49\u90fd\u4e3b\u8981\u5728\u6570\u7ec4\u4e0a\u8fdb\u884c\u3002</li> <li>\u67e5\u627e\u8868\uff1a\u5f53\u9700\u8981\u5feb\u901f\u67e5\u627e\u4e00\u4e2a\u5143\u7d20\u6216\u5176\u5bf9\u5e94\u5173\u7cfb\u65f6\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u4f5c\u4e3a\u67e5\u627e\u8868\u3002\u5047\u5982\u6211\u4eec\u60f3\u5b9e\u73b0\u5b57\u7b26\u5230 ASCII \u7801\u7684\u6620\u5c04\uff0c\u5219\u53ef\u4ee5\u5c06\u5b57\u7b26\u7684 ASCII \u7801\u503c\u4f5c\u4e3a\u7d22\u5f15\uff0c\u5bf9\u5e94\u7684\u5143\u7d20\u5b58\u653e\u5728\u6570\u7ec4\u4e2d\u7684\u5bf9\u5e94\u4f4d\u7f6e\u3002</li> <li>\u673a\u5668\u5b66\u4e60\uff1a\u795e\u7ecf\u7f51\u7edc\u4e2d\u5927\u91cf\u4f7f\u7528\u4e86\u5411\u91cf\u3001\u77e9\u9635\u3001\u5f20\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u4ee3\u6570\u8fd0\u7b97\uff0c\u8fd9\u4e9b\u6570\u636e\u90fd\u662f\u4ee5\u6570\u7ec4\u7684\u5f62\u5f0f\u6784\u5efa\u7684\u3002\u6570\u7ec4\u662f\u795e\u7ecf\u7f51\u7edc\u7f16\u7a0b\u4e2d\u6700\u5e38\u4f7f\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff1a\u6570\u7ec4\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002\u4f8b\u5982\uff0c\u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/linked_list/","title":"4.2 \u00a0 \u94fe\u8868","text":"<p>\u5185\u5b58\u7a7a\u95f4\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u516c\u5171\u8d44\u6e90\uff0c\u5728\u4e00\u4e2a\u590d\u6742\u7684\u7cfb\u7edf\u8fd0\u884c\u73af\u5883\u4e0b\uff0c\u7a7a\u95f2\u7684\u5185\u5b58\u7a7a\u95f4\u53ef\u80fd\u6563\u843d\u5728\u5185\u5b58\u5404\u5904\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5b58\u50a8\u6570\u7ec4\u7684\u5185\u5b58\u7a7a\u95f4\u5fc5\u987b\u662f\u8fde\u7eed\u7684\uff0c\u800c\u5f53\u6570\u7ec4\u975e\u5e38\u5927\u65f6\uff0c\u5185\u5b58\u53ef\u80fd\u65e0\u6cd5\u63d0\u4f9b\u5982\u6b64\u5927\u7684\u8fde\u7eed\u7a7a\u95f4\u3002\u6b64\u65f6\u94fe\u8868\u7684\u7075\u6d3b\u6027\u4f18\u52bf\u5c31\u4f53\u73b0\u51fa\u6765\u4e86\u3002</p> <p>\u300c\u94fe\u8868 linked list\u300d\u662f\u4e00\u79cd\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f\u4e00\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u5404\u4e2a\u8282\u70b9\u901a\u8fc7\u201c\u5f15\u7528\u201d\u76f8\u8fde\u63a5\u3002\u5f15\u7528\u8bb0\u5f55\u4e86\u4e0b\u4e00\u4e2a\u8282\u70b9\u7684\u5185\u5b58\u5730\u5740\uff0c\u901a\u8fc7\u5b83\u53ef\u4ee5\u4ece\u5f53\u524d\u8282\u70b9\u8bbf\u95ee\u5230\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</p> <p>\u94fe\u8868\u7684\u8bbe\u8ba1\u4f7f\u5f97\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\uff0c\u5b83\u4eec\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p></p> <p> \u56fe 4-5 \u00a0 \u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f </p> <p>\u89c2\u5bdf\u56fe 4-5 \uff0c\u94fe\u8868\u7684\u7ec4\u6210\u5355\u4f4d\u662f\u300c\u8282\u70b9 node\u300d\u5bf9\u8c61\u3002\u6bcf\u4e2a\u8282\u70b9\u90fd\u5305\u542b\u4e24\u9879\u6570\u636e\uff1a\u8282\u70b9\u7684\u201c\u503c\u201d\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u201c\u5f15\u7528\u201d\u3002</p> <ul> <li>\u94fe\u8868\u7684\u9996\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5934\u8282\u70b9\u201d\uff0c\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u88ab\u79f0\u4e3a\u201c\u5c3e\u8282\u70b9\u201d\u3002</li> <li>\u5c3e\u8282\u70b9\u6307\u5411\u7684\u662f\u201c\u7a7a\u201d\uff0c\u5b83\u5728 Java\u3001C++ \u548c Python \u4e2d\u5206\u522b\u88ab\u8bb0\u4e3a \\(\\text{null}\\)\u3001\\(\\text{nullptr}\\) \u548c \\(\\text{None}\\) \u3002</li> <li>\u5728 C\u3001C++\u3001Go \u548c Rust \u7b49\u652f\u6301\u6307\u9488\u7684\u8bed\u8a00\u4e2d\uff0c\u4e0a\u8ff0\u7684\u201c\u5f15\u7528\u201d\u5e94\u88ab\u66ff\u6362\u4e3a\u201c\u6307\u9488\u201d\u3002</li> </ul> <p>\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u94fe\u8868\u8282\u70b9 <code>ListNode</code> \u9664\u4e86\u5305\u542b\u503c\uff0c\u8fd8\u9700\u989d\u5916\u4fdd\u5b58\u4e00\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u56e0\u6b64\u5728\u76f8\u540c\u6570\u636e\u91cf\u4e0b\uff0c\u94fe\u8868\u6bd4\u6570\u7ec4\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class ListNode:\n    \"\"\"\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val               # \u8282\u70b9\u503c\n        self.next: ListNode | None = None # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n    int val;         // \u8282\u70b9\u503c\n    ListNode *next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n    ListNode(int x) : val(x), next(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  //\u6784\u9020\u51fd\u6570\n    int val = x;         // \u8282\u70b9\u503c\n    ListNode? next;      // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype ListNode struct {\n    Val  int       // \u8282\u70b9\u503c\n    Next *ListNode // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewListNode \u6784\u9020\u51fd\u6570\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u94fe\u8868\nfunc NewListNode(val int) *ListNode {\n    return &amp;ListNode{\n        Val:  val,\n        Next: nil,\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\n    init(x: Int) { // \u6784\u9020\u51fd\u6570\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    constructor(val, next) {\n        this.val = (val === undefined ? 0 : val);       // \u8282\u70b9\u503c\n        this.next = (next === undefined ? null : next); // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    constructor(val?: number, next?: ListNode | null) {\n        this.val = val === undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next === undefined ? null : next;  // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n  ListNode(this.val, [this.next]); // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n/* \u94fe\u8868\u8282\u70b9\u7c7b */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u8282\u70b9\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n}\n</code></pre> <pre><code>/* \u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n    int val;               // \u8282\u70b9\u503c\n    struct ListNode *next; // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    return node;\n}\n</code></pre> <pre><code>// \u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u8282\u70b9\u503c\n        next: ?*Self = null, // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u6307\u9488\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#421","title":"4.2.1 \u00a0 \u94fe\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/linked_list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u94fe\u8868","text":"<p>\u5efa\u7acb\u94fe\u8868\u5206\u4e3a\u4e24\u6b65\uff0c\u7b2c\u4e00\u6b65\u662f\u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\u5bf9\u8c61\uff0c\u7b2c\u4e8c\u6b65\u662f\u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u5173\u7cfb\u3002\u521d\u59cb\u5316\u5b8c\u6210\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u94fe\u8868\u7684\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u901a\u8fc7\u5f15\u7528\u6307\u5411 <code>next</code> \u4f9d\u6b21\u8bbf\u95ee\u6240\u6709\u8282\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code># \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4\n# \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 = ListNode(1)\nn1 = ListNode(3)\nn2 = ListNode(2)\nn3 = ListNode(5)\nn4 = ListNode(4)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.cpp<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = new ListNode(1);\nListNode* n1 = new ListNode(3);\nListNode* n2 = new ListNode(2);\nListNode* n3 = new ListNode(5);\nListNode* n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.java<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new ListNode(1);\nListNode n1 = new ListNode(3);\nListNode n2 = new ListNode(2);\nListNode n3 = new ListNode(5);\nListNode n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.cs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = new(1);\nListNode n1 = new(3);\nListNode n2 = new(2);\nListNode n3 = new(5);\nListNode n4 = new(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.go<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nn0 := NewListNode(1)\nn1 := NewListNode(3)\nn2 := NewListNode(2)\nn3 := NewListNode(5)\nn4 := NewListNode(4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.Next = n1\nn1.Next = n2\nn2.Next = n3\nn3.Next = n4\n</code></pre> linked_list.swift<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = ListNode(x: 1)\nlet n1 = ListNode(x: 3)\nlet n2 = ListNode(x: 2)\nlet n3 = ListNode(x: 5)\nlet n4 = ListNode(x: 4)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1\nn1.next = n2\nn2.next = n3\nn3.next = n4\n</code></pre> linked_list.js<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.ts<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nconst n0 = new ListNode(1);\nconst n1 = new ListNode(3);\nconst n2 = new ListNode(2);\nconst n3 = new ListNode(5);\nconst n4 = new ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.dart<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\\\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode n0 = ListNode(1);\nListNode n1 = ListNode(3);\nListNode n2 = ListNode(2);\nListNode n3 = ListNode(5);\nListNode n4 = ListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = n1;\nn1.next = n2;\nn2.next = n3;\nn3.next = n4;\n</code></pre> linked_list.rs<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nlet n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));\nlet n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));\nlet n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));\nlet n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));\nlet n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));\n\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.borrow_mut().next = Some(n1.clone());\nn1.borrow_mut().next = Some(n2.clone());\nn2.borrow_mut().next = Some(n3.clone());\nn3.borrow_mut().next = Some(n4.clone());\n</code></pre> linked_list.c<pre><code>/* \u521d\u59cb\u5316\u94fe\u8868 1 -&gt; 3 -&gt; 2 -&gt; 5 -&gt; 4 */\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nListNode* n0 = newListNode(1);\nListNode* n1 = newListNode(3);\nListNode* n2 = newListNode(2);\nListNode* n3 = newListNode(5);\nListNode* n4 = newListNode(4);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0-&gt;next = n1;\nn1-&gt;next = n2;\nn2-&gt;next = n3;\nn3-&gt;next = n4;\n</code></pre> linked_list.zig<pre><code>// \u521d\u59cb\u5316\u94fe\u8868\n// \u521d\u59cb\u5316\u5404\u4e2a\u8282\u70b9\nvar n0 = inc.ListNode(i32){.val = 1};\nvar n1 = inc.ListNode(i32){.val = 3};\nvar n2 = inc.ListNode(i32){.val = 2};\nvar n3 = inc.ListNode(i32){.val = 5};\nvar n4 = inc.ListNode(i32){.val = 4};\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\nn0.next = &amp;n1;\nn1.next = &amp;n2;\nn2.next = &amp;n3;\nn3.next = &amp;n4;\n</code></pre> <p>\u6570\u7ec4\u6574\u4f53\u662f\u4e00\u4e2a\u53d8\u91cf\uff0c\u6bd4\u5982\u6570\u7ec4 <code>nums</code> \u5305\u542b\u5143\u7d20 <code>nums[0]</code> \u548c <code>nums[1]</code> \u7b49\uff0c\u800c\u94fe\u8868\u662f\u7531\u591a\u4e2a\u72ec\u7acb\u7684\u8282\u70b9\u5bf9\u8c61\u7ec4\u6210\u7684\u3002\u6211\u4eec\u901a\u5e38\u5c06\u5934\u8282\u70b9\u5f53\u4f5c\u94fe\u8868\u7684\u4ee3\u79f0\uff0c\u6bd4\u5982\u4ee5\u4e0a\u4ee3\u7801\u4e2d\u7684\u94fe\u8868\u53ef\u8bb0\u4f5c\u94fe\u8868 <code>n0</code> \u3002</p>"},{"location":"chapter_array_and_linkedlist/linked_list/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u8282\u70b9\u975e\u5e38\u5bb9\u6613\u3002\u5982\u56fe 4-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u60f3\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u8282\u70b9 <code>n0</code> \u548c <code>n1</code> \u4e4b\u95f4\u63d2\u5165\u4e00\u4e2a\u65b0\u8282\u70b9 <code>P</code> \uff0c\u5219\u53ea\u9700\u6539\u53d8\u4e24\u4e2a\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> <p>\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5728\u6570\u7ec4\u4e2d\u63d2\u5165\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u6548\u7387\u8f83\u4f4e\u3002</p> <p></p> <p> \u56fe 4-6 \u00a0 \u94fe\u8868\u63d2\u5165\u8282\u70b9\u793a\u4f8b </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def insert(n0: ListNode, P: ListNode):\n    \"\"\"\u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\"\"\"\n    n1 = n0.next\n    P.next = n1\n    n0.next = P\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n    ListNode n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid Insert(ListNode n0, ListNode P) {\n    ListNode? n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insertNode(n0 *ListNode, P *ListNode) {\n    n1 := n0.Next\n    P.Next = n1\n    n0.Next = P\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunc insert(n0: ListNode, P: ListNode) {\n    let n1 = n0.next\n    P.next = n1\n    n0.next = P\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0, P) {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nfunction insert(n0: ListNode, P: ListNode): void {\n    const n1 = n0.next;\n    P.next = n1;\n    n0.next = P;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode n0, ListNode P) {\n  ListNode? n1 = n0.next;\n  P.next = n1;\n  n0.next = P;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\n#[allow(non_snake_case)]\npub fn insert&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, P: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    let n1 =  n0.borrow_mut().next.take();\n    P.borrow_mut().next = n1;\n    n0.borrow_mut().next = Some(P);\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P */\nvoid insert(ListNode *n0, ListNode *P) {\n    ListNode *n1 = n0-&gt;next;\n    P-&gt;next = n1;\n    n0-&gt;next = P;\n}\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u63d2\u5165\u8282\u70b9 P\nfn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {\n    var n1 = n0.?.next;\n    P.?.next = n1;\n    n0.?.next = P;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5982\u56fe 4-7 \u6240\u793a\uff0c\u5728\u94fe\u8868\u4e2d\u5220\u9664\u8282\u70b9\u4e5f\u975e\u5e38\u65b9\u4fbf\uff0c\u53ea\u9700\u6539\u53d8\u4e00\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5c3d\u7ba1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\u8282\u70b9 <code>P</code> \u4ecd\u7136\u6307\u5411 <code>n1</code> \uff0c\u4f46\u5b9e\u9645\u4e0a\u904d\u5386\u6b64\u94fe\u8868\u5df2\u7ecf\u65e0\u6cd5\u8bbf\u95ee\u5230 <code>P</code> \uff0c\u8fd9\u610f\u5473\u7740 <code>P</code> \u5df2\u7ecf\u4e0d\u518d\u5c5e\u4e8e\u8be5\u94fe\u8868\u4e86\u3002</p> <p></p> <p> \u56fe 4-7 \u00a0 \u94fe\u8868\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def remove(n0: ListNode):\n    \"\"\"\u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    if not n0.next:\n        return\n    # n0 -&gt; P -&gt; n1\n    P = n0.next\n    n1 = P.next\n    n0.next = n1\n</code></pre> linked_list.cpp<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode *n0) {\n    if (n0-&gt;next == nullptr)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    delete P;\n}\n</code></pre> linked_list.java<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid Remove(ListNode n0) {\n    if (n0.next == null)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode P = n0.next;\n    ListNode? n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc removeItem(n0 *ListNode) {\n    if n0.Next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    P := n0.Next\n    n1 := P.Next\n    n0.Next = n1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunc remove(n0: ListNode) {\n    if n0.next == nil {\n        return\n    }\n    // n0 -&gt; P -&gt; n1\n    let P = n0.next\n    let n1 = P?.next\n    n0.next = n1\n    P?.next = nil\n}\n</code></pre> linked_list.js<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0) {\n    if (!n0.next) return;\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nfunction remove(n0: ListNode): void {\n    if (!n0.next) {\n        return;\n    }\n    // n0 -&gt; P -&gt; n1\n    const P = n0.next;\n    const n1 = P.next;\n    n0.next = n1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\nvoid remove(ListNode n0) {\n  if (n0.next == null) return;\n  // n0 -&gt; P -&gt; n1\n  ListNode P = n0.next!;\n  ListNode? n1 = P.next;\n  n0.next = n1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n#[allow(non_snake_case)]\npub fn remove&lt;T&gt;(n0: &amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;) {\n    if n0.borrow().next.is_none() {return};\n    // n0 -&gt; P -&gt; n1\n    let P = n0.borrow_mut().next.take();\n    if let Some(node) = P {\n        let n1 = node.borrow_mut().next.take();\n        n0.borrow_mut().next = n1;\n    }\n}\n</code></pre> linked_list.c<pre><code>/* \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nvoid removeItem(ListNode *n0) {\n    if (!n0-&gt;next)\n        return;\n    // n0 -&gt; P -&gt; n1\n    ListNode *P = n0-&gt;next;\n    ListNode *n1 = P-&gt;next;\n    n0-&gt;next = n1;\n    // \u91ca\u653e\u5185\u5b58\n    free(P);\n}\n</code></pre> linked_list.zig<pre><code>// \u5220\u9664\u94fe\u8868\u7684\u8282\u70b9 n0 \u4e4b\u540e\u7684\u9996\u4e2a\u8282\u70b9\nfn remove(n0: ?*inc.ListNode(i32)) void {\n    if (n0.?.next == null) return;\n    // n0 -&gt; P -&gt; n1\n    var P = n0.?.next;\n    var n1 = P.?.next;\n    n0.?.next = n1;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#4","title":"4. \u00a0 \u8bbf\u95ee\u8282\u70b9","text":"<p>\u5728\u94fe\u8868\u4e2d\u8bbf\u95ee\u8282\u70b9\u7684\u6548\u7387\u8f83\u4f4e\u3002\u5982\u4e0a\u4e00\u8282\u6240\u8ff0\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u4e0b\u8bbf\u95ee\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u5143\u7d20\u3002\u94fe\u8868\u5219\u4e0d\u7136\uff0c\u7a0b\u5e8f\u9700\u8981\u4ece\u5934\u8282\u70b9\u51fa\u53d1\uff0c\u9010\u4e2a\u5411\u540e\u904d\u5386\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u8282\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8bbf\u95ee\u94fe\u8868\u7684\u7b2c \\(i\\) \u4e2a\u8282\u70b9\u9700\u8981\u5faa\u73af \\(i - 1\\) \u8f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def access(head: ListNode, index: int) -&gt; ListNode | None:\n    \"\"\"\u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\"\"\"\n    for _ in range(index):\n        if not head:\n            return None\n        head = head.next\n    return head\n</code></pre> linked_list.cpp<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == nullptr)\n            return nullptr;\n        head = head-&gt;next;\n    }\n    return head;\n}\n</code></pre> linked_list.java<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode access(ListNode head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.cs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? Access(ListNode? head, int index) {\n    for (int i = 0; i &lt; index; i++) {\n        if (head == null)\n            return null;\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.go<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head *ListNode, index int) *ListNode {\n    for i := 0; i &lt; index; i++ {\n        if head == nil {\n            return nil\n        }\n        head = head.Next\n    }\n    return head\n}\n</code></pre> linked_list.swift<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunc access(head: ListNode, index: Int) -&gt; ListNode? {\n    var head: ListNode? = head\n    for _ in 0 ..&lt; index {\n        if head == nil {\n            return nil\n        }\n        head = head?.next\n    }\n    return head\n}\n</code></pre> linked_list.js<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head, index) {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.ts<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nfunction access(head: ListNode | null, index: number): ListNode | null {\n    for (let i = 0; i &lt; index; i++) {\n        if (!head) {\n            return null;\n        }\n        head = head.next;\n    }\n    return head;\n}\n</code></pre> linked_list.dart<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode? access(ListNode? head, int index) {\n  for (var i = 0; i &lt; index; i++) {\n    if (head == null) return null;\n    head = head.next;\n  }\n  return head;\n}\n</code></pre> linked_list.rs<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\npub fn access&lt;T&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, index: i32) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n    if index &lt;= 0 {return head};\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return access(node.clone(), index - 1);\n    }\n    return head;\n}\n</code></pre> linked_list.c<pre><code>/* \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9 */\nListNode *access(ListNode *head, int index) {\n    while (head &amp;&amp; head-&gt;next &amp;&amp; index) {\n        head = head-&gt;next;\n        index--;\n    }\n    return head;\n}\n</code></pre> linked_list.zig<pre><code>// \u8bbf\u95ee\u94fe\u8868\u4e2d\u7d22\u5f15\u4e3a index \u7684\u8282\u70b9\nfn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {\n    var head = node;\n    var i: i32 = 0;\n    while (i &lt; index) : (i += 1) {\n        head = head.?.next;\n        if (head == null) return null;\n    }\n    return head;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#5","title":"5. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u904d\u5386\u94fe\u8868\uff0c\u67e5\u627e\u5176\u4e2d\u503c\u4e3a <code>target</code> \u7684\u8282\u70b9\uff0c\u8f93\u51fa\u8be5\u8282\u70b9\u5728\u94fe\u8868\u4e2d\u7684\u7d22\u5f15\u3002\u6b64\u8fc7\u7a0b\u4e5f\u5c5e\u4e8e\u7ebf\u6027\u67e5\u627e\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linked_list.py<pre><code>def find(head: ListNode, target: int) -&gt; int:\n    \"\"\"\u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\"\"\"\n    index = 0\n    while head:\n        if head.val == target:\n            return index\n        head = head.next\n        index += 1\n    return -1\n</code></pre> linked_list.cpp<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head != nullptr) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.java<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.cs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint Find(ListNode? head, int target) {\n    int index = 0;\n    while (head != null) {\n        if (head.val == target)\n            return index;\n        head = head.next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.go<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc findNode(head *ListNode, target int) int {\n    index := 0\n    for head != nil {\n        if head.Val == target {\n            return index\n        }\n        head = head.Next\n        index++\n    }\n    return -1\n}\n</code></pre> linked_list.swift<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunc find(head: ListNode, target: Int) -&gt; Int {\n    var head: ListNode? = head\n    var index = 0\n    while head != nil {\n        if head?.val == target {\n            return index\n        }\n        head = head?.next\n        index += 1\n    }\n    return -1\n}\n</code></pre> linked_list.js<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head, target) {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.ts<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nfunction find(head: ListNode | null, target: number): number {\n    let index = 0;\n    while (head !== null) {\n        if (head.val === target) {\n            return index;\n        }\n        head = head.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre> linked_list.dart<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode? head, int target) {\n  int index = 0;\n  while (head != null) {\n    if (head.val == target) {\n      return index;\n    }\n    head = head.next;\n    index++;\n  }\n  return -1;\n}\n</code></pre> linked_list.rs<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\npub fn find&lt;T: PartialEq&gt;(head: Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;, target: T, index: i32) -&gt; i32 {\n    if head.borrow().val == target {return index};\n    if let Some(node) = &amp;head.borrow_mut().next {\n        return find(node.clone(), target, index + 1);\n    }\n    return -1;\n}\n</code></pre> linked_list.c<pre><code>/* \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9 */\nint find(ListNode *head, int target) {\n    int index = 0;\n    while (head) {\n        if (head-&gt;val == target)\n            return index;\n        head = head-&gt;next;\n        index++;\n    }\n    return -1;\n}\n</code></pre> linked_list.zig<pre><code>// \u5728\u94fe\u8868\u4e2d\u67e5\u627e\u503c\u4e3a target \u7684\u9996\u4e2a\u8282\u70b9\nfn find(node: ?*inc.ListNode(i32), target: i32) i32 {\n    var head = node;\n    var index: i32 = 0;\n    while (head != null) {\n        if (head.?.val == target) return index;\n        head = head.?.next;\n        index += 1;\n    }\n    return -1;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/linked_list/#422-vs","title":"4.2.2 \u00a0 \u6570\u7ec4 vs. \u94fe\u8868","text":"<p>\u8868 4-1 \u603b\u7ed3\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u7684\u5404\u9879\u7279\u70b9\u5e76\u5bf9\u6bd4\u4e86\u64cd\u4f5c\u6548\u7387\u3002\u7531\u4e8e\u5b83\u4eec\u91c7\u7528\u4e24\u79cd\u76f8\u53cd\u7684\u5b58\u50a8\u7b56\u7565\uff0c\u56e0\u6b64\u5404\u79cd\u6027\u8d28\u548c\u64cd\u4f5c\u6548\u7387\u4e5f\u5448\u73b0\u5bf9\u7acb\u7684\u7279\u70b9\u3002</p> <p> \u8868 4-1 \u00a0 \u6570\u7ec4\u4e0e\u94fe\u8868\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u5b58\u50a8\u65b9\u5f0f \u8fde\u7eed\u5185\u5b58\u7a7a\u95f4 \u5206\u6563\u5185\u5b58\u7a7a\u95f4 \u5bb9\u91cf\u6269\u5c55 \u957f\u5ea6\u4e0d\u53ef\u53d8 \u53ef\u7075\u6d3b\u6269\u5c55 \u5185\u5b58\u6548\u7387 \u5143\u7d20\u5360\u7528\u5185\u5b58\u5c11\u3001\u4f46\u53ef\u80fd\u6d6a\u8d39\u7a7a\u95f4 \u5143\u7d20\u5360\u7528\u5185\u5b58\u591a \u8bbf\u95ee\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(1)\\)"},{"location":"chapter_array_and_linkedlist/linked_list/#423","title":"4.2.3 \u00a0 \u5e38\u89c1\u94fe\u8868\u7c7b\u578b","text":"<p>\u5982\u56fe 4-8 \u6240\u793a\uff0c\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u4e09\u79cd\u3002</p> <ul> <li>\u5355\u5411\u94fe\u8868\uff1a\u5373\u524d\u9762\u4ecb\u7ecd\u7684\u666e\u901a\u94fe\u8868\u3002\u5355\u5411\u94fe\u8868\u7684\u8282\u70b9\u5305\u542b\u503c\u548c\u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\u4e24\u9879\u6570\u636e\u3002\u6211\u4eec\u5c06\u9996\u4e2a\u8282\u70b9\u79f0\u4e3a\u5934\u8282\u70b9\uff0c\u5c06\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u79f0\u4e3a\u5c3e\u8282\u70b9\uff0c\u5c3e\u8282\u70b9\u6307\u5411\u7a7a \\(\\text{None}\\) \u3002</li> <li>\u73af\u5f62\u94fe\u8868\uff1a\u5982\u679c\u6211\u4eec\u4ee4\u5355\u5411\u94fe\u8868\u7684\u5c3e\u8282\u70b9\u6307\u5411\u5934\u8282\u70b9\uff08\u9996\u5c3e\u76f8\u63a5\uff09\uff0c\u5219\u5f97\u5230\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\u3002\u5728\u73af\u5f62\u94fe\u8868\u4e2d\uff0c\u4efb\u610f\u8282\u70b9\u90fd\u53ef\u4ee5\u89c6\u4f5c\u5934\u8282\u70b9\u3002</li> <li>\u53cc\u5411\u94fe\u8868\uff1a\u4e0e\u5355\u5411\u94fe\u8868\u76f8\u6bd4\uff0c\u53cc\u5411\u94fe\u8868\u8bb0\u5f55\u4e86\u4e24\u4e2a\u65b9\u5411\u7684\u5f15\u7528\u3002\u53cc\u5411\u94fe\u8868\u7684\u8282\u70b9\u5b9a\u4e49\u540c\u65f6\u5305\u542b\u6307\u5411\u540e\u7ee7\u8282\u70b9\uff08\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\u548c\u524d\u9a71\u8282\u70b9\uff08\u4e0a\u4e00\u4e2a\u8282\u70b9\uff09\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\u3002\u76f8\u8f83\u4e8e\u5355\u5411\u94fe\u8868\uff0c\u53cc\u5411\u94fe\u8868\u66f4\u5177\u7075\u6d3b\u6027\uff0c\u53ef\u4ee5\u671d\u4e24\u4e2a\u65b9\u5411\u904d\u5386\u94fe\u8868\uff0c\u4f46\u76f8\u5e94\u5730\u4e5f\u9700\u8981\u5360\u7528\u66f4\u591a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u8282\u70b9\u503c\n        self.next: ListNode | None = None  # \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        self.prev: ListNode | None = None  # \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct ListNode {\n    int val;         // \u8282\u70b9\u503c\n    ListNode *next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    ListNode *prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n    ListNode(int x) : val(x), next(nullptr), prev(nullptr) {}  // \u6784\u9020\u51fd\u6570\n};\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(int x) { val = x; }  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode(int x) {  // \u6784\u9020\u51fd\u6570\n    int val = x;    // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype DoublyListNode struct {\n    Val  int             // \u8282\u70b9\u503c\n    Next *DoublyListNode // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    Prev *DoublyListNode // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n// NewDoublyListNode \u521d\u59cb\u5316\nfunc NewDoublyListNode(val int) *DoublyListNode {\n    return &amp;DoublyListNode{\n        Val:  val,\n        Next: nil,\n        Prev: nil,\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    var prev: ListNode? // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n\n    init(x: Int) { // \u6784\u9020\u51fd\u6570\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    constructor(val, next, prev) {\n        this.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    val: number;\n    next: ListNode | null;\n    prev: ListNode | null;\n    constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {\n        this.val = val  ===  undefined ? 0 : val;        // \u8282\u70b9\u503c\n        this.next = next  ===  undefined ? null : next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n        this.prev = prev  ===  undefined ? null : prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b */\nclass ListNode {\n    int val;        // \u8282\u70b9\u503c\n    ListNode next;  // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u5f15\u7528\n    ListNode prev;  // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u5f15\u7528\n    ListNode(this.val, [this.next, this.prev]);  // \u6784\u9020\u51fd\u6570\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\u578b */\n#[derive(Debug)]\nstruct ListNode {\n    val: i32, // \u8282\u70b9\u503c\n    next: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    prev: Option&lt;Rc&lt;RefCell&lt;ListNode&gt;&gt;&gt;, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nimpl ListNode {\n    fn new(val: i32) -&gt; Self {\n        ListNode {\n            val,\n            next: None,\n            prev: None,\n        }\n    }\n}\n</code></pre> <pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct ListNode {\n    int val;               // \u8282\u70b9\u503c\n    struct ListNode *next; // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n    struct ListNode *prev; // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n} ListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nListNode *newListNode(int val) {\n    ListNode *node;\n    node = (ListNode *) malloc(sizeof(ListNode));\n    node-&gt;val = val;\n    node-&gt;next = NULL;\n    node-&gt;prev = NULL;\n    return node;\n}\n</code></pre> <pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\u7c7b\npub fn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = 0, // \u8282\u70b9\u503c\n        next: ?*Self = null, // \u6307\u5411\u540e\u7ee7\u8282\u70b9\u7684\u6307\u9488\n        prev: ?*Self = null, // \u6307\u5411\u524d\u9a71\u8282\u70b9\u7684\u6307\u9488\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n</code></pre> <p></p> <p> \u56fe 4-8 \u00a0 \u5e38\u89c1\u94fe\u8868\u79cd\u7c7b </p>"},{"location":"chapter_array_and_linkedlist/linked_list/#424","title":"4.2.4 \u00a0 \u94fe\u8868\u5178\u578b\u5e94\u7528","text":"<p>\u5355\u5411\u94fe\u8868\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u548c\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u3002</p> <ul> <li>\u6808\u4e0e\u961f\u5217\uff1a\u5f53\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u90fd\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\u65f6\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u540e\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u6808\uff1b\u5f53\u63d2\u5165\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u4e00\u7aef\u8fdb\u884c\uff0c\u5220\u9664\u64cd\u4f5c\u5728\u94fe\u8868\u7684\u53e6\u4e00\u7aef\u8fdb\u884c\uff0c\u5b83\u8868\u73b0\u51fa\u5148\u8fdb\u5148\u51fa\u7684\u7279\u6027\uff0c\u5bf9\u5e94\u961f\u5217\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u94fe\u5f0f\u5730\u5740\u662f\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\u7684\u4e3b\u6d41\u65b9\u6848\u4e4b\u4e00\uff0c\u5728\u8be5\u65b9\u6848\u4e2d\uff0c\u6240\u6709\u51b2\u7a81\u7684\u5143\u7d20\u90fd\u4f1a\u88ab\u653e\u5230\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002</li> <li>\u56fe\uff1a\u90bb\u63a5\u8868\u662f\u8868\u793a\u56fe\u7684\u4e00\u79cd\u5e38\u7528\u65b9\u5f0f\uff0c\u5176\u4e2d\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u90fd\u4e0e\u4e00\u4e2a\u94fe\u8868\u76f8\u5173\u8054\uff0c\u94fe\u8868\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u4ee3\u8868\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u5176\u4ed6\u9876\u70b9\u3002</li> </ul> <p>\u53cc\u5411\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5feb\u901f\u67e5\u627e\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u5143\u7d20\u7684\u573a\u666f\u3002</p> <ul> <li>\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\uff1a\u6bd4\u5982\u5728\u7ea2\u9ed1\u6811\u3001B \u6811\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u8bbf\u95ee\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5728\u8282\u70b9\u4e2d\u4fdd\u5b58\u4e00\u4e2a\u6307\u5411\u7236\u8282\u70b9\u7684\u5f15\u7528\u6765\u5b9e\u73b0\uff0c\u7c7b\u4f3c\u4e8e\u53cc\u5411\u94fe\u8868\u3002</li> <li>\u6d4f\u89c8\u5668\u5386\u53f2\uff1a\u5728\u7f51\u9875\u6d4f\u89c8\u5668\u4e2d\uff0c\u5f53\u7528\u6237\u70b9\u51fb\u524d\u8fdb\u6216\u540e\u9000\u6309\u94ae\u65f6\uff0c\u6d4f\u89c8\u5668\u9700\u8981\u77e5\u9053\u7528\u6237\u8bbf\u95ee\u8fc7\u7684\u524d\u4e00\u4e2a\u548c\u540e\u4e00\u4e2a\u7f51\u9875\u3002\u53cc\u5411\u94fe\u8868\u7684\u7279\u6027\u4f7f\u5f97\u8fd9\u79cd\u64cd\u4f5c\u53d8\u5f97\u7b80\u5355\u3002</li> <li>LRU \u7b97\u6cd5\uff1a\u5728\u7f13\u5b58\u6dd8\u6c70\uff08LRU\uff09\u7b97\u6cd5\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5feb\u901f\u627e\u5230\u6700\u8fd1\u6700\u5c11\u4f7f\u7528\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u652f\u6301\u5feb\u901f\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u3002\u8fd9\u65f6\u5019\u4f7f\u7528\u53cc\u5411\u94fe\u8868\u5c31\u975e\u5e38\u5408\u9002\u3002</li> </ul> <p>\u73af\u5f62\u94fe\u8868\u5e38\u7528\u4e8e\u9700\u8981\u5468\u671f\u6027\u64cd\u4f5c\u7684\u573a\u666f\uff0c\u6bd4\u5982\u64cd\u4f5c\u7cfb\u7edf\u7684\u8d44\u6e90\u8c03\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\uff1a\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5\u662f\u4e00\u79cd\u5e38\u89c1\u7684 CPU \u8c03\u5ea6\u7b97\u6cd5\uff0c\u5b83\u9700\u8981\u5bf9\u4e00\u7ec4\u8fdb\u7a0b\u8fdb\u884c\u5faa\u73af\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u88ab\u8d4b\u4e88\u4e00\u4e2a\u65f6\u95f4\u7247\uff0c\u5f53\u65f6\u95f4\u7247\u7528\u5b8c\u65f6\uff0cCPU \u5c06\u5207\u6362\u5230\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u3002\u8fd9\u79cd\u5faa\u73af\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u73af\u5f62\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u6570\u636e\u7f13\u51b2\u533a\uff1a\u5728\u67d0\u4e9b\u6570\u636e\u7f13\u51b2\u533a\u7684\u5b9e\u73b0\u4e2d\uff0c\u4e5f\u53ef\u80fd\u4f1a\u4f7f\u7528\u73af\u5f62\u94fe\u8868\u3002\u6bd4\u5982\u5728\u97f3\u9891\u3001\u89c6\u9891\u64ad\u653e\u5668\u4e2d\uff0c\u6570\u636e\u6d41\u53ef\u80fd\u4f1a\u88ab\u5206\u6210\u591a\u4e2a\u7f13\u51b2\u5757\u5e76\u653e\u5165\u4e00\u4e2a\u73af\u5f62\u94fe\u8868\uff0c\u4ee5\u4fbf\u5b9e\u73b0\u65e0\u7f1d\u64ad\u653e\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/list/","title":"4.3 \u00a0 \u5217\u8868","text":"<p>\u300c\u5217\u8868 list\u300d\u662f\u4e00\u4e2a\u62bd\u8c61\u7684\u6570\u636e\u7ed3\u6784\u6982\u5ff5\uff0c\u5b83\u8868\u793a\u5143\u7d20\u7684\u6709\u5e8f\u96c6\u5408\uff0c\u652f\u6301\u5143\u7d20\u8bbf\u95ee\u3001\u4fee\u6539\u3001\u6dfb\u52a0\u3001\u5220\u9664\u548c\u904d\u5386\u7b49\u64cd\u4f5c\uff0c\u65e0\u987b\u4f7f\u7528\u8005\u8003\u8651\u5bb9\u91cf\u9650\u5236\u7684\u95ee\u9898\u3002\u5217\u8868\u53ef\u4ee5\u57fa\u4e8e\u94fe\u8868\u6216\u6570\u7ec4\u5b9e\u73b0\u3002</p> <ul> <li>\u94fe\u8868\u5929\u7136\u53ef\u4ee5\u88ab\u770b\u4f5c\u4e00\u4e2a\u5217\u8868\uff0c\u5176\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\u64cd\u4f5c\uff0c\u5e76\u4e14\u53ef\u4ee5\u7075\u6d3b\u52a8\u6001\u6269\u5bb9\u3002</li> <li>\u6570\u7ec4\u4e5f\u652f\u6301\u5143\u7d20\u589e\u5220\u67e5\u6539\uff0c\u4f46\u7531\u4e8e\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\uff0c\u56e0\u6b64\u53ea\u80fd\u88ab\u770b\u4f5c\u4e00\u4e2a\u5177\u6709\u957f\u5ea6\u9650\u5236\u7684\u5217\u8868\u3002</li> </ul> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u5217\u8868\u65f6\uff0c\u957f\u5ea6\u4e0d\u53ef\u53d8\u7684\u6027\u8d28\u4f1a\u5bfc\u81f4\u5217\u8868\u7684\u5b9e\u7528\u6027\u964d\u4f4e\u3002\u8fd9\u662f\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u65e0\u6cd5\u4e8b\u5148\u786e\u5b9a\u9700\u8981\u5b58\u50a8\u591a\u5c11\u6570\u636e\uff0c\u4ece\u800c\u96be\u4ee5\u9009\u62e9\u5408\u9002\u7684\u5217\u8868\u957f\u5ea6\u3002\u82e5\u957f\u5ea6\u8fc7\u5c0f\uff0c\u5219\u5f88\u53ef\u80fd\u65e0\u6cd5\u6ee1\u8db3\u4f7f\u7528\u9700\u6c42\uff1b\u82e5\u957f\u5ea6\u8fc7\u5927\uff0c\u5219\u4f1a\u9020\u6210\u5185\u5b58\u7a7a\u95f4\u7684\u6d6a\u8d39\u3002</p> <p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u300c\u52a8\u6001\u6570\u7ec4 dynamic array\u300d\u6765\u5b9e\u73b0\u5217\u8868\u3002\u5b83\u7ee7\u627f\u4e86\u6570\u7ec4\u7684\u5404\u9879\u4f18\u70b9\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u8fdb\u884c\u52a8\u6001\u6269\u5bb9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u5217\u8868\u90fd\u662f\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982 Python \u4e2d\u7684 <code>list</code> \u3001Java \u4e2d\u7684 <code>ArrayList</code> \u3001C++ \u4e2d\u7684 <code>vector</code> \u548c C# \u4e2d\u7684 <code>List</code> \u7b49\u3002\u5728\u63a5\u4e0b\u6765\u7684\u8ba8\u8bba\u4e2d\uff0c\u6211\u4eec\u5c06\u628a\u201c\u5217\u8868\u201d\u548c\u201c\u52a8\u6001\u6570\u7ec4\u201d\u89c6\u4e3a\u7b49\u540c\u7684\u6982\u5ff5\u3002</p>"},{"location":"chapter_array_and_linkedlist/list/#431","title":"4.3.1 \u00a0 \u5217\u8868\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_array_and_linkedlist/list/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u5217\u8868","text":"<p>\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u201c\u65e0\u521d\u59cb\u503c\u201d\u548c\u201c\u6709\u521d\u59cb\u503c\u201d\u8fd9\u4e24\u79cd\u521d\u59cb\u5316\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u521d\u59cb\u5316\u5217\u8868\n# \u65e0\u521d\u59cb\u503c\nnums1: list[int] = []\n# \u6709\u521d\u59cb\u503c\nnums: list[int] = [1, 3, 2, 5, 4]\n</code></pre> list.cpp<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u9700\u6ce8\u610f\uff0cC++ \u4e2d vector \u5373\u662f\u672c\u6587\u63cf\u8ff0\u7684 nums\n// \u65e0\u521d\u59cb\u503c\nvector&lt;int&gt; nums1;\n// \u6709\u521d\u59cb\u503c\nvector&lt;int&gt; nums = { 1, 3, 2, 5, 4 };\n</code></pre> list.java<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;();\n// \u6709\u521d\u59cb\u503c\uff08\u6ce8\u610f\u6570\u7ec4\u7684\u5143\u7d20\u7c7b\u578b\u9700\u4e3a int[] \u7684\u5305\u88c5\u7c7b Integer[]\uff09\nInteger[] numbers = new Integer[] { 1, 3, 2, 5, 4 };\nList&lt;Integer&gt; nums = new ArrayList&lt;&gt;(Arrays.asList(numbers));\n</code></pre> list.cs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nint[] numbers = [1, 3, 2, 5, 4];\nList&lt;int&gt; nums = [.. numbers];\n</code></pre> list_test.go<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nnums1 := []int{}\n// \u6709\u521d\u59cb\u503c\nnums := []int{1, 3, 2, 5, 4}\n</code></pre> list.swift<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: [Int] = []\n// \u6709\u521d\u59cb\u503c\nvar nums = [1, 3, 2, 5, 4]\n</code></pre> list.js<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1 = [];\n// \u6709\u521d\u59cb\u503c\nconst nums = [1, 3, 2, 5, 4];\n</code></pre> list.ts<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nconst nums1: number[] = [];\n// \u6709\u521d\u59cb\u503c\nconst nums: number[] = [1, 3, 2, 5, 4];\n</code></pre> list.dart<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nList&lt;int&gt; nums1 = [];\n// \u6709\u521d\u59cb\u503c\nList&lt;int&gt; nums = [1, 3, 2, 5, 4];\n</code></pre> list.rs<pre><code>/* \u521d\u59cb\u5316\u5217\u8868 */\n// \u65e0\u521d\u59cb\u503c\nlet nums1: Vec&lt;i32&gt; = Vec::new();\n// \u6709\u521d\u59cb\u503c\nlet nums: Vec&lt;i32&gt; = vec![1, 3, 2, 5, 4];\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u521d\u59cb\u5316\u5217\u8868\nvar nums = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums.deinit();\ntry nums.appendSlice(&amp;[_]i32{ 1, 3, 2, 5, 4 });\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#2","title":"2. \u00a0 \u8bbf\u95ee\u5143\u7d20","text":"<p>\u5217\u8868\u672c\u8d28\u4e0a\u662f\u6570\u7ec4\uff0c\u56e0\u6b64\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u8bbf\u95ee\u548c\u66f4\u65b0\u5143\u7d20\uff0c\u6548\u7387\u5f88\u9ad8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u8bbf\u95ee\u5143\u7d20\nnum: int = nums[1]  # \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n# \u66f4\u65b0\u5143\u7d20\nnums[1] = 0    # \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cpp<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.java<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums.get(1);  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums.set(1, 0);  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.cs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list_test.go<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nnum := nums[1]  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0     // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.swift<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num = nums[1] // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0 // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.js<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.ts<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nconst num: number = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.dart<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nint num = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;  // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.rs<pre><code>/* \u8bbf\u95ee\u5143\u7d20 */\nlet num: i32 = nums[1];  // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n/* \u66f4\u65b0\u5143\u7d20 */\nnums[1] = 0;             // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u8bbf\u95ee\u5143\u7d20\nvar num = nums.items[1]; // \u8bbf\u95ee\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\n\n// \u66f4\u65b0\u5143\u7d20\nnums.items[1] = 0; // \u5c06\u7d22\u5f15 1 \u5904\u7684\u5143\u7d20\u66f4\u65b0\u4e3a 0  \n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#3","title":"3. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u5143\u7d20","text":"<p>\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5730\u6dfb\u52a0\u4e0e\u5220\u9664\u5143\u7d20\u3002\u5728\u5217\u8868\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u7684\u6548\u7387\u4ecd\u4e0e\u6570\u7ec4\u76f8\u540c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u6e05\u7a7a\u5217\u8868\nnums.clear()\n\n# \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n# \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\nnums.insert(3, 6)  # \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n# \u5220\u9664\u5143\u7d20\nnums.pop(3)        # \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cpp<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push_back(1);\nnums.push_back(3);\nnums.push_back(2);\nnums.push_back(5);\nnums.push_back(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(nums.begin() + 3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.erase(nums.begin() + 3);      // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.java<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.add(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);  // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.cs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.Clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.Add(1);\nnums.Add(3);\nnums.Add(2);\nnums.Add(5);\nnums.Add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.Insert(3, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.RemoveAt(3);\n</code></pre> list_test.go<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums = nil\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums = append(nums, 1)\nnums = append(nums, 3)\nnums = append(nums, 2)\nnums = append(nums, 5)\nnums = append(nums, 4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums = append(nums[:3], append([]int{6}, nums[3:]...)...) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums = append(nums[:3], nums[4:]...) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.swift<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.removeAll()\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.append(1)\nnums.append(3)\nnums.append(2)\nnums.append(5)\nnums.append(4)\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(6, at: 3) // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(at: 3) // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.js<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.ts<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.length = 0;\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.splice(3, 0, 6);\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.splice(3, 1);\n</code></pre> list.dart<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.add(1);\nnums.add(3);\nnums.add(2);\nnums.add(5);\nnums.add(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.removeAt(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.rs<pre><code>/* \u6e05\u7a7a\u5217\u8868 */\nnums.clear();\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nnums.push(1);\nnums.push(3);\nnums.push(2);\nnums.push(5);\nnums.push(4);\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nnums.insert(3, 6);  // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n/* \u5220\u9664\u5143\u7d20 */\nnums.remove(3);    // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u6e05\u7a7a\u5217\u8868\nnums.clearRetainingCapacity();\n\n// \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\ntry nums.append(1);\ntry nums.append(3);\ntry nums.append(2);\ntry nums.append(5);\ntry nums.append(4);\n\n// \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\ntry nums.insert(3, 6); // \u5728\u7d22\u5f15 3 \u5904\u63d2\u5165\u6570\u5b57 6\n\n// \u5220\u9664\u5143\u7d20\n_ = nums.orderedRemove(3); // \u5220\u9664\u7d22\u5f15 3 \u5904\u7684\u5143\u7d20\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#4","title":"4. \u00a0 \u904d\u5386\u5217\u8868","text":"<p>\u4e0e\u6570\u7ec4\u4e00\u6837\uff0c\u5217\u8868\u53ef\u4ee5\u6839\u636e\u7d22\u5f15\u904d\u5386\uff0c\u4e5f\u53ef\u4ee5\u76f4\u63a5\u904d\u5386\u5404\u5143\u7d20\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\ncount = 0\nfor i in range(len(nums)):\n    count += nums[i]\n\n# \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\nfor num in nums:\n    count += num\n</code></pre> list.cpp<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.java<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.size(); i++) {\n    count += nums.get(i);\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\nfor (int num : nums) {\n    count += num;\n}\n</code></pre> list.cs<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (int i = 0; i &lt; nums.Count; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nforeach (int num in nums) {\n    count += num;\n}\n</code></pre> list_test.go<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\ncount := 0\nfor i := 0; i &lt; len(nums); i++ {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor _, num := range nums {\n    count += num\n}\n</code></pre> list.swift<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nvar count = 0\nfor i in nums.indices {\n    count += nums[i]\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0\nfor num in nums {\n    count += num\n}\n</code></pre> list.js<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.ts<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nlet count = 0;\nfor (let i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (const num of nums) {\n    count += num;\n}\n</code></pre> list.dart<pre><code>/* \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868 */\nint count = 0;\nfor (var i = 0; i &lt; nums.length; i++) {\n    count += nums[i];\n}\n\n/* \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20 */\ncount = 0;\nfor (var num in nums) {\n    count += num;\n}\n</code></pre> list.rs<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nlet mut _count = 0;\nfor i in 0..nums.len() {\n    _count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\n_count = 0;\nfor num in &amp;nums {\n    _count += num;\n}\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u901a\u8fc7\u7d22\u5f15\u904d\u5386\u5217\u8868\nvar count: i32 = 0;\nvar i: i32 = 0;\nwhile (i &lt; nums.items.len) : (i += 1) {\n    count += nums[i];\n}\n\n// \u76f4\u63a5\u904d\u5386\u5217\u8868\u5143\u7d20\ncount = 0;\nfor (nums.items) |num| {\n    count += num;\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#5","title":"5. \u00a0 \u62fc\u63a5\u5217\u8868","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u65b0\u5217\u8868 <code>nums1</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u62fc\u63a5\u5230\u539f\u5217\u8868\u7684\u5c3e\u90e8\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nnums1: list[int] = [6, 8, 7, 10, 9]\nnums += nums1  # \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cpp<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nvector&lt;int&gt; nums1 = { 6, 8, 7, 10, 9 };\n// \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\nnums.insert(nums.end(), nums1.begin(), nums1.end());\n</code></pre> list.java<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;Integer&gt; nums1 = new ArrayList&lt;&gt;(Arrays.asList(new Integer[] { 6, 8, 7, 10, 9 }));\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.cs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.AddRange(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list_test.go<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nnums1 := []int{6, 8, 7, 10, 9}\nnums = append(nums, nums1...)  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.swift<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1 = [6, 8, 7, 10, 9]\nnums.append(contentsOf: nums1) // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.js<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1 = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.ts<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nconst nums1: number[] = [6, 8, 7, 10, 9];\nnums.push(...nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.dart<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nList&lt;int&gt; nums1 = [6, 8, 7, 10, 9];\nnums.addAll(nums1);  // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre> list.rs<pre><code>/* \u62fc\u63a5\u4e24\u4e2a\u5217\u8868 */\nlet nums1: Vec&lt;i32&gt; = vec![6, 8, 7, 10, 9];\nnums.extend(nums1);\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u62fc\u63a5\u4e24\u4e2a\u5217\u8868\nvar nums1 = std.ArrayList(i32).init(std.heap.page_allocator);\ndefer nums1.deinit();\ntry nums1.appendSlice(&amp;[_]i32{ 6, 8, 7, 10, 9 });\ntry nums.insertSlice(nums.items.len, nums1.items); // \u5c06\u5217\u8868 nums1 \u62fc\u63a5\u5230 nums \u4e4b\u540e\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#6","title":"6. \u00a0 \u6392\u5e8f\u5217\u8868","text":"<p>\u5b8c\u6210\u5217\u8868\u6392\u5e8f\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u4ee5\u4f7f\u7528\u5728\u6570\u7ec4\u7c7b\u7b97\u6cd5\u9898\u4e2d\u7ecf\u5e38\u8003\u67e5\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u548c\u201c\u53cc\u6307\u9488\u201d\u7b97\u6cd5\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig list.py<pre><code># \u6392\u5e8f\u5217\u8868\nnums.sort()  # \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cpp<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort(nums.begin(), nums.end());  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.java<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nCollections.sort(nums);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.cs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.Sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list_test.go<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nsort.Ints(nums)  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.swift<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort() // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.js<pre><code>/* \u6392\u5e8f\u5217\u8868 */  \nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.ts<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort((a, b) =&gt; a - b);  // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.dart<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.rs<pre><code>/* \u6392\u5e8f\u5217\u8868 */\nnums.sort(); // \u6392\u5e8f\u540e\uff0c\u5217\u8868\u5143\u7d20\u4ece\u5c0f\u5230\u5927\u6392\u5217\n</code></pre> list.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u52a8\u6001\u6570\u7ec4\n</code></pre> list.zig<pre><code>// \u6392\u5e8f\u5217\u8868\nstd.sort.sort(i32, nums.items, {}, comptime std.sort.asc(i32));\n</code></pre>"},{"location":"chapter_array_and_linkedlist/list/#432","title":"4.3.2 \u00a0 \u5217\u8868\u5b9e\u73b0","text":"<p>\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u4e86\u5217\u8868\uff0c\u4f8b\u5982 Java\u3001C++\u3001Python \u7b49\u3002\u5b83\u4eec\u7684\u5b9e\u73b0\u6bd4\u8f83\u590d\u6742\uff0c\u5404\u4e2a\u53c2\u6570\u7684\u8bbe\u5b9a\u4e5f\u975e\u5e38\u8003\u7a76\uff0c\u4f8b\u5982\u521d\u59cb\u5bb9\u91cf\u3001\u6269\u5bb9\u500d\u6570\u7b49\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u6e90\u7801\u8fdb\u884c\u5b66\u4e60\u3002</p> <p>\u4e3a\u4e86\u52a0\u6df1\u5bf9\u5217\u8868\u5de5\u4f5c\u539f\u7406\u7684\u7406\u89e3\uff0c\u6211\u4eec\u5c1d\u8bd5\u5b9e\u73b0\u4e00\u4e2a\u7b80\u6613\u7248\u5217\u8868\uff0c\u5305\u62ec\u4ee5\u4e0b\u4e09\u4e2a\u91cd\u70b9\u8bbe\u8ba1\u3002</p> <ul> <li>\u521d\u59cb\u5bb9\u91cf\uff1a\u9009\u53d6\u4e00\u4e2a\u5408\u7406\u7684\u6570\u7ec4\u521d\u59cb\u5bb9\u91cf\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u9009\u62e9 10 \u4f5c\u4e3a\u521d\u59cb\u5bb9\u91cf\u3002</li> <li>\u6570\u91cf\u8bb0\u5f55\uff1a\u58f0\u660e\u4e00\u4e2a\u53d8\u91cf <code>size</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5217\u8868\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff0c\u5e76\u968f\u7740\u5143\u7d20\u63d2\u5165\u548c\u5220\u9664\u5b9e\u65f6\u66f4\u65b0\u3002\u6839\u636e\u6b64\u53d8\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4f4d\u5217\u8868\u5c3e\u90e8\uff0c\u4ee5\u53ca\u5224\u65ad\u662f\u5426\u9700\u8981\u6269\u5bb9\u3002</li> <li>\u6269\u5bb9\u673a\u5236\uff1a\u82e5\u63d2\u5165\u5143\u7d20\u65f6\u5217\u8868\u5bb9\u91cf\u5df2\u6ee1\uff0c\u5219\u9700\u8981\u8fdb\u884c\u6269\u5bb9\u3002\u5148\u6839\u636e\u6269\u5bb9\u500d\u6570\u521b\u5efa\u4e00\u4e2a\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u518d\u5c06\u5f53\u524d\u6570\u7ec4\u7684\u6240\u6709\u5143\u7d20\u4f9d\u6b21\u79fb\u52a8\u81f3\u65b0\u6570\u7ec4\u3002\u5728\u672c\u793a\u4f8b\u4e2d\uff0c\u6211\u4eec\u89c4\u5b9a\u6bcf\u6b21\u5c06\u6570\u7ec4\u6269\u5bb9\u81f3\u4e4b\u524d\u7684 2 \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_list.py<pre><code>class MyList:\n    \"\"\"\u5217\u8868\u7c7b\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._capacity: int = 10  # \u5217\u8868\u5bb9\u91cf\n        self._arr: list[int] = [0] * self._capacity  # \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        self._size: int = 0  # \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        self._extend_ratio: int = 2  # \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\"\"\"\n        return self._size\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u5217\u8868\u5bb9\u91cf\"\"\"\n        return self._capacity\n\n    def get(self, index: int) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u5143\u7d20\"\"\"\n        # \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        return self._arr[index]\n\n    def set(self, num: int, index: int):\n        \"\"\"\u66f4\u65b0\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        self._arr[index] = num\n\n    def add(self, num: int):\n        \"\"\"\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\"\"\"\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size() == self.capacity():\n            self.extend_capacity()\n        self._arr[self._size] = num\n        self._size += 1\n\n    def insert(self, num: int, index: int):\n        \"\"\"\u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        # \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self._size == self.capacity():\n            self.extend_capacity()\n        # \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in range(self._size - 1, index - 1, -1):\n            self._arr[j + 1] = self._arr[j]\n        self._arr[index] = num\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size += 1\n\n    def remove(self, index: int) -&gt; int:\n        \"\"\"\u5220\u9664\u5143\u7d20\"\"\"\n        if index &lt; 0 or index &gt;= self._size:\n            raise IndexError(\"\u7d22\u5f15\u8d8a\u754c\")\n        num = self._arr[index]\n        # \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in range(index, self._size - 1):\n            self._arr[j] = self._arr[j + 1]\n        # \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self._size -= 1\n        # \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num\n\n    def extend_capacity(self):\n        \"\"\"\u5217\u8868\u6269\u5bb9\"\"\"\n        # \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 __extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n        self._arr = self._arr + [0] * self.capacity() * (self._extend_ratio - 1)\n        # \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self._capacity = len(self._arr)\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868\"\"\"\n        return self._arr[: self._size]\n</code></pre> my_list.cpp<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  private:\n    int *arr;             // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int arrCapacity = 10; // \u5217\u8868\u5bb9\u91cf\n    int arrSize = 0;      // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    int extendRatio = 2;   // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~MyList() {\n        delete[] arr;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    int size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    int capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        arr[size()] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size() == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size() - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    int remove(int index) {\n        if (index &lt; 0 || index &gt;= size())\n            throw out_of_range(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size() - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n        int newCapacity = capacity() * extendRatio;\n        int *tmp = arr;\n        arr = new int[newCapacity];\n        // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            arr[i] = tmp[i];\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete[] tmp;\n        arrCapacity = newCapacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Vector \u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; vec(size());\n        for (int i = 0; i &lt; size(); i++) {\n            vec[i] = arr[i];\n        }\n        return vec;\n    }\n};\n</code></pre> my_list.java<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    private int size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private int extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[capacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\n    public int size() {\n        return size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int capacity() {\n        return capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void set(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        arr[size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (size == capacity())\n            extendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = size - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int remove(int index) {\n        if (index &lt; 0 || index &gt;= size)\n            throw new IndexOutOfBoundsException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; size - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n        arr = Arrays.copyOf(arr, capacity() * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        capacity = arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] toArray() {\n        int size = size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[size];\n        for (int i = 0; i &lt; size; i++) {\n            arr[i] = get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.cs<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private int[] arr;           // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private int arrCapacity = 10;    // \u5217\u8868\u5bb9\u91cf\n    private int arrSize = 0;         // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private readonly int extendRatio = 2;  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public MyList() {\n        arr = new int[arrCapacity];\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public int Size() {\n        return arrSize;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public int Capacity() {\n        return arrCapacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public int Get(int index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        return arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public void Set(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public void Add(int num) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        arr[arrSize] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public void Insert(int index, int num) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (arrSize == arrCapacity)\n            ExtendCapacity();\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (int j = arrSize - 1; j &gt;= index; j--) {\n            arr[j + 1] = arr[j];\n        }\n        arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public int Remove(int index) {\n        if (index &lt; 0 || index &gt;= arrSize)\n            throw new IndexOutOfRangeException(\"\u7d22\u5f15\u8d8a\u754c\");\n        int num = arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (int j = index; j &lt; arrSize - 1; j++) {\n            arr[j] = arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        arrSize--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public void ExtendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a arrCapacity * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n        Array.Resize(ref arr, arrCapacity * extendRatio);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        arrCapacity = arr.Length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] arr = new int[arrSize];\n        for (int i = 0; i &lt; arrSize; i++) {\n            arr[i] = Get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.go<pre><code>/* \u5217\u8868\u7c7b */\ntype myList struct {\n    arrCapacity int\n    arr         []int\n    arrSize     int\n    extendRatio int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newMyList() *myList {\n    return &amp;myList{\n        arrCapacity: 10,              // \u5217\u8868\u5bb9\u91cf\n        arr:         make([]int, 10), // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrSize:     0,               // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: 2,               // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n    }\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09 */\nfunc (l *myList) size() int {\n    return l.arrSize\n}\n\n/*  \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nfunc (l *myList) capacity() int {\n    return l.arrCapacity\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfunc (l *myList) get(index int) int {\n    // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    return l.arr[index]\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nfunc (l *myList) set(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    l.arr[index] = num\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nfunc (l *myList) add(num int) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    l.arr[l.arrSize] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nfunc (l *myList) insert(num, index int) {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if l.arrSize == l.arrCapacity {\n        l.extendCapacity()\n    }\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for j := l.arrSize - 1; j &gt;= index; j-- {\n        l.arr[j+1] = l.arr[j]\n    }\n    l.arr[index] = num\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize++\n}\n\n/* \u5220\u9664\u5143\u7d20 */\nfunc (l *myList) remove(index int) int {\n    if index &lt; 0 || index &gt;= l.arrSize {\n        panic(\"\u7d22\u5f15\u8d8a\u754c\")\n    }\n    num := l.arr[index]\n    // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for j := index; j &lt; l.arrSize-1; j++ {\n        l.arr[j] = l.arr[j+1]\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    l.arrSize--\n    // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n    return num\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nfunc (l *myList) extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n    l.arr = append(l.arr, make([]int, l.arrCapacity*(l.extendRatio-1))...)\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    l.arrCapacity = len(l.arr)\n}\n\n/* \u8fd4\u56de\u6709\u6548\u957f\u5ea6\u7684\u5217\u8868 */\nfunc (l *myList) toArray() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    return l.arr[:l.arrSize]\n}\n</code></pre> my_list.swift<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private var arr: [Int] // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private var _capacity = 10 // \u5217\u8868\u5bb9\u91cf\n    private var _size = 0 // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private let extendRatio = 2 // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        arr = Array(repeating: 0, count: _capacity)\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        _capacity\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    func get(index: Int) -&gt; Int {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u9519\u8bef\uff0c\u4e0b\u540c\n        if index &lt; 0 || index &gt;= _size {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        return arr[index]\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    func set(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= _size {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        arr[index] = num\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    func add(num: Int) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if _size == _capacity {\n            extendCapacity()\n        }\n        arr[_size] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    func insert(index: Int, num: Int) {\n        if index &lt; 0 || index &gt;= _size {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if _size == _capacity {\n            extendCapacity()\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in sequence(first: _size - 1, next: { $0 &gt;= index + 1 ? $0 - 1 : nil }) {\n            arr[j + 1] = arr[j]\n        }\n        arr[index] = num\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size += 1\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    @discardableResult\n    func remove(index: Int) -&gt; Int {\n        if index &lt; 0 || index &gt;= _size {\n            fatalError(\"\u7d22\u5f15\u8d8a\u754c\")\n        }\n        let num = arr[index]\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in index ..&lt; (_size - 1) {\n            arr[j] = arr[j + 1]\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        _size -= 1\n        // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    func extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n        arr = arr + Array(repeating: 0, count: _capacity * (extendRatio - 1))\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        _capacity = arr.count\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        var arr = Array(repeating: 0, count: _size)\n        for i in 0 ..&lt; _size {\n            arr[i] = get(index: i)\n        }\n        return arr\n    }\n}\n</code></pre> my_list.js<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    #arr = new Array(); // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    #capacity = 10; // \u5217\u8868\u5bb9\u91cf\n    #size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    #extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#arr = new Array(this.#capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    size() {\n        return this.#size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    capacity() {\n        return this.#capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    get(index) {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.#arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    set(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.#arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    add(num) {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.#arr[this.#size] = num;\n        this.#size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    insert(index, num) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this.#size === this.#capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this.#size - 1; j &gt;= index; j--) {\n            this.#arr[j + 1] = this.#arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#arr[index] = num;\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    remove(index) {\n        if (index &lt; 0 || index &gt;= this.#size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.#arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this.#size - 1; j++) {\n            this.#arr[j] = this.#arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.#size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    extendCapacity() {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extendRatio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n        this.#arr = this.#arr.concat(\n            new Array(this.capacity() * (this.#extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this.#capacity = this.#arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    toArray() {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.ts<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n    private arr: Array&lt;number&gt;; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    private _capacity: number = 10; // \u5217\u8868\u5bb9\u91cf\n    private _size: number = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    private extendRatio: number = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.arr = new Array(this._capacity);\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    public size(): number {\n        return this._size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    public capacity(): number {\n        return this._capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    public get(index: number): number {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        return this.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    public set(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        this.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    public add(num: number): void {\n        // \u5982\u679c\u957f\u5ea6\u7b49\u4e8e\u5bb9\u91cf\uff0c\u5219\u9700\u8981\u6269\u5bb9\n        if (this._size === this._capacity) this.extendCapacity();\n        // \u5c06\u65b0\u5143\u7d20\u6dfb\u52a0\u5230\u5217\u8868\u5c3e\u90e8\n        this.arr[this._size] = num;\n        this._size++;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    public insert(index: number, num: number): void {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if (this._size === this._capacity) {\n            this.extendCapacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for (let j = this._size - 1; j &gt;= index; j--) {\n            this.arr[j + 1] = this.arr[j];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this.arr[index] = num;\n        this._size++;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    public remove(index: number): number {\n        if (index &lt; 0 || index &gt;= this._size) throw new Error('\u7d22\u5f15\u8d8a\u754c');\n        let num = this.arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for (let j = index; j &lt; this._size - 1; j++) {\n            this.arr[j] = this.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        this._size--;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    public extendCapacity(): void {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n        this.arr = this.arr.concat(\n            new Array(this.capacity() * (this.extendRatio - 1))\n        );\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        this._capacity = this.arr.length;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    public toArray(): number[] {\n        let size = this.size();\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(size);\n        for (let i = 0; i &lt; size; i++) {\n            arr[i] = this.get(i);\n        }\n        return arr;\n    }\n}\n</code></pre> my_list.dart<pre><code>/* \u5217\u8868\u7c7b */\nclass MyList {\n  late List&lt;int&gt; _arr; // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n  int _capacity = 10; // \u5217\u8868\u5bb9\u91cf\n  int _size = 0; // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n  int _extendRatio = 2; // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  MyList() {\n    _arr = List.filled(_capacity, 0);\n  }\n\n  /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n  int size() =&gt; _size;\n\n  /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n  int capacity() =&gt; _capacity;\n\n  /* \u8bbf\u95ee\u5143\u7d20 */\n  int get(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    return _arr[index];\n  }\n\n  /* \u66f4\u65b0\u5143\u7d20 */\n  void set(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    _arr[index] = _num;\n  }\n\n  /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n  void add(int _num) {\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    _arr[_size] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n  void insert(int index, int _num) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (_size == _capacity) extendCapacity();\n    // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    for (var j = _size - 1; j &gt;= index; j--) {\n      _arr[j + 1] = _arr[j];\n    }\n    _arr[index] = _num;\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size++;\n  }\n\n  /* \u5220\u9664\u5143\u7d20 */\n  int remove(int index) {\n    if (index &gt;= _size) throw RangeError('\u7d22\u5f15\u8d8a\u754c');\n    int _num = _arr[index];\n    // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n    for (var j = index; j &lt; _size - 1; j++) {\n      _arr[j] = _arr[j + 1];\n    }\n    // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n    _size--;\n    // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n    return _num;\n  }\n\n  /* \u5217\u8868\u6269\u5bb9 */\n  void extendCapacity() {\n    // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 _extendRatio \u500d\u7684\u65b0\u6570\u7ec4\n    final _newNums = List.filled(_capacity * _extendRatio, 0);\n    // \u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n    List.copyRange(_newNums, 0, _arr);\n    // \u66f4\u65b0 _arr \u7684\u5f15\u7528\n    _arr = _newNums;\n    // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n    _capacity = _arr.length;\n  }\n\n  /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n  List&lt;int&gt; toArray() {\n    List&lt;int&gt; arr = [];\n    for (var i = 0; i &lt; _size; i++) {\n      arr.add(get(i));\n    }\n    return arr;\n  }\n}\n</code></pre> my_list.rs<pre><code>/* \u5217\u8868\u7c7b */\n#[allow(dead_code)]\nstruct MyList {\n    arr: Vec&lt;i32&gt;,       // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    capacity: usize,      // \u5217\u8868\u5bb9\u91cf\n    size: usize,          // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n    extend_ratio: usize,  // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n}\n\n#[allow(unused,unused_comparisons)]\nimpl MyList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        let mut vec = Vec::new(); \n        vec.resize(capacity, 0);\n        Self {\n            arr: vec,\n            capacity,\n            size: 0,\n            extend_ratio: 2,\n        }\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09*/\n    pub fn size(&amp;self) -&gt; usize {\n        return self.size;\n    }\n\n    /* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        return self.capacity;\n    }\n\n    /* \u8bbf\u95ee\u5143\u7d20 */\n    pub fn get(&amp;self, index: usize) -&gt; i32 {\n        // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n        if index &gt;= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n        return self.arr[index];\n    }\n\n    /* \u66f4\u65b0\u5143\u7d20 */\n    pub fn set(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n        self.arr[index] = num;\n    }\n\n    /* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\n    pub fn add(&amp;mut self, num: i32) {\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        self.arr[self.size] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\n    pub fn insert(&amp;mut self, index: usize, num: i32) {\n        if index &gt;= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n        // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n        if self.size == self.capacity() {\n            self.extend_capacity();\n        }\n        // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size).rev() {\n            self.arr[j + 1] = self.arr[j];\n        }\n        self.arr[index] = num;\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u5143\u7d20 */\n    pub fn remove(&amp;mut self, index: usize) -&gt; i32 {\n        if index &gt;= self.size() {panic!(\"\u7d22\u5f15\u8d8a\u754c\")};\n        let num = self.arr[index];\n        // \u5c06\u7d22\u5f15 index \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n        for j in (index..self.size - 1) {\n            self.arr[j] = self.arr[j + 1];\n        }\n        // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n        self.size -= 1;\n        // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n        return num;\n    }\n\n    /* \u5217\u8868\u6269\u5bb9 */\n    pub fn extend_capacity(&amp;mut self) {\n        // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a\u539f\u6570\u7ec4 extend_ratio \u500d\u7684\u65b0\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n        let new_capacity = self.capacity * self.extend_ratio;\n        self.arr.resize(new_capacity, 0);\n        // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n        self.capacity = new_capacity;\n    }\n\n    /* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4 */\n    pub fn to_array(&amp;mut self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut arr = Vec::new();\n        for i in 0..self.size {\n            arr.push(self.get(i));\n        }\n        arr\n    }\n}\n</code></pre> my_list.c<pre><code>/* \u5217\u8868\u7c7b */\ntypedef struct {\n    int *arr;        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n    int capacity;    // \u5217\u8868\u5bb9\u91cf\n    int size;        // \u5217\u8868\u5927\u5c0f\n    int extendRatio; // \u5217\u8868\u6bcf\u6b21\u6269\u5bb9\u7684\u500d\u6570\n} MyList;\n\n/* \u6784\u9020\u51fd\u6570 */\nMyList *newMyList() {\n    MyList *nums = malloc(sizeof(MyList));\n    nums-&gt;capacity = 10;\n    nums-&gt;arr = malloc(sizeof(int) * nums-&gt;capacity);\n    nums-&gt;size = 0;\n    nums-&gt;extendRatio = 2;\n    return nums;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delMyList(MyList *nums) {\n    free(nums-&gt;arr);\n    free(nums);\n}\n\n/* \u83b7\u53d6\u5217\u8868\u957f\u5ea6 */\nint size(MyList *nums) {\n    return nums-&gt;size;\n}\n\n/* \u83b7\u53d6\u5217\u8868\u5bb9\u91cf */\nint capacity(MyList *nums) {\n    return nums-&gt;capacity;\n}\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint get(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    return nums-&gt;arr[index];\n}\n\n/* \u66f4\u65b0\u5143\u7d20 */\nvoid set(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; nums-&gt;size);\n    nums-&gt;arr[index] = num;\n}\n\n/* \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20 */\nvoid add(MyList *nums, int num) {\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    nums-&gt;arr[size(nums)] = num;\n    nums-&gt;size++;\n}\n\n/* \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20 */\nvoid insert(MyList *nums, int index, int num) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n    if (size(nums) == capacity(nums)) {\n        extendCapacity(nums); // \u6269\u5bb9\n    }\n    for (int i = size(nums); i &gt; index; --i) {\n        nums-&gt;arr[i] = nums-&gt;arr[i - 1];\n    }\n    nums-&gt;arr[index] = num;\n    nums-&gt;size++;\n}\n\n/* \u5220\u9664\u5143\u7d20 */\n// \u6ce8\u610f\uff1astdio.h \u5360\u7528\u4e86 remove \u5173\u952e\u8bcd\nint removeItem(MyList *nums, int index) {\n    assert(index &gt;= 0 &amp;&amp; index &lt; size(nums));\n    int num = nums-&gt;arr[index];\n    for (int i = index; i &lt; size(nums) - 1; i++) {\n        nums-&gt;arr[i] = nums-&gt;arr[i + 1];\n    }\n    nums-&gt;size--;\n    return num;\n}\n\n/* \u5217\u8868\u6269\u5bb9 */\nvoid extendCapacity(MyList *nums) {\n    // \u5148\u5206\u914d\u7a7a\u95f4\n    int newCapacity = capacity(nums) * nums-&gt;extendRatio;\n    int *extend = (int *)malloc(sizeof(int) * newCapacity);\n    int *temp = nums-&gt;arr;\n\n    // \u62f7\u8d1d\u65e7\u6570\u636e\u5230\u65b0\u6570\u636e\n    for (int i = 0; i &lt; size(nums); i++)\n        extend[i] = nums-&gt;arr[i];\n\n    // \u91ca\u653e\u65e7\u6570\u636e\n    free(temp);\n\n    // \u66f4\u65b0\u65b0\u6570\u636e\n    nums-&gt;arr = extend;\n    nums-&gt;capacity = newCapacity;\n}\n\n/* \u5c06\u5217\u8868\u8f6c\u6362\u4e3a Array \u7528\u4e8e\u6253\u5370 */\nint *toArray(MyList *nums) {\n    return nums-&gt;arr;\n}\n</code></pre> my_list.zig<pre><code>// \u5217\u8868\u7c7b\nfn MyList(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        arr: []T = undefined,                        // \u6570\u7ec4\uff08\u5b58\u50a8\u5217\u8868\u5143\u7d20\uff09\n        arrCapacity: usize = 10,                     // \u5217\u8868\u5bb9\u91cf\n        numSize: usize = 0,                           // \u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        extendRatio: usize = 2,                       // \u6bcf\u6b21\u5217\u8868\u6269\u5bb9\u7684\u500d\u6570\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined, // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u5217\u8868\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.arr = try self.mem_allocator.alloc(T, self.arrCapacity);\n            @memset(self.arr, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u957f\u5ea6\uff08\u5f53\u524d\u5143\u7d20\u6570\u91cf\uff09\n        pub fn size(self: *Self) usize {\n            return self.numSize;\n        }\n\n        // \u83b7\u53d6\u5217\u8868\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.arrCapacity;\n        }\n\n        // \u8bbf\u95ee\u5143\u7d20\n        pub fn get(self: *Self, index: usize) T {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            return self.arr[index];\n        }  \n\n        // \u66f4\u65b0\u5143\u7d20\n        pub fn set(self: *Self, index: usize, num: T) void {\n            // \u7d22\u5f15\u5982\u679c\u8d8a\u754c\u5219\u629b\u51fa\u5f02\u5e38\uff0c\u4e0b\u540c\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            self.arr[index] = num;\n        }  \n\n        // \u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\n        pub fn add(self: *Self, num: T) !void {\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            self.arr[self.size()] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }  \n\n        // \u5728\u4e2d\u95f4\u63d2\u5165\u5143\u7d20\n        pub fn insert(self: *Self, index: usize, num: T) !void {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            // \u5143\u7d20\u6570\u91cf\u8d85\u51fa\u5bb9\u91cf\u65f6\uff0c\u89e6\u53d1\u6269\u5bb9\u673a\u5236\n            if (self.size() == self.capacity()) try self.extendCapacity();\n            // \u5c06\u7d22\u5f15 index \u4ee5\u53ca\u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n            var j = self.size() - 1;\n            while (j &gt;= index) : (j -= 1) {\n                self.arr[j + 1] = self.arr[j];\n            }\n            self.arr[index] = num;\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize += 1;\n        }\n\n        // \u5220\u9664\u5143\u7d20\n        pub fn remove(self: *Self, index: usize) T {\n            if (index &lt; 0 or index &gt;= self.size()) @panic(\"\u7d22\u5f15\u8d8a\u754c\");\n            var num = self.arr[index];\n            // \u7d22\u5f15 i \u4e4b\u540e\u7684\u5143\u7d20\u90fd\u5411\u524d\u79fb\u52a8\u4e00\u4f4d\n            var j = index;\n            while (j &lt; self.size() - 1) : (j += 1) {\n                self.arr[j] = self.arr[j + 1];\n            }\n            // \u66f4\u65b0\u5143\u7d20\u6570\u91cf\n            self.numSize -= 1;\n            // \u8fd4\u56de\u88ab\u5220\u9664\u5143\u7d20\n            return num;\n        }\n\n        // \u5217\u8868\u6269\u5bb9\n        pub fn extendCapacity(self: *Self) !void {\n            // \u65b0\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a size * extendRatio \u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u539f\u6570\u7ec4\u62f7\u8d1d\u5230\u65b0\u6570\u7ec4\n            var newCapacity = self.capacity() * self.extendRatio;\n            var extend = try self.mem_allocator.alloc(T, newCapacity);\n            @memset(extend, @as(T, 0));\n            // \u5c06\u539f\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\u590d\u5236\u5230\u65b0\u6570\u7ec4\n            std.mem.copy(T, extend, self.arr);\n            self.arr = extend;\n            // \u66f4\u65b0\u5217\u8868\u5bb9\u91cf\n            self.arrCapacity = newCapacity;\n        }\n\n        // \u5c06\u5217\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var arr = try self.mem_allocator.alloc(T, self.size());\n           @memset(arr, @as(T, 0));\n            for (arr, 0..) |*num, i| {\n                num.* = self.get(i);\n            }\n            return arr;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/","title":"4.4 \u00a0 \u5185\u5b58\u4e0e\u7f13\u5b58 *","text":"<p>\u5728\u672c\u7ae0\u7684\u524d\u4e24\u8282\u4e2d\uff0c\u6211\u4eec\u63a2\u8ba8\u4e86\u6570\u7ec4\u548c\u94fe\u8868\u8fd9\u4e24\u79cd\u57fa\u7840\u4e14\u91cd\u8981\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u4eec\u5206\u522b\u4ee3\u8868\u4e86\u201c\u8fde\u7eed\u5b58\u50a8\u201d\u548c\u201c\u5206\u6563\u5b58\u50a8\u201d\u8fd9\u4e24\u79cd\u4e0d\u540c\u7684\u7269\u7406\u7ed3\u6784\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u7269\u7406\u7ed3\u6784\u5728\u5f88\u5927\u7a0b\u5ea6\u4e0a\u51b3\u5b9a\u4e86\u7a0b\u5e8f\u5bf9\u5185\u5b58\u548c\u7f13\u5b58\u7684\u4f7f\u7528\u6548\u7387\uff0c\u8fdb\u800c\u5f71\u54cd\u7b97\u6cd5\u7a0b\u5e8f\u7684\u6574\u4f53\u6027\u80fd\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#441","title":"4.4.1 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u8bbe\u5907","text":"<p>\u8ba1\u7b97\u673a\u4e2d\u5305\u62ec\u4e09\u79cd\u4e0d\u540c\u7c7b\u578b\u7684\u5b58\u50a8\u8bbe\u5907\uff1a\u300c\u786c\u76d8 hard disk\u300d\u3001\u300c\u5185\u5b58 random-access memory, RAM\u300d\u3001\u300c\u7f13\u5b58 cache memory\u300d\u3002\u8868 4-2 \u5c55\u793a\u4e86\u5b83\u4eec\u5728\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e0d\u540c\u89d2\u8272\u548c\u6027\u80fd\u7279\u70b9\u3002</p> <p> \u8868 4-2 \u00a0 \u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u8bbe\u5907 </p> \u786c\u76d8 \u5185\u5b58 \u7f13\u5b58 \u7528\u9014 \u957f\u671f\u5b58\u50a8\u6570\u636e\uff0c\u5305\u62ec\u64cd\u4f5c\u7cfb\u7edf\u3001\u7a0b\u5e8f\u3001\u6587\u4ef6\u7b49 \u4e34\u65f6\u5b58\u50a8\u5f53\u524d\u8fd0\u884c\u7684\u7a0b\u5e8f\u548c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e \u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u51cf\u5c11 CPU \u8bbf\u95ee\u5185\u5b58\u7684\u6b21\u6570 \u6613\u5931\u6027 \u65ad\u7535\u540e\u6570\u636e\u4e0d\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u65ad\u7535\u540e\u6570\u636e\u4f1a\u4e22\u5931 \u5bb9\u91cf \u8f83\u5927\uff0cTB \u7ea7\u522b \u8f83\u5c0f\uff0cGB \u7ea7\u522b \u975e\u5e38\u5c0f\uff0cMB \u7ea7\u522b \u901f\u5ea6 \u8f83\u6162\uff0c\u51e0\u767e\u5230\u51e0\u5343 MB/s \u8f83\u5feb\uff0c\u51e0\u5341 GB/s \u975e\u5e38\u5feb\uff0c\u51e0\u5341\u5230\u51e0\u767e GB/s \u4ef7\u683c \u8f83\u4fbf\u5b9c\uff0c\u51e0\u6bdb\u5230\u51e0\u5143 / GB \u8f83\u8d35\uff0c\u51e0\u5341\u5230\u51e0\u767e\u5143 / GB \u975e\u5e38\u8d35\uff0c\u968f CPU \u6253\u5305\u8ba1\u4ef7 <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf\u60f3\u8c61\u4e3a\u56fe 4-9 \u6240\u793a\u7684\u91d1\u5b57\u5854\u7ed3\u6784\u3002\u8d8a\u9760\u8fd1\u91d1\u5b57\u5854\u9876\u7aef\u7684\u5b58\u50a8\u8bbe\u5907\u7684\u901f\u5ea6\u8d8a\u5feb\u3001\u5bb9\u91cf\u8d8a\u5c0f\u3001\u6210\u672c\u8d8a\u9ad8\u3002\u8fd9\u79cd\u591a\u5c42\u7ea7\u7684\u8bbe\u8ba1\u5e76\u975e\u5076\u7136\uff0c\u800c\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u5de5\u7a0b\u5e08\u4eec\u7ecf\u8fc7\u6df1\u601d\u719f\u8651\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u786c\u76d8\u96be\u4ee5\u88ab\u5185\u5b58\u53d6\u4ee3\u3002\u9996\u5148\uff0c\u5185\u5b58\u4e2d\u7684\u6570\u636e\u5728\u65ad\u7535\u540e\u4f1a\u4e22\u5931\uff0c\u56e0\u6b64\u5b83\u4e0d\u9002\u5408\u957f\u671f\u5b58\u50a8\u6570\u636e\uff1b\u5176\u6b21\uff0c\u5185\u5b58\u7684\u6210\u672c\u5927\u7ea6\u662f\u786c\u76d8\u7684\u51e0\u5341\u500d\uff0c\u8fd9\u4f7f\u5f97\u5b83\u96be\u4ee5\u5728\u6d88\u8d39\u8005\u5e02\u573a\u666e\u53ca\u3002</li> <li>\u7f13\u5b58\u7684\u5927\u5bb9\u91cf\u548c\u9ad8\u901f\u5ea6\u96be\u4ee5\u517c\u5f97\u3002\u968f\u7740 L1\u3001L2\u3001L3 \u7f13\u5b58\u7684\u5bb9\u91cf\u9010\u6b65\u589e\u5927\uff0c\u5176\u7269\u7406\u5c3a\u5bf8\u4f1a\u53d8\u5927\uff0c\u4e0e CPU \u6838\u5fc3\u4e4b\u95f4\u7684\u7269\u7406\u8ddd\u79bb\u4f1a\u53d8\u8fdc\uff0c\u4ece\u800c\u5bfc\u81f4\u6570\u636e\u4f20\u8f93\u65f6\u95f4\u589e\u52a0\uff0c\u5143\u7d20\u8bbf\u95ee\u5ef6\u8fdf\u53d8\u9ad8\u3002\u5728\u5f53\u524d\u6280\u672f\u4e0b\uff0c\u591a\u5c42\u7ea7\u7684\u7f13\u5b58\u7ed3\u6784\u662f\u5bb9\u91cf\u3001\u901f\u5ea6\u548c\u6210\u672c\u4e4b\u95f4\u7684\u6700\u4f73\u5e73\u8861\u70b9\u3002</li> </ul> <p></p> <p> \u56fe 4-9 \u00a0 \u8ba1\u7b97\u673a\u5b58\u50a8\u7cfb\u7edf </p> <p>Note</p> <p>\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u5c42\u6b21\u7ed3\u6784\u4f53\u73b0\u4e86\u901f\u5ea6\u3001\u5bb9\u91cf\u548c\u6210\u672c\u4e09\u8005\u4e4b\u95f4\u7684\u7cbe\u5999\u5e73\u8861\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u79cd\u6743\u8861\u666e\u904d\u5b58\u5728\u4e8e\u6240\u6709\u5de5\u4e1a\u9886\u57df\uff0c\u5b83\u8981\u6c42\u6211\u4eec\u5728\u4e0d\u540c\u7684\u4f18\u52bf\u548c\u9650\u5236\u4e4b\u95f4\u627e\u5230\u6700\u4f73\u7684\u5e73\u8861\u70b9\u3002</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u786c\u76d8\u7528\u4e8e\u957f\u671f\u5b58\u50a8\u5927\u91cf\u6570\u636e\uff0c\u5185\u5b58\u7528\u4e8e\u4e34\u65f6\u5b58\u50a8\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\uff0c\u800c\u7f13\u5b58\u5219\u7528\u4e8e\u5b58\u50a8\u7ecf\u5e38\u8bbf\u95ee\u7684\u6570\u636e\u548c\u6307\u4ee4\uff0c\u4ee5\u63d0\u9ad8\u7a0b\u5e8f\u8fd0\u884c\u6548\u7387\u3002\u8fd9\u4e09\u8005\u5171\u540c\u534f\u4f5c\uff0c\u786e\u4fdd\u8ba1\u7b97\u673a\u7cfb\u7edf\u7684\u9ad8\u6548\u8fd0\u884c\u3002</p> <p>\u5982\u56fe 4-10 \u6240\u793a\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4f1a\u4ece\u786c\u76d8\u4e2d\u88ab\u8bfb\u53d6\u5230\u5185\u5b58\u4e2d\uff0c\u4f9b\u7ed9 CPU \u8ba1\u7b97\u4f7f\u7528\u3002\u7f13\u5b58\u53ef\u4ee5\u770b\u4f5c CPU \u7684\u4e00\u90e8\u5206\uff0c\u5b83\u901a\u8fc7\u667a\u80fd\u5730\u4ece\u5185\u5b58\u52a0\u8f7d\u6570\u636e\uff0c\u7ed9 CPU \u63d0\u4f9b\u9ad8\u901f\u7684\u6570\u636e\u8bfb\u53d6\uff0c\u4ece\u800c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\uff0c\u51cf\u5c11\u5bf9\u8f83\u6162\u7684\u5185\u5b58\u7684\u4f9d\u8d56\u3002</p> <p></p> <p> \u56fe 4-10 \u00a0 \u786c\u76d8\u3001\u5185\u5b58\u548c\u7f13\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u6d41\u901a </p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#442","title":"4.4.2 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u5185\u5b58\u6548\u7387","text":"<p>\u5728\u5185\u5b58\u7a7a\u95f4\u5229\u7528\u65b9\u9762\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5177\u6709\u5404\u81ea\u7684\u4f18\u52bf\u548c\u5c40\u9650\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u5185\u5b58\u662f\u6709\u9650\u7684\uff0c\u4e14\u540c\u4e00\u5757\u5185\u5b58\u4e0d\u80fd\u88ab\u591a\u4e2a\u7a0b\u5e8f\u5171\u4eab\uff0c\u56e0\u6b64\u6211\u4eec\u5e0c\u671b\u6570\u636e\u7ed3\u6784\u80fd\u591f\u5c3d\u53ef\u80fd\u9ad8\u6548\u5730\u5229\u7528\u7a7a\u95f4\u3002\u6570\u7ec4\u7684\u5143\u7d20\u7d27\u5bc6\u6392\u5217\uff0c\u4e0d\u9700\u8981\u989d\u5916\u7684\u7a7a\u95f4\u6765\u5b58\u50a8\u94fe\u8868\u8282\u70b9\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u56e0\u6b64\u7a7a\u95f4\u6548\u7387\u66f4\u9ad8\u3002\u7136\u800c\uff0c\u6570\u7ec4\u9700\u8981\u4e00\u6b21\u6027\u5206\u914d\u8db3\u591f\u7684\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u8fd9\u53ef\u80fd\u5bfc\u81f4\u5185\u5b58\u7684\u6d6a\u8d39\uff0c\u6570\u7ec4\u6269\u5bb9\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u6210\u672c\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u94fe\u8868\u4ee5\u201c\u8282\u70b9\u201d\u4e3a\u5355\u4f4d\u8fdb\u884c\u52a8\u6001\u5185\u5b58\u5206\u914d\u548c\u56de\u6536\uff0c\u8fd9\u79cd\u65b9\u5f0f\u63d0\u4f9b\u4e86\u66f4\u5927\u7684\u7075\u6d3b\u6027\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u968f\u7740\u53cd\u590d\u7533\u8bf7\u4e0e\u91ca\u653e\u5185\u5b58\uff0c\u7a7a\u95f2\u5185\u5b58\u7684\u788e\u7247\u5316\u7a0b\u5ea6\u4f1a\u8d8a\u6765\u8d8a\u9ad8\uff0c\u4ece\u800c\u5bfc\u81f4\u5185\u5b58\u7684\u5229\u7528\u6548\u7387\u964d\u4f4e\u3002\u6570\u7ec4\u7531\u4e8e\u5176\u8fde\u7eed\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u76f8\u5bf9\u4e0d\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002\u76f8\u53cd\uff0c\u94fe\u8868\u7684\u5143\u7d20\u662f\u5206\u6563\u5b58\u50a8\u7684\uff0c\u5728\u9891\u7e41\u7684\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u4e2d\uff0c\u66f4\u5bb9\u6613\u5bfc\u81f4\u5185\u5b58\u788e\u7247\u5316\u3002</p>"},{"location":"chapter_array_and_linkedlist/ram_and_cache/#443","title":"4.4.3 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u7f13\u5b58\u6548\u7387","text":"<p>\u7f13\u5b58\u867d\u7136\u5728\u7a7a\u95f4\u5bb9\u91cf\u4e0a\u8fdc\u5c0f\u4e8e\u5185\u5b58\uff0c\u4f46\u5b83\u6bd4\u5185\u5b58\u5feb\u5f97\u591a\uff0c\u5728\u7a0b\u5e8f\u6267\u884c\u901f\u5ea6\u4e0a\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\u3002\u7531\u4e8e\u7f13\u5b58\u7684\u5bb9\u91cf\u6709\u9650\uff0c\u5b83\u53ea\u80fd\u5b58\u50a8\u4e00\u5c0f\u90e8\u5206\u9891\u7e41\u8bbf\u95ee\u7684\u6570\u636e\u3002\u56e0\u6b64\uff0c\u5f53 CPU \u5c1d\u8bd5\u8bbf\u95ee\u7684\u6570\u636e\u4e0d\u5728\u7f13\u5b58\u4e2d\u65f6\uff0c\u5c31\u4f1a\u53d1\u751f\u300c\u7f13\u5b58\u672a\u547d\u4e2d cache miss\u300d\uff0c\u6b64\u65f6 CPU \u4e0d\u5f97\u4e0d\u4ece\u901f\u5ea6\u8f83\u6162\u7684\u5185\u5b58\u4e2d\u52a0\u8f7d\u6240\u9700\u6570\u636e\u3002</p> <p>\u663e\u7136\uff0c\u201c\u7f13\u5b58\u672a\u547d\u4e2d\u201d\u8d8a\u5c11\uff0cCPU \u8bfb\u5199\u6570\u636e\u7684\u6548\u7387\u5c31\u8d8a\u9ad8\uff0c\u7a0b\u5e8f\u6027\u80fd\u4e5f\u5c31\u8d8a\u597d\u3002\u6211\u4eec\u5c06 CPU \u4ece\u7f13\u5b58\u4e2d\u6210\u529f\u83b7\u53d6\u6570\u636e\u7684\u6bd4\u4f8b\u79f0\u4e3a\u300c\u7f13\u5b58\u547d\u4e2d\u7387 cache hit rate\u300d\uff0c\u8fd9\u4e2a\u6307\u6807\u901a\u5e38\u7528\u6765\u8861\u91cf\u7f13\u5b58\u6548\u7387\u3002</p> <p>\u4e3a\u4e86\u5c3d\u53ef\u80fd\u8fbe\u5230\u66f4\u9ad8\u6548\u7387\uff0c\u7f13\u5b58\u4f1a\u91c7\u53d6\u4ee5\u4e0b\u6570\u636e\u52a0\u8f7d\u673a\u5236\u3002</p> <ul> <li>\u7f13\u5b58\u884c\uff1a\u7f13\u5b58\u4e0d\u662f\u5355\u4e2a\u5b57\u8282\u5730\u5b58\u50a8\u4e0e\u52a0\u8f7d\u6570\u636e\uff0c\u800c\u662f\u4ee5\u7f13\u5b58\u884c\u4e3a\u5355\u4f4d\u3002\u76f8\u6bd4\u4e8e\u5355\u4e2a\u5b57\u8282\u7684\u4f20\u8f93\uff0c\u7f13\u5b58\u884c\u7684\u4f20\u8f93\u5f62\u5f0f\u66f4\u52a0\u9ad8\u6548\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u5904\u7406\u5668\u4f1a\u5c1d\u8bd5\u9884\u6d4b\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\uff08\u4f8b\u5982\u987a\u5e8f\u8bbf\u95ee\u3001\u56fa\u5b9a\u6b65\u957f\u8df3\u8dc3\u8bbf\u95ee\u7b49\uff09\uff0c\u5e76\u6839\u636e\u7279\u5b9a\u6a21\u5f0f\u5c06\u6570\u636e\u52a0\u8f7d\u81f3\u7f13\u5b58\u4e4b\u4e2d\uff0c\u4ece\u800c\u63d0\u5347\u547d\u4e2d\u7387\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u9644\u8fd1\u7684\u6570\u636e\u53ef\u80fd\u4e5f\u4f1a\u8fd1\u671f\u88ab\u8bbf\u95ee\u3002\u56e0\u6b64\uff0c\u7f13\u5b58\u5728\u52a0\u8f7d\u67d0\u4e00\u6570\u636e\u65f6\uff0c\u4e5f\u4f1a\u5c06\u5176\u9644\u8fd1\u7684\u6570\u636e\u52a0\u8f7d\u8fdb\u6765\uff0c\u4ee5\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> <li>\u65f6\u95f4\u5c40\u90e8\u6027\uff1a\u5982\u679c\u4e00\u4e2a\u6570\u636e\u88ab\u8bbf\u95ee\uff0c\u90a3\u4e48\u5b83\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u5f88\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\u3002\u7f13\u5b58\u5229\u7528\u8fd9\u4e00\u539f\u7406\uff0c\u901a\u8fc7\u4fdd\u7559\u6700\u8fd1\u8bbf\u95ee\u8fc7\u7684\u6570\u636e\u6765\u63d0\u9ad8\u547d\u4e2d\u7387\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u5bf9\u7f13\u5b58\u7684\u5229\u7528\u6548\u7387\u4e5f\u662f\u4e0d\u540c\u7684\uff0c\u4e3b\u8981\u4f53\u73b0\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\uff1a\u94fe\u8868\u5143\u7d20\u6bd4\u6570\u7ec4\u5143\u7d20\u5360\u7528\u7a7a\u95f4\u66f4\u591a\uff0c\u5bfc\u81f4\u7f13\u5b58\u4e2d\u5bb9\u7eb3\u7684\u6709\u6548\u6570\u636e\u91cf\u66f4\u5c11\u3002</li> <li>\u7f13\u5b58\u884c\uff1a\u94fe\u8868\u6570\u636e\u5206\u6563\u5728\u5185\u5b58\u5404\u5904\uff0c\u800c\u7f13\u5b58\u662f\u201c\u6309\u884c\u52a0\u8f7d\u201d\u7684\uff0c\u56e0\u6b64\u52a0\u8f7d\u5230\u7684\u65e0\u6548\u6570\u636e\u7684\u6bd4\u4f8b\u66f4\u9ad8\u3002</li> <li>\u9884\u53d6\u673a\u5236\uff1a\u6570\u7ec4\u6bd4\u94fe\u8868\u7684\u6570\u636e\u8bbf\u95ee\u6a21\u5f0f\u66f4\u5177\u201c\u53ef\u9884\u6d4b\u6027\u201d\uff0c\u5373\u7cfb\u7edf\u66f4\u5bb9\u6613\u731c\u51fa\u5373\u5c06\u88ab\u52a0\u8f7d\u7684\u6570\u636e\u3002</li> <li>\u7a7a\u95f4\u5c40\u90e8\u6027\uff1a\u6570\u7ec4\u88ab\u5b58\u50a8\u5728\u96c6\u4e2d\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u56e0\u6b64\u88ab\u52a0\u8f7d\u6570\u636e\u7684\u9644\u8fd1\u6570\u636e\u66f4\u6709\u53ef\u80fd\u5373\u5c06\u88ab\u8bbf\u95ee\u3002</li> </ul> <p>\u603b\u4f53\u800c\u8a00\uff0c\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u5728\u64cd\u4f5c\u6548\u7387\u4e0a\u901a\u5e38\u4f18\u4e8e\u94fe\u8868\u3002\u8fd9\u4f7f\u5f97\u5728\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u5f80\u5f80\u66f4\u53d7\u6b22\u8fce\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9ad8\u7f13\u5b58\u6548\u7387\u5e76\u4e0d\u610f\u5473\u7740\u6570\u7ec4\u5728\u6240\u6709\u60c5\u51b5\u4e0b\u90fd\u4f18\u4e8e\u94fe\u8868\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\u9009\u62e9\u54ea\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u6765\u51b3\u5b9a\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5b9e\u73b0\u201c\u6808\u201d\u6570\u636e\u7ed3\u6784\uff08\u4e0b\u4e00\u7ae0\u4f1a\u8be6\u7ec6\u4ecb\u7ecd\uff09\uff0c\u4f46\u5b83\u4eec\u9002\u7528\u4e8e\u4e0d\u540c\u573a\u666f\u3002</p> <ul> <li>\u5728\u505a\u7b97\u6cd5\u9898\u65f6\uff0c\u6211\u4eec\u4f1a\u503e\u5411\u4e8e\u9009\u62e9\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\uff0c\u56e0\u4e3a\u5b83\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u64cd\u4f5c\u6548\u7387\u548c\u968f\u673a\u8bbf\u95ee\u7684\u80fd\u529b\uff0c\u4ee3\u4ef7\u4ec5\u662f\u9700\u8981\u9884\u5148\u4e3a\u6570\u7ec4\u5206\u914d\u4e00\u5b9a\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u5982\u679c\u6570\u636e\u91cf\u975e\u5e38\u5927\u3001\u52a8\u6001\u6027\u5f88\u9ad8\u3001\u6808\u7684\u9884\u671f\u5927\u5c0f\u96be\u4ee5\u4f30\u8ba1\uff0c\u90a3\u4e48\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u5c31\u66f4\u52a0\u5408\u9002\u3002\u94fe\u8868\u80fd\u591f\u5c06\u5927\u91cf\u6570\u636e\u5206\u6563\u5b58\u50a8\u4e8e\u5185\u5b58\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u5e76\u4e14\u907f\u514d\u4e86\u6570\u7ec4\u6269\u5bb9\u4ea7\u751f\u7684\u989d\u5916\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/","title":"4.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_array_and_linkedlist/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u7ec4\u548c\u94fe\u8868\u662f\u4e24\u79cd\u57fa\u672c\u7684\u6570\u636e\u7ed3\u6784\uff0c\u5206\u522b\u4ee3\u8868\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u4e24\u79cd\u5b58\u50a8\u65b9\u5f0f\uff1a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\u3002\u4e24\u8005\u7684\u7279\u70b9\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u6027\u3002</li> <li>\u6570\u7ec4\u652f\u6301\u968f\u673a\u8bbf\u95ee\u3001\u5360\u7528\u5185\u5b58\u8f83\u5c11\uff1b\u4f46\u63d2\u5165\u548c\u5220\u9664\u5143\u7d20\u6548\u7387\u4f4e\uff0c\u4e14\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002</li> <li>\u94fe\u8868\u901a\u8fc7\u66f4\u6539\u5f15\u7528\uff08\u6307\u9488\uff09\u5b9e\u73b0\u9ad8\u6548\u7684\u8282\u70b9\u63d2\u5165\u4e0e\u5220\u9664\uff0c\u4e14\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\uff1b\u4f46\u8282\u70b9\u8bbf\u95ee\u6548\u7387\u4f4e\u3001\u5360\u7528\u5185\u5b58\u8f83\u591a\u3002</li> <li>\u5e38\u89c1\u7684\u94fe\u8868\u7c7b\u578b\u5305\u62ec\u5355\u5411\u94fe\u8868\u3001\u73af\u5f62\u94fe\u8868\u3001\u53cc\u5411\u94fe\u8868\uff0c\u5b83\u4eec\u5206\u522b\u5177\u6709\u5404\u81ea\u7684\u5e94\u7528\u573a\u666f\u3002</li> <li>\u5217\u8868\u662f\u4e00\u79cd\u652f\u6301\u589e\u5220\u67e5\u6539\u7684\u5143\u7d20\u6709\u5e8f\u96c6\u5408\uff0c\u901a\u5e38\u57fa\u4e8e\u52a8\u6001\u6570\u7ec4\u5b9e\u73b0\uff0c\u5176\u4fdd\u7559\u4e86\u6570\u7ec4\u7684\u4f18\u52bf\uff0c\u540c\u65f6\u53ef\u4ee5\u7075\u6d3b\u8c03\u6574\u957f\u5ea6\u3002</li> <li>\u5217\u8868\u7684\u51fa\u73b0\u5927\u5e45\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u3002</li> <li>\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u4e3b\u8981\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u6570\u7ec4\u63d0\u4f9b\u66f4\u9ad8\u7684\u5185\u5b58\u7a7a\u95f4\u6548\u7387\uff0c\u800c\u94fe\u8868\u5219\u5728\u5185\u5b58\u4f7f\u7528\u4e0a\u66f4\u52a0\u7075\u6d3b\u3002</li> <li>\u7f13\u5b58\u901a\u8fc7\u7f13\u5b58\u884c\u3001\u9884\u53d6\u673a\u5236\u4ee5\u53ca\u7a7a\u95f4\u548c\u65f6\u95f4\u5c40\u90e8\u6027\u7b49\u6570\u636e\u52a0\u8f7d\u673a\u5236\uff0c\u4e3a CPU \u63d0\u4f9b\u5feb\u901f\u6570\u636e\u8bbf\u95ee\uff0c\u663e\u8457\u63d0\u5347\u7a0b\u5e8f\u7684\u6267\u884c\u6548\u7387\u3002</li> <li>\u7531\u4e8e\u6570\u7ec4\u5177\u6709\u66f4\u9ad8\u7684\u7f13\u5b58\u547d\u4e2d\u7387\uff0c\u56e0\u6b64\u5b83\u901a\u5e38\u6bd4\u94fe\u8868\u66f4\u9ad8\u6548\u3002\u5728\u9009\u62e9\u6570\u636e\u7ed3\u6784\u65f6\uff0c\u5e94\u6839\u636e\u5177\u4f53\u9700\u6c42\u548c\u573a\u666f\u505a\u51fa\u6070\u5f53\u9009\u62e9\u3002</li> </ul>"},{"location":"chapter_array_and_linkedlist/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6570\u7ec4\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5b58\u50a8\u5728\u5806\u4e0a\uff0c\u5bf9\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u5426\u6709\u5f71\u54cd\uff1f</p> <p>\u5b58\u50a8\u5728\u6808\u4e0a\u548c\u5806\u4e0a\u7684\u6570\u7ec4\u90fd\u88ab\u5b58\u50a8\u5728\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\u5185\uff0c\u6570\u636e\u64cd\u4f5c\u6548\u7387\u662f\u57fa\u672c\u4e00\u81f4\u7684\u3002\u7136\u800c\uff0c\u6808\u548c\u5806\u5177\u6709\u5404\u81ea\u7684\u7279\u70b9\uff0c\u4ece\u800c\u5bfc\u81f4\u4ee5\u4e0b\u4e0d\u540c\u70b9\u3002</p> <ol> <li>\u5206\u914d\u548c\u91ca\u653e\u6548\u7387\uff1a\u6808\u662f\u4e00\u5757\u8f83\u5c0f\u7684\u5185\u5b58\uff0c\u5206\u914d\u7531\u7f16\u8bd1\u5668\u81ea\u52a8\u5b8c\u6210\uff1b\u800c\u5806\u5185\u5b58\u76f8\u5bf9\u66f4\u5927\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u52a8\u6001\u5206\u914d\uff0c\u66f4\u5bb9\u6613\u788e\u7247\u5316\u3002\u56e0\u6b64\uff0c\u5806\u4e0a\u7684\u5206\u914d\u548c\u91ca\u653e\u64cd\u4f5c\u901a\u5e38\u6bd4\u6808\u4e0a\u7684\u6162\u3002</li> <li>\u5927\u5c0f\u9650\u5236\uff1a\u6808\u5185\u5b58\u76f8\u5bf9\u8f83\u5c0f\uff0c\u5806\u7684\u5927\u5c0f\u4e00\u822c\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\u3002\u56e0\u6b64\u5806\u66f4\u52a0\u9002\u5408\u5b58\u50a8\u5927\u578b\u6570\u7ec4\u3002</li> <li>\u7075\u6d3b\u6027\uff1a\u6808\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u9700\u8981\u5728\u7f16\u8bd1\u65f6\u786e\u5b9a\uff0c\u800c\u5806\u4e0a\u7684\u6570\u7ec4\u7684\u5927\u5c0f\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u786e\u5b9a\u3002</li> </ol> <p>\u4e3a\u4ec0\u4e48\u6570\u7ec4\u8981\u6c42\u76f8\u540c\u7c7b\u578b\u7684\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u5374\u6ca1\u6709\u5f3a\u8c03\u540c\u7c7b\u578b\u5462\uff1f</p> <p>\u94fe\u8868\u7531\u8282\u70b9\u7ec4\u6210\uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u5f15\u7528\uff08\u6307\u9488\uff09\u8fde\u63a5\uff0c\u5404\u4e2a\u8282\u70b9\u53ef\u4ee5\u5b58\u50a8\u4e0d\u540c\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f8b\u5982 int\u3001double\u3001string\u3001object \u7b49\u3002</p> <p>\u76f8\u5bf9\u5730\uff0c\u6570\u7ec4\u5143\u7d20\u5219\u5fc5\u987b\u662f\u76f8\u540c\u7c7b\u578b\u7684\uff0c\u8fd9\u6837\u624d\u80fd\u901a\u8fc7\u8ba1\u7b97\u504f\u79fb\u91cf\u6765\u83b7\u53d6\u5bf9\u5e94\u5143\u7d20\u4f4d\u7f6e\u3002\u4f8b\u5982\uff0c\u6570\u7ec4\u540c\u65f6\u5305\u542b int \u548c long \u4e24\u79cd\u7c7b\u578b\uff0c\u5355\u4e2a\u5143\u7d20\u5206\u522b\u5360\u7528 4 bytes \u548c 8 bytes \uff0c\u6b64\u65f6\u5c31\u4e0d\u80fd\u7528\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u504f\u79fb\u91cf\u4e86\uff0c\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5305\u542b\u4e86\u4e24\u79cd\u957f\u5ea6\u7684\u5143\u7d20\u3002</p> <pre><code># \u5143\u7d20\u5185\u5b58\u5730\u5740 = \u6570\u7ec4\u5185\u5b58\u5730\u5740 + \u5143\u7d20\u957f\u5ea6 * \u5143\u7d20\u7d22\u5f15\n</code></pre> <p>\u5220\u9664\u8282\u70b9\u540e\uff0c\u662f\u5426\u9700\u8981\u628a <code>P.next</code> \u8bbe\u4e3a \\(\\text{None}\\) \u5462\uff1f</p> <p>\u4e0d\u4fee\u6539 <code>P.next</code> \u4e5f\u53ef\u4ee5\u3002\u4ece\u8be5\u94fe\u8868\u7684\u89d2\u5ea6\u770b\uff0c\u4ece\u5934\u8282\u70b9\u904d\u5386\u5230\u5c3e\u8282\u70b9\u5df2\u7ecf\u4e0d\u4f1a\u9047\u5230 <code>P</code> \u4e86\u3002\u8fd9\u610f\u5473\u7740\u8282\u70b9 <code>P</code> \u5df2\u7ecf\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u4e86\uff0c\u6b64\u65f6\u8282\u70b9 <code>P</code> \u6307\u5411\u54ea\u91cc\u90fd\u4e0d\u4f1a\u5bf9\u8be5\u94fe\u8868\u4ea7\u751f\u5f71\u54cd\u3002</p> <p>\u4ece\u5783\u573e\u56de\u6536\u7684\u89d2\u5ea6\u770b\uff0c\u5bf9\u4e8e Java\u3001Python\u3001Go \u7b49\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\u7684\u8bed\u8a00\u6765\u8bf4\uff0c\u8282\u70b9 <code>P</code> \u662f\u5426\u88ab\u56de\u6536\u53d6\u51b3\u4e8e\u662f\u5426\u4ecd\u5b58\u5728\u6307\u5411\u5b83\u7684\u5f15\u7528\uff0c\u800c\u4e0d\u662f <code>P.next</code> \u7684\u503c\u3002\u5728 C \u548c C++ \u7b49\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u624b\u52a8\u91ca\u653e\u8282\u70b9\u5185\u5b58\u3002</p> <p>\u5728\u94fe\u8868\u4e2d\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u4f46\u662f\u589e\u5220\u4e4b\u524d\u90fd\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u67e5\u627e\u5143\u7d20\uff0c\u90a3\u4e3a\u4ec0\u4e48\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u662f \\(O(n)\\) \u5462\uff1f</p> <p>\u5982\u679c\u662f\u5148\u67e5\u627e\u5143\u7d20\u3001\u518d\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u786e\u5b9e\u662f \\(O(n)\\) \u3002\u7136\u800c\uff0c\u94fe\u8868\u7684 \\(O(1)\\) \u589e\u5220\u7684\u4f18\u52bf\u53ef\u4ee5\u5728\u5176\u4ed6\u5e94\u7528\u4e0a\u5f97\u5230\u4f53\u73b0\u3002\u4f8b\u5982\uff0c\u53cc\u5411\u961f\u5217\u9002\u5408\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff0c\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6307\u9488\u53d8\u91cf\u59cb\u7ec8\u6307\u5411\u5934\u8282\u70b9\u3001\u5c3e\u8282\u70b9\uff0c\u6bcf\u6b21\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u90fd\u662f \\(O(1)\\) \u3002</p> <p>\u56fe\u201c\u94fe\u8868\u5b9a\u4e49\u4e0e\u5b58\u50a8\u65b9\u5f0f\u201d\u4e2d\uff0c\u6d45\u84dd\u8272\u7684\u5b58\u50a8\u8282\u70b9\u6307\u9488\u662f\u5360\u7528\u4e00\u5757\u5185\u5b58\u5730\u5740\u5417\uff1f\u8fd8\u662f\u548c\u8282\u70b9\u503c\u5404\u5360\u4e00\u534a\u5462\uff1f</p> <p>\u8be5\u793a\u610f\u56fe\u53ea\u662f\u5b9a\u6027\u8868\u793a\uff0c\u5b9a\u91cf\u8868\u793a\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p> <ul> <li>\u4e0d\u540c\u7c7b\u578b\u7684\u8282\u70b9\u503c\u5360\u7528\u7684\u7a7a\u95f4\u662f\u4e0d\u540c\u7684\uff0c\u6bd4\u5982 int\u3001long\u3001double \u548c\u5b9e\u4f8b\u5bf9\u8c61\u7b49\u3002</li> <li>\u6307\u9488\u53d8\u91cf\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u5927\u5c0f\u6839\u636e\u6240\u4f7f\u7528\u7684\u64cd\u4f5c\u7cfb\u7edf\u53ca\u7f16\u8bd1\u73af\u5883\u800c\u5b9a\uff0c\u5927\u591a\u4e3a 8 \u5b57\u8282\u6216 4 \u5b57\u8282\u3002</li> </ul> <p>\u5728\u5217\u8868\u672b\u5c3e\u6dfb\u52a0\u5143\u7d20\u662f\u5426\u65f6\u65f6\u523b\u523b\u90fd\u4e3a \\(O(1)\\) \uff1f</p> <p>\u5982\u679c\u6dfb\u52a0\u5143\u7d20\u65f6\u8d85\u51fa\u5217\u8868\u957f\u5ea6\uff0c\u5219\u9700\u8981\u5148\u6269\u5bb9\u5217\u8868\u518d\u6dfb\u52a0\u3002\u7cfb\u7edf\u4f1a\u7533\u8bf7\u4e00\u5757\u65b0\u7684\u5185\u5b58\uff0c\u5e76\u5c06\u539f\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u642c\u8fd0\u8fc7\u53bb\uff0c\u8fd9\u65f6\u5019\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u4f1a\u662f \\(O(n)\\) \u3002</p> <p>\u201c\u5217\u8868\u7684\u51fa\u73b0\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u6570\u7ec4\u7684\u5b9e\u7528\u6027\uff0c\u4f46\u53ef\u80fd\u5bfc\u81f4\u90e8\u5206\u5185\u5b58\u7a7a\u95f4\u6d6a\u8d39\u201d\uff0c\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u662f\u6307\u989d\u5916\u589e\u52a0\u7684\u53d8\u91cf\u5982\u5bb9\u91cf\u3001\u957f\u5ea6\u3001\u6269\u5bb9\u500d\u6570\u6240\u5360\u7684\u5185\u5b58\u5417\uff1f</p> <p>\u8fd9\u91cc\u7684\u7a7a\u95f4\u6d6a\u8d39\u4e3b\u8981\u6709\u4e24\u65b9\u9762\u542b\u4e49\uff1a\u4e00\u65b9\u9762\uff0c\u5217\u8868\u90fd\u4f1a\u8bbe\u5b9a\u4e00\u4e2a\u521d\u59cb\u957f\u5ea6\uff0c\u6211\u4eec\u4e0d\u4e00\u5b9a\u9700\u8981\u7528\u8fd9\u4e48\u591a\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u4e3a\u4e86\u9632\u6b62\u9891\u7e41\u6269\u5bb9\uff0c\u6269\u5bb9\u4e00\u822c\u4f1a\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u6bd4\u5982 \\(\\times 1.5\\) \u3002\u8fd9\u6837\u4e00\u6765\uff0c\u4e5f\u4f1a\u51fa\u73b0\u5f88\u591a\u7a7a\u4f4d\uff0c\u6211\u4eec\u901a\u5e38\u4e0d\u80fd\u5b8c\u5168\u586b\u6ee1\u5b83\u4eec\u3002</p> <p>\u5728 Python \u4e2d\u521d\u59cb\u5316 <code>n = [1, 2, 3]</code> \u540e\uff0c\u8fd9 3 \u4e2a\u5143\u7d20\u7684\u5730\u5740\u662f\u76f8\u8fde\u7684\uff0c\u4f46\u662f\u521d\u59cb\u5316 <code>m = [2, 1, 3]</code> \u4f1a\u53d1\u73b0\u5b83\u4eec\u6bcf\u4e2a\u5143\u7d20\u7684 id \u5e76\u4e0d\u662f\u8fde\u7eed\u7684\uff0c\u800c\u662f\u5206\u522b\u8ddf <code>n</code> \u4e2d\u7684\u76f8\u540c\u3002\u8fd9\u4e9b\u5143\u7d20\u7684\u5730\u5740\u4e0d\u8fde\u7eed\uff0c\u90a3\u4e48 <code>m</code> \u8fd8\u662f\u6570\u7ec4\u5417\uff1f</p> <p>\u5047\u5982\u628a\u5217\u8868\u5143\u7d20\u6362\u6210\u94fe\u8868\u8282\u70b9 <code>n = [n1, n2, n3, n4, n5]</code> \uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u8fd9 5 \u4e2a\u8282\u70b9\u5bf9\u8c61\u4e5f\u5206\u6563\u5b58\u50a8\u5728\u5185\u5b58\u5404\u5904\u3002\u7136\u800c\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u5217\u8868\u7d22\u5f15\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u8282\u70b9\u5185\u5b58\u5730\u5740\uff0c\u4ece\u800c\u8bbf\u95ee\u5230\u5bf9\u5e94\u7684\u8282\u70b9\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u4e2d\u5b58\u50a8\u7684\u662f\u8282\u70b9\u7684\u5f15\u7528\uff0c\u800c\u975e\u8282\u70b9\u672c\u8eab\u3002</p> <p>\u4e0e\u8bb8\u591a\u8bed\u8a00\u4e0d\u540c\uff0cPython \u4e2d\u7684\u6570\u5b57\u4e5f\u88ab\u5305\u88c5\u4e3a\u5bf9\u8c61\uff0c\u5217\u8868\u4e2d\u5b58\u50a8\u7684\u4e0d\u662f\u6570\u5b57\u672c\u8eab\uff0c\u800c\u662f\u5bf9\u6570\u5b57\u7684\u5f15\u7528\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u4e24\u4e2a\u6570\u7ec4\u4e2d\u7684\u76f8\u540c\u6570\u5b57\u62e5\u6709\u540c\u4e00\u4e2a id \uff0c\u5e76\u4e14\u8fd9\u4e9b\u6570\u5b57\u7684\u5185\u5b58\u5730\u5740\u65e0\u987b\u8fde\u7eed\u3002</p> <p>C++ STL \u91cc\u9762\u7684 <code>std::list</code> \u5df2\u7ecf\u5b9e\u73b0\u4e86\u53cc\u5411\u94fe\u8868\uff0c\u4f46\u597d\u50cf\u4e00\u4e9b\u7b97\u6cd5\u4e66\u4e0a\u4e0d\u600e\u4e48\u76f4\u63a5\u4f7f\u7528\u5b83\uff0c\u662f\u4e0d\u662f\u56e0\u4e3a\u6709\u4ec0\u4e48\u5c40\u9650\u6027\u5462\uff1f</p> <p>\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u5f80\u5f80\u66f4\u9752\u7750\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u7b97\u6cd5\uff0c\u800c\u53ea\u5728\u5fc5\u8981\u65f6\u624d\u4f7f\u7528\u94fe\u8868\uff0c\u4e3b\u8981\u6709\u4e24\u4e2a\u539f\u56e0\u3002</p> <ul> <li>\u7a7a\u95f4\u5f00\u9500\uff1a\u7531\u4e8e\u6bcf\u4e2a\u5143\u7d20\u9700\u8981\u4e24\u4e2a\u989d\u5916\u7684\u6307\u9488\uff08\u4e00\u4e2a\u7528\u4e8e\u524d\u4e00\u4e2a\u5143\u7d20\uff0c\u4e00\u4e2a\u7528\u4e8e\u540e\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u6240\u4ee5 <code>std::list</code> \u901a\u5e38\u6bd4 <code>std::vector</code> \u66f4\u5360\u7528\u7a7a\u95f4\u3002</li> <li>\u7f13\u5b58\u4e0d\u53cb\u597d\uff1a\u7531\u4e8e\u6570\u636e\u4e0d\u662f\u8fde\u7eed\u5b58\u653e\u7684\uff0c\u56e0\u6b64 <code>std::list</code> \u5bf9\u7f13\u5b58\u7684\u5229\u7528\u7387\u8f83\u4f4e\u3002\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c<code>std::vector</code> \u7684\u6027\u80fd\u4f1a\u66f4\u597d\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5fc5\u8981\u4f7f\u7528\u94fe\u8868\u7684\u60c5\u51b5\u4e3b\u8981\u662f\u4e8c\u53c9\u6811\u548c\u56fe\u3002\u6808\u548c\u961f\u5217\u5f80\u5f80\u4f1a\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684 <code>stack</code> \u548c <code>queue</code> \uff0c\u800c\u975e\u94fe\u8868\u3002</p> <p>\u521d\u59cb\u5316\u5217\u8868 <code>res = [0] * self.size()</code> \u64cd\u4f5c\uff0c\u4f1a\u5bfc\u81f4 <code>res</code> \u7684\u6bcf\u4e2a\u5143\u7d20\u5f15\u7528\u76f8\u540c\u7684\u5730\u5740\u5417\uff1f</p> <p>\u4e0d\u4f1a\u3002\u4f46\u4e8c\u7ef4\u6570\u7ec4\u4f1a\u6709\u8fd9\u4e2a\u95ee\u9898\uff0c\u4f8b\u5982\u521d\u59cb\u5316\u4e8c\u7ef4\u5217\u8868 <code>res = [[0] * self.size()]</code> \uff0c\u5219\u591a\u6b21\u5f15\u7528\u4e86\u540c\u4e00\u4e2a\u5217\u8868 <code>[0]</code> \u3002</p> <p>\u5728\u5220\u9664\u8282\u70b9\u4e2d\uff0c\u9700\u8981\u65ad\u5f00\u8be5\u8282\u70b9\u4e0e\u5176\u540e\u7ee7\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\u6307\u5411\u5417\uff1f</p> <p>\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\uff08\u505a\u9898\uff09\u7684\u89d2\u5ea6\u770b\uff0c\u4e0d\u65ad\u5f00\u6ca1\u6709\u5173\u7cfb\uff0c\u53ea\u8981\u4fdd\u8bc1\u7a0b\u5e8f\u7684\u903b\u8f91\u662f\u6b63\u786e\u7684\u5c31\u884c\u3002\u4ece\u6807\u51c6\u5e93\u7684\u89d2\u5ea6\u770b\uff0c\u65ad\u5f00\u66f4\u52a0\u5b89\u5168\u3001\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3002\u5982\u679c\u4e0d\u65ad\u5f00\uff0c\u5047\u8bbe\u88ab\u5220\u9664\u8282\u70b9\u672a\u88ab\u6b63\u5e38\u56de\u6536\uff0c\u90a3\u4e48\u5b83\u4e5f\u4f1a\u5f71\u54cd\u540e\u7ee7\u8282\u70b9\u7684\u5185\u5b58\u56de\u6536\u3002</p>"},{"location":"chapter_backtracking/","title":"\u7b2c 13 \u7ae0 \u00a0 \u56de\u6eaf","text":"<p>Abstract</p> <p>\u6211\u4eec\u5982\u540c\u8ff7\u5bab\u4e2d\u7684\u63a2\u7d22\u8005\uff0c\u5728\u524d\u8fdb\u7684\u9053\u8def\u4e0a\u53ef\u80fd\u4f1a\u9047\u5230\u56f0\u96be\u3002</p> <p>\u56de\u6eaf\u7684\u529b\u91cf\u8ba9\u6211\u4eec\u80fd\u591f\u91cd\u65b0\u5f00\u59cb\uff0c\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u6700\u7ec8\u627e\u5230\u901a\u5f80\u5149\u660e\u7684\u51fa\u53e3\u3002</p>"},{"location":"chapter_backtracking/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5</li> <li>13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898</li> <li>13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898</li> <li>13.4 \u00a0 N \u7687\u540e\u95ee\u9898</li> <li>13.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/","title":"13.1 \u00a0 \u56de\u6eaf\u7b97\u6cd5","text":"<p>\u300c\u56de\u6eaf\u7b97\u6cd5 backtracking algorithm\u300d\u662f\u4e00\u79cd\u901a\u8fc7\u7a77\u4e3e\u6765\u89e3\u51b3\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u4ece\u4e00\u4e2a\u521d\u59cb\u72b6\u6001\u51fa\u53d1\uff0c\u66b4\u529b\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5f53\u9047\u5230\u6b63\u786e\u7684\u89e3\u5219\u5c06\u5176\u8bb0\u5f55\uff0c\u76f4\u5230\u627e\u5230\u89e3\u6216\u8005\u5c1d\u8bd5\u4e86\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u90fd\u65e0\u6cd5\u627e\u5230\u89e3\u4e3a\u6b62\u3002</p> <p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u6765\u904d\u5386\u89e3\u7a7a\u95f4\u3002\u5728\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5229\u7528\u524d\u5e8f\u904d\u5386\u6784\u9020\u4e00\u4e2a\u56de\u6eaf\u95ee\u9898\uff0c\u9010\u6b65\u4e86\u89e3\u56de\u6eaf\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p>\u4f8b\u9898\u4e00</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u641c\u7d22\u5e76\u8bb0\u5f55\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u8282\u70b9\u5217\u8868\u3002</p> <p>\u5bf9\u4e8e\u6b64\u9898\uff0c\u6211\u4eec\u524d\u5e8f\u904d\u5386\u8fd9\u68f5\u6811\uff0c\u5e76\u5224\u65ad\u5f53\u524d\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \uff0c\u82e5\u662f\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u7684\u503c\u52a0\u5165\u7ed3\u679c\u5217\u8868 <code>res</code> \u4e4b\u4e2d\u3002\u76f8\u5173\u8fc7\u7a0b\u5b9e\u73b0\u5982\u56fe 13-1 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_i_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00\"\"\"\n    if root is None:\n        return\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(root)\n    pre_order(root.left)\n    pre_order(root.right)\n</code></pre> preorder_traversal_i_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(root);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(root);\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(root);\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n</code></pre> preorder_traversal_i_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrderI(root *TreeNode, res *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    if (root.Val).(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, root)\n    }\n    preOrderI(root.Left, res)\n    preOrderI(root.Right, res)\n}\n</code></pre> preorder_traversal_i_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(root)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n</code></pre> preorder_traversal_i_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root, res) {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfunction preOrder(root: TreeNode | null, res: TreeNode[]): void {\n    if (root === null) {\n        return;\n    }\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push(root);\n    }\n    preOrder(root.left, res);\n    preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode? root, List&lt;TreeNode&gt; res) {\n  if (root == null) {\n    return;\n  }\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(root);\n  }\n  preOrder(root.left, res);\n  preOrder(root.right, res);\n}\n</code></pre> preorder_traversal_i_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nfn pre_order(res: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(node.clone());\n        }\n        pre_order(res, node.borrow().left.clone());\n        pre_order(res, node.borrow().right.clone());\n    }\n}\n</code></pre> preorder_traversal_i_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e00 */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res[resSize++] = root;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n</code></pre> preorder_traversal_i_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> <p></p> <p> \u56fe 13-1 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u4e2d\u641c\u7d22\u8282\u70b9 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1311","title":"13.1.1 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000","text":"<p>\u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u56de\u6eaf\u7b97\u6cd5\uff0c\u662f\u56e0\u4e3a\u8be5\u7b97\u6cd5\u5728\u641c\u7d22\u89e3\u7a7a\u95f4\u65f6\u4f1a\u91c7\u7528\u201c\u5c1d\u8bd5\u201d\u4e0e\u201c\u56de\u9000\u201d\u7684\u7b56\u7565\u3002\u5f53\u7b97\u6cd5\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u9047\u5230\u67d0\u4e2a\u72b6\u6001\u65e0\u6cd5\u7ee7\u7eed\u524d\u8fdb\u6216\u65e0\u6cd5\u5f97\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u65f6\uff0c\u5b83\u4f1a\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u5c1d\u8bd5\u5176\u4ed6\u53ef\u80fd\u7684\u9009\u62e9\u3002</p> <p>\u5bf9\u4e8e\u4f8b\u9898\u4e00\uff0c\u8bbf\u95ee\u6bcf\u4e2a\u8282\u70b9\u90fd\u4ee3\u8868\u4e00\u6b21\u201c\u5c1d\u8bd5\u201d\uff0c\u800c\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u8fd4\u56de\u7236\u8282\u70b9\u7684 <code>return</code> \u5219\u8868\u793a\u201c\u56de\u9000\u201d\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u56de\u9000\u5e76\u4e0d\u4ec5\u4ec5\u5305\u62ec\u51fd\u6570\u8fd4\u56de\u3002\u4e3a\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5bf9\u4f8b\u9898\u4e00\u7a0d\u4f5c\u62d3\u5c55\u3002</p> <p>\u4f8b\u9898\u4e8c</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\u3002</p> <p>\u5728\u4f8b\u9898\u4e00\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u5217\u8868 <code>path</code> \u8bb0\u5f55\u8bbf\u95ee\u8fc7\u7684\u8282\u70b9\u8def\u5f84\u3002\u5f53\u8bbf\u95ee\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u590d\u5236 <code>path</code> \u5e76\u6dfb\u52a0\u8fdb\u7ed3\u679c\u5217\u8868 <code>res</code> \u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c<code>res</code> \u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6240\u6709\u7684\u89e3\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_ii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c\"\"\"\n    if root is None:\n        return\n    # \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_ii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_ii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode root) {\n    if (root == null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_ii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_ii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    if root == nil {\n        return\n    }\n    // \u5c1d\u8bd5\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, *path)\n    }\n    preOrderII(root.Left, res, path)\n    preOrderII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_ii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_ii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(root, path, res) {\n    if (root === null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    if (root === null) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_ii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null) {\n    return;\n  }\n\n  // \u5c1d\u8bd5\n  path.add(root);\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_ii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nfn pre_order(res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;, path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    if root.is_none() {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5c1d\u8bd5\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.clone());\n        pre_order(res, path, node.borrow().right.clone());\n        // \u56de\u9000\n        path.remove(path.len() -  1);\n    }\n}\n</code></pre> preorder_traversal_ii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e8c */\nvoid preOrder(TreeNode *root) {\n    if (root == NULL) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        for (int i = 0; i &lt; pathSize; ++i) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_ii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> <p>\u5728\u6bcf\u6b21\u201c\u5c1d\u8bd5\u201d\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u5f53\u524d\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \u6765\u8bb0\u5f55\u8def\u5f84\uff1b\u800c\u5728\u201c\u56de\u9000\u201d\u524d\uff0c\u6211\u4eec\u9700\u8981\u5c06\u8be5\u8282\u70b9\u4ece <code>path</code> \u4e2d\u5f39\u51fa\uff0c\u4ee5\u6062\u590d\u672c\u6b21\u5c1d\u8bd5\u4e4b\u524d\u7684\u72b6\u6001\u3002</p> <p>\u89c2\u5bdf\u56fe 13-2 \u6240\u793a\u7684\u8fc7\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5c1d\u8bd5\u548c\u56de\u9000\u7406\u89e3\u4e3a\u201c\u524d\u8fdb\u201d\u4e0e\u201c\u64a4\u9500\u201d\uff0c\u4e24\u4e2a\u64cd\u4f5c\u4e92\u4e3a\u9006\u5411\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 13-2 \u00a0 \u5c1d\u8bd5\u4e0e\u56de\u9000 </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1312","title":"13.1.2 \u00a0 \u526a\u679d","text":"<p>\u590d\u6742\u7684\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u4e00\u4e2a\u6216\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u7ea6\u675f\u6761\u4ef6\u901a\u5e38\u53ef\u7528\u4e8e\u201c\u526a\u679d\u201d\u3002</p> <p>\u4f8b\u9898\u4e09</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\u641c\u7d22\u6240\u6709\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\uff0c\u8bf7\u8fd4\u56de\u6839\u8282\u70b9\u5230\u8fd9\u4e9b\u8282\u70b9\u7684\u8def\u5f84\uff0c\u5e76\u8981\u6c42\u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u3002</p> <p>\u4e3a\u4e86\u6ee1\u8db3\u4ee5\u4e0a\u7ea6\u675f\u6761\u4ef6\uff0c\u6211\u4eec\u9700\u8981\u6dfb\u52a0\u526a\u679d\u64cd\u4f5c\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\uff0c\u5219\u63d0\u524d\u8fd4\u56de\uff0c\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_compact.py<pre><code>def pre_order(root: TreeNode):\n    \"\"\"\u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09\"\"\"\n    # \u526a\u679d\n    if root is None or root.val == 3:\n        return\n    # \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7:\n        # \u8bb0\u5f55\u89e3\n        res.append(list(path))\n    pre_order(root.left)\n    pre_order(root.right)\n    # \u56de\u9000\n    path.pop()\n</code></pre> preorder_traversal_iii_compact.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == nullptr || root-&gt;val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push_back(root);\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push_back(path);\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    path.pop_back();\n}\n</code></pre> preorder_traversal_iii_compact.java<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.add(new ArrayList&lt;&gt;(path));\n    }\n    preOrder(root.left);\n    preOrder(root.right);\n    // \u56de\u9000\n    path.remove(path.size() - 1);\n}\n</code></pre> preorder_traversal_iii_compact.cs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid PreOrder(TreeNode? root) {\n    // \u526a\u679d\n    if (root == null || root.val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.Add(root);\n    if (root.val == 7) {\n        // \u8bb0\u5f55\u89e3\n        res.Add(new List&lt;TreeNode&gt;(path));\n    }\n    PreOrder(root.left);\n    PreOrder(root.right);\n    // \u56de\u9000\n    path.RemoveAt(path.Count - 1);\n}\n</code></pre> preorder_traversal_iii_compact.go<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {\n    // \u526a\u679d\n    if root == nil || root.Val == 3 {\n        return\n    }\n    // \u5c1d\u8bd5\n    *path = append(*path, root)\n    if root.Val.(int) == 7 {\n        // \u8bb0\u5f55\u89e3\n        *res = append(*res, *path)\n    }\n    preOrderIII(root.Left, res, path)\n    preOrderIII(root.Right, res, path)\n    // \u56de\u9000\n    *path = (*path)[:len(*path)-1]\n}\n</code></pre> preorder_traversal_iii_compact.swift<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunc preOrder(root: TreeNode?) {\n    // \u526a\u679d\n    guard let root = root, root.val != 3 else {\n        return\n    }\n    // \u5c1d\u8bd5\n    path.append(root)\n    if root.val == 7 {\n        // \u8bb0\u5f55\u89e3\n        res.append(path)\n    }\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n    // \u56de\u9000\n    path.removeLast()\n}\n</code></pre> preorder_traversal_iii_compact.js<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(root, path, res) {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.ts<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfunction preOrder(\n    root: TreeNode | null,\n    path: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u526a\u679d\n    if (root === null || root.val === 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path.push(root);\n    if (root.val === 7) {\n        // \u8bb0\u5f55\u89e3\n        res.push([...path]);\n    }\n    preOrder(root.left, path, res);\n    preOrder(root.right, path, res);\n    // \u56de\u9000\n    path.pop();\n}\n</code></pre> preorder_traversal_iii_compact.dart<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(\n  TreeNode? root,\n  List&lt;TreeNode&gt; path,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  if (root == null || root.val == 3) {\n    return;\n  }\n\n  // \u5c1d\u8bd5\n  path.add(root);\n  if (root.val == 7) {\n    // \u8bb0\u5f55\u89e3\n    res.add(List.from(path));\n  }\n  preOrder(root.left, path, res);\n  preOrder(root.right, path, res);\n  // \u56de\u9000\n  path.removeLast();\n}\n</code></pre> preorder_traversal_iii_compact.rs<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nfn pre_order(res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;, path: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, root: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) {\n    // \u526a\u679d\n    if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {\n        return;\n    }\n    if let Some(node) = root {\n        // \u5c1d\u8bd5\n        path.push(node.clone());\n        if node.borrow().val == 7 {\n            // \u8bb0\u5f55\u89e3\n            res.push(path.clone());\n        }\n        pre_order(res, path, node.borrow().left.clone());\n        pre_order(res, path, node.borrow().right.clone());\n        // \u56de\u9000\n        path.remove(path.len() -  1);\n    }\n}\n</code></pre> preorder_traversal_iii_compact.c<pre><code>/* \u524d\u5e8f\u904d\u5386\uff1a\u4f8b\u9898\u4e09 */\nvoid preOrder(TreeNode *root) {\n    // \u526a\u679d\n    if (root == NULL || root-&gt;val == 3) {\n        return;\n    }\n    // \u5c1d\u8bd5\n    path[pathSize++] = root;\n    if (root-&gt;val == 7) {\n        // \u8bb0\u5f55\u89e3\n        for (int i = 0; i &lt; pathSize; i++) {\n            res[resSize][i] = path[i];\n        }\n        resSize++;\n    }\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n    // \u56de\u9000\n    pathSize--;\n}\n</code></pre> preorder_traversal_iii_compact.zig<pre><code>[class]{}-[func]{preOrder}\n</code></pre> <p>\u201c\u526a\u679d\u201d\u662f\u4e00\u4e2a\u975e\u5e38\u5f62\u8c61\u7684\u540d\u8bcd\u3002\u5982\u56fe 13-3 \u6240\u793a\uff0c\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u201c\u526a\u6389\u201d\u4e86\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u641c\u7d22\u5206\u652f\uff0c\u907f\u514d\u8bb8\u591a\u65e0\u610f\u4e49\u7684\u5c1d\u8bd5\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u641c\u7d22\u6548\u7387\u3002</p> <p></p> <p> \u56fe 13-3 \u00a0 \u6839\u636e\u7ea6\u675f\u6761\u4ef6\u526a\u679d </p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1313","title":"13.1.3 \u00a0 \u6846\u67b6\u4ee3\u7801","text":"<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c1d\u8bd5\u5c06\u56de\u6eaf\u7684\u201c\u5c1d\u8bd5\u3001\u56de\u9000\u3001\u526a\u679d\u201d\u7684\u4e3b\u4f53\u6846\u67b6\u63d0\u70bc\u51fa\u6765\uff0c\u63d0\u5347\u4ee3\u7801\u7684\u901a\u7528\u6027\u3002</p> <p>\u5728\u4ee5\u4e0b\u6846\u67b6\u4ee3\u7801\u4e2d\uff0c<code>state</code> \u8868\u793a\u95ee\u9898\u7684\u5f53\u524d\u72b6\u6001\uff0c<code>choices</code> \u8868\u793a\u5f53\u524d\u72b6\u6001\u4e0b\u53ef\u4ee5\u505a\u51fa\u7684\u9009\u62e9\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def backtrack(state: State, choices: list[choice], res: list[state]):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\"\"\"\n    # \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if is_solution(state):\n        # \u8bb0\u5f55\u89e3\n        record_solution(state, res)\n        # \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice)\n            backtrack(state, choices, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice)\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, vector&lt;Choice *&gt; &amp;choices, vector&lt;State *&gt; &amp;res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Choice choice : choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid Backtrack(State state, List&lt;Choice&gt; choices, List&lt;State&gt; res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (IsSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        RecordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (Choice choice in choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            MakeChoice(state, choice);\n            Backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state *State, choices []Choice, res *[]State) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if isSolution(state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            backtrack(state, choices, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunc backtrack(state: inout State, choices: [Choice], res: inout [State]) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if isSolution(state: state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state: state, res: &amp;res)\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state: &amp;state, choice: choice)\n            backtrack(state: &amp;state, choices: choices, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state, choices, res) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfunction backtrack(state: State, choices: Choice[], res: State[]): void {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let choice of choices) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State state, List&lt;Choice&gt;, List&lt;State&gt; res) {\n  // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n  if (isSolution(state)) {\n    // \u8bb0\u5f55\u89e3\n    recordSolution(state, res);\n    // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (Choice choice in choices) {\n    // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      makeChoice(state, choice);\n      backtrack(state, choices, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nfn backtrack(state: &amp;mut State, choices: &amp;Vec&lt;Choice&gt;, res: &amp;mut Vec&lt;State&gt;) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if is_solution(state) {\n        // \u8bb0\u5f55\u89e3\n        record_solution(state, res);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice);\n            backtrack(state, choices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice);\n        }\n    }\n}\n</code></pre> <pre><code>/* \u56de\u6eaf\u7b97\u6cd5\u6846\u67b6 */\nvoid backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {\n    // \u5224\u65ad\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res, numRes);\n        // \u4e0d\u518d\u7ee7\u7eed\u641c\u7d22\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; numChoices; i++) {\n        // \u526a\u679d\uff1a\u5224\u65ad\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, &amp;choices[i])) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, &amp;choices[i]);\n            backtrack(state, choices, numChoices, res, numRes);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, &amp;choices[i]);\n        }\n    }\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u57fa\u4e8e\u6846\u67b6\u4ee3\u7801\u6765\u89e3\u51b3\u4f8b\u9898\u4e09\u3002\u72b6\u6001 <code>state</code> \u4e3a\u8282\u70b9\u904d\u5386\u8def\u5f84\uff0c\u9009\u62e9 <code>choices</code> \u4e3a\u5f53\u524d\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\uff0c\u7ed3\u679c <code>res</code> \u662f\u8def\u5f84\u5217\u8868\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig preorder_traversal_iii_template.py<pre><code>def is_solution(state: list[TreeNode]) -&gt; bool:\n    \"\"\"\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3\"\"\"\n    return state and state[-1].val == 7\n\ndef record_solution(state: list[TreeNode], res: list[list[TreeNode]]):\n    \"\"\"\u8bb0\u5f55\u89e3\"\"\"\n    res.append(list(state))\n\ndef is_valid(state: list[TreeNode], choice: TreeNode) -&gt; bool:\n    \"\"\"\u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5\"\"\"\n    return choice is not None and choice.val != 3\n\ndef make_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u66f4\u65b0\u72b6\u6001\"\"\"\n    state.append(choice)\n\ndef undo_choice(state: list[TreeNode], choice: TreeNode):\n    \"\"\"\u6062\u590d\u72b6\u6001\"\"\"\n    state.pop()\n\ndef backtrack(\n    state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09\"\"\"\n    # \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if is_solution(state):\n        # \u8bb0\u5f55\u89e3\n        record_solution(state, res)\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice):\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice)\n</code></pre> preorder_traversal_iii_template.cpp<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(vector&lt;TreeNode *&gt; &amp;state) {\n    return !state.empty() &amp;&amp; state.back()-&gt;val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(vector&lt;TreeNode *&gt; &amp;state, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    res.push_back(state);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    return choice != nullptr &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.push_back(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(vector&lt;TreeNode *&gt; &amp;state, TreeNode *choice) {\n    state.pop_back();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(vector&lt;TreeNode *&gt; &amp;state, vector&lt;TreeNode *&gt; &amp;choices, vector&lt;vector&lt;TreeNode *&gt;&gt; &amp;res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (TreeNode *choice : choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            vector&lt;TreeNode *&gt; nextChoices{choice-&gt;left, choice-&gt;right};\n            backtrack(state, nextChoices, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.java<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nboolean isSolution(List&lt;TreeNode&gt; state) {\n    return !state.isEmpty() &amp;&amp; state.get(state.size() - 1).val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.add(new ArrayList&lt;&gt;(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nboolean isValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.remove(state.size() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (TreeNode choice : choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, Arrays.asList(choice.left, choice.right), res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.cs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool IsSolution(List&lt;TreeNode&gt; state) {\n    return state.Count != 0 &amp;&amp; state[^1].val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid RecordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    res.Add(new List&lt;TreeNode&gt;(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool IsValid(List&lt;TreeNode&gt; state, TreeNode choice) {\n    return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid MakeChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.Add(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid UndoChoice(List&lt;TreeNode&gt; state, TreeNode choice) {\n    state.RemoveAt(state.Count - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid Backtrack(List&lt;TreeNode&gt; state, List&lt;TreeNode&gt; choices, List&lt;List&lt;TreeNode&gt;&gt; res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (IsSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        RecordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (TreeNode choice in choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (IsValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            MakeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, [choice.left!, choice.right!], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            UndoChoice(state, choice);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.go<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state *[]*TreeNode) bool {\n    return len(*state) != 0 &amp;&amp; (*state)[len(*state)-1].Val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {\n    *res = append(*res, *state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state *[]*TreeNode, choice *TreeNode) bool {\n    return choice != nil &amp;&amp; choice.Val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = append(*state, choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state *[]*TreeNode, choice *TreeNode) {\n    *state = (*state)[:len(*state)-1]\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if isSolution(state) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range *choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state, choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            temp := make([]*TreeNode, 0)\n            temp = append(temp, choice.Left, choice.Right)\n            backtrackIII(state, &amp;temp, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state, choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.swift<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunc isSolution(state: [TreeNode]) -&gt; Bool {\n    !state.isEmpty &amp;&amp; state.last!.val == 7\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunc recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {\n    res.append(state)\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunc isValid(state: [TreeNode], choice: TreeNode?) -&gt; Bool {\n    choice != nil &amp;&amp; choice!.val != 3\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunc makeChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.append(choice)\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunc undoChoice(state: inout [TreeNode], choice: TreeNode) {\n    state.removeLast()\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunc backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if isSolution(state: state) {\n        recordSolution(state: state, res: &amp;res)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if isValid(state: state, choice: choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state: &amp;state, choice: choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: [choice.left, choice.right].compactMap { $0 }, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state: &amp;state, choice: choice)\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.js<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state) {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state, res) {\n    res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state, choice) {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state, choice) {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state) {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(state, choices, res) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.ts<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfunction isSolution(state: TreeNode[]): boolean {\n    return state &amp;&amp; state[state.length - 1]?.val === 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfunction recordSolution(state: TreeNode[], res: TreeNode[][]): void {\n    res.push([...state]);\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfunction isValid(state: TreeNode[], choice: TreeNode): boolean {\n    return choice !== null &amp;&amp; choice.val !== 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfunction makeChoice(state: TreeNode[], choice: TreeNode): void {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfunction undoChoice(state: TreeNode[]): void {\n    state.pop();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfunction backtrack(\n    state: TreeNode[],\n    choices: TreeNode[],\n    res: TreeNode[][]\n): void {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution(state)) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(state, choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(state, choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, [choice.left, choice.right], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice(state);\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.dart<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(List&lt;TreeNode&gt; state) {\n  return state.isNotEmpty &amp;&amp; state.last.val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(List&lt;TreeNode&gt; state, List&lt;List&lt;TreeNode&gt;&gt; res) {\n  res.add(List.from(state));\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  return choice != null &amp;&amp; choice.val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.add(choice!);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(List&lt;TreeNode&gt; state, TreeNode? choice) {\n  state.removeLast();\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(\n  List&lt;TreeNode&gt; state,\n  List&lt;TreeNode?&gt; choices,\n  List&lt;List&lt;TreeNode&gt;&gt; res,\n) {\n  // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n  if (isSolution(state)) {\n    // \u8bb0\u5f55\u89e3\n    recordSolution(state, res);\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (TreeNode? choice in choices) {\n    // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n    if (isValid(state, choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      makeChoice(state, choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, [choice!.left, choice.right], res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      undoChoice(state, choice);\n    }\n  }\n}\n</code></pre> preorder_traversal_iii_template.rs<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nfn is_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; bool {\n    return !state.is_empty() &amp;&amp; state.get(state.len() - 1).unwrap().borrow().val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nfn record_solution(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;) {\n    res.push(state.clone());\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nfn is_valid(_: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; bool {\n    return choice.borrow().val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nfn make_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choice: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.push(choice);\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nfn undo_choice(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, _: Rc&lt;RefCell&lt;TreeNode&gt;&gt;) {\n    state.remove(state.len() - 1);\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nfn backtrack(state: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, choices: &amp;mut Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;, res: &amp;mut Vec&lt;Vec&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;&gt;) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if is_solution(state) {\n        // \u8bb0\u5f55\u89e3\n        record_solution(state, res);\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if is_valid(state, choice.clone()) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            make_choice(state, choice.clone());\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, &amp;mut vec![choice.borrow().left.clone().unwrap(), choice.borrow().right.clone().unwrap()], res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undo_choice(state, choice.clone());\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.c<pre><code>/* \u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u5426\u4e3a\u89e3 */\nbool isSolution(void) {\n    return pathSize &gt; 0 &amp;&amp; path[pathSize - 1]-&gt;val == 7;\n}\n\n/* \u8bb0\u5f55\u89e3 */\nvoid recordSolution(void) {\n    for (int i = 0; i &lt; pathSize; i++) {\n        res[resSize][i] = path[i];\n    }\n    resSize++;\n}\n\n/* \u5224\u65ad\u5728\u5f53\u524d\u72b6\u6001\u4e0b\uff0c\u8be5\u9009\u62e9\u662f\u5426\u5408\u6cd5 */\nbool isValid(TreeNode *choice) {\n    return choice != NULL &amp;&amp; choice-&gt;val != 3;\n}\n\n/* \u66f4\u65b0\u72b6\u6001 */\nvoid makeChoice(TreeNode *choice) {\n    path[pathSize++] = choice;\n}\n\n/* \u6062\u590d\u72b6\u6001 */\nvoid undoChoice(void) {\n    pathSize--;\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u4f8b\u9898\u4e09 */\nvoid backtrack(TreeNode *choices[2]) {\n    // \u68c0\u67e5\u662f\u5426\u4e3a\u89e3\n    if (isSolution()) {\n        // \u8bb0\u5f55\u89e3\n        recordSolution();\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; 2; i++) {\n        TreeNode *choice = choices[i];\n        // \u526a\u679d\uff1a\u68c0\u67e5\u9009\u62e9\u662f\u5426\u5408\u6cd5\n        if (isValid(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            makeChoice(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            TreeNode *nextChoices[2] = {choice-&gt;left, choice-&gt;right};\n            backtrack(nextChoices);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            undoChoice();\n        }\n    }\n}\n</code></pre> preorder_traversal_iii_template.zig<pre><code>[class]{}-[func]{isSolution}\n\n[class]{}-[func]{recordSolution}\n\n[class]{}-[func]{isValid}\n\n[class]{}-[func]{makeChoice}\n\n[class]{}-[func]{undoChoice}\n\n[class]{}-[func]{backtrack}\n</code></pre> <p>\u6839\u636e\u9898\u610f\uff0c\u6211\u4eec\u5728\u627e\u5230\u503c\u4e3a \\(7\\) \u7684\u8282\u70b9\u540e\u5e94\u8be5\u7ee7\u7eed\u641c\u7d22\uff0c\u56e0\u6b64\u9700\u8981\u5c06\u8bb0\u5f55\u89e3\u4e4b\u540e\u7684 <code>return</code> \u8bed\u53e5\u5220\u9664\u3002\u56fe 13-4 \u5bf9\u6bd4\u4e86\u4fdd\u7559\u6216\u5220\u9664 <code>return</code> \u8bed\u53e5\u7684\u641c\u7d22\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-4 \u00a0 \u4fdd\u7559\u4e0e\u5220\u9664 return \u7684\u641c\u7d22\u8fc7\u7a0b\u5bf9\u6bd4 </p> <p>\u76f8\u6bd4\u57fa\u4e8e\u524d\u5e8f\u904d\u5386\u7684\u4ee3\u7801\u5b9e\u73b0\uff0c\u57fa\u4e8e\u56de\u6eaf\u7b97\u6cd5\u6846\u67b6\u7684\u4ee3\u7801\u5b9e\u73b0\u867d\u7136\u663e\u5f97\u5570\u5506\uff0c\u4f46\u901a\u7528\u6027\u66f4\u597d\u3002\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u56de\u6eaf\u95ee\u9898\u53ef\u4ee5\u5728\u8be5\u6846\u67b6\u4e0b\u89e3\u51b3\u3002\u6211\u4eec\u53ea\u9700\u6839\u636e\u5177\u4f53\u95ee\u9898\u6765\u5b9a\u4e49 <code>state</code> \u548c <code>choices</code> \uff0c\u5e76\u5b9e\u73b0\u6846\u67b6\u4e2d\u7684\u5404\u4e2a\u65b9\u6cd5\u5373\u53ef\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1314","title":"13.1.4 \u00a0 \u5e38\u7528\u672f\u8bed","text":"<p>\u4e3a\u4e86\u66f4\u6e05\u6670\u5730\u5206\u6790\u7b97\u6cd5\u95ee\u9898\uff0c\u6211\u4eec\u603b\u7ed3\u4e00\u4e0b\u56de\u6eaf\u7b97\u6cd5\u4e2d\u5e38\u7528\u672f\u8bed\u7684\u542b\u4e49\uff0c\u5e76\u5bf9\u7167\u4f8b\u9898\u4e09\u7ed9\u51fa\u5bf9\u5e94\u793a\u4f8b\uff0c\u5982\u8868 13-1 \u6240\u793a\u3002</p> <p> \u8868 13-1 \u00a0 \u5e38\u89c1\u7684\u56de\u6eaf\u7b97\u6cd5\u672f\u8bed </p> \u540d\u8bcd \u5b9a\u4e49 \u4f8b\u9898\u4e09 \u89e3\uff08solution\uff09 \u89e3\u662f\u6ee1\u8db3\u95ee\u9898\u7279\u5b9a\u6761\u4ef6\u7684\u7b54\u6848\uff0c\u53ef\u80fd\u6709\u4e00\u4e2a\u6216\u591a\u4e2a \u6839\u8282\u70b9\u5230\u8282\u70b9 \\(7\\) \u7684\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u6240\u6709\u8def\u5f84 \u7ea6\u675f\u6761\u4ef6\uff08constraint\uff09 \u7ea6\u675f\u6761\u4ef6\u662f\u95ee\u9898\u4e2d\u9650\u5236\u89e3\u7684\u53ef\u884c\u6027\u7684\u6761\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u526a\u679d \u8def\u5f84\u4e2d\u4e0d\u5305\u542b\u8282\u70b9 \\(3\\) \u72b6\u6001\uff08state\uff09 \u72b6\u6001\u8868\u793a\u95ee\u9898\u5728\u67d0\u4e00\u65f6\u523b\u7684\u60c5\u51b5\uff0c\u5305\u62ec\u5df2\u7ecf\u505a\u51fa\u7684\u9009\u62e9 \u5f53\u524d\u5df2\u8bbf\u95ee\u7684\u8282\u70b9\u8def\u5f84\uff0c\u5373 <code>path</code> \u8282\u70b9\u5217\u8868 \u5c1d\u8bd5\uff08attempt\uff09 \u5c1d\u8bd5\u662f\u6839\u636e\u53ef\u7528\u9009\u62e9\u6765\u63a2\u7d22\u89e3\u7a7a\u95f4\u7684\u8fc7\u7a0b\uff0c\u5305\u62ec\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\uff0c\u68c0\u67e5\u662f\u5426\u4e3a\u89e3 \u9012\u5f52\u8bbf\u95ee\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\uff0c\u5c06\u8282\u70b9\u6dfb\u52a0\u8fdb <code>path</code> \uff0c\u5224\u65ad\u8282\u70b9\u7684\u503c\u662f\u5426\u4e3a \\(7\\) \u56de\u9000\uff08backtracking\uff09 \u56de\u9000\u6307\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u72b6\u6001\u65f6\uff0c\u64a4\u9500\u524d\u9762\u505a\u51fa\u7684\u9009\u62e9\uff0c\u56de\u5230\u4e0a\u4e00\u4e2a\u72b6\u6001 \u5f53\u8d8a\u8fc7\u53f6\u8282\u70b9\u3001\u7ed3\u675f\u8282\u70b9\u8bbf\u95ee\u3001\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\u7ec8\u6b62\u641c\u7d22\uff0c\u51fd\u6570\u8fd4\u56de \u526a\u679d\uff08pruning\uff09 \u526a\u679d\u662f\u6839\u636e\u95ee\u9898\u7279\u6027\u548c\u7ea6\u675f\u6761\u4ef6\u907f\u514d\u65e0\u610f\u4e49\u7684\u641c\u7d22\u8def\u5f84\u7684\u65b9\u6cd5\uff0c\u53ef\u63d0\u9ad8\u641c\u7d22\u6548\u7387 \u5f53\u9047\u5230\u503c\u4e3a \\(3\\) \u7684\u8282\u70b9\u65f6\uff0c\u5219\u4e0d\u518d\u7ee7\u7eed\u641c\u7d22 <p>Tip</p> <p>\u95ee\u9898\u3001\u89e3\u3001\u72b6\u6001\u7b49\u6982\u5ff5\u662f\u901a\u7528\u7684\uff0c\u5728\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</p>"},{"location":"chapter_backtracking/backtracking_algorithm/#1315","title":"13.1.5 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u4e0a\u662f\u4e00\u79cd\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u7b97\u6cd5\uff0c\u5b83\u5c1d\u8bd5\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u4f18\u70b9\u5728\u4e8e\u80fd\u591f\u627e\u5230\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5408\u7406\u7684\u526a\u679d\u64cd\u4f5c\u4e0b\uff0c\u5177\u6709\u5f88\u9ad8\u7684\u6548\u7387\u3002</p> <p>\u7136\u800c\uff0c\u5728\u5904\u7406\u5927\u89c4\u6a21\u6216\u8005\u590d\u6742\u95ee\u9898\u65f6\uff0c\u56de\u6eaf\u7b97\u6cd5\u7684\u8fd0\u884c\u6548\u7387\u53ef\u80fd\u96be\u4ee5\u63a5\u53d7\u3002</p> <ul> <li>\u65f6\u95f4\uff1a\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u9700\u8981\u904d\u5386\u72b6\u6001\u7a7a\u95f4\u7684\u6240\u6709\u53ef\u80fd\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u6307\u6570\u9636\u6216\u9636\u4e58\u9636\u3002</li> <li>\u7a7a\u95f4\uff1a\u5728\u9012\u5f52\u8c03\u7528\u4e2d\u9700\u8981\u4fdd\u5b58\u5f53\u524d\u7684\u72b6\u6001\uff08\u4f8b\u5982\u8def\u5f84\u3001\u7528\u4e8e\u526a\u679d\u7684\u8f85\u52a9\u53d8\u91cf\u7b49\uff09\uff0c\u5f53\u6df1\u5ea6\u5f88\u5927\u65f6\uff0c\u7a7a\u95f4\u9700\u6c42\u53ef\u80fd\u4f1a\u53d8\u5f97\u5f88\u5927\u3002</li> </ul> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u56de\u6eaf\u7b97\u6cd5\u4ecd\u7136\u662f\u67d0\u4e9b\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u7684\u6700\u4f73\u89e3\u51b3\u65b9\u6848\u3002\u5bf9\u4e8e\u8fd9\u4e9b\u95ee\u9898\uff0c\u7531\u4e8e\u65e0\u6cd5\u9884\u6d4b\u54ea\u4e9b\u9009\u62e9\u53ef\u751f\u6210\u6709\u6548\u7684\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5bf9\u6240\u6709\u53ef\u80fd\u7684\u9009\u62e9\u8fdb\u884c\u904d\u5386\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5173\u952e\u662f\u5982\u4f55\u4f18\u5316\u6548\u7387\uff0c\u5e38\u89c1\u7684\u6548\u7387\u4f18\u5316\u65b9\u6cd5\u6709\u4e24\u79cd\u3002</p> <ul> <li>\u526a\u679d\uff1a\u907f\u514d\u641c\u7d22\u90a3\u4e9b\u80af\u5b9a\u4e0d\u4f1a\u4ea7\u751f\u89e3\u7684\u8def\u5f84\uff0c\u4ece\u800c\u8282\u7701\u65f6\u95f4\u548c\u7a7a\u95f4\u3002</li> <li>\u542f\u53d1\u5f0f\u641c\u7d22\uff1a\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5f15\u5165\u4e00\u4e9b\u7b56\u7565\u6216\u8005\u4f30\u8ba1\u503c\uff0c\u4ece\u800c\u4f18\u5148\u641c\u7d22\u6700\u6709\u53ef\u80fd\u4ea7\u751f\u6709\u6548\u89e3\u7684\u8def\u5f84\u3002</li> </ul>"},{"location":"chapter_backtracking/backtracking_algorithm/#1316","title":"13.1.6 \u00a0 \u56de\u6eaf\u5178\u578b\u4f8b\u9898","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u641c\u7d22\u95ee\u9898\u3001\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u548c\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u3002</p> <p>\u641c\u7d22\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>\u5168\u6392\u5217\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff0c\u6c42\u51fa\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u7ec4\u5408\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\u548c\u4e00\u4e2a\u76ee\u6807\u548c\uff0c\u627e\u5230\u96c6\u5408\u4e2d\u6240\u6709\u548c\u4e3a\u76ee\u6807\u548c\u7684\u5b50\u96c6\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\u548c\u4e00\u7cfb\u5217\u5927\u5c0f\u4e0d\u540c\u7684\u5706\u76d8\uff0c\u8981\u6c42\u5c06\u6240\u6709\u5706\u76d8\u4ece\u4e00\u6839\u67f1\u5b50\u79fb\u52a8\u5230\u53e6\u4e00\u6839\u67f1\u5b50\uff0c\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\uff0c\u4e14\u4e0d\u80fd\u5c06\u5927\u5706\u76d8\u653e\u5728\u5c0f\u5706\u76d8\u4e0a\u3002</li> </ul> <p>\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u5230\u6ee1\u8db3\u6240\u6709\u7ea6\u675f\u6761\u4ef6\u7684\u89e3\u3002</p> <ul> <li>\\(n\\) \u7687\u540e\uff1a\u5728 \\(n \\times n\\) \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e \\(n\\) \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u4e92\u4e0d\u653b\u51fb\u3002</li> <li>\u6570\u72ec\uff1a\u5728 \\(9 \\times 9\\) \u7684\u7f51\u683c\u4e2d\u586b\u5165\u6570\u5b57 \\(1\\) ~ \\(9\\) \uff0c\u4f7f\u5f97\u6bcf\u884c\u3001\u6bcf\u5217\u548c\u6bcf\u4e2a \\(3 \\times 3\\) \u5b50\u7f51\u683c\u4e2d\u7684\u6570\u5b57\u4e0d\u91cd\u590d\u3002</li> <li>\u56fe\u7740\u8272\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u7528\u6700\u5c11\u7684\u989c\u8272\u7ed9\u56fe\u7684\u6bcf\u4e2a\u9876\u70b9\u7740\u8272\uff0c\u4f7f\u5f97\u76f8\u90bb\u9876\u70b9\u989c\u8272\u4e0d\u540c\u3002</li> </ul> <p>\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff1a\u8fd9\u7c7b\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u4e00\u4e2a\u7ec4\u5408\u7a7a\u95f4\u4e2d\u627e\u5230\u6ee1\u8db3\u67d0\u4e9b\u6761\u4ef6\u7684\u6700\u4f18\u89e3\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u80cc\u5305\uff0c\u6bcf\u4e2a\u7269\u54c1\u6709\u4e00\u5b9a\u7684\u4ef7\u503c\u548c\u91cd\u91cf\uff0c\u8981\u6c42\u5728\u80cc\u5305\u5bb9\u91cf\u9650\u5236\u5185\uff0c\u9009\u62e9\u7269\u54c1\u4f7f\u5f97\u603b\u4ef7\u503c\u6700\u5927\u3002</li> <li>\u65c5\u884c\u5546\u95ee\u9898\uff1a\u5728\u4e00\u4e2a\u56fe\u4e2d\uff0c\u4ece\u4e00\u4e2a\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u6240\u6709\u5176\u4ed6\u70b9\u6070\u597d\u4e00\u6b21\u540e\u8fd4\u56de\u8d77\u70b9\uff0c\u6c42\u6700\u77ed\u8def\u5f84\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\uff0c\u627e\u5230\u6700\u5927\u7684\u5b8c\u5168\u5b50\u56fe\uff0c\u5373\u5b50\u56fe\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u90fd\u6709\u8fb9\u76f8\u8fde\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u56de\u6eaf\u4e0d\u662f\u6700\u4f18\u89e3\u51b3\u65b9\u6848\u3002</p> <ul> <li>0-1 \u80cc\u5305\u95ee\u9898\u901a\u5e38\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\uff0c\u4ee5\u8fbe\u5230\u66f4\u9ad8\u7684\u65f6\u95f4\u6548\u7387\u3002</li> <li>\u65c5\u884c\u5546\u662f\u4e00\u4e2a\u8457\u540d\u7684 NP-Hard \u95ee\u9898\uff0c\u5e38\u7528\u89e3\u6cd5\u6709\u9057\u4f20\u7b97\u6cd5\u548c\u8681\u7fa4\u7b97\u6cd5\u7b49\u3002</li> <li>\u6700\u5927\u56e2\u95ee\u9898\u662f\u56fe\u8bba\u4e2d\u7684\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\uff0c\u53ef\u7528\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u542f\u53d1\u5f0f\u7b97\u6cd5\u6765\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_backtracking/n_queens_problem/","title":"13.4 \u00a0 N \u7687\u540e\u95ee\u9898","text":"<p>Question</p> <p>\u6839\u636e\u56fd\u9645\u8c61\u68cb\u7684\u89c4\u5219\uff0c\u7687\u540e\u53ef\u4ee5\u653b\u51fb\u4e0e\u540c\u5904\u4e00\u884c\u3001\u4e00\u5217\u6216\u4e00\u6761\u659c\u7ebf\u4e0a\u7684\u68cb\u5b50\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u7687\u540e\u548c\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5bfb\u627e\u4f7f\u5f97\u6240\u6709\u7687\u540e\u4e4b\u95f4\u65e0\u6cd5\u76f8\u4e92\u653b\u51fb\u7684\u6446\u653e\u65b9\u6848\u3002</p> <p>\u5982\u56fe 13-15 \u6240\u793a\uff0c\u5f53 \\(n = 4\\) \u65f6\uff0c\u5171\u53ef\u4ee5\u627e\u5230\u4e24\u4e2a\u89e3\u3002\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\\(n \\times n\\) \u5927\u5c0f\u7684\u68cb\u76d8\u5171\u6709 \\(n^2\\) \u4e2a\u683c\u5b50\uff0c\u7ed9\u51fa\u4e86\u6240\u6709\u7684\u9009\u62e9 <code>choices</code> \u3002\u5728\u9010\u4e2a\u653e\u7f6e\u7687\u540e\u7684\u8fc7\u7a0b\u4e2d\uff0c\u68cb\u76d8\u72b6\u6001\u5728\u4e0d\u65ad\u5730\u53d8\u5316\uff0c\u6bcf\u4e2a\u65f6\u523b\u7684\u68cb\u76d8\u5c31\u662f\u72b6\u6001 <code>state</code> \u3002</p> <p></p> <p> \u56fe 13-15 \u00a0 4 \u7687\u540e\u95ee\u9898\u7684\u89e3 </p> <p>\u56fe 13-16 \u5c55\u793a\u4e86\u672c\u9898\u7684\u4e09\u4e2a\u7ea6\u675f\u6761\u4ef6\uff1a\u591a\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u3001\u540c\u4e00\u5bf9\u89d2\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u89d2\u7ebf\u5206\u4e3a\u4e3b\u5bf9\u89d2\u7ebf <code>\\</code> \u548c\u6b21\u5bf9\u89d2\u7ebf <code>/</code> \u4e24\u79cd\u3002</p> <p></p> <p> \u56fe 13-16 \u00a0 n \u7687\u540e\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6 </p>"},{"location":"chapter_backtracking/n_queens_problem/#1","title":"1. \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565","text":"<p>\u7687\u540e\u7684\u6570\u91cf\u548c\u68cb\u76d8\u7684\u884c\u6570\u90fd\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u6211\u4eec\u5bb9\u6613\u5f97\u5230\u4e00\u4e2a\u63a8\u8bba\uff1a\u68cb\u76d8\u6bcf\u884c\u90fd\u5141\u8bb8\u4e14\u53ea\u5141\u8bb8\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u9010\u884c\u653e\u7f6e\u7b56\u7565\uff1a\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb\uff0c\u5728\u6bcf\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\uff0c\u76f4\u81f3\u6700\u540e\u4e00\u884c\u7ed3\u675f\u3002</p> <p>\u56fe 13-17 \u6240\u793a\u4e3a \\(4\\) \u7687\u540e\u95ee\u9898\u7684\u9010\u884c\u653e\u7f6e\u8fc7\u7a0b\u3002\u53d7\u753b\u5e45\u9650\u5236\uff0c\u56fe 13-17 \u4ec5\u5c55\u5f00\u4e86\u7b2c\u4e00\u884c\u7684\u5176\u4e2d\u4e00\u4e2a\u641c\u7d22\u5206\u652f\uff0c\u5e76\u4e14\u5c06\u4e0d\u6ee1\u8db3\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u65b9\u6848\u90fd\u8fdb\u884c\u4e86\u526a\u679d\u3002</p> <p></p> <p> \u56fe 13-17 \u00a0 \u9010\u884c\u653e\u7f6e\u7b56\u7565 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9010\u884c\u653e\u7f6e\u7b56\u7565\u8d77\u5230\u4e86\u526a\u679d\u7684\u4f5c\u7528\uff0c\u5b83\u907f\u514d\u4e86\u540c\u4e00\u884c\u51fa\u73b0\u591a\u4e2a\u7687\u540e\u7684\u6240\u6709\u641c\u7d22\u5206\u652f\u3002</p>"},{"location":"chapter_backtracking/n_queens_problem/#2","title":"2. \u00a0 \u5217\u4e0e\u5bf9\u89d2\u7ebf\u526a\u679d","text":"<p>\u4e3a\u4e86\u6ee1\u8db3\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5e03\u5c14\u578b\u6570\u7ec4 <code>cols</code> \u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\u3002\u5728\u6bcf\u6b21\u51b3\u5b9a\u653e\u7f6e\u524d\uff0c\u6211\u4eec\u901a\u8fc7 <code>cols</code> \u5c06\u5df2\u6709\u7687\u540e\u7684\u5217\u8fdb\u884c\u526a\u679d\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u52a8\u6001\u66f4\u65b0 <code>cols</code> \u7684\u72b6\u6001\u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u5904\u7406\u5bf9\u89d2\u7ebf\u7ea6\u675f\u5462\uff1f\u8bbe\u68cb\u76d8\u4e2d\u67d0\u4e2a\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\((row, col)\\) \uff0c\u9009\u5b9a\u77e9\u9635\u4e2d\u7684\u67d0\u6761\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u6211\u4eec\u53d1\u73b0\u8be5\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684\u884c\u7d22\u5f15\u51cf\u5217\u7d22\u5f15\u90fd\u76f8\u7b49\uff0c\u5373\u5bf9\u89d2\u7ebf\u4e0a\u6240\u6709\u683c\u5b50\u7684 \\(row - col\\) \u4e3a\u6052\u5b9a\u503c\u3002</p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u4e24\u4e2a\u683c\u5b50\u6ee1\u8db3 \\(row_1 - col_1 = row_2 - col_2\\) \uff0c\u5219\u5b83\u4eec\u4e00\u5b9a\u5904\u5728\u540c\u4e00\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u3002\u5229\u7528\u8be5\u89c4\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u56fe 13-18 \u6240\u793a\u7684\u6570\u7ec4 <code>diags1</code> \u8bb0\u5f55\u6bcf\u6761\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\u3002</p> <p>\u540c\u7406\uff0c\u6b21\u5bf9\u89d2\u7ebf\u4e0a\u7684\u6240\u6709\u683c\u5b50\u7684 \\(row + col\\) \u662f\u6052\u5b9a\u503c\u3002\u6211\u4eec\u540c\u6837\u4e5f\u53ef\u4ee5\u501f\u52a9\u6570\u7ec4 <code>diags2</code> \u6765\u5904\u7406\u6b21\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002</p> <p></p> <p> \u56fe 13-18 \u00a0 \u5904\u7406\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f </p>"},{"location":"chapter_backtracking/n_queens_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u8bf7\u6ce8\u610f\uff0c\\(n\\) \u7ef4\u65b9\u9635\u4e2d \\(row - col\\) \u7684\u8303\u56f4\u662f \\([-n + 1, n - 1]\\) \uff0c\\(row + col\\) \u7684\u8303\u56f4\u662f \\([0, 2n - 2]\\) \uff0c\u6240\u4ee5\u4e3b\u5bf9\u89d2\u7ebf\u548c\u6b21\u5bf9\u89d2\u7ebf\u7684\u6570\u91cf\u90fd\u4e3a \\(2n - 1\\) \uff0c\u5373\u6570\u7ec4 <code>diags1</code> \u548c <code>diags2</code> \u7684\u957f\u5ea6\u90fd\u4e3a \\(2n - 1\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig n_queens.py<pre><code>def backtrack(\n    row: int,\n    n: int,\n    state: list[list[str]],\n    res: list[list[list[str]]],\n    cols: list[bool],\n    diags1: list[bool],\n    diags2: list[bool],\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e\"\"\"\n    # \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n:\n        res.append([list(row) for row in state])\n        return\n    # \u904d\u5386\u6240\u6709\u5217\n    for col in range(n):\n        # \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        diag1 = row - col + n - 1\n        diag2 = row + col\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if not cols[col] and not diags1[diag1] and not diags2[diag2]:\n            # \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = diags1[diag1] = diags2[diag2] = True\n            # \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2)\n            # \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = diags1[diag1] = diags2[diag2] = False\n\ndef n_queens(n: int) -&gt; list[list[list[str]]]:\n    \"\"\"\u6c42\u89e3 N \u7687\u540e\"\"\"\n    # \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state = [[\"#\" for _ in range(n)] for _ in range(n)]\n    cols = [False] * n  # \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    diags1 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    diags2 = [False] * (2 * n - 1)  # \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    res = []\n    backtrack(0, n, state, res, cols, diags1, diags2)\n\n    return res\n</code></pre> n_queens.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, vector&lt;vector&lt;string&gt;&gt; &amp;state, vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; &amp;res, vector&lt;bool&gt; &amp;cols,\n               vector&lt;bool&gt; &amp;diags1, vector&lt;bool&gt; &amp;diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nvector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    vector&lt;vector&lt;string&gt;&gt; state(n, vector&lt;string&gt;(n, \"#\"));\n    vector&lt;bool&gt; cols(n, false);           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags1(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;bool&gt; diags2(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    vector&lt;vector&lt;vector&lt;string&gt;&gt;&gt; res;\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, List&lt;List&lt;String&gt;&gt; state, List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n        boolean[] cols, boolean[] diags1, boolean[] diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        List&lt;List&lt;String&gt;&gt; copyState = new ArrayList&lt;&gt;();\n        for (List&lt;String&gt; sRow : state) {\n            copyState.add(new ArrayList&lt;&gt;(sRow));\n        }\n        res.add(copyState);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state.get(row).set(col, \"Q\");\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state.get(row).set(col, \"#\");\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;String&gt;&gt; state = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;String&gt; row = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            row.add(\"#\");\n        }\n        state.add(row);\n    }\n    boolean[] cols = new boolean[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags1 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    boolean[] diags2 = new boolean[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = new ArrayList&lt;&gt;();\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid Backtrack(int row, int n, List&lt;List&lt;string&gt;&gt; state, List&lt;List&lt;List&lt;string&gt;&gt;&gt; res,\n        bool[] cols, bool[] diags1, bool[] diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        List&lt;List&lt;string&gt;&gt; copyState = [];\n        foreach (List&lt;string&gt; sRow in state) {\n            copyState.Add(new List&lt;string&gt;(sRow));\n        }\n        res.Add(copyState);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\";\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            Backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\";\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList&lt;List&lt;List&lt;string&gt;&gt;&gt; NQueens(int n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    List&lt;List&lt;string&gt;&gt; state = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;string&gt; row = [];\n        for (int j = 0; j &lt; n; j++) {\n            row.Add(\"#\");\n        }\n        state.Add(row);\n    }\n    bool[] cols = new bool[n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    bool[] diags1 = new bool[2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool[] diags2 = new bool[2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    List&lt;List&lt;List&lt;string&gt;&gt;&gt; res = [];\n\n    Backtrack(0, n, state, res, cols, diags1, diags2);\n\n    return res;\n}\n</code></pre> n_queens.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col := 0; col &lt; n; col++ {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\n/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row, n int, state *[][]string, res *[][][]string, cols, diags1, diags2 *[]bool) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        newState := make([][]string, len(*state))\n        for i, _ := range newState {\n            newState[i] = make([]string, len((*state)[0]))\n            copy(newState[i], (*state)[i])\n\n        }\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col := 0; col &lt; n; col++ {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        diag1 := row - col + n - 1\n        diag2 := row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !(*cols)[col] &amp;&amp; !(*diags1)[diag1] &amp;&amp; !(*diags2)[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            (*state)[row][col] = \"Q\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = true, true, true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row+1, n, state, res, cols, diags1, diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            (*state)[row][col] = \"#\"\n            (*cols)[col], (*diags1)[diag1], (*diags2)[diag2] = false, false, false\n        }\n    }\n}\n\nfunc nQueens(n int) [][][]string {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    state := make([][]string, n)\n    for i := 0; i &lt; n; i++ {\n        row := make([]string, n)\n        for i := 0; i &lt; n; i++ {\n            row[i] = \"#\"\n        }\n        state[i] = row\n    }\n    // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    cols := make([]bool, n)\n    diags1 := make([]bool, 2*n-1)\n    diags2 := make([]bool, 2*n-1)\n    res := make([][][]string, 0)\n    backtrack(0, n, &amp;state, &amp;res, &amp;cols, &amp;diags1, &amp;diags2)\n    return res\n}\n</code></pre> n_queens.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunc backtrack(row: Int, n: Int, state: inout [[String]], res: inout [[[String]]], cols: inout [Bool], diags1: inout [Bool], diags2: inout [Bool]) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col in 0 ..&lt; n {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        let diag1 = row - col + n - 1\n        let diag2 = row + col\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = \"Q\"\n            cols[col] = true\n            diags1[diag1] = true\n            diags2[diag2] = true\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row: row + 1, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = \"#\"\n            cols[col] = false\n            diags1[diag1] = false\n            diags2[diag2] = false\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunc nQueens(n: Int) -&gt; [[[String]]] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    var state = Array(repeating: Array(repeating: \"#\", count: n), count: n)\n    var cols = Array(repeating: false, count: n) // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    var diags1 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var diags2 = Array(repeating: false, count: 2 * n - 1) // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    var res: [[[String]]] = []\n\n    backtrack(row: 0, n: n, state: &amp;state, res: &amp;res, cols: &amp;cols, diags1: &amp;diags1, diags2: &amp;diags2)\n\n    return res\n}\n</code></pre> n_queens.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(row, n, state, res, cols, diags1, diags2) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n) {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfunction backtrack(\n    row: number,\n    n: number,\n    state: string[][],\n    res: string[][][],\n    cols: boolean[],\n    diags1: boolean[],\n    diags2: boolean[]\n): void {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row === n) {\n        res.push(state.map((row) =&gt; row.slice()));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (let col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        const diag1 = row - col + n - 1;\n        const diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfunction nQueens(n: number): string[][][] {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    const state = Array.from({ length: n }, () =&gt; Array(n).fill('#'));\n    const cols = Array(n).fill(false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    const diags1 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const diags2 = Array(2 * n - 1).fill(false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    const res: string[][][] = [];\n\n    backtrack(0, n, state, res, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(\n  int row,\n  int n,\n  List&lt;List&lt;String&gt;&gt; state,\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res,\n  List&lt;bool&gt; cols,\n  List&lt;bool&gt; diags1,\n  List&lt;bool&gt; diags2,\n) {\n  // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (row == n) {\n    List&lt;List&lt;String&gt;&gt; copyState = [];\n    for (List&lt;String&gt; sRow in state) {\n      copyState.add(List.from(sRow));\n    }\n    res.add(copyState);\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u5217\n  for (int col = 0; col &lt; n; col++) {\n    // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n    int diag1 = row - col + n - 1;\n    int diag2 = row + col;\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n    if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n      // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n      state[row][col] = \"Q\";\n      cols[col] = true;\n      diags1[diag1] = true;\n      diags2[diag2] = true;\n      // \u653e\u7f6e\u4e0b\u4e00\u884c\n      backtrack(row + 1, n, state, res, cols, diags1, diags2);\n      // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n      state[row][col] = \"#\";\n      cols[col] = false;\n      diags1[diag1] = false;\n      diags2[diag2] = false;\n    }\n  }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nList&lt;List&lt;List&lt;String&gt;&gt;&gt; nQueens(int n) {\n  // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n  List&lt;List&lt;String&gt;&gt; state = List.generate(n, (index) =&gt; List.filled(n, \"#\"));\n  List&lt;bool&gt; cols = List.filled(n, false); // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags1 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;bool&gt; diags2 = List.filled(2 * n - 1, false); // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n  List&lt;List&lt;List&lt;String&gt;&gt;&gt; res = [];\n\n  backtrack(0, n, state, res, cols, diags1, diags2);\n\n  return res;\n}\n</code></pre> n_queens.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nfn backtrack(row: usize, n: usize, state: &amp;mut Vec&lt;Vec&lt;String&gt;&gt;, res: &amp;mut Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt;,\n    cols: &amp;mut [bool], diags1: &amp;mut [bool], diags2: &amp;mut [bool]) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if row == n {\n        let mut copy_state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\n        for s_row in state.clone() {\n            copy_state.push(s_row);\n        }\n        res.push(copy_state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for col in 0..n {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        let diag1 = row + n - 1 - col;\n        let diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if !cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2] {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state.get_mut(row).unwrap()[col] = \"Q\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (true, true, true);\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state.get_mut(row).unwrap()[col] = \"#\".into();\n            (cols[col], diags1[diag1], diags2[diag2]) = (false, false, false);\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nfn n_queens(n: usize) -&gt; Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    let mut state: Vec&lt;Vec&lt;String&gt;&gt; = Vec::new();\n    for _ in 0..n {\n        let mut row: Vec&lt;String&gt; = Vec::new();\n        for _ in 0..n {\n            row.push(\"#\".into());\n        }\n        state.push(row);\n    }\n    let mut cols = vec![false; n]; // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    let mut diags1 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut diags2 = vec![false; 2 * n - 1]; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    let mut res: Vec&lt;Vec&lt;Vec&lt;String&gt;&gt;&gt; = Vec::new();\n\n    backtrack(0, n, &amp;mut state, &amp;mut res, &amp;mut cols, &amp;mut diags1, &amp;mut diags2);\n\n    res\n}\n</code></pre> n_queens.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1aN \u7687\u540e */\nvoid backtrack(int row, int n, char state[MAX_SIZE][MAX_SIZE], char ***res, int *resSize, bool cols[MAX_SIZE],\n               bool diags1[2 * MAX_SIZE - 1], bool diags2[2 * MAX_SIZE - 1]) {\n    // \u5f53\u653e\u7f6e\u5b8c\u6240\u6709\u884c\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (row == n) {\n        res[*resSize] = (char **)malloc(sizeof(char *) * n);\n        for (int i = 0; i &lt; n; ++i) {\n            res[*resSize][i] = (char *)malloc(sizeof(char) * (n + 1));\n            strcpy(res[*resSize][i], state[i]);\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u5217\n    for (int col = 0; col &lt; n; col++) {\n        // \u8ba1\u7b97\u8be5\u683c\u5b50\u5bf9\u5e94\u7684\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\n        int diag1 = row - col + n - 1;\n        int diag2 = row + col;\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8be5\u683c\u5b50\u6240\u5728\u5217\u3001\u4e3b\u5bf9\u89d2\u7ebf\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u5b58\u5728\u7687\u540e\n        if (!cols[col] &amp;&amp; !diags1[diag1] &amp;&amp; !diags2[diag2]) {\n            // \u5c1d\u8bd5\uff1a\u5c06\u7687\u540e\u653e\u7f6e\u5728\u8be5\u683c\u5b50\n            state[row][col] = 'Q';\n            cols[col] = diags1[diag1] = diags2[diag2] = true;\n            // \u653e\u7f6e\u4e0b\u4e00\u884c\n            backtrack(row + 1, n, state, res, resSize, cols, diags1, diags2);\n            // \u56de\u9000\uff1a\u5c06\u8be5\u683c\u5b50\u6062\u590d\u4e3a\u7a7a\u4f4d\n            state[row][col] = '#';\n            cols[col] = diags1[diag1] = diags2[diag2] = false;\n        }\n    }\n}\n\n/* \u6c42\u89e3 N \u7687\u540e */\nchar ***nQueens(int n, int *returnSize) {\n    char state[MAX_SIZE][MAX_SIZE];\n    // \u521d\u59cb\u5316 n*n \u5927\u5c0f\u7684\u68cb\u76d8\uff0c\u5176\u4e2d 'Q' \u4ee3\u8868\u7687\u540e\uff0c'#' \u4ee3\u8868\u7a7a\u4f4d\n    for (int i = 0; i &lt; n; ++i) {\n        for (int j = 0; j &lt; n; ++j) {\n            state[i][j] = '#';\n        }\n        state[i][n] = '\\0';\n    }\n    bool cols[MAX_SIZE] = {false};           // \u8bb0\u5f55\u5217\u662f\u5426\u6709\u7687\u540e\n    bool diags1[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n    bool diags2[2 * MAX_SIZE - 1] = {false}; // \u8bb0\u5f55\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u6709\u7687\u540e\n\n    char ***res = (char ***)malloc(sizeof(char **) * MAX_SIZE);\n    *returnSize = 0;\n    backtrack(0, n, state, res, returnSize, cols, diags1, diags2);\n    return res;\n}\n</code></pre> n_queens.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{nQueens}\n</code></pre> <p>\u9010\u884c\u653e\u7f6e \\(n\\) \u6b21\uff0c\u8003\u8651\u5217\u7ea6\u675f\uff0c\u5219\u4ece\u7b2c\u4e00\u884c\u5230\u6700\u540e\u4e00\u884c\u5206\u522b\u6709 \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u4e2a\u9009\u62e9\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!)\\) \u3002\u5b9e\u9645\u4e0a\uff0c\u6839\u636e\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u526a\u679d\u4e5f\u80fd\u591f\u5927\u5e45\u7f29\u5c0f\u641c\u7d22\u7a7a\u95f4\uff0c\u56e0\u800c\u641c\u7d22\u6548\u7387\u5f80\u5f80\u4f18\u4e8e\u4ee5\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\u6570\u7ec4 <code>state</code> \u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff0c\u6570\u7ec4 <code>cols</code>\u3001<code>diags1</code> \u548c <code>diags2</code> \u7686\u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/","title":"13.2 \u00a0 \u5168\u6392\u5217\u95ee\u9898","text":"<p>\u5168\u6392\u5217\u95ee\u9898\u662f\u56de\u6eaf\u7b97\u6cd5\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u7684\u5b9a\u4e49\u662f\u5728\u7ed9\u5b9a\u4e00\u4e2a\u96c6\u5408\uff08\u5982\u4e00\u4e2a\u6570\u7ec4\u6216\u5b57\u7b26\u4e32\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u627e\u51fa\u5176\u4e2d\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u8868 13-2 \u5217\u4e3e\u4e86\u51e0\u4e2a\u793a\u4f8b\u6570\u636e\uff0c\u5305\u62ec\u8f93\u5165\u6570\u7ec4\u548c\u5bf9\u5e94\u7684\u6240\u6709\u6392\u5217\u3002</p> <p> \u8868 13-2 \u00a0 \u5168\u6392\u5217\u793a\u4f8b </p> \u8f93\u5165\u6570\u7ec4 \u6240\u6709\u6392\u5217 \\([1]\\) \\([1]\\) \\([1, 2]\\) \\([1, 2], [2, 1]\\) \\([1, 2, 3]\\) \\([1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]\\)"},{"location":"chapter_backtracking/permutations_problem/#1321","title":"13.2.1 \u00a0 \u65e0\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u5176\u4e2d\u4e0d\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002</p> <p>\u4ece\u56de\u6eaf\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u751f\u6210\u6392\u5217\u7684\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\u3002\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 2, 3]\\) \uff0c\u5982\u679c\u6211\u4eec\u5148\u9009\u62e9 \\(1\\) \uff0c\u518d\u9009\u62e9 \\(3\\) \uff0c\u6700\u540e\u9009\u62e9 \\(2\\) \uff0c\u5219\u83b7\u5f97\u6392\u5217 \\([1, 3, 2]\\) \u3002\u56de\u9000\u8868\u793a\u64a4\u9500\u4e00\u4e2a\u9009\u62e9\uff0c\u4e4b\u540e\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002</p> <p>\u4ece\u56de\u6eaf\u4ee3\u7801\u7684\u89d2\u5ea6\u770b\uff0c\u5019\u9009\u96c6\u5408 <code>choices</code> \u662f\u8f93\u5165\u6570\u7ec4\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u72b6\u6001 <code>state</code> \u662f\u76f4\u81f3\u76ee\u524d\u5df2\u88ab\u9009\u62e9\u7684\u5143\u7d20\u3002\u8bf7\u6ce8\u610f\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u5141\u8bb8\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u56e0\u6b64 <code>state</code> \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5e94\u8be5\u662f\u552f\u4e00\u7684\u3002</p> <p>\u5982\u56fe 13-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u641c\u7d22\u8fc7\u7a0b\u5c55\u5f00\u6210\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u5f53\u524d\u72b6\u6001 <code>state</code> \u3002\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u7ecf\u8fc7\u4e09\u8f6e\u9009\u62e9\u540e\u5230\u8fbe\u53f6\u8282\u70b9\uff0c\u6bcf\u4e2a\u53f6\u8282\u70b9\u90fd\u5bf9\u5e94\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-5 \u00a0 \u5168\u6392\u5217\u7684\u9012\u5f52\u6811 </p>"},{"location":"chapter_backtracking/permutations_problem/#1","title":"1. \u00a0 \u91cd\u590d\u9009\u62e9\u526a\u679d","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u6211\u4eec\u8003\u8651\u5f15\u5165\u4e00\u4e2a\u5e03\u5c14\u578b\u6570\u7ec4 <code>selected</code> \uff0c\u5176\u4e2d <code>selected[i]</code> \u8868\u793a <code>choices[i]</code> \u662f\u5426\u5df2\u88ab\u9009\u62e9\uff0c\u5e76\u57fa\u4e8e\u5b83\u5b9e\u73b0\u4ee5\u4e0b\u526a\u679d\u64cd\u4f5c\u3002</p> <ul> <li>\u5728\u505a\u51fa\u9009\u62e9 <code>choice[i]</code> \u540e\uff0c\u6211\u4eec\u5c31\u5c06 <code>selected[i]</code> \u8d4b\u503c\u4e3a \\(\\text{True}\\) \uff0c\u4ee3\u8868\u5b83\u5df2\u88ab\u9009\u62e9\u3002</li> <li>\u904d\u5386\u9009\u62e9\u5217\u8868 <code>choices</code> \u65f6\uff0c\u8df3\u8fc7\u6240\u6709\u5df2\u88ab\u9009\u62e9\u7684\u8282\u70b9\uff0c\u5373\u526a\u679d\u3002</li> </ul> <p>\u5982\u56fe 13-6 \u6240\u793a\uff0c\u5047\u8bbe\u6211\u4eec\u7b2c\u4e00\u8f6e\u9009\u62e9 1 \uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9 3 \uff0c\u7b2c\u4e09\u8f6e\u9009\u62e9 2 \uff0c\u5219\u9700\u8981\u5728\u7b2c\u4e8c\u8f6e\u526a\u6389\u5143\u7d20 1 \u7684\u5206\u652f\uff0c\u5728\u7b2c\u4e09\u8f6e\u526a\u6389\u5143\u7d20 1 \u548c\u5143\u7d20 3 \u7684\u5206\u652f\u3002</p> <p></p> <p> \u56fe 13-6 \u00a0 \u5168\u6392\u5217\u526a\u679d\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 13-6 \u53d1\u73b0\uff0c\u8be5\u526a\u679d\u64cd\u4f5c\u5c06\u641c\u7d22\u7a7a\u95f4\u5927\u5c0f\u4ece \\(O(n^n)\\) \u51cf\u5c0f\u81f3 \\(O(n!)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u60f3\u6e05\u695a\u4ee5\u4e0a\u4fe1\u606f\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5728\u6846\u67b6\u4ee3\u7801\u4e2d\u505a\u201c\u5b8c\u5f62\u586b\u7a7a\u201d\u4e86\u3002\u4e3a\u4e86\u7f29\u77ed\u6574\u4f53\u4ee3\u7801\uff0c\u6211\u4eec\u4e0d\u5355\u72ec\u5b9e\u73b0\u6846\u67b6\u4ee3\u7801\u4e2d\u7684\u5404\u4e2a\u51fd\u6570\uff0c\u800c\u662f\u5c06\u5b83\u4eec\u5c55\u5f00\u5728 <code>backtrack()</code> \u51fd\u6570\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_i.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I\"\"\"\n    # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if not selected[i]:\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = True\n            state.append(choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = False\n            state.pop()\n\ndef permutations_i(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 I\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push_back(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nvector&lt;vector&lt;int&gt;&gt; permutationsI(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;Integer&gt;&gt; permutationsI(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.Add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; PermutationsI(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrackI(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !(*selected)[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackI(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !selected[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true\n            state.append(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfunc permutationsI(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(state, choices, selected, res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 I */\nfunction permutationsI(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n    if (!selected[i]) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      selected[i] = true;\n      state.add(choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 I */\nList&lt;List&lt;int&gt;&gt; permutationsI(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if !selected[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.len() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nfn permutations_i(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 I */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20\n        if (!selected[i]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 I */\nint **permutationsI(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsI}\n</code></pre>"},{"location":"chapter_backtracking/permutations_problem/#1322","title":"13.2.2 \u00a0 \u8003\u8651\u76f8\u7b49\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd4\u56de\u6240\u6709\u4e0d\u91cd\u590d\u7684\u6392\u5217\u3002</p> <p>\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u4e3a \\([1, 1, 2]\\) \u3002\u4e3a\u4e86\u65b9\u4fbf\u533a\u5206\u4e24\u4e2a\u91cd\u590d\u5143\u7d20 \\(1\\) \uff0c\u6211\u4eec\u5c06\u7b2c\u4e8c\u4e2a \\(1\\) \u8bb0\u4e3a \\(\\hat{1}\\) \u3002</p> <p>\u5982\u56fe 13-7 \u6240\u793a\uff0c\u4e0a\u8ff0\u65b9\u6cd5\u751f\u6210\u7684\u6392\u5217\u6709\u4e00\u534a\u662f\u91cd\u590d\u7684\u3002</p> <p></p> <p> \u56fe 13-7 \u00a0 \u91cd\u590d\u6392\u5217 </p> <p>\u90a3\u4e48\u5982\u4f55\u53bb\u9664\u91cd\u590d\u7684\u6392\u5217\u5462\uff1f\u6700\u76f4\u63a5\u5730\uff0c\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u76f4\u63a5\u5bf9\u6392\u5217\u7ed3\u679c\u8fdb\u884c\u53bb\u91cd\u3002\u7136\u800c\u8fd9\u6837\u505a\u4e0d\u591f\u4f18\u96c5\uff0c\u56e0\u4e3a\u751f\u6210\u91cd\u590d\u6392\u5217\u7684\u641c\u7d22\u5206\u652f\u6ca1\u6709\u5fc5\u8981\uff0c\u5e94\u5f53\u63d0\u524d\u8bc6\u522b\u5e76\u526a\u679d\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u89c2\u5bdf\u56fe 13-8 \uff0c\u5728\u7b2c\u4e00\u8f6e\u4e2d\uff0c\u9009\u62e9 \\(1\\) \u6216\u9009\u62e9 \\(\\hat{1}\\) \u662f\u7b49\u4ef7\u7684\uff0c\u5728\u8fd9\u4e24\u4e2a\u9009\u62e9\u4e4b\u4e0b\u751f\u6210\u7684\u6240\u6709\u6392\u5217\u90fd\u662f\u91cd\u590d\u7684\u3002\u56e0\u6b64\u5e94\u8be5\u628a \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u540c\u7406\uff0c\u5728\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(2\\) \u4e4b\u540e\uff0c\u7b2c\u4e8c\u8f6e\u9009\u62e9\u4e2d\u7684 \\(1\\) \u548c \\(\\hat{1}\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5206\u652f\uff0c\u56e0\u6b64\u4e5f\u5e94\u5c06\u7b2c\u4e8c\u8f6e\u7684 \\(\\hat{1}\\) \u526a\u679d\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u5728\u67d0\u4e00\u8f6e\u9009\u62e9\u4e2d\uff0c\u4fdd\u8bc1\u591a\u4e2a\u76f8\u7b49\u7684\u5143\u7d20\u4ec5\u88ab\u9009\u62e9\u4e00\u6b21\u3002</p> <p></p> <p> \u56fe 13-8 \u00a0 \u91cd\u590d\u6392\u5217\u526a\u679d </p>"},{"location":"chapter_backtracking/permutations_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4e0a\u4e00\u9898\u7684\u4ee3\u7801\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u8003\u8651\u5728\u6bcf\u4e00\u8f6e\u9009\u62e9\u4e2d\u5f00\u542f\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>duplicated</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u8be5\u8f6e\u4e2d\u5df2\u7ecf\u5c1d\u8bd5\u8fc7\u7684\u5143\u7d20\uff0c\u5e76\u5c06\u91cd\u590d\u5143\u7d20\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig permutations_ii.py<pre><code>def backtrack(\n    state: list[int], choices: list[int], selected: list[bool], res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II\"\"\"\n    # \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(state) == len(choices):\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    duplicated = set[int]()\n    for i, choice in enumerate(choices):\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if not selected[i] and choice not in duplicated:\n            # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice)  # \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = True\n            state.append(choice)\n            # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res)\n            # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = False\n            state.pop()\n\ndef permutations_ii(nums: list[int]) -&gt; list[list[int]]:\n    \"\"\"\u5168\u6392\u5217 II\"\"\"\n    res = []\n    backtrack(state=[], choices=nums, selected=[False] * len(nums), res=res)\n    return res\n</code></pre> permutations_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(vector&lt;int&gt; &amp;state, const vector&lt;int&gt; &amp;choices, vector&lt;bool&gt; &amp;selected, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.size()) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    unordered_set&lt;int&gt; duplicated;\n    for (int i = 0; i &lt; choices.size(); i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; duplicated.find(choice) == duplicated.end()) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.emplace(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push_back(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop_back();\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nvector&lt;vector&lt;int&gt;&gt; permutationsII(vector&lt;int&gt; nums) {\n    vector&lt;int&gt; state;\n    vector&lt;bool&gt; selected(nums.size(), false);\n    vector&lt;vector&lt;int&gt;&gt; res;\n    backtrack(state, nums, selected, res);\n    return res;\n}\n</code></pre> permutations_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(List&lt;Integer&gt; state, int[] choices, boolean[] selected, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.size() == choices.length) {\n        res.add(new ArrayList&lt;Integer&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    Set&lt;Integer&gt; duplicated = new HashSet&lt;Integer&gt;();\n    for (int i = 0; i &lt; choices.length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.size() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;Integer&gt;&gt; permutationsII(int[] nums) {\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;List&lt;Integer&gt;&gt;();\n    backtrack(new ArrayList&lt;Integer&gt;(), nums, new boolean[nums.length], res);\n    return res;\n}\n</code></pre> permutations_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid Backtrack(List&lt;int&gt; state, int[] choices, bool[] selected, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.Count == choices.Length) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    HashSet&lt;int&gt; duplicated = [];\n    for (int i = 0; i &lt; choices.Length; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.Contains(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.Add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.Add(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            Backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.RemoveAt(state.Count - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; PermutationsII(int[] nums) {\n    List&lt;List&lt;int&gt;&gt; res = [];\n    Backtrack([], nums, new bool[nums.Length], res);\n    return res;\n}\n</code></pre> permutations_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrackII(state *[]int, choices *[]int, selected *[]bool, res *[][]int) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if len(*state) == len(*choices) {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    duplicated := make(map[int]struct{}, 0)\n    for i := 0; i &lt; len(*choices); i++ {\n        choice := (*choices)[i]\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if _, ok := duplicated[choice]; !ok &amp;&amp; !(*selected)[i] {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            duplicated[choice] = struct{}{}\n            (*selected)[i] = true\n            *state = append(*state, choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrackI(state, choices, selected, res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            (*selected)[i] = false\n            *state = (*state)[:len(*state)-1]\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums []int) [][]int {\n    res := make([][]int, 0)\n    state := make([]int, 0)\n    selected := make([]bool, len(nums))\n    backtrackII(&amp;state, &amp;nums, &amp;selected, &amp;res)\n    return res\n}\n</code></pre> permutations_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunc backtrack(state: inout [Int], choices: [Int], selected: inout [Bool], res: inout [[Int]]) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.count == choices.count {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    var duplicated: Set&lt;Int&gt; = []\n    for (i, choice) in choices.enumerated() {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if !selected[i], !duplicated.contains(choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.insert(choice) // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true\n            state.append(choice)\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state: &amp;state, choices: choices, selected: &amp;selected, res: &amp;res)\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false\n            state.removeLast()\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfunc permutationsII(nums: [Int]) -&gt; [[Int]] {\n    var state: [Int] = []\n    var selected = Array(repeating: false, count: nums.count)\n    var res: [[Int]] = []\n    backtrack(state: &amp;state, choices: nums, selected: &amp;selected, res: &amp;res)\n    return res\n}\n</code></pre> permutations_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(state, choices, selected, res) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums) {\n    const res = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfunction backtrack(\n    state: number[],\n    choices: number[],\n    selected: boolean[],\n    res: number[][]\n): void {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (state.length === choices.length) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    const duplicated = new Set();\n    choices.forEach((choice, i) =&gt; {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated.has(choice)) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.pop();\n        }\n    });\n}\n\n/* \u5168\u6392\u5217 II */\nfunction permutationsII(nums: number[]): number[][] {\n    const res: number[][] = [];\n    backtrack([], nums, Array(nums.length).fill(false), res);\n    return res;\n}\n</code></pre> permutations_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  List&lt;int&gt; choices,\n  List&lt;bool&gt; selected,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (state.length == choices.length) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  Set&lt;int&gt; duplicated = {};\n  for (int i = 0; i &lt; choices.length; i++) {\n    int choice = choices[i];\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n    if (!selected[i] &amp;&amp; !duplicated.contains(choice)) {\n      // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n      duplicated.add(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n      selected[i] = true;\n      state.add(choice);\n      // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n      backtrack(state, choices, selected, res);\n      // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n      selected[i] = false;\n      state.removeLast();\n    }\n  }\n}\n\n/* \u5168\u6392\u5217 II */\nList&lt;List&lt;int&gt;&gt; permutationsII(List&lt;int&gt; nums) {\n  List&lt;List&lt;int&gt;&gt; res = [];\n  backtrack([], nums, List.filled(nums.length, false), res);\n  return res;\n}\n</code></pre> permutations_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nfn backtrack(mut state: Vec&lt;i32&gt;, choices: &amp;[i32], selected: &amp;mut [bool], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if state.len() == choices.len() {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    let mut duplicated = HashSet::&lt;i32&gt;::new();\n    for i in 0..choices.len() {\n        let choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if !selected[i] &amp;&amp; !duplicated.contains(&amp;choice) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated.insert(choice); // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state.push(choice);\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state.clone(), choices, selected, res);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n            state.remove(state.len() - 1);\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nfn permutations_ii(nums: &amp;mut [i32]) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let mut res = Vec::new();\n    backtrack(Vec::new(), nums, &amp;mut vec![false; nums.len()], &amp;mut res);\n    res\n}\n</code></pre> permutations_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5168\u6392\u5217 II */\nvoid backtrack(int *state, int stateSize, int *choices, int choicesSize, bool *selected, int **res, int *resSize) {\n    // \u5f53\u72b6\u6001\u957f\u5ea6\u7b49\u4e8e\u5143\u7d20\u6570\u91cf\u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (stateSize == choicesSize) {\n        res[*resSize] = (int *)malloc(choicesSize * sizeof(int));\n        for (int i = 0; i &lt; choicesSize; i++) {\n            res[*resSize][i] = state[i];\n        }\n        (*resSize)++;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    bool duplicated[MAX_SIZE] = {false};\n    for (int i = 0; i &lt; choicesSize; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u5143\u7d20 \u4e14 \u4e0d\u5141\u8bb8\u91cd\u590d\u9009\u62e9\u76f8\u7b49\u5143\u7d20\n        if (!selected[i] &amp;&amp; !duplicated[choice]) {\n            // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n            duplicated[choice] = true; // \u8bb0\u5f55\u9009\u62e9\u8fc7\u7684\u5143\u7d20\u503c\n            selected[i] = true;\n            state[stateSize] = choice;\n            // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n            backtrack(state, stateSize + 1, choices, choicesSize, selected, res, resSize);\n            // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n            selected[i] = false;\n        }\n    }\n}\n\n/* \u5168\u6392\u5217 II */\nint **permutationsII(int *nums, int numsSize, int *returnSize) {\n    int *state = (int *)malloc(numsSize * sizeof(int));\n    bool *selected = (bool *)malloc(numsSize * sizeof(bool));\n    for (int i = 0; i &lt; numsSize; i++) {\n        selected[i] = false;\n    }\n    int **res = (int **)malloc(MAX_SIZE * sizeof(int *));\n    *returnSize = 0;\n\n    backtrack(state, 0, nums, numsSize, selected, res, returnSize);\n\n    free(state);\n    free(selected);\n\n    return res;\n}\n</code></pre> permutations_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{permutationsII}\n</code></pre> <p>\u5047\u8bbe\u5143\u7d20\u4e24\u4e24\u4e4b\u95f4\u4e92\u4e0d\u76f8\u540c\uff0c\u5219 \\(n\\) \u4e2a\u5143\u7d20\u5171\u6709 \\(n!\\)  \u79cd\u6392\u5217\uff08\u9636\u4e58\uff09\uff1b\u5728\u8bb0\u5f55\u7ed3\u679c\u65f6\uff0c\u9700\u8981\u590d\u5236\u957f\u5ea6\u4e3a \\(n\\) \u7684\u5217\u8868\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n!n)\\) \u3002</p> <p>\u6700\u5927\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002<code>selected</code> \u4f7f\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002\u540c\u4e00\u65f6\u523b\u6700\u591a\u5171\u6709 \\(n\\) \u4e2a <code>duplicated</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u7a7a\u95f4\u3002\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_backtracking/permutations_problem/#3","title":"3. \u00a0 \u4e24\u79cd\u526a\u679d\u5bf9\u6bd4","text":"<p>\u8bf7\u6ce8\u610f\uff0c\u867d\u7136 <code>selected</code> \u548c <code>duplicated</code> \u90fd\u7528\u4e8e\u526a\u679d\uff0c\u4f46\u4e24\u8005\u7684\u76ee\u6807\u4e0d\u540c\u3002</p> <ul> <li>\u91cd\u590d\u9009\u62e9\u526a\u679d\uff1a\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\u4e2d\u53ea\u6709\u4e00\u4e2a <code>selected</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5f53\u524d\u72b6\u6001\u4e2d\u5305\u542b\u54ea\u4e9b\u5143\u7d20\uff0c\u5176\u4f5c\u7528\u662f\u9632\u6b62 <code>choices</code> \u4e2d\u7684\u4efb\u4e00\u5143\u7d20\u5728 <code>state</code> \u4e2d\u91cd\u590d\u51fa\u73b0\u3002</li> <li>\u76f8\u7b49\u5143\u7d20\u526a\u679d\uff1a\u6bcf\u8f6e\u9009\u62e9\uff08\u6bcf\u4e2a\u8c03\u7528\u7684 <code>backtrack</code> \u51fd\u6570\uff09\u90fd\u5305\u542b\u4e00\u4e2a <code>duplicated</code> \u3002\u5b83\u8bb0\u5f55\u7684\u662f\u5728\u672c\u8f6e\u904d\u5386\uff08<code>for</code> \u5faa\u73af\uff09\u4e2d\u54ea\u4e9b\u5143\u7d20\u5df2\u88ab\u9009\u62e9\u8fc7\uff0c\u5176\u4f5c\u7528\u662f\u4fdd\u8bc1\u76f8\u7b49\u7684\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> </ul> <p>\u56fe 13-9 \u5c55\u793a\u4e86\u4e24\u4e2a\u526a\u679d\u6761\u4ef6\u7684\u751f\u6548\u8303\u56f4\u3002\u6ce8\u610f\uff0c\u6811\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u9009\u62e9\uff0c\u4ece\u6839\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\u6784\u6210\u4e00\u4e2a\u6392\u5217\u3002</p> <p></p> <p> \u56fe 13-9 \u00a0 \u4e24\u79cd\u526a\u679d\u6761\u4ef6\u7684\u4f5c\u7528\u8303\u56f4 </p>"},{"location":"chapter_backtracking/subset_sum_problem/","title":"13.3 \u00a0 \u5b50\u96c6\u548c\u95ee\u9898","text":""},{"location":"chapter_backtracking/subset_sum_problem/#1331","title":"13.3.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u65e0\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u88ab\u9009\u53d6\u591a\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u4f8b\u5982\uff0c\u8f93\u5165\u96c6\u5408 \\(\\{3, 4, 5\\}\\) \u548c\u76ee\u6807\u6574\u6570 \\(9\\) \uff0c\u89e3\u4e3a \\(\\{3, 3, 3\\}, \\{4, 5\\}\\) \u3002\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u8f93\u5165\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u91cd\u590d\u9009\u53d6\u3002</li> <li>\u5b50\u96c6\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u6bd4\u5982 \\(\\{4, 5\\}\\) \u548c \\(\\{5, 4\\}\\) \u662f\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#1","title":"1. \u00a0 \u53c2\u8003\u5168\u6392\u5217\u89e3\u6cd5","text":"<p>\u7c7b\u4f3c\u4e8e\u5168\u6392\u5217\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b50\u96c6\u7684\u751f\u6210\u8fc7\u7a0b\u60f3\u8c61\u6210\u4e00\u7cfb\u5217\u9009\u62e9\u7684\u7ed3\u679c\uff0c\u5e76\u5728\u9009\u62e9\u8fc7\u7a0b\u4e2d\u5b9e\u65f6\u66f4\u65b0\u201c\u5143\u7d20\u548c\u201d\uff0c\u5f53\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u65f6\uff0c\u5c31\u5c06\u5b50\u96c6\u8bb0\u5f55\u81f3\u7ed3\u679c\u5217\u8868\u3002</p> <p>\u800c\u4e0e\u5168\u6392\u5217\u95ee\u9898\u4e0d\u540c\u7684\u662f\uff0c\u672c\u9898\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u53ef\u4ee5\u88ab\u65e0\u9650\u6b21\u9009\u53d6\uff0c\u56e0\u6b64\u65e0\u987b\u501f\u52a9 <code>selected</code> \u5e03\u5c14\u5217\u8868\u6765\u8bb0\u5f55\u5143\u7d20\u662f\u5426\u5df2\u88ab\u9009\u62e9\u3002\u6211\u4eec\u53ef\u4ee5\u5bf9\u5168\u6392\u5217\u4ee3\u7801\u8fdb\u884c\u5c0f\u5e45\u4fee\u6539\uff0c\u521d\u6b65\u5f97\u5230\u89e3\u9898\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_i_naive.py<pre><code>def backtrack(\n    state: list[int],\n    target: int,\n    total: int,\n    choices: list[int],\n    res: list[list[int]],\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in range(len(choices)):\n        # \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_i_naive(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    total = 0  # \u5b50\u96c6\u548c\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res)\n    return res\n</code></pre> subset_sum_i_naive.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, int total, vector&lt;int&gt; &amp;choices, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (size_t i = 0; i &lt; choices.size(); i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvector&lt;vector&lt;int&gt;&gt; subsetSumINaive(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;       // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0;           // \u5b50\u96c6\u548c\n    vector&lt;vector&lt;int&gt;&gt; res; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int total, int[] choices, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;Integer&gt;&gt; subsetSumINaive(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int total, int[] choices, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choices.Length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; SubsetSumINaive(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    int total = 0; // \u5b50\u96c6\u548c\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumINaive(total, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == total {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i := 0; i &lt; len(*choices); i++ {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total+(*choices)[i] &gt; target {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumINaive(total+(*choices)[i], target, state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    total := 0              // \u5b50\u96c6\u548c\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumINaive(total, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, total: Int, choices: [Int], res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in stride(from: 0, to: choices.count, by: 1) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target, total: total + choices[i], choices: choices, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunc subsetSumINaive(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let total = 0 // \u5b50\u96c6\u548c\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, total: total, choices: nums, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i_naive.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, total, choices, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    total: number,\n    choices: number[],\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total === target) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (let i = 0; i &lt; choices.length; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfunction subsetSumINaive(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    const total = 0; // \u5b50\u96c6\u548c\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, res);\n    return res;\n}\n</code></pre> subset_sum_i_naive.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  int total,\n  List&lt;int&gt; choices,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (total == target) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int i = 0; i &lt; choices.length; i++) {\n    // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n    if (total + choices[i] &gt; target) {\n      continue;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target, total + choices[i], choices, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nList&lt;List&lt;int&gt;&gt; subsetSumINaive(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  int total = 0; // \u5143\u7d20\u548c\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, total, nums, res);\n  return res;\n}\n</code></pre> subset_sum_i_naive.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec&lt;i32&gt;, target: i32, total: i32, choices: &amp;[i32], res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if total == target {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for i in 0..choices.len() {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if total + choices[i] &gt; target {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target, total + choices[i], choices, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nfn subset_sum_i_naive(nums: &amp;[i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let total = 0; // \u5b50\u96c6\u548c\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, total, nums, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i_naive.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int total, int *choices, int choicesSize) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (total == target) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; choicesSize; i++) {\n        // \u526a\u679d\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u8df3\u8fc7\u8be5\u9009\u62e9\n        if (total + choices[i] &gt; target) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u5143\u7d20\u548c total\n        state[stateSize++] = choices[i];\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target, total + choices[i], choices, choicesSize);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I\uff08\u5305\u542b\u91cd\u590d\u5b50\u96c6\uff09 */\nvoid subsetSumINaive(int *nums, int numsSize, int target) {\n    resSize = 0; // \u521d\u59cb\u5316\u89e3\u7684\u6570\u91cf\u4e3a0\n    backtrack(target, 0, nums, numsSize);\n}\n</code></pre> subset_sum_i_naive.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumINaive}\n</code></pre> <p>\u5411\u4ee5\u4e0a\u4ee3\u7801\u8f93\u5165\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u8f93\u51fa\u7ed3\u679c\u4e3a \\([3, 3, 3], [4, 5], [5, 4]\\) \u3002\u867d\u7136\u6210\u529f\u627e\u51fa\u4e86\u6240\u6709\u548c\u4e3a \\(9\\) \u7684\u5b50\u96c6\uff0c\u4f46\u5176\u4e2d\u5b58\u5728\u91cd\u590d\u7684\u5b50\u96c6 \\([4, 5]\\) \u548c \\([5, 4]\\) \u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u641c\u7d22\u8fc7\u7a0b\u662f\u533a\u5206\u9009\u62e9\u987a\u5e8f\u7684\uff0c\u7136\u800c\u5b50\u96c6\u4e0d\u533a\u5206\u9009\u62e9\u987a\u5e8f\u3002\u5982\u56fe 13-10 \u6240\u793a\uff0c\u5148\u9009 \\(4\\) \u540e\u9009 \\(5\\) \u4e0e\u5148\u9009 \\(5\\) \u540e\u9009 \\(4\\) \u662f\u4e0d\u540c\u7684\u5206\u652f\uff0c\u4f46\u5bf9\u5e94\u540c\u4e00\u4e2a\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-10 \u00a0 \u5b50\u96c6\u641c\u7d22\u4e0e\u8d8a\u754c\u526a\u679d </p> <p>\u4e3a\u4e86\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u601d\u8def\u662f\u5bf9\u7ed3\u679c\u5217\u8868\u8fdb\u884c\u53bb\u91cd\u3002\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\uff0c\u6709\u4e24\u65b9\u9762\u539f\u56e0\u3002</p> <ul> <li>\u5f53\u6570\u7ec4\u5143\u7d20\u8f83\u591a\uff0c\u5c24\u5176\u662f\u5f53 <code>target</code> \u8f83\u5927\u65f6\uff0c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u4ea7\u751f\u5927\u91cf\u7684\u91cd\u590d\u5b50\u96c6\u3002</li> <li>\u6bd4\u8f83\u5b50\u96c6\uff08\u6570\u7ec4\uff09\u7684\u5f02\u540c\u975e\u5e38\u8017\u65f6\uff0c\u9700\u8981\u5148\u6392\u5e8f\u6570\u7ec4\uff0c\u518d\u6bd4\u8f83\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u5143\u7d20\u7684\u5f02\u540c\u3002</li> </ul>"},{"location":"chapter_backtracking/subset_sum_problem/#2","title":"2. \u00a0 \u91cd\u590d\u5b50\u96c6\u526a\u679d","text":"<p>\u6211\u4eec\u8003\u8651\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u901a\u8fc7\u526a\u679d\u8fdb\u884c\u53bb\u91cd\u3002\u89c2\u5bdf\u56fe 13-11 \uff0c\u91cd\u590d\u5b50\u96c6\u662f\u5728\u4ee5\u4e0d\u540c\u987a\u5e8f\u9009\u62e9\u6570\u7ec4\u5143\u7d20\u65f6\u4ea7\u751f\u7684\uff0c\u4f8b\u5982\u4ee5\u4e0b\u60c5\u51b5\u3002</p> <ol> <li>\u5f53\u7b2c\u4e00\u8f6e\u548c\u7b2c\u4e8c\u8f6e\u5206\u522b\u9009\u62e9 \\(3\\) \u548c \\(4\\) \u65f6\uff0c\u4f1a\u751f\u6210\u5305\u542b\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u6240\u6709\u5b50\u96c6\uff0c\u8bb0\u4e3a \\([3, 4, \\dots]\\) \u3002</li> <li>\u4e4b\u540e\uff0c\u5f53\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(4\\) \u65f6\uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \uff0c\u56e0\u4e3a\u8be5\u9009\u62e9\u4ea7\u751f\u7684\u5b50\u96c6 \\([4, 3, \\dots]\\) \u548c\u7b2c <code>1.</code> \u6b65\u4e2d\u751f\u6210\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p>\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e00\u5c42\u7684\u9009\u62e9\u90fd\u662f\u4ece\u5de6\u5230\u53f3\u88ab\u9010\u4e2a\u5c1d\u8bd5\u7684\uff0c\u56e0\u6b64\u8d8a\u9760\u53f3\u7684\u5206\u652f\u88ab\u526a\u6389\u7684\u8d8a\u591a\u3002</p> <ol> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(3\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([3, 5, \\dots]\\) \u3002</li> <li>\u524d\u4e24\u8f6e\u9009\u62e9 \\(4\\) \u548c \\(5\\) \uff0c\u751f\u6210\u5b50\u96c6 \\([4, 5, \\dots]\\) \u3002</li> <li>\u82e5\u7b2c\u4e00\u8f6e\u9009\u62e9 \\(5\\) \uff0c\u5219\u7b2c\u4e8c\u8f6e\u5e94\u8be5\u8df3\u8fc7 \\(3\\) \u548c \\(4\\) \uff0c\u56e0\u4e3a\u5b50\u96c6 \\([5, 3, \\dots]\\) \u548c \\([5, 4, \\dots]\\) \u4e0e\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\u4e2d\u63cf\u8ff0\u7684\u5b50\u96c6\u5b8c\u5168\u91cd\u590d\u3002</li> </ol> <p></p> <p> \u56fe 13-11 \u00a0 \u4e0d\u540c\u9009\u62e9\u987a\u5e8f\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p> <p>\u603b\u7ed3\u6765\u770b\uff0c\u7ed9\u5b9a\u8f93\u5165\u6570\u7ec4 \\([x_1, x_2, \\dots, x_n]\\) \uff0c\u8bbe\u641c\u7d22\u8fc7\u7a0b\u4e2d\u7684\u9009\u62e9\u5e8f\u5217\u4e3a \\([x_{i_1}, x_{i_2}, \\dots, x_{i_m}]\\) \uff0c\u5219\u8be5\u9009\u62e9\u5e8f\u5217\u9700\u8981\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4e0d\u6ee1\u8db3\u8be5\u6761\u4ef6\u7684\u9009\u62e9\u5e8f\u5217\u90fd\u4f1a\u9020\u6210\u91cd\u590d\uff0c\u5e94\u5f53\u526a\u679d\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u5b9e\u73b0\u8be5\u526a\u679d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u53d8\u91cf <code>start</code> \uff0c\u7528\u4e8e\u6307\u793a\u904d\u5386\u8d77\u59cb\u70b9\u3002\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i\\) \u5f00\u59cb\u904d\u5386\u3002\u8fd9\u6837\u505a\u5c31\u53ef\u4ee5\u8ba9\u9009\u62e9\u5e8f\u5217\u6ee1\u8db3 \\(i_1 \\leq i_2 \\leq \\dots \\leq i_m\\) \uff0c\u4ece\u800c\u4fdd\u8bc1\u5b50\u96c6\u552f\u4e00\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u8fd8\u5bf9\u4ee3\u7801\u8fdb\u884c\u4e86\u4ee5\u4e0b\u4e24\u9879\u4f18\u5316\u3002</p> <ul> <li>\u5728\u5f00\u542f\u641c\u7d22\u524d\uff0c\u5148\u5c06\u6570\u7ec4 <code>nums</code> \u6392\u5e8f\u3002\u5728\u904d\u5386\u6240\u6709\u9009\u62e9\u65f6\uff0c\u5f53\u5b50\u96c6\u548c\u8d85\u8fc7 <code>target</code> \u65f6\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\uff0c\u56e0\u4e3a\u540e\u8fb9\u7684\u5143\u7d20\u66f4\u5927\uff0c\u5176\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 <code>target</code> \u3002</li> <li>\u7701\u53bb\u5143\u7d20\u548c\u53d8\u91cf <code>total</code> \uff0c\u901a\u8fc7\u5728 <code>target</code> \u4e0a\u6267\u884c\u51cf\u6cd5\u6765\u7edf\u8ba1\u5143\u7d20\u548c\uff0c\u5f53 <code>target</code> \u7b49\u4e8e \\(0\\) \u65f6\u8bb0\u5f55\u89e3\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_i.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0:\n            break\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_i(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c I\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start = 0  # \u904d\u5386\u8d77\u59cb\u70b9\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_i.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvector&lt;vector&lt;int&gt;&gt; subsetSumI(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;Integer&gt;&gt; subsetSumI(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; SubsetSumI(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrackSubsetSumI(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumI(i, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start := 0              // \u904d\u5386\u8d77\u59cb\u70b9\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumI(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in stride(from: start, to: choices.count, by: 1) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunc subsetSumI(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_i.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfunction subsetSumI(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_i.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n    // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target - choices[i], choices, i, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nList&lt;List&lt;int&gt;&gt; subsetSumI(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n  int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_i.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nfn backtrack(mut state: Vec&lt;i32&gt;, target: i32, choices: &amp;[i32], start: usize, res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nfn subset_sum_i(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_i.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c I */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; ++i) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target - choices[i], choices, choicesSize, i);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c I */\nvoid subsetSumI(int *nums, int numsSize, int target) {\n    qsort(nums, numsSize, sizeof(int), cmp); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                           // \u904d\u5386\u8d77\u59cb\u70b9\n    backtrack(target, nums, numsSize, start);\n}\n</code></pre> subset_sum_i.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumI}\n</code></pre> <p>\u56fe 13-12 \u6240\u793a\u4e3a\u5c06\u6570\u7ec4 \\([3, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u8f93\u5165\u4ee5\u4e0a\u4ee3\u7801\u540e\u7684\u6574\u4f53\u56de\u6eaf\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 13-12 \u00a0 \u5b50\u96c6\u548c I \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1332","title":"13.3.2 \u00a0 \u8003\u8651\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u6b63\u6574\u6570 <code>target</code> \uff0c\u8bf7\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\uff0c\u4f7f\u5f97\u7ec4\u5408\u4e2d\u7684\u5143\u7d20\u548c\u7b49\u4e8e <code>target</code> \u3002\u7ed9\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u6bcf\u4e2a\u5143\u7d20\u53ea\u53ef\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u8bf7\u4ee5\u5217\u8868\u5f62\u5f0f\u8fd4\u56de\u8fd9\u4e9b\u7ec4\u5408\uff0c\u5217\u8868\u4e2d\u4e0d\u5e94\u5305\u542b\u91cd\u590d\u7ec4\u5408\u3002</p> <p>\u76f8\u6bd4\u4e8e\u4e0a\u9898\uff0c\u672c\u9898\u7684\u8f93\u5165\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u8fd9\u5f15\u5165\u4e86\u65b0\u7684\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7ed9\u5b9a\u6570\u7ec4 \\([4, \\hat{4}, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \uff0c\u5219\u73b0\u6709\u4ee3\u7801\u7684\u8f93\u51fa\u7ed3\u679c\u4e3a \\([4, 5], [\\hat{4}, 5]\\) \uff0c\u51fa\u73b0\u4e86\u91cd\u590d\u5b50\u96c6\u3002</p> <p>\u9020\u6210\u8fd9\u79cd\u91cd\u590d\u7684\u539f\u56e0\u662f\u76f8\u7b49\u5143\u7d20\u5728\u67d0\u8f6e\u4e2d\u88ab\u591a\u6b21\u9009\u62e9\u3002\u5728\u56fe 13-13 \u4e2d\uff0c\u7b2c\u4e00\u8f6e\u5171\u6709\u4e09\u4e2a\u9009\u62e9\uff0c\u5176\u4e2d\u4e24\u4e2a\u90fd\u4e3a \\(4\\) \uff0c\u4f1a\u4ea7\u751f\u4e24\u4e2a\u91cd\u590d\u7684\u641c\u7d22\u5206\u652f\uff0c\u4ece\u800c\u8f93\u51fa\u91cd\u590d\u5b50\u96c6\uff1b\u540c\u7406\uff0c\u7b2c\u4e8c\u8f6e\u7684\u4e24\u4e2a \\(4\\) \u4e5f\u4f1a\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002</p> <p></p> <p> \u56fe 13-13 \u00a0 \u76f8\u7b49\u5143\u7d20\u5bfc\u81f4\u7684\u91cd\u590d\u5b50\u96c6 </p>"},{"location":"chapter_backtracking/subset_sum_problem/#1_1","title":"1. \u00a0 \u76f8\u7b49\u5143\u7d20\u526a\u679d","text":"<p>\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u9650\u5236\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u4e00\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5b9e\u73b0\u65b9\u5f0f\u6bd4\u8f83\u5de7\u5999\uff1a\u7531\u4e8e\u6570\u7ec4\u662f\u5df2\u6392\u5e8f\u7684\uff0c\u56e0\u6b64\u76f8\u7b49\u5143\u7d20\u90fd\u662f\u76f8\u90bb\u7684\u3002\u8fd9\u610f\u5473\u7740\u5728\u67d0\u8f6e\u9009\u62e9\u4e2d\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u4e0e\u5176\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u5219\u8bf4\u660e\u5b83\u5df2\u7ecf\u88ab\u9009\u62e9\u8fc7\uff0c\u56e0\u6b64\u76f4\u63a5\u8df3\u8fc7\u5f53\u524d\u5143\u7d20\u3002</p> <p>\u4e0e\u6b64\u540c\u65f6\uff0c\u672c\u9898\u89c4\u5b9a\u6bcf\u4e2a\u6570\u7ec4\u5143\u7d20\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\u3002\u5e78\u8fd0\u7684\u662f\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5229\u7528\u53d8\u91cf <code>start</code> \u6765\u6ee1\u8db3\u8be5\u7ea6\u675f\uff1a\u5f53\u505a\u51fa\u9009\u62e9 \\(x_{i}\\) \u540e\uff0c\u8bbe\u5b9a\u4e0b\u4e00\u8f6e\u4ece\u7d22\u5f15 \\(i + 1\\) \u5f00\u59cb\u5411\u540e\u904d\u5386\u3002\u8fd9\u6837\u65e2\u80fd\u53bb\u9664\u91cd\u590d\u5b50\u96c6\uff0c\u4e5f\u80fd\u907f\u514d\u91cd\u590d\u9009\u62e9\u5143\u7d20\u3002</p>"},{"location":"chapter_backtracking/subset_sum_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig subset_sum_ii.py<pre><code>def backtrack(\n    state: list[int], target: int, choices: list[int], start: int, res: list[list[int]]\n):\n    \"\"\"\u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II\"\"\"\n    # \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0:\n        res.append(list(state))\n        return\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    # \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    # \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in range(start, len(choices)):\n        # \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        # \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0:\n            break\n        # \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start and choices[i] == choices[i - 1]:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        # \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res)\n        # \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop()\n\ndef subset_sum_ii(nums: list[int], target: int) -&gt; list[list[int]]:\n    \"\"\"\u6c42\u89e3\u5b50\u96c6\u548c II\"\"\"\n    state = []  # \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort()  # \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start = 0  # \u904d\u5386\u8d77\u59cb\u70b9\n    res = []  # \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res)\n    return res\n</code></pre> subset_sum_ii.cpp<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(vector&lt;int&gt; &amp;state, int target, vector&lt;int&gt; &amp;choices, int start, vector&lt;vector&lt;int&gt;&gt; &amp;res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.push_back(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.size(); i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push_back(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop_back();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvector&lt;vector&lt;int&gt;&gt; subsetSumII(vector&lt;int&gt; &amp;nums, int target) {\n    vector&lt;int&gt; state;              // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort(nums.begin(), nums.end()); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0;                  // \u904d\u5386\u8d77\u59cb\u70b9\n    vector&lt;vector&lt;int&gt;&gt; res;        // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.java<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(List&lt;Integer&gt; state, int target, int[] choices, int start, List&lt;List&lt;Integer&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.add(new ArrayList&lt;&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.remove(state.size() - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;Integer&gt;&gt; subsetSumII(int[] nums, int target) {\n    List&lt;Integer&gt; state = new ArrayList&lt;&gt;(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Arrays.sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;Integer&gt;&gt; res = new ArrayList&lt;&gt;(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.cs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid Backtrack(List&lt;int&gt; state, int target, int[] choices, int start, List&lt;List&lt;int&gt;&gt; res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        res.Add(new List&lt;int&gt;(state));\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choices.Length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.Add(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        Backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.RemoveAt(state.Count - 1);\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; SubsetSumII(int[] nums, int target) {\n    List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    Array.Sort(nums); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    Backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.go<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrackSubsetSumII(start, target int, state, choices *[]int, res *[][]int) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        newState := append([]int{}, *state...)\n        *res = append(*res, newState)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i := start; i &lt; len(*choices); i++ {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target-(*choices)[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start &amp;&amp; (*choices)[i] == (*choices)[i-1] {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        *state = append(*state, (*choices)[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrackSubsetSumII(i+1, target-(*choices)[i], state, choices, res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        *state = (*state)[:len(*state)-1]\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums []int, target int) [][]int {\n    state := make([]int, 0) // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    sort.Ints(nums)         // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    start := 0              // \u904d\u5386\u8d77\u59cb\u70b9\n    res := make([][]int, 0) // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrackSubsetSumII(start, target, &amp;state, &amp;nums, &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.swift<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunc backtrack(state: inout [Int], target: Int, choices: [Int], start: Int, res: inout [[Int]]) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.append(state)\n        return\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in stride(from: start, to: choices.count, by: 1) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start, choices[i] == choices[i - 1] {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.append(choices[i])\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state: &amp;state, target: target - choices[i], choices: choices, start: i + 1, res: &amp;res)\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.removeLast()\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunc subsetSumII(nums: [Int], target: Int) -&gt; [[Int]] {\n    var state: [Int] = [] // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    let nums = nums.sorted() // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0 // \u904d\u5386\u8d77\u59cb\u70b9\n    var res: [[Int]] = [] // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state: &amp;state, target: target, choices: nums, start: start, res: &amp;res)\n    return res\n}\n</code></pre> subset_sum_ii.js<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(state, target, choices, start, res) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums, target) {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.ts<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfunction backtrack(\n    state: number[],\n    target: number,\n    choices: number[],\n    start: number,\n    res: number[][]\n): void {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target === 0) {\n        res.push([...state]);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (let i = start; i &lt; choices.length; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if (target - choices[i] &lt; 0) {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] === choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state, target - choices[i], choices, i + 1, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfunction subsetSumII(nums: number[], target: number): number[][] {\n    const state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort((a, b) =&gt; a - b); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    const start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    const res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, res);\n    return res;\n}\n</code></pre> subset_sum_ii.dart<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(\n  List&lt;int&gt; state,\n  int target,\n  List&lt;int&gt; choices,\n  int start,\n  List&lt;List&lt;int&gt;&gt; res,\n) {\n  // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n  if (target == 0) {\n    res.add(List.from(state));\n    return;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n  // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n  for (int i = start; i &lt; choices.length; i++) {\n    // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n    // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n    if (target - choices[i] &lt; 0) {\n      break;\n    }\n    // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n    if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n      continue;\n    }\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n    state.add(choices[i]);\n    // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n    backtrack(state, target - choices[i], choices, i + 1, res);\n    // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n    state.removeLast();\n  }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nList&lt;List&lt;int&gt;&gt; subsetSumII(List&lt;int&gt; nums, int target) {\n  List&lt;int&gt; state = []; // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n  nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n  int start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n  List&lt;List&lt;int&gt;&gt; res = []; // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n  backtrack(state, target, nums, start, res);\n  return res;\n}\n</code></pre> subset_sum_ii.rs<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nfn backtrack(mut state: Vec&lt;i32&gt;, target: i32, choices: &amp;[i32], start: usize, res: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if target == 0 {\n        res.push(state);\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for i in start..choices.len() {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u7ed3\u675f\u5faa\u73af\n        // \u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u5df2\u6392\u5e8f\uff0c\u540e\u8fb9\u5143\u7d20\u66f4\u5927\uff0c\u5b50\u96c6\u548c\u4e00\u5b9a\u8d85\u8fc7 target\n        if target - choices[i] &lt; 0 {\n            break;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if i &gt; start &amp;&amp; choices[i] == choices[i - 1] {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state.push(choices[i]);\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(state.clone(), target - choices[i], choices, i, res);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        state.pop();\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nfn subset_sum_ii(nums: &amp;mut [i32], target: i32) -&gt; Vec&lt;Vec&lt;i32&gt;&gt; {\n    let state = Vec::new(); // \u72b6\u6001\uff08\u5b50\u96c6\uff09\n    nums.sort(); // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    let start = 0; // \u904d\u5386\u8d77\u59cb\u70b9\n    let mut res = Vec::new(); // \u7ed3\u679c\u5217\u8868\uff08\u5b50\u96c6\u5217\u8868\uff09\n    backtrack(state, target, nums, start, &amp;mut res);\n    res\n}\n</code></pre> subset_sum_ii.c<pre><code>/* \u56de\u6eaf\u7b97\u6cd5\uff1a\u5b50\u96c6\u548c II */\nvoid backtrack(int target, int *choices, int choicesSize, int start) {\n    // \u5b50\u96c6\u548c\u7b49\u4e8e target \u65f6\uff0c\u8bb0\u5f55\u89e3\n    if (target == 0) {\n        for (int i = 0; i &lt; stateSize; i++) {\n            res[resSize][i] = state[i];\n        }\n        resColSizes[resSize++] = stateSize;\n        return;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    // \u526a\u679d\u4e8c\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u751f\u6210\u91cd\u590d\u5b50\u96c6\n    // \u526a\u679d\u4e09\uff1a\u4ece start \u5f00\u59cb\u904d\u5386\uff0c\u907f\u514d\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\n    for (int i = start; i &lt; choicesSize; i++) {\n        // \u526a\u679d\u4e00\uff1a\u82e5\u5b50\u96c6\u548c\u8d85\u8fc7 target \uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\n        if (target - choices[i] &lt; 0) {\n            continue;\n        }\n        // \u526a\u679d\u56db\uff1a\u5982\u679c\u8be5\u5143\u7d20\u4e0e\u5de6\u8fb9\u5143\u7d20\u76f8\u7b49\uff0c\u8bf4\u660e\u8be5\u641c\u7d22\u5206\u652f\u91cd\u590d\uff0c\u76f4\u63a5\u8df3\u8fc7\n        if (i &gt; start &amp;&amp; choices[i] == choices[i - 1]) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0 target, start\n        state[stateSize] = choices[i];\n        stateSize++;\n        // \u8fdb\u884c\u4e0b\u4e00\u8f6e\u9009\u62e9\n        backtrack(target - choices[i], choices, choicesSize, i + 1);\n        // \u56de\u9000\uff1a\u64a4\u9500\u9009\u62e9\uff0c\u6062\u590d\u5230\u4e4b\u524d\u7684\u72b6\u6001\n        stateSize--;\n    }\n}\n\n/* \u6c42\u89e3\u5b50\u96c6\u548c II */\nvoid subsetSumII(int *nums, int numsSize, int target) {\n    // \u5bf9 nums \u8fdb\u884c\u6392\u5e8f\n    qsort(nums, numsSize, sizeof(int), cmp);\n    // \u5f00\u59cb\u56de\u6eaf\n    backtrack(target, nums, numsSize, 0);\n}\n</code></pre> subset_sum_ii.zig<pre><code>[class]{}-[func]{backtrack}\n\n[class]{}-[func]{subsetSumII}\n</code></pre> <p>\u56fe 13-14 \u5c55\u793a\u4e86\u6570\u7ec4 \\([4, 4, 5]\\) \u548c\u76ee\u6807\u5143\u7d20 \\(9\\) \u7684\u56de\u6eaf\u8fc7\u7a0b\uff0c\u5171\u5305\u542b\u56db\u79cd\u526a\u679d\u64cd\u4f5c\u3002\u8bf7\u4f60\u5c06\u56fe\u793a\u4e0e\u4ee3\u7801\u6ce8\u91ca\u76f8\u7ed3\u5408\uff0c\u7406\u89e3\u6574\u4e2a\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ee5\u53ca\u6bcf\u79cd\u526a\u679d\u64cd\u4f5c\u662f\u5982\u4f55\u5de5\u4f5c\u7684\u3002</p> <p></p> <p> \u56fe 13-14 \u00a0 \u5b50\u96c6\u548c II \u56de\u6eaf\u8fc7\u7a0b </p>"},{"location":"chapter_backtracking/summary/","title":"13.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_backtracking/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56de\u6eaf\u7b97\u6cd5\u672c\u8d28\u662f\u7a77\u4e3e\u6cd5\uff0c\u901a\u8fc7\u5bf9\u89e3\u7a7a\u95f4\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6765\u5bfb\u627e\u7b26\u5408\u6761\u4ef6\u7684\u89e3\u3002\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u6ee1\u8db3\u6761\u4ef6\u7684\u89e3\u5219\u8bb0\u5f55\uff0c\u76f4\u81f3\u627e\u5230\u6240\u6709\u89e3\u6216\u904d\u5386\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u7684\u641c\u7d22\u8fc7\u7a0b\u5305\u62ec\u5c1d\u8bd5\u4e0e\u56de\u9000\u4e24\u4e2a\u90e8\u5206\u3002\u5b83\u901a\u8fc7\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u6765\u5c1d\u8bd5\u5404\u79cd\u9009\u62e9\uff0c\u5f53\u9047\u5230\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\u7684\u60c5\u51b5\u65f6\uff0c\u5219\u64a4\u9500\u4e0a\u4e00\u6b65\u7684\u9009\u62e9\uff0c\u9000\u56de\u5230\u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5e76\u7ee7\u7eed\u5c1d\u8bd5\u5176\u4ed6\u9009\u62e9\u3002\u5c1d\u8bd5\u4e0e\u56de\u9000\u662f\u4e24\u4e2a\u65b9\u5411\u76f8\u53cd\u7684\u64cd\u4f5c\u3002</li> <li>\u56de\u6eaf\u95ee\u9898\u901a\u5e38\u5305\u542b\u591a\u4e2a\u7ea6\u675f\u6761\u4ef6\uff0c\u5b83\u4eec\u53ef\u7528\u4e8e\u5b9e\u73b0\u526a\u679d\u64cd\u4f5c\u3002\u526a\u679d\u53ef\u4ee5\u63d0\u524d\u7ed3\u675f\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\uff0c\u5927\u5e45\u63d0\u5347\u641c\u7d22\u6548\u7387\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u4e3b\u8981\u53ef\u7528\u4e8e\u89e3\u51b3\u641c\u7d22\u95ee\u9898\u548c\u7ea6\u675f\u6ee1\u8db3\u95ee\u9898\u3002\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u867d\u7136\u53ef\u4ee5\u7528\u56de\u6eaf\u7b97\u6cd5\u89e3\u51b3\uff0c\u4f46\u5f80\u5f80\u5b58\u5728\u6548\u7387\u66f4\u9ad8\u6216\u6548\u679c\u66f4\u597d\u7684\u89e3\u6cd5\u3002</li> <li>\u5168\u6392\u5217\u95ee\u9898\u65e8\u5728\u641c\u7d22\u7ed9\u5b9a\u96c6\u5408\u5143\u7d20\u7684\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e2a\u5143\u7d20\u662f\u5426\u88ab\u9009\u62e9\uff0c\u526a\u6389\u91cd\u590d\u9009\u62e9\u540c\u4e00\u5143\u7d20\u7684\u641c\u7d22\u5206\u652f\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5143\u7d20\u53ea\u88ab\u9009\u62e9\u4e00\u6b21\u3002</li> <li>\u5728\u5168\u6392\u5217\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u96c6\u5408\u4e2d\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff0c\u5219\u6700\u7ec8\u7ed3\u679c\u4f1a\u51fa\u73b0\u91cd\u590d\u6392\u5217\u3002\u6211\u4eec\u9700\u8981\u7ea6\u675f\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u62e9\u4e00\u6b21\uff0c\u8fd9\u901a\u5e38\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5b50\u96c6\u548c\u95ee\u9898\u7684\u76ee\u6807\u662f\u5728\u7ed9\u5b9a\u96c6\u5408\u4e2d\u627e\u5230\u548c\u4e3a\u76ee\u6807\u503c\u7684\u6240\u6709\u5b50\u96c6\u3002\u96c6\u5408\u4e0d\u533a\u5206\u5143\u7d20\u987a\u5e8f\uff0c\u800c\u641c\u7d22\u8fc7\u7a0b\u4f1a\u8f93\u51fa\u6240\u6709\u987a\u5e8f\u7684\u7ed3\u679c\uff0c\u4ea7\u751f\u91cd\u590d\u5b50\u96c6\u3002\u6211\u4eec\u5728\u56de\u6eaf\u524d\u5c06\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u5e76\u8bbe\u7f6e\u4e00\u4e2a\u53d8\u91cf\u6765\u6307\u793a\u6bcf\u4e00\u8f6e\u7684\u904d\u5386\u8d77\u59cb\u70b9\uff0c\u4ece\u800c\u5c06\u751f\u6210\u91cd\u590d\u5b50\u96c6\u7684\u641c\u7d22\u5206\u652f\u8fdb\u884c\u526a\u679d\u3002</li> <li>\u5bf9\u4e8e\u5b50\u96c6\u548c\u95ee\u9898\uff0c\u6570\u7ec4\u4e2d\u7684\u76f8\u7b49\u5143\u7d20\u4f1a\u4ea7\u751f\u91cd\u590d\u96c6\u5408\u3002\u6211\u4eec\u5229\u7528\u6570\u7ec4\u5df2\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\uff0c\u901a\u8fc7\u5224\u65ad\u76f8\u90bb\u5143\u7d20\u662f\u5426\u76f8\u7b49\u5b9e\u73b0\u526a\u679d\uff0c\u4ece\u800c\u786e\u4fdd\u76f8\u7b49\u5143\u7d20\u5728\u6bcf\u8f6e\u4e2d\u53ea\u80fd\u88ab\u9009\u4e2d\u4e00\u6b21\u3002</li> <li>\\(n\\) \u7687\u540e\u95ee\u9898\u65e8\u5728\u5bfb\u627e\u5c06 \\(n\\) \u4e2a\u7687\u540e\u653e\u7f6e\u5230 \\(n \\times n\\) \u5c3a\u5bf8\u68cb\u76d8\u4e0a\u7684\u65b9\u6848\uff0c\u8981\u6c42\u6240\u6709\u7687\u540e\u4e24\u4e24\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u5bf9\u65b9\u3002\u8be5\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\u6709\u884c\u7ea6\u675f\u3001\u5217\u7ea6\u675f\u3001\u4e3b\u5bf9\u89d2\u7ebf\u548c\u526f\u5bf9\u89d2\u7ebf\u7ea6\u675f\u3002\u4e3a\u6ee1\u8db3\u884c\u7ea6\u675f\uff0c\u6211\u4eec\u91c7\u7528\u6309\u884c\u653e\u7f6e\u7684\u7b56\u7565\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u884c\u653e\u7f6e\u4e00\u4e2a\u7687\u540e\u3002</li> <li>\u5217\u7ea6\u675f\u548c\u5bf9\u89d2\u7ebf\u7ea6\u675f\u7684\u5904\u7406\u65b9\u5f0f\u7c7b\u4f3c\u3002\u5bf9\u4e8e\u5217\u7ea6\u675f\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u8bb0\u5f55\u6bcf\u4e00\u5217\u662f\u5426\u6709\u7687\u540e\uff0c\u4ece\u800c\u6307\u793a\u9009\u4e2d\u7684\u683c\u5b50\u662f\u5426\u5408\u6cd5\u3002\u5bf9\u4e8e\u5bf9\u89d2\u7ebf\u7ea6\u675f\uff0c\u6211\u4eec\u501f\u52a9\u4e24\u4e2a\u6570\u7ec4\u6765\u5206\u522b\u8bb0\u5f55\u8be5\u4e3b\u3001\u526f\u5bf9\u89d2\u7ebf\u4e0a\u662f\u5426\u5b58\u5728\u7687\u540e\uff1b\u96be\u70b9\u5728\u4e8e\u627e\u5904\u5728\u5230\u540c\u4e00\u4e3b\uff08\u526f\uff09\u5bf9\u89d2\u7ebf\u4e0a\u683c\u5b50\u6ee1\u8db3\u7684\u884c\u5217\u7d22\u5f15\u89c4\u5f8b\u3002</li> </ul>"},{"location":"chapter_backtracking/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u600e\u4e48\u7406\u89e3\u56de\u6eaf\u548c\u9012\u5f52\u7684\u5173\u7cfb\uff1f</p> <p>\u603b\u7684\u6765\u770b\uff0c\u56de\u6eaf\u662f\u4e00\u79cd\u201c\u7b97\u6cd5\u7b56\u7565\u201d\uff0c\u800c\u9012\u5f52\u66f4\u50cf\u662f\u4e00\u4e2a\u201c\u5de5\u5177\u201d\u3002</p> <ul> <li>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002\u7136\u800c\uff0c\u56de\u6eaf\u662f\u9012\u5f52\u7684\u5e94\u7528\u573a\u666f\u4e4b\u4e00\uff0c\u662f\u9012\u5f52\u5728\u641c\u7d22\u95ee\u9898\u4e2d\u7684\u5e94\u7528\u3002</li> <li>\u9012\u5f52\u7684\u7ed3\u6784\u4f53\u73b0\u4e86\u201c\u5b50\u95ee\u9898\u5206\u89e3\u201d\u7684\u89e3\u9898\u8303\u5f0f\uff0c\u5e38\u7528\u4e8e\u89e3\u51b3\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\uff08\u8bb0\u5fc6\u5316\u9012\u5f52\uff09\u7b49\u95ee\u9898\u3002</li> </ul>"},{"location":"chapter_computational_complexity/","title":"\u7b2c 2 \u7ae0 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>Abstract</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u72b9\u5982\u6d69\u701a\u7684\u7b97\u6cd5\u5b87\u5b99\u4e2d\u7684\u65f6\u7a7a\u5411\u5bfc\u3002</p> <p>\u5b83\u5e26\u9886\u6211\u4eec\u5728\u65f6\u95f4\u4e0e\u7a7a\u95f4\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u4e0a\u6df1\u5165\u63a2\u7d22\uff0c\u5bfb\u627e\u66f4\u4f18\u96c5\u7684\u89e3\u51b3\u65b9\u6848\u3002</p>"},{"location":"chapter_computational_complexity/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</li> <li>2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52</li> <li>2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6</li> <li>2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6</li> <li>2.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/","title":"2.2 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52","text":"<p>\u5728\u7b97\u6cd5\u4e2d\uff0c\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u662f\u5f88\u5e38\u89c1\u7684\uff0c\u5176\u4e0e\u590d\u6742\u5ea6\u5206\u6790\u606f\u606f\u76f8\u5173\u3002\u56e0\u6b64\uff0c\u5728\u5c55\u5f00\u4ecb\u7ecd\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e4b\u524d\uff0c\u6211\u4eec\u5148\u6765\u4e86\u89e3\u5982\u4f55\u5728\u7a0b\u5e8f\u4e2d\u5b9e\u73b0\u91cd\u590d\u6267\u884c\u4efb\u52a1\uff0c\u5373\u4e24\u79cd\u57fa\u672c\u7684\u7a0b\u5e8f\u63a7\u5236\u7ed3\u6784\uff1a\u8fed\u4ee3\u3001\u9012\u5f52\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#221","title":"2.2.1 \u00a0 \u8fed\u4ee3","text":"<p>\u300c\u8fed\u4ee3 iteration\u300d\u662f\u4e00\u79cd\u91cd\u590d\u6267\u884c\u67d0\u4e2a\u4efb\u52a1\u7684\u63a7\u5236\u7ed3\u6784\u3002\u5728\u8fed\u4ee3\u4e2d\uff0c\u7a0b\u5e8f\u4f1a\u5728\u6ee1\u8db3\u4e00\u5b9a\u7684\u6761\u4ef6\u4e0b\u91cd\u590d\u6267\u884c\u67d0\u6bb5\u4ee3\u7801\uff0c\u76f4\u5230\u8fd9\u4e2a\u6761\u4ef6\u4e0d\u518d\u6ee1\u8db3\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1-for","title":"1. \u00a0 for \u5faa\u73af","text":"<p><code>for</code> \u5faa\u73af\u662f\u6700\u5e38\u89c1\u7684\u8fed\u4ee3\u5f62\u5f0f\u4e4b\u4e00\uff0c\u9002\u5408\u5728\u9884\u5148\u77e5\u9053\u8fed\u4ee3\u6b21\u6570\u65f6\u4f7f\u7528\u3002</p> <p>\u4ee5\u4e0b\u51fd\u6570\u57fa\u4e8e <code>for</code> \u5faa\u73af\u5b9e\u73b0\u4e86\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff0c\u6c42\u548c\u7ed3\u679c\u4f7f\u7528\u53d8\u91cf <code>res</code> \u8bb0\u5f55\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0cPython \u4e2d <code>range(a, b)</code> \u5bf9\u5e94\u7684\u533a\u95f4\u662f\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u7684\uff0c\u5bf9\u5e94\u7684\u904d\u5386\u8303\u56f4\u4e3a \\(a, a + 1, \\dots, b-1\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def for_loop(n: int) -&gt; int:\n    \"\"\"for \u5faa\u73af\"\"\"\n    res = 0\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        res += i\n    return res\n</code></pre> iteration.cpp<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* for \u5faa\u73af */\nint ForLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n int) int {\n    res := 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* for \u5faa\u73af */\nfunc forLoop(n: Int) -&gt; Int {\n    var res = 0\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        res += i\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n) {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* for \u5faa\u73af */\nfunction forLoop(n: number): number {\n    let res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n  int res = 0;\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    res += i;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* for \u5faa\u73af */\nfn for_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i in 1..=n {\n        res += i;\n    }\n    res\n} \n</code></pre> iteration.c<pre><code>/* for \u5faa\u73af */\nint forLoop(int n) {\n    int res = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        res += i;\n    }\n    return res;\n}\n</code></pre> iteration.zig<pre><code>// for \u5faa\u73af\nfn forLoop(n: usize) i32 {\n    var res: i32 = 0;\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        res = res + @as(i32, @intCast(i));\n    }\n    return res;\n} \n</code></pre> <p>\u56fe 2-1 \u662f\u8be5\u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-1 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u6b64\u6c42\u548c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u6210\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u63cf\u8ff0\u7684\u5c31\u662f\u8fd9\u4e2a\u201c\u7ebf\u6027\u5173\u7cfb\u201d\u3002\u76f8\u5173\u5185\u5bb9\u5c06\u4f1a\u5728\u4e0b\u4e00\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2-while","title":"2. \u00a0 while \u5faa\u73af","text":"<p>\u4e0e <code>for</code> \u5faa\u73af\u7c7b\u4f3c\uff0c<code>while</code> \u5faa\u73af\u4e5f\u662f\u4e00\u79cd\u5b9e\u73b0\u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u7a0b\u5e8f\u6bcf\u8f6e\u90fd\u4f1a\u5148\u68c0\u67e5\u6761\u4ef6\uff0c\u5982\u679c\u6761\u4ef6\u4e3a\u771f\uff0c\u5219\u7ee7\u7eed\u6267\u884c\uff0c\u5426\u5219\u5c31\u7ed3\u675f\u5faa\u73af\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u7528 <code>while</code> \u5faa\u73af\u6765\u5b9e\u73b0\u6c42\u548c \\(1 + 2 + \\dots + n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def while_loop(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n:\n        res += i\n        i += 1  # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af */\nint WhileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af */\nfunc whileLoop(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i\n        i += 1 // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af */\nfunction whileLoop(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n  while (i &lt;= n) {\n    res += i;\n    i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af */\nfn while_loop(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while i &lt;= n {\n        res += i;\n        i += 1; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af */\nint whileLoop(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += i;\n        i++; // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    }\n    return res;\n}\n</code></pre> iteration.zig<pre><code>// while \u5faa\u73af\nfn whileLoop(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, ..., n-1, n\n    while (i &lt;= n) {\n        res += @intCast(i);\n        i += 1;\n    }\n    return res;\n}\n</code></pre> <p><code>while</code> \u5faa\u73af\u6bd4 <code>for</code> \u5faa\u73af\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\u3002\u5728 <code>while</code> \u5faa\u73af\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7531\u5730\u8bbe\u8ba1\u6761\u4ef6\u53d8\u91cf\u7684\u521d\u59cb\u5316\u548c\u66f4\u65b0\u6b65\u9aa4\u3002</p> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6761\u4ef6\u53d8\u91cf \\(i\\) \u6bcf\u8f6e\u8fdb\u884c\u4e24\u6b21\u66f4\u65b0\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u4e0d\u592a\u65b9\u4fbf\u7528 <code>for</code> \u5faa\u73af\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def while_loop_ii(n: int) -&gt; int:\n    \"\"\"while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\"\"\"\n    res = 0\n    i = 1  # \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    # \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n:\n        res += i\n        # \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    return res\n</code></pre> iteration.cpp<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.java<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.cs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint WhileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 2, 4, 5...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1; \n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.go<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n int) int {\n    res := 0\n    // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    i := 1\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    for i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunc whileLoopII(n: Int) -&gt; Int {\n    var res = 0\n    var i = 1 // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1\n        i *= 2\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n) {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfunction whileLoopII(n: number): number {\n    let res = 0;\n    let i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n  int res = 0;\n  int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n  // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n  while (i &lt;= n) {\n    res += i;\n    // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n    i++;\n    i *= 2;\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nfn while_loop_ii(n: i32) -&gt; i32 {\n    let mut res = 0;\n    let mut i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while i &lt;= n {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    res\n}\n</code></pre> iteration.c<pre><code>/* while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09 */\nint whileLoopII(int n) {\n    int res = 0;\n    int i = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += i;\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i++;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> iteration.zig<pre><code>//  while \u5faa\u73af\uff08\u4e24\u6b21\u66f4\u65b0\uff09\nfn whileLoopII(n: i32) i32 {\n    var res: i32 = 0;\n    var i: i32 = 1; // \u521d\u59cb\u5316\u6761\u4ef6\u53d8\u91cf\n    // \u5faa\u73af\u6c42\u548c 1, 4, 10, ...\n    while (i &lt;= n) {\n        res += @intCast(i);\n        // \u66f4\u65b0\u6761\u4ef6\u53d8\u91cf\n        i += 1;\n        i *= 2;\n    }\n    return res;\n}\n</code></pre> <p>\u603b\u7684\u6765\u8bf4\uff0c<code>for</code> \u5faa\u73af\u7684\u4ee3\u7801\u66f4\u52a0\u7d27\u51d1\uff0c<code>while</code> \u5faa\u73af\u66f4\u52a0\u7075\u6d3b\uff0c\u4e24\u8005\u90fd\u53ef\u4ee5\u5b9e\u73b0\u8fed\u4ee3\u7ed3\u6784\u3002\u9009\u62e9\u4f7f\u7528\u54ea\u4e00\u4e2a\u5e94\u8be5\u6839\u636e\u7279\u5b9a\u95ee\u9898\u7684\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3","title":"3. \u00a0 \u5d4c\u5957\u5faa\u73af","text":"<p>\u6211\u4eec\u53ef\u4ee5\u5728\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\u5185\u5d4c\u5957\u53e6\u4e00\u4e2a\u5faa\u73af\u7ed3\u6784\uff0c\u4e0b\u9762\u4ee5 <code>for</code> \u5faa\u73af\u4e3a\u4f8b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig iteration.py<pre><code>def nested_for_loop(n: int) -&gt; str:\n    \"\"\"\u53cc\u5c42 for \u5faa\u73af\"\"\"\n    res = \"\"\n    # \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in range(1, n + 1):\n        # \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in range(1, n + 1):\n            res += f\"({i}, {j}), \"\n    return res\n</code></pre> iteration.cpp<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring nestedForLoop(int n) {\n    ostringstream res;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; ++i) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; ++j) {\n            res &lt;&lt; \"(\" &lt;&lt; i &lt;&lt; \", \" &lt;&lt; j &lt;&lt; \"), \";\n        }\n    }\n    return res.str();\n}\n</code></pre> iteration.java<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n    StringBuilder res = new StringBuilder();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.append(\"(\" + i + \", \" + j + \"), \");\n        }\n    }\n    return res.toString();\n}\n</code></pre> iteration.cs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nstring NestedForLoop(int n) {\n    StringBuilder res = new();\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            res.Append($\"({i}, {j}), \");\n        }\n    }\n    return res.ToString();\n}\n</code></pre> iteration.go<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n int) string {\n    res := \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= n; j++ {\n            // \u5faa\u73af j = 1, 2, ..., n-1, n\n            res += fmt.Sprintf(\"(%d, %d), \", i, j)\n        }\n    }\n    return res\n}\n</code></pre> iteration.swift<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunc nestedForLoop(n: Int) -&gt; String {\n    var res = \"\"\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1 ... n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1 ... n {\n            res.append(\"(\\(i), \\(j)), \")\n        }\n    }\n    return res\n}\n</code></pre> iteration.js<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n) {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.ts<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfunction nestedForLoop(n: number): string {\n    let res = '';\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (let j = 1; j &lt;= n; j++) {\n            res += `(${i}, ${j}), `;\n        }\n    }\n    return res;\n}\n</code></pre> iteration.dart<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nString nestedForLoop(int n) {\n  String res = \"\";\n  // \u5faa\u73af i = 1, 2, ..., n-1, n\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5faa\u73af j = 1, 2, ..., n-1, n\n    for (int j = 1; j &lt;= n; j++) {\n      res += \"($i, $j), \";\n    }\n  }\n  return res;\n}\n</code></pre> iteration.rs<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nfn nested_for_loop(n: i32) -&gt; String {\n    let mut res = vec![];\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for i in 1..=n {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for j in 1..=n {\n            res.push(format!(\"({}, {}), \", i, j));\n        }\n    }\n    res.join(\"\")\n}\n</code></pre> iteration.c<pre><code>/* \u53cc\u5c42 for \u5faa\u73af */\nchar *nestedForLoop(int n) {\n    // n * n \u4e3a\u5bf9\u5e94\u70b9\u6570\u91cf\uff0c\"(i, j), \" \u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u6700\u5927\u4e3a 6+10*2\uff0c\u52a0\u4e0a\u6700\u540e\u4e00\u4e2a\u7a7a\u5b57\u7b26 \\0 \u7684\u989d\u5916\u7a7a\u95f4\n    int size = n * n * 26 + 1;\n    char *res = malloc(size * sizeof(char));\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (int j = 1; j &lt;= n; j++) {\n            char tmp[26];\n            snprintf(tmp, sizeof(tmp), \"(%d, %d), \", i, j);\n            strncat(res, tmp, size - strlen(res) - 1);\n        }\n    }\n    return res;\n}\n</code></pre> iteration.zig<pre><code>// \u53cc\u5c42 for \u5faa\u73af\nfn nestedForLoop(allocator: Allocator, n: usize) ![]const u8 {\n    var res = std.ArrayList(u8).init(allocator);\n    defer res.deinit();\n    var buffer: [20]u8 = undefined;\n    // \u5faa\u73af i = 1, 2, ..., n-1, n\n    for (1..n+1) |i| {\n        // \u5faa\u73af j = 1, 2, ..., n-1, n\n        for (1..n+1) |j| {\n            var _str = try std.fmt.bufPrint(&amp;buffer, \"({d}, {d}), \", .{i, j});\n            try res.appendSlice(_str);\n        }\n    }\n    return res.toOwnedSlice();\n}\n</code></pre> <p>\u56fe 2-2 \u662f\u8be5\u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe\u3002</p> <p></p> <p> \u56fe 2-2 \u00a0 \u5d4c\u5957\u5faa\u73af\u7684\u6d41\u7a0b\u6846\u56fe </p> <p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e \\(n^2\\) \u6210\u6b63\u6bd4\uff0c\u6216\u8005\u8bf4\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u548c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u6210\u201c\u5e73\u65b9\u5173\u7cfb\u201d\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u6dfb\u52a0\u5d4c\u5957\u5faa\u73af\uff0c\u6bcf\u4e00\u6b21\u5d4c\u5957\u90fd\u662f\u4e00\u6b21\u201c\u5347\u7ef4\u201d\uff0c\u5c06\u4f1a\u4f7f\u65f6\u95f4\u590d\u6742\u5ea6\u63d0\u9ad8\u81f3\u201c\u7acb\u65b9\u5173\u7cfb\u201d\u201c\u56db\u6b21\u65b9\u5173\u7cfb\u201d\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#222","title":"2.2.2 \u00a0 \u9012\u5f52","text":"<p>\u300c\u9012\u5f52 recursion\u300d\u662f\u4e00\u79cd\u7b97\u6cd5\u7b56\u7565\uff0c\u901a\u8fc7\u51fd\u6570\u8c03\u7528\u81ea\u8eab\u6765\u89e3\u51b3\u95ee\u9898\u3002\u5b83\u4e3b\u8981\u5305\u542b\u4e24\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9012\uff1a\u7a0b\u5e8f\u4e0d\u65ad\u6df1\u5165\u5730\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u4f20\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\uff0c\u76f4\u5230\u8fbe\u5230\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u3002</li> <li>\u5f52\uff1a\u89e6\u53d1\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u540e\uff0c\u7a0b\u5e8f\u4ece\u6700\u6df1\u5c42\u7684\u9012\u5f52\u51fd\u6570\u5f00\u59cb\u9010\u5c42\u8fd4\u56de\uff0c\u6c47\u805a\u6bcf\u4e00\u5c42\u7684\u7ed3\u679c\u3002</li> </ol> <p>\u800c\u4ece\u5b9e\u73b0\u7684\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u4ee3\u7801\u4e3b\u8981\u5305\u542b\u4e09\u4e2a\u8981\u7d20\u3002</p> <ol> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u7528\u4e8e\u51b3\u5b9a\u4ec0\u4e48\u65f6\u5019\u7531\u201c\u9012\u201d\u8f6c\u201c\u5f52\u201d\u3002</li> <li>\u9012\u5f52\u8c03\u7528\uff1a\u5bf9\u5e94\u201c\u9012\u201d\uff0c\u51fd\u6570\u8c03\u7528\u81ea\u8eab\uff0c\u901a\u5e38\u8f93\u5165\u66f4\u5c0f\u6216\u66f4\u7b80\u5316\u7684\u53c2\u6570\u3002</li> <li>\u8fd4\u56de\u7ed3\u679c\uff1a\u5bf9\u5e94\u201c\u5f52\u201d\uff0c\u5c06\u5f53\u524d\u9012\u5f52\u5c42\u7ea7\u7684\u7ed3\u679c\u8fd4\u56de\u81f3\u4e0a\u4e00\u5c42\u3002</li> </ol> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u53ea\u9700\u8c03\u7528\u51fd\u6570 <code>recur(n)</code>  \uff0c\u5c31\u53ef\u4ee5\u5b8c\u6210 \\(1 + 2 + \\dots + n\\) \u7684\u8ba1\u7b97\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def recur(n: int) -&gt; int:\n    \"\"\"\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 1:\n        return 1\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res = recur(n - 1)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n</code></pre> recursion.cpp<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.java<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.cs<pre><code>/* \u9012\u5f52 */\nint Recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = Recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.go<pre><code>/* \u9012\u5f52 */\nfunc recur(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    res := recur(n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.swift<pre><code>/* \u9012\u5f52 */\nfunc recur(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n: n - 1)\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res\n}\n</code></pre> recursion.js<pre><code>/* \u9012\u5f52 */\nfunction recur(n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.ts<pre><code>/* \u9012\u5f52 */\nfunction recur(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 1) return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    const res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.dart<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 1) return 1;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  int res = recur(n - 1);\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  return n + res;\n}\n</code></pre> recursion.rs<pre><code>/* \u9012\u5f52 */\nfn recur(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 1 {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    let res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    n + res\n}\n</code></pre> recursion.c<pre><code>/* \u9012\u5f52 */\nint recur(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1)\n        return 1;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    int res = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> recursion.zig<pre><code>// \u9012\u5f52\u51fd\u6570\nfn recur(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 1) {\n        return 1;\n    }\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var res: i32 = recur(n - 1);\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    return n + res;\n}\n</code></pre> <p>\u56fe 2-3 \u5c55\u793a\u4e86\u8be5\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 2-3 \u00a0 \u6c42\u548c\u51fd\u6570\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u867d\u7136\u4ece\u8ba1\u7b97\u89d2\u5ea6\u770b\uff0c\u8fed\u4ee3\u4e0e\u9012\u5f52\u53ef\u4ee5\u5f97\u5230\u76f8\u540c\u7684\u7ed3\u679c\uff0c\u4f46\u5b83\u4eec\u4ee3\u8868\u4e86\u4e24\u79cd\u5b8c\u5168\u4e0d\u540c\u7684\u601d\u8003\u548c\u89e3\u51b3\u95ee\u9898\u7684\u8303\u5f0f\u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u4ece\u6700\u57fa\u7840\u7684\u6b65\u9aa4\u5f00\u59cb\uff0c\u7136\u540e\u4e0d\u65ad\u91cd\u590d\u6216\u7d2f\u52a0\u8fd9\u4e9b\u6b65\u9aa4\uff0c\u76f4\u5230\u4efb\u52a1\u5b8c\u6210\u3002</li> <li>\u9012\u5f52\uff1a\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u5730\u89e3\u51b3\u95ee\u9898\u3002\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u8fd9\u4e9b\u5b50\u95ee\u9898\u548c\u539f\u95ee\u9898\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f\u3002\u63a5\u4e0b\u6765\u5c06\u5b50\u95ee\u9898\u7ee7\u7eed\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u57fa\u672c\u60c5\u51b5\u65f6\u505c\u6b62\uff08\u57fa\u672c\u60c5\u51b5\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff09\u3002</li> </ul> <p>\u4ee5\u4e0a\u8ff0\u6c42\u548c\u51fd\u6570\u4e3a\u4f8b\uff0c\u8bbe\u95ee\u9898 \\(f(n) = 1 + 2 + \\dots + n\\) \u3002</p> <ul> <li>\u8fed\u4ee3\uff1a\u5728\u5faa\u73af\u4e2d\u6a21\u62df\u6c42\u548c\u8fc7\u7a0b\uff0c\u4ece \\(1\\) \u904d\u5386\u5230 \\(n\\) \uff0c\u6bcf\u8f6e\u6267\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5373\u53ef\u6c42\u5f97 \\(f(n)\\) \u3002</li> <li>\u9012\u5f52\uff1a\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u5b50\u95ee\u9898 \\(f(n) = n + f(n-1)\\) \uff0c\u4e0d\u65ad\uff08\u9012\u5f52\u5730\uff09\u5206\u89e3\u4e0b\u53bb\uff0c\u76f4\u81f3\u57fa\u672c\u60c5\u51b5 \\(f(1) = 1\\) \u65f6\u7ec8\u6b62\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#1","title":"1. \u00a0 \u8c03\u7528\u6808","text":"<p>\u9012\u5f52\u51fd\u6570\u6bcf\u6b21\u8c03\u7528\u81ea\u8eab\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u4e3a\u65b0\u5f00\u542f\u7684\u51fd\u6570\u5206\u914d\u5185\u5b58\uff0c\u4ee5\u5b58\u50a8\u5c40\u90e8\u53d8\u91cf\u3001\u8c03\u7528\u5730\u5740\u548c\u5176\u4ed6\u4fe1\u606f\u7b49\u3002\u8fd9\u5c06\u5bfc\u81f4\u4e24\u65b9\u9762\u7684\u7ed3\u679c\u3002</p> <ul> <li>\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u90fd\u5b58\u50a8\u5728\u79f0\u4e3a\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u7684\u5185\u5b58\u533a\u57df\u4e2d\uff0c\u76f4\u81f3\u51fd\u6570\u8fd4\u56de\u540e\u624d\u4f1a\u88ab\u91ca\u653e\u3002\u56e0\u6b64\uff0c\u9012\u5f52\u901a\u5e38\u6bd4\u8fed\u4ee3\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u9012\u5f52\u8c03\u7528\u51fd\u6570\u4f1a\u4ea7\u751f\u989d\u5916\u7684\u5f00\u9500\u3002\u56e0\u6b64\u9012\u5f52\u901a\u5e38\u6bd4\u5faa\u73af\u7684\u65f6\u95f4\u6548\u7387\u66f4\u4f4e\u3002</li> </ul> <p>\u5982\u56fe 2-4 \u6240\u793a\uff0c\u5728\u89e6\u53d1\u7ec8\u6b62\u6761\u4ef6\u524d\uff0c\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684\u9012\u5f52\u51fd\u6570\uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002</p> <p></p> <p> \u56fe 2-4 \u00a0 \u9012\u5f52\u8c03\u7528\u6df1\u5ea6 </p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u7f16\u7a0b\u8bed\u8a00\u5141\u8bb8\u7684\u9012\u5f52\u6df1\u5ea6\u901a\u5e38\u662f\u6709\u9650\u7684\uff0c\u8fc7\u6df1\u7684\u9012\u5f52\u53ef\u80fd\u5bfc\u81f4\u6808\u6ea2\u51fa\u9519\u8bef\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#2","title":"2. \u00a0 \u5c3e\u9012\u5f52","text":"<p>\u6709\u8da3\u7684\u662f\uff0c\u5982\u679c\u51fd\u6570\u5728\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u6b65\u624d\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5219\u8be5\u51fd\u6570\u53ef\u4ee5\u88ab\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u4f18\u5316\uff0c\u4f7f\u5176\u5728\u7a7a\u95f4\u6548\u7387\u4e0a\u4e0e\u8fed\u4ee3\u76f8\u5f53\u3002\u8fd9\u79cd\u60c5\u51b5\u88ab\u79f0\u4e3a\u300c\u5c3e\u9012\u5f52 tail recursion\u300d\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u5f53\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u7684\u51fd\u6570\u540e\uff0c\u9700\u8981\u7ee7\u7eed\u6267\u884c\u4ee3\u7801\uff0c\u56e0\u6b64\u7cfb\u7edf\u9700\u8981\u4fdd\u5b58\u4e0a\u4e00\u5c42\u8c03\u7528\u7684\u4e0a\u4e0b\u6587\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u9012\u5f52\u8c03\u7528\u662f\u51fd\u6570\u8fd4\u56de\u524d\u7684\u6700\u540e\u4e00\u4e2a\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u51fd\u6570\u8fd4\u56de\u5230\u4e0a\u4e00\u5c42\u7ea7\u540e\uff0c\u65e0\u987b\u7ee7\u7eed\u6267\u884c\u5176\u4ed6\u64cd\u4f5c\uff0c\u56e0\u6b64\u7cfb\u7edf\u65e0\u987b\u4fdd\u5b58\u4e0a\u4e00\u5c42\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u3002</li> </ul> <p>\u4ee5\u8ba1\u7b97 \\(1 + 2 + \\dots + n\\) \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u7ed3\u679c\u53d8\u91cf <code>res</code> \u8bbe\u4e3a\u51fd\u6570\u53c2\u6570\uff0c\u4ece\u800c\u5b9e\u73b0\u5c3e\u9012\u5f52\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def tail_recur(n, res):\n    \"\"\"\u5c3e\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if n == 0:\n        return res\n    # \u5c3e\u9012\u5f52\u8c03\u7528\n    return tail_recur(n - 1, res + n)\n</code></pre> recursion.cpp<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.java<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.cs<pre><code>/* \u5c3e\u9012\u5f52 */\nint TailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return TailRecur(n - 1, res + n);\n}\n</code></pre> recursion.go<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n int, res int) int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n-1, res+n)\n}\n</code></pre> recursion.swift<pre><code>/* \u5c3e\u9012\u5f52 */\nfunc tailRecur(n: Int, res: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n: n - 1, res: res + n)\n}\n</code></pre> recursion.js<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n, res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.ts<pre><code>/* \u5c3e\u9012\u5f52 */\nfunction tailRecur(n: number, res: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n === 0) return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.dart<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (n == 0) return res;\n  // \u5c3e\u9012\u5f52\u8c03\u7528\n  return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.rs<pre><code>/* \u5c3e\u9012\u5f52 */\nfn tail_recur(n: i32, res: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if n == 0 {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    tail_recur(n - 1, res + n)\n}\n</code></pre> recursion.c<pre><code>/* \u5c3e\u9012\u5f52 */\nint tailRecur(int n, int res) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0)\n        return res;\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> recursion.zig<pre><code>// \u5c3e\u9012\u5f52\u51fd\u6570\nfn tailRecur(n: i32, res: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (n == 0) {\n        return res;\n    }\n    // \u5c3e\u9012\u5f52\u8c03\u7528\n    return tailRecur(n - 1, res + n);\n}\n</code></pre> <p>\u5c3e\u9012\u5f52\u7684\u6267\u884c\u8fc7\u7a0b\u5982\u56fe 2-5 \u6240\u793a\u3002\u5bf9\u6bd4\u666e\u901a\u9012\u5f52\u548c\u5c3e\u9012\u5f52\uff0c\u4e24\u8005\u7684\u6c42\u548c\u64cd\u4f5c\u7684\u6267\u884c\u70b9\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li>\u666e\u901a\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u6bcf\u5c42\u8fd4\u56de\u540e\u90fd\u8981\u518d\u6267\u884c\u4e00\u6b21\u6c42\u548c\u64cd\u4f5c\u3002</li> <li>\u5c3e\u9012\u5f52\uff1a\u6c42\u548c\u64cd\u4f5c\u662f\u5728\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\uff0c\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u53ea\u9700\u5c42\u5c42\u8fd4\u56de\u3002</li> </ul> <p></p> <p> \u56fe 2-5 \u00a0 \u5c3e\u9012\u5f52\u8fc7\u7a0b </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8bb8\u591a\u7f16\u8bd1\u5668\u6216\u89e3\u91ca\u5668\u5e76\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\u3002\u4f8b\u5982\uff0cPython \u9ed8\u8ba4\u4e0d\u652f\u6301\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u56e0\u6b64\u5373\u4f7f\u51fd\u6570\u662f\u5c3e\u9012\u5f52\u5f62\u5f0f\uff0c\u4ecd\u7136\u53ef\u80fd\u4f1a\u9047\u5230\u6808\u6ea2\u51fa\u95ee\u9898\u3002</p>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#3_1","title":"3. \u00a0 \u9012\u5f52\u6811","text":"<p>\u5f53\u5904\u7406\u4e0e\u201c\u5206\u6cbb\u201d\u76f8\u5173\u7684\u7b97\u6cd5\u95ee\u9898\u65f6\uff0c\u9012\u5f52\u5f80\u5f80\u6bd4\u8fed\u4ee3\u7684\u601d\u8def\u66f4\u52a0\u76f4\u89c2\u3001\u4ee3\u7801\u66f4\u52a0\u6613\u8bfb\u3002\u4ee5\u201c\u6590\u6ce2\u90a3\u5951\u6570\u5217\u201d\u4e3a\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5217 \\(0, 1, 1, 2, 3, 5, 8, 13, \\dots\\) \uff0c\u6c42\u8be5\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u3002</p> <p>\u8bbe\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\u4e3a \\(f(n)\\) \uff0c\u6613\u5f97\u4e24\u4e2a\u7ed3\u8bba\u3002</p> <ul> <li>\u6570\u5217\u7684\u524d\u4e24\u4e2a\u6570\u5b57\u4e3a \\(f(1) = 0\\) \u548c \\(f(2) = 1\\) \u3002</li> <li>\u6570\u5217\u4e2d\u7684\u6bcf\u4e2a\u6570\u5b57\u662f\u524d\u4e24\u4e2a\u6570\u5b57\u7684\u548c\uff0c\u5373 \\(f(n) = f(n - 1) + f(n - 2)\\) \u3002</li> </ul> <p>\u6309\u7167\u9012\u63a8\u5173\u7cfb\u8fdb\u884c\u9012\u5f52\u8c03\u7528\uff0c\u5c06\u524d\u4e24\u4e2a\u6570\u5b57\u4f5c\u4e3a\u7ec8\u6b62\u6761\u4ef6\uff0c\u4fbf\u53ef\u5199\u51fa\u9012\u5f52\u4ee3\u7801\u3002\u8c03\u7528 <code>fib(n)</code> \u5373\u53ef\u5f97\u5230\u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u7b2c \\(n\\) \u4e2a\u6570\u5b57\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def fib(n: int) -&gt; int:\n    \"\"\"\u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 or n == 2:\n        return n - 1\n    # \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res = fib(n - 1) + fib(n - 2)\n    # \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint Fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = Fib(n - 1) + Fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n int) int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    res := fib(n-1) + fib(n-2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunc fib(n: Int) -&gt; Int {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n: n - 1) + fib(n: n - 2)\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfunction fib(n: number): number {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n === 1 || n === 2) return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    const res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n  // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n  if (n == 1 || n == 2) return n - 1;\n  // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n  int res = fib(n - 1) + fib(n - 2);\n  // \u8fd4\u56de\u7ed3\u679c f(n)\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nfn fib(n: i32) -&gt; i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if n == 1 || n == 2 {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    let res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u6590\u6ce2\u90a3\u5951\u6570\u5217\uff1a\u9012\u5f52 */\nint fib(int n) {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 || n == 2)\n        return n - 1;\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    int res = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> recursion.zig<pre><code>// \u6590\u6ce2\u90a3\u5951\u6570\u5217\nfn fib(n: i32) i32 {\n    // \u7ec8\u6b62\u6761\u4ef6 f(1) = 0, f(2) = 1\n    if (n == 1 or n == 2) {\n        return n - 1;\n    }\n    // \u9012\u5f52\u8c03\u7528 f(n) = f(n-1) + f(n-2)\n    var res: i32 = fib(n - 1) + fib(n - 2);\n    // \u8fd4\u56de\u7ed3\u679c f(n)\n    return res;\n}\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u6211\u4eec\u5728\u51fd\u6570\u5185\u9012\u5f52\u8c03\u7528\u4e86\u4e24\u4e2a\u51fd\u6570\uff0c\u8fd9\u610f\u5473\u7740\u4ece\u4e00\u4e2a\u8c03\u7528\u4ea7\u751f\u4e86\u4e24\u4e2a\u8c03\u7528\u5206\u652f\u3002\u5982\u56fe 2-6 \u6240\u793a\uff0c\u8fd9\u6837\u4e0d\u65ad\u9012\u5f52\u8c03\u7528\u4e0b\u53bb\uff0c\u6700\u7ec8\u5c06\u4ea7\u751f\u4e00\u68f5\u5c42\u6570\u4e3a \\(n\\) \u7684\u300c\u9012\u5f52\u6811 recursion tree\u300d\u3002</p> <p></p> <p> \u56fe 2-6 \u00a0 \u6590\u6ce2\u90a3\u5951\u6570\u5217\u7684\u9012\u5f52\u6811 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9012\u5f52\u4f53\u73b0\u4e86\u201c\u5c06\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u5c0f\u5b50\u95ee\u9898\u201d\u7684\u601d\u7ef4\u8303\u5f0f\uff0c\u8fd9\u79cd\u5206\u6cbb\u7b56\u7565\u81f3\u5173\u91cd\u8981\u3002</p> <ul> <li>\u4ece\u7b97\u6cd5\u89d2\u5ea6\u770b\uff0c\u641c\u7d22\u3001\u6392\u5e8f\u3001\u56de\u6eaf\u3001\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u7b49\u8bb8\u591a\u91cd\u8981\u7b97\u6cd5\u7b56\u7565\u76f4\u63a5\u6216\u95f4\u63a5\u5730\u5e94\u7528\u4e86\u8fd9\u79cd\u601d\u7ef4\u65b9\u5f0f\u3002</li> <li>\u4ece\u6570\u636e\u7ed3\u6784\u89d2\u5ea6\u770b\uff0c\u9012\u5f52\u5929\u7136\u9002\u5408\u5904\u7406\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u76f8\u5173\u95ee\u9898\uff0c\u56e0\u4e3a\u5b83\u4eec\u975e\u5e38\u9002\u5408\u7528\u5206\u6cbb\u601d\u60f3\u8fdb\u884c\u5206\u6790\u3002</li> </ul>"},{"location":"chapter_computational_complexity/iteration_and_recursion/#223","title":"2.2.3 \u00a0 \u4e24\u8005\u5bf9\u6bd4","text":"<p>\u603b\u7ed3\u4ee5\u4e0a\u5185\u5bb9\uff0c\u5982\u8868 2-1 \u6240\u793a\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5b9e\u73b0\u3001\u6027\u80fd\u548c\u9002\u7528\u6027\u4e0a\u6709\u6240\u4e0d\u540c\u3002</p> <p> \u8868 2-1 \u00a0 \u8fed\u4ee3\u4e0e\u9012\u5f52\u7279\u70b9\u5bf9\u6bd4 </p> \u8fed\u4ee3 \u9012\u5f52 \u5b9e\u73b0\u65b9\u5f0f \u5faa\u73af\u7ed3\u6784 \u51fd\u6570\u8c03\u7528\u81ea\u8eab \u65f6\u95f4\u6548\u7387 \u6548\u7387\u901a\u5e38\u8f83\u9ad8\uff0c\u65e0\u51fd\u6570\u8c03\u7528\u5f00\u9500 \u6bcf\u6b21\u51fd\u6570\u8c03\u7528\u90fd\u4f1a\u4ea7\u751f\u5f00\u9500 \u5185\u5b58\u4f7f\u7528 \u901a\u5e38\u4f7f\u7528\u56fa\u5b9a\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4 \u7d2f\u79ef\u51fd\u6570\u8c03\u7528\u53ef\u80fd\u4f7f\u7528\u5927\u91cf\u7684\u6808\u5e27\u7a7a\u95f4 \u9002\u7528\u95ee\u9898 \u9002\u7528\u4e8e\u7b80\u5355\u5faa\u73af\u4efb\u52a1\uff0c\u4ee3\u7801\u76f4\u89c2\u3001\u53ef\u8bfb\u6027\u597d \u9002\u7528\u4e8e\u5b50\u95ee\u9898\u5206\u89e3\uff0c\u5982\u6811\u3001\u56fe\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u7b49\uff0c\u4ee3\u7801\u7ed3\u6784\u7b80\u6d01\u3001\u6e05\u6670 <p>Tip</p> <p>\u5982\u679c\u611f\u89c9\u4ee5\u4e0b\u5185\u5bb9\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5728\u8bfb\u5b8c\u201c\u6808\u201d\u7ae0\u8282\u540e\u518d\u6765\u590d\u4e60\u3002</p> <p>\u90a3\u4e48\uff0c\u8fed\u4ee3\u548c\u9012\u5f52\u5177\u6709\u4ec0\u4e48\u5185\u5728\u8054\u7cfb\u5462\uff1f\u4ee5\u4e0a\u8ff0\u7684\u9012\u5f52\u51fd\u6570\u4e3a\u4f8b\uff0c\u6c42\u548c\u64cd\u4f5c\u5728\u9012\u5f52\u7684\u201c\u5f52\u201d\u9636\u6bb5\u8fdb\u884c\u3002\u8fd9\u610f\u5473\u7740\u6700\u521d\u88ab\u8c03\u7528\u7684\u51fd\u6570\u5b9e\u9645\u4e0a\u662f\u6700\u540e\u5b8c\u6210\u5176\u6c42\u548c\u64cd\u4f5c\u7684\uff0c\u8fd9\u79cd\u5de5\u4f5c\u673a\u5236\u4e0e\u6808\u7684\u201c\u5148\u5165\u540e\u51fa\u201d\u539f\u5219\u662f\u5f02\u66f2\u540c\u5de5\u7684\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u201c\u8c03\u7528\u6808\u201d\u548c\u201c\u6808\u5e27\u7a7a\u95f4\u201d\u8fd9\u7c7b\u9012\u5f52\u672f\u8bed\u5df2\u7ecf\u6697\u793a\u4e86\u9012\u5f52\u4e0e\u6808\u4e4b\u95f4\u7684\u5bc6\u5207\u5173\u7cfb\u3002</p> <ol> <li>\u9012\uff1a\u5f53\u51fd\u6570\u88ab\u8c03\u7528\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5728\u201c\u8c03\u7528\u6808\u201d\u4e0a\u4e3a\u8be5\u51fd\u6570\u5206\u914d\u65b0\u7684\u6808\u5e27\uff0c\u7528\u4e8e\u5b58\u50a8\u51fd\u6570\u7684\u5c40\u90e8\u53d8\u91cf\u3001\u53c2\u6570\u3001\u8fd4\u56de\u5730\u5740\u7b49\u6570\u636e\u3002</li> <li>\u5f52\uff1a\u5f53\u51fd\u6570\u5b8c\u6210\u6267\u884c\u5e76\u8fd4\u56de\u65f6\uff0c\u5bf9\u5e94\u7684\u6808\u5e27\u4f1a\u4ece\u201c\u8c03\u7528\u6808\u201d\u4e0a\u88ab\u79fb\u9664\uff0c\u6062\u590d\u4e4b\u524d\u51fd\u6570\u7684\u6267\u884c\u73af\u5883\u3002</li> </ol> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u8c03\u7528\u6808\u7684\u884c\u4e3a\uff0c\u4ece\u800c\u5c06\u9012\u5f52\u8f6c\u5316\u4e3a\u8fed\u4ee3\u5f62\u5f0f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig recursion.py<pre><code>def for_loop_recur(n: int) -&gt; int:\n    \"\"\"\u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\"\"\"\n    # \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack = []\n    res = 0\n    # \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in range(n, 0, -1):\n        # \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    # \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while stack:\n        # \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop()\n    # res = 1+2+3+...+n\n    return res\n</code></pre> recursion.cpp<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack&lt;int&gt; stack;\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.empty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.top();\n        stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.java<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (!stack.isEmpty()) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.cs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint ForLoopRecur(int n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    Stack&lt;int&gt; stack = new();\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.Push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.Count &gt; 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.go<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n int) int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    stack := list.New()\n    res := 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i := n; i &gt; 0; i-- {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.PushBack(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    for stack.Len() != 0 {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.Back().Value.(int)\n        stack.Remove(stack.Back())\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.swift<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunc forLoopRecur(n: Int) -&gt; Int {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [Int] = []\n    var res = 0\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in stride(from: n, to: 0, by: -1) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.append(i)\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.isEmpty {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.removeLast()\n    }\n    // res = 1+2+3+...+n\n    return res\n}\n</code></pre> recursion.js<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n) {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    const stack = [];\n    let res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = 1; i &lt;= n; i++) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) { \n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.ts<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfunction forLoopRecur(n: number): number {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808 \n    const stack: number[] = [];\n    let res: number = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (let i = 1; i &lt;= n; i++) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (stack.length) { \n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop();\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.dart<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n  // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n  List&lt;int&gt; stack = [];\n  int res = 0;\n  // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n  for (int i = n; i &gt; 0; i--) {\n    // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n    stack.add(i);\n  }\n  // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n  while (!stack.isEmpty) {\n    // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n    res += stack.removeLast();\n  }\n  // res = 1+2+3+...+n\n  return res;\n}\n</code></pre> recursion.rs<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nfn for_loop_recur(n: i32) -&gt; i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    let mut stack = Vec::new();\n    let mut res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for i in (1..=n).rev() {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack.push(i);\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while !stack.is_empty() {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack.pop().unwrap();\n    }\n    // res = 1+2+3+...+n\n    res\n}\n</code></pre> recursion.c<pre><code>/* \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52 */\nint forLoopRecur(int n) {\n    int stack[1000]; // \u501f\u52a9\u4e00\u4e2a\u5927\u6570\u7ec4\u6765\u6a21\u62df\u6808\n    int top = -1;    // \u6808\u9876\u7d22\u5f15\n    int res = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    for (int i = n; i &gt; 0; i--) {\n        // \u901a\u8fc7\u201c\u5165\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u9012\u201d\n        stack[1 + top++] = i;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    while (top &gt;= 0) {\n        // \u901a\u8fc7\u201c\u51fa\u6808\u64cd\u4f5c\u201d\u6a21\u62df\u201c\u5f52\u201d\n        res += stack[top--];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> recursion.zig<pre><code>// \u4f7f\u7528\u8fed\u4ee3\u6a21\u62df\u9012\u5f52\nfn forLoopRecur(comptime n: i32) i32 {\n    // \u4f7f\u7528\u4e00\u4e2a\u663e\u5f0f\u7684\u6808\u6765\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\n    var stack: [n]i32 = undefined;\n    var res: i32 = 0;\n    // \u9012\uff1a\u9012\u5f52\u8c03\u7528\n    var i: usize = n;\n    while (i &gt; 0) {\n        stack[i - 1] = @intCast(i);\n        i -= 1;\n    }\n    // \u5f52\uff1a\u8fd4\u56de\u7ed3\u679c\n    var index: usize = n;\n    while (index &gt; 0) {\n        index -= 1;\n        res += stack[index];\n    }\n    // res = 1+2+3+...+n\n    return res;\n}\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u5f53\u9012\u5f52\u88ab\u8f6c\u6362\u4e3a\u8fed\u4ee3\u540e\uff0c\u4ee3\u7801\u53d8\u5f97\u66f4\u52a0\u590d\u6742\u4e86\u3002\u5c3d\u7ba1\u8fed\u4ee3\u548c\u9012\u5f52\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\uff0c\u4f46\u4e5f\u4e0d\u4e00\u5b9a\u503c\u5f97\u8fd9\u6837\u505a\uff0c\u6709\u4ee5\u4e0b\u4e24\u70b9\u539f\u56e0\u3002</p> <ul> <li>\u8f6c\u5316\u540e\u7684\u4ee3\u7801\u53ef\u80fd\u66f4\u52a0\u96be\u4ee5\u7406\u89e3\uff0c\u53ef\u8bfb\u6027\u66f4\u5dee\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u6a21\u62df\u7cfb\u7edf\u8c03\u7528\u6808\u7684\u884c\u4e3a\u53ef\u80fd\u975e\u5e38\u56f0\u96be\u3002</li> </ul> <p>\u603b\u4e4b\uff0c\u9009\u62e9\u8fed\u4ee3\u8fd8\u662f\u9012\u5f52\u53d6\u51b3\u4e8e\u7279\u5b9a\u95ee\u9898\u7684\u6027\u8d28\u3002\u5728\u7f16\u7a0b\u5b9e\u8df5\u4e2d\uff0c\u6743\u8861\u4e24\u8005\u7684\u4f18\u52a3\u5e76\u6839\u636e\u60c5\u5883\u9009\u62e9\u5408\u9002\u7684\u65b9\u6cd5\u662f\u81f3\u5173\u91cd\u8981\u7684\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/","title":"2.1 \u00a0 \u7b97\u6cd5\u6548\u7387\u8bc4\u4f30","text":"<p>\u5728\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u8ffd\u6c42\u4ee5\u4e0b\u4e24\u4e2a\u5c42\u9762\u7684\u76ee\u6807\u3002</p> <ol> <li>\u627e\u5230\u95ee\u9898\u89e3\u6cd5\uff1a\u7b97\u6cd5\u9700\u8981\u5728\u89c4\u5b9a\u7684\u8f93\u5165\u8303\u56f4\u5185\u53ef\u9760\u5730\u6c42\u5f97\u95ee\u9898\u7684\u6b63\u786e\u89e3\u3002</li> <li>\u5bfb\u6c42\u6700\u4f18\u89e3\u6cd5\uff1a\u540c\u4e00\u4e2a\u95ee\u9898\u53ef\u80fd\u5b58\u5728\u591a\u79cd\u89e3\u6cd5\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u5c3d\u53ef\u80fd\u9ad8\u6548\u7684\u7b97\u6cd5\u3002</li> </ol> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5728\u80fd\u591f\u89e3\u51b3\u95ee\u9898\u7684\u524d\u63d0\u4e0b\uff0c\u7b97\u6cd5\u6548\u7387\u5df2\u6210\u4e3a\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\uff0c\u5b83\u5305\u62ec\u4ee5\u4e0b\u4e24\u4e2a\u7ef4\u5ea6\u3002</p> <ul> <li>\u65f6\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u5feb\u6162\u3002</li> <li>\u7a7a\u95f4\u6548\u7387\uff1a\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u7684\u5927\u5c0f\u3002</li> </ul> <p>\u7b80\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bbe\u8ba1\u201c\u65e2\u5feb\u53c8\u7701\u201d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u3002\u800c\u6709\u6548\u5730\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u53ea\u6709\u8fd9\u6837\u6211\u4eec\u624d\u80fd\u5c06\u5404\u79cd\u7b97\u6cd5\u8fdb\u884c\u5bf9\u6bd4\uff0c\u8fdb\u800c\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u4f18\u5316\u8fc7\u7a0b\u3002</p> <p>\u6548\u7387\u8bc4\u4f30\u65b9\u6cd5\u4e3b\u8981\u5206\u4e3a\u4e24\u79cd\uff1a\u5b9e\u9645\u6d4b\u8bd5\u3001\u7406\u8bba\u4f30\u7b97\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#211","title":"2.1.1 \u00a0 \u5b9e\u9645\u6d4b\u8bd5","text":"<p>\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u7b97\u6cd5 <code>A</code> \u548c\u7b97\u6cd5 <code>B</code> \uff0c\u5b83\u4eec\u90fd\u80fd\u89e3\u51b3\u540c\u4e00\u95ee\u9898\uff0c\u73b0\u5728\u9700\u8981\u5bf9\u6bd4\u8fd9\u4e24\u4e2a\u7b97\u6cd5\u7684\u6548\u7387\u3002\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u627e\u4e00\u53f0\u8ba1\u7b97\u673a\uff0c\u8fd0\u884c\u8fd9\u4e24\u4e2a\u7b97\u6cd5\uff0c\u5e76\u76d1\u63a7\u8bb0\u5f55\u5b83\u4eec\u7684\u8fd0\u884c\u65f6\u95f4\u548c\u5185\u5b58\u5360\u7528\u60c5\u51b5\u3002\u8fd9\u79cd\u8bc4\u4f30\u65b9\u5f0f\u80fd\u591f\u53cd\u6620\u771f\u5b9e\u60c5\u51b5\uff0c\u4f46\u4e5f\u5b58\u5728\u8f83\u5927\u7684\u5c40\u9650\u6027\u3002</p> <p>\u4e00\u65b9\u9762\uff0c\u96be\u4ee5\u6392\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5e72\u6270\u56e0\u7d20\u3002\u786c\u4ef6\u914d\u7f6e\u4f1a\u5f71\u54cd\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u6bd4\u5982\u5728\u67d0\u53f0\u8ba1\u7b97\u673a\u4e2d\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u4f46\u5728\u53e6\u4e00\u53f0\u914d\u7f6e\u4e0d\u540c\u7684\u8ba1\u7b97\u673a\u4e2d\uff0c\u53ef\u80fd\u5f97\u5230\u76f8\u53cd\u7684\u6d4b\u8bd5\u7ed3\u679c\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u9700\u8981\u5728\u5404\u79cd\u673a\u5668\u4e0a\u8fdb\u884c\u6d4b\u8bd5\uff0c\u7edf\u8ba1\u5e73\u5747\u6548\u7387\uff0c\u800c\u8fd9\u662f\u4e0d\u73b0\u5b9e\u7684\u3002</p> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5c55\u5f00\u5b8c\u6574\u6d4b\u8bd5\u975e\u5e38\u8017\u8d39\u8d44\u6e90\u3002\u968f\u7740\u8f93\u5165\u6570\u636e\u91cf\u7684\u53d8\u5316\uff0c\u7b97\u6cd5\u4f1a\u8868\u73b0\u51fa\u4e0d\u540c\u7684\u6548\u7387\u3002\u4f8b\u5982\uff0c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>A</code> \u7684\u8fd0\u884c\u65f6\u95f4\u6bd4\u7b97\u6cd5 <code>B</code> \u77ed\uff1b\u800c\u5728\u8f93\u5165\u6570\u636e\u91cf\u8f83\u5927\u65f6\uff0c\u6d4b\u8bd5\u7ed3\u679c\u53ef\u80fd\u6070\u6070\u76f8\u53cd\u3002\u56e0\u6b64\uff0c\u4e3a\u4e86\u5f97\u5230\u6709\u8bf4\u670d\u529b\u7684\u7ed3\u8bba\uff0c\u6211\u4eec\u9700\u8981\u6d4b\u8bd5\u5404\u79cd\u89c4\u6a21\u7684\u8f93\u5165\u6570\u636e\uff0c\u800c\u8fd9\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002</p>"},{"location":"chapter_computational_complexity/performance_evaluation/#212","title":"2.1.2 \u00a0 \u7406\u8bba\u4f30\u7b97","text":"<p>\u7531\u4e8e\u5b9e\u9645\u6d4b\u8bd5\u5177\u6709\u8f83\u5927\u7684\u5c40\u9650\u6027\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u4ec5\u901a\u8fc7\u4e00\u4e9b\u8ba1\u7b97\u6765\u8bc4\u4f30\u7b97\u6cd5\u7684\u6548\u7387\u3002\u8fd9\u79cd\u4f30\u7b97\u65b9\u6cd5\u88ab\u79f0\u4e3a\u300c\u6e10\u8fd1\u590d\u6742\u5ea6\u5206\u6790 asymptotic complexity analysis\u300d\uff0c\u7b80\u79f0\u300c\u590d\u6742\u5ea6\u5206\u6790\u300d\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u80fd\u591f\u4f53\u73b0\u7b97\u6cd5\u8fd0\u884c\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u5b83\u63cf\u8ff0\u4e86\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\uff0c\u7b97\u6cd5\u6267\u884c\u6240\u9700\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u5b9a\u4e49\u6709\u4e9b\u62d7\u53e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5206\u4e3a\u4e09\u4e2a\u91cd\u70b9\u6765\u7406\u89e3\u3002</p> <ul> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\u201d\u5206\u522b\u5bf9\u5e94\u300c\u65f6\u95f4\u590d\u6742\u5ea6 time complexity\u300d\u548c\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u3002</li> <li>\u201c\u968f\u7740\u8f93\u5165\u6570\u636e\u5927\u5c0f\u7684\u589e\u52a0\u201d\u610f\u5473\u7740\u590d\u6742\u5ea6\u53cd\u6620\u4e86\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u4e0e\u8f93\u5165\u6570\u636e\u4f53\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002</li> <li>\u201c\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u201d\u8868\u793a\u590d\u6742\u5ea6\u5206\u6790\u5173\u6ce8\u7684\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4\u6216\u5360\u7528\u7a7a\u95f4\u7684\u5177\u4f53\u503c\uff0c\u800c\u662f\u65f6\u95f4\u6216\u7a7a\u95f4\u589e\u957f\u7684\u201c\u5feb\u6162\u201d\u3002</li> </ul> <p>\u590d\u6742\u5ea6\u5206\u6790\u514b\u670d\u4e86\u5b9e\u9645\u6d4b\u8bd5\u65b9\u6cd5\u7684\u5f0a\u7aef\uff0c\u4f53\u73b0\u5728\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u5b83\u72ec\u7acb\u4e8e\u6d4b\u8bd5\u73af\u5883\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\u3002</li> <li>\u5b83\u53ef\u4ee5\u4f53\u73b0\u4e0d\u540c\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6548\u7387\uff0c\u5c24\u5176\u662f\u5728\u5927\u6570\u636e\u91cf\u4e0b\u7684\u7b97\u6cd5\u6027\u80fd\u3002</li> </ul> <p>Tip</p> <p>\u5982\u679c\u4f60\u4ecd\u5bf9\u590d\u6742\u5ea6\u7684\u6982\u5ff5\u611f\u5230\u56f0\u60d1\uff0c\u65e0\u987b\u62c5\u5fc3\uff0c\u6211\u4eec\u4f1a\u5728\u540e\u7eed\u7ae0\u8282\u4e2d\u8be6\u7ec6\u4ecb\u7ecd\u3002</p> <p>\u590d\u6742\u5ea6\u5206\u6790\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u628a\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u7684\u201c\u6807\u5c3a\u201d\uff0c\u4f7f\u6211\u4eec\u53ef\u4ee5\u8861\u91cf\u6267\u884c\u67d0\u4e2a\u7b97\u6cd5\u6240\u9700\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u8d44\u6e90\uff0c\u5bf9\u6bd4\u4e0d\u540c\u7b97\u6cd5\u4e4b\u95f4\u7684\u6548\u7387\u3002</p> <p>\u590d\u6742\u5ea6\u662f\u4e2a\u6570\u5b66\u6982\u5ff5\uff0c\u5bf9\u4e8e\u521d\u5b66\u8005\u53ef\u80fd\u6bd4\u8f83\u62bd\u8c61\uff0c\u5b66\u4e60\u96be\u5ea6\u76f8\u5bf9\u8f83\u9ad8\u3002\u4ece\u8fd9\u4e2a\u89d2\u5ea6\u770b\uff0c\u590d\u6742\u5ea6\u5206\u6790\u53ef\u80fd\u4e0d\u592a\u9002\u5408\u4f5c\u4e3a\u6700\u5148\u4ecb\u7ecd\u7684\u5185\u5bb9\u3002\u7136\u800c\uff0c\u5f53\u6211\u4eec\u8ba8\u8bba\u67d0\u4e2a\u6570\u636e\u7ed3\u6784\u6216\u7b97\u6cd5\u7684\u7279\u70b9\u65f6\uff0c\u96be\u4ee5\u907f\u514d\u8981\u5206\u6790\u5176\u8fd0\u884c\u901f\u5ea6\u548c\u7a7a\u95f4\u4f7f\u7528\u60c5\u51b5\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5efa\u8bae\u4f60\u5728\u6df1\u5165\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u524d\uff0c\u5148\u5bf9\u590d\u6742\u5ea6\u5206\u6790\u5efa\u7acb\u521d\u6b65\u7684\u4e86\u89e3\uff0c\u4ee5\u4fbf\u80fd\u591f\u5b8c\u6210\u7b80\u5355\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/","title":"2.4 \u00a0 \u7a7a\u95f4\u590d\u6742\u5ea6","text":"<p>\u300c\u7a7a\u95f4\u590d\u6742\u5ea6 space complexity\u300d\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002\u8fd9\u4e2a\u6982\u5ff5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u5e38\u7c7b\u4f3c\uff0c\u53ea\u9700\u5c06\u201c\u8fd0\u884c\u65f6\u95f4\u201d\u66ff\u6362\u4e3a\u201c\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u201d\u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#241","title":"2.4.1 \u00a0 \u7b97\u6cd5\u76f8\u5173\u7a7a\u95f4","text":"<p>\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f7f\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u8f93\u5165\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u5165\u6570\u636e\u3002</li> <li>\u6682\u5b58\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u53d8\u91cf\u3001\u5bf9\u8c61\u3001\u51fd\u6570\u4e0a\u4e0b\u6587\u7b49\u6570\u636e\u3002</li> <li>\u8f93\u51fa\u7a7a\u95f4\uff1a\u7528\u4e8e\u5b58\u50a8\u7b97\u6cd5\u7684\u8f93\u51fa\u6570\u636e\u3002</li> </ul> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u7edf\u8ba1\u8303\u56f4\u662f\u201c\u6682\u5b58\u7a7a\u95f4\u201d\u52a0\u4e0a\u201c\u8f93\u51fa\u7a7a\u95f4\u201d\u3002</p> <p>\u6682\u5b58\u7a7a\u95f4\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u6682\u5b58\u6570\u636e\uff1a\u7528\u4e8e\u4fdd\u5b58\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u5404\u79cd\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u7b49\u3002</li> <li>\u6808\u5e27\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u8c03\u7528\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u6570\u636e\u3002\u7cfb\u7edf\u5728\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\u90fd\u4f1a\u5728\u6808\u9876\u90e8\u521b\u5efa\u4e00\u4e2a\u6808\u5e27\uff0c\u51fd\u6570\u8fd4\u56de\u540e\uff0c\u6808\u5e27\u7a7a\u95f4\u4f1a\u88ab\u91ca\u653e\u3002</li> <li>\u6307\u4ee4\u7a7a\u95f4\uff1a\u7528\u4e8e\u4fdd\u5b58\u7f16\u8bd1\u540e\u7684\u7a0b\u5e8f\u6307\u4ee4\uff0c\u5728\u5b9e\u9645\u7edf\u8ba1\u4e2d\u901a\u5e38\u5ffd\u7565\u4e0d\u8ba1\u3002</li> </ul> <p>\u5728\u5206\u6790\u4e00\u6bb5\u7a0b\u5e8f\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u7edf\u8ba1\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u8f93\u51fa\u6570\u636e\u4e09\u90e8\u5206\uff0c\u5982\u56fe 2-15 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 2-15 \u00a0 \u7b97\u6cd5\u4f7f\u7528\u7684\u76f8\u5173\u7a7a\u95f4 </p> <p>\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class Node:\n    \"\"\"\u7c7b\"\"\"\n    def __init__(self, x: int):\n        self.val: int = x              # \u8282\u70b9\u503c\n        self.next: Node | None = None  # \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n\ndef function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n\ndef algorithm(n) -&gt; int:  # \u8f93\u5165\u6570\u636e\n    A = 0                 # \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff0c\u4e00\u822c\u7528\u5927\u5199\u5b57\u6bcd\u8868\u793a\uff09\n    b = 0                 # \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    node = Node(0)        # \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    c = function()        # \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return A + b + c      # \u8f93\u51fa\u6570\u636e\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\nstruct Node {\n    int val;\n    Node *next;\n    Node(int x) : val(x), next(nullptr) {}\n};\n\n/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    const int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node* node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = func();           // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    int val;\n    Node next;\n    Node(int x) { val = x; }\n}\n\n/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    final int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node node = new Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node(int x) {\n    int val = x;\n    Node next;\n}\n\n/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint Algorithm(int n) {        // \u8f93\u5165\u6570\u636e\n    const int a = 0;          // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    Node node = new(0);       // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    int c = Function();       // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7ed3\u6784\u4f53 */\ntype node struct {\n    val  int\n    next *node\n}\n\n/* \u521b\u5efa node \u7ed3\u6784\u4f53  */\nfunc newNode(val int) *node {\n    return &amp;node{val: val}\n}\n\n/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n int) int { // \u8f93\u5165\u6570\u636e\n    const a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    b := 0                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    newNode(0)              // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    c := function()         // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c        // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    var val: Int\n    var next: Node?\n\n    init(x: Int) {\n        val = x\n    }\n}\n\n/* \u51fd\u6570 */\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\nfunc algorithm(n: Int) -&gt; Int { // \u8f93\u5165\u6570\u636e\n    let a = 0             // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    var b = 0             // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    let node = Node(x: 0) // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    let c = function()    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c      // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    val;\n    next;\n    constructor(val) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n) {       // \u8f93\u5165\u6570\u636e\n    const a = 0;              // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let b = 0;                // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    const node = new Node(0); // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    const c = constFunc();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;         // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n    val: number;\n    next: Node | null;\n    constructor(val?: number) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.next = null;                       // \u6307\u5411\u4e0b\u4e00\u8282\u70b9\u7684\u5f15\u7528\n    }\n}\n\n/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\nfunction algorithm(n: number): number { // \u8f93\u5165\u6570\u636e\n    const a = 0;                        // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let b = 0;                          // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    const node = new Node(0);           // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    const c = constFunc();              // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;                   // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u7c7b */\nclass Node {\n  int val;\n  Node next;\n  Node(this.val, [this.next]);\n}\n\n/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n  return 0;\n}\n\nint algorithm(int n) {  // \u8f93\u5165\u6570\u636e\n  const int a = 0;      // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n  int b = 0;            // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n  Node node = Node(0);  // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n  int c = function();   // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n  return a + b + c;     // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u7ed3\u6784\u4f53 */\nstruct Node {\n    val: i32,\n    next: Option&lt;Rc&lt;RefCell&lt;Node&gt;&gt;&gt;,\n}\n\n/* \u521b\u5efa Node \u7ed3\u6784\u4f53 */\nimpl Node {\n    fn new(val: i32) -&gt; Self {\n        Self { val: val, next: None }\n    }\n}\n\n/* \u51fd\u6570 */\nfn function() -&gt; i32 {      \n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nfn algorithm(n: i32) -&gt; i32 {       // \u8f93\u5165\u6570\u636e\n    const a: i32 = 0;               // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    let mut b = 0;                  // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    let node = Node::new(0);        // \u6682\u5b58\u6570\u636e\uff08\u5bf9\u8c61\uff09\n    let c = function();             // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;               // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0;\n}\n\nint algorithm(int n) { // \u8f93\u5165\u6570\u636e\n    const int a = 0;   // \u6682\u5b58\u6570\u636e\uff08\u5e38\u91cf\uff09\n    int b = 0;         // \u6682\u5b58\u6570\u636e\uff08\u53d8\u91cf\uff09\n    int c = func();    // \u6808\u5e27\u7a7a\u95f4\uff08\u8c03\u7528\u51fd\u6570\uff09\n    return a + b + c;  // \u8f93\u51fa\u6570\u636e\n}\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#242","title":"2.4.2 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u5927\u81f4\u76f8\u540c\uff0c\u53ea\u9700\u5c06\u7edf\u8ba1\u5bf9\u8c61\u4ece\u201c\u64cd\u4f5c\u6570\u91cf\u201d\u8f6c\u4e3a\u201c\u4f7f\u7528\u7a7a\u95f4\u5927\u5c0f\u201d\u3002</p> <p>\u800c\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u7684\u662f\uff0c\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u3002\u8fd9\u662f\u56e0\u4e3a\u5185\u5b58\u7a7a\u95f4\u662f\u4e00\u9879\u786c\u6027\u8981\u6c42\uff0c\u6211\u4eec\u5fc5\u987b\u786e\u4fdd\u5728\u6240\u6709\u8f93\u5165\u6570\u636e\u4e0b\u90fd\u6709\u8db3\u591f\u7684\u5185\u5b58\u7a7a\u95f4\u9884\u7559\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u201c\u6700\u5dee\u201d\u6709\u4e24\u5c42\u542b\u4e49\u3002</p> <ol> <li>\u4ee5\u6700\u5dee\u8f93\u5165\u6570\u636e\u4e3a\u51c6\uff1a\u5f53 \\(n &lt; 10\\) \u65f6\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff1b\u4f46\u5f53 \\(n &gt; 10\\) \u65f6\uff0c\u521d\u59cb\u5316\u7684\u6570\u7ec4 <code>nums</code> \u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u4ee5\u7b97\u6cd5\u8fd0\u884c\u4e2d\u7684\u5cf0\u503c\u5185\u5b58\u4e3a\u51c6\uff1a\u4f8b\u5982\uff0c\u7a0b\u5e8f\u5728\u6267\u884c\u6700\u540e\u4e00\u884c\u4e4b\u524d\uff0c\u5360\u7528 \\(O(1)\\) \u7a7a\u95f4\uff1b\u5f53\u521d\u59cb\u5316\u6570\u7ec4 <code>nums</code> \u65f6\uff0c\u7a0b\u5e8f\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\uff0c\u56e0\u6b64\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> </ol> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def algorithm(n: int):\n    a = 0               # O(1)\n    b = [0] * 10000     # O(1)\n    if n &gt; 10:\n        nums = [0] * n  # O(n)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    vector&lt;int&gt; b(10000);    // O(1)\n    if (n &gt; 10)\n        vector&lt;int&gt; nums(n); // O(n)\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10)\n        int[] nums = new int[n]; // O(n)\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 0;                   // O(1)\n    int[] b = new int[10000];    // O(1)\n    if (n &gt; 10) {\n        int[] nums = new int[n]; // O(n)\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 0                      // O(1)\n    b := make([]int, 10000)     // O(1)\n    var nums []int\n    if n &gt; 10 {\n        nums := make([]int, n)  // O(n)\n    }\n    fmt.Println(a, b, nums)\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    let a = 0 // O(1)\n    let b = Array(repeating: 0, count: 10000) // O(1)\n    if n &gt; 10 {\n        let nums = Array(repeating: 0, count: n) // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    const a = 0;                   // O(1)\n    const b = new Array(10000);    // O(1)\n    if (n &gt; 10) {\n        const nums = new Array(n); // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 0;                            // O(1)\n  List&lt;int&gt; b = List.filled(10000, 0);  // O(1)\n  if (n &gt; 10) {\n    List&lt;int&gt; nums = List.filled(n, 0); // O(n)\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let a = 0;                              // O(1)\n    let b = [0; 10000];                     // O(1)\n    if n &gt; 10 {\n        let nums = vec![0; n as usize];     // O(n)\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 0;               // O(1)\n    int b[10000];            // O(1)\n    if (n &gt; 10)\n        int nums[n] = {0};   // O(n)\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u7edf\u8ba1\u6808\u5e27\u7a7a\u95f4\u3002\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def function() -&gt; int:\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef loop(n: int):\n    \"\"\"\u5faa\u73af O(1)\"\"\"\n    for _ in range(n):\n        function()\n\ndef recur(n: int) -&gt; int:\n    \"\"\"\u9012\u5f52 O(n)\"\"\"\n    if n == 1: return\n    return recur(n - 1)\n</code></pre> <pre><code>int func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid Loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nint Recur(int n) {\n    if (n == 1) return 1;\n    return Recur(n - 1);\n}\n</code></pre> <pre><code>func function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n int) {\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n int) {\n    if n == 1 {\n        return\n    }\n    recur(n - 1)\n}\n</code></pre> <pre><code>@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5faa\u73af O(1) */\nfunc loop(n: Int) {\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n\n/* \u9012\u5f52 O(n) */\nfunc recur(n: Int) {\n    if n == 1 {\n        return\n    }\n    recur(n: n - 1)\n}\n</code></pre> <pre><code>function constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n) {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n) {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>function constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfunction loop(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n/* \u9012\u5f52 O(n) */\nfunction recur(n: number): void {\n    if (n === 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>int function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n  if (n == 1) return;\n  return recur(n - 1);\n}\n</code></pre> <pre><code>fn function() -&gt; i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nfn loop(n: i32) {\n    for i in 0..n {\n        function();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(n: i32) {\n    if n == 1 {\n        return;\n    }\n    recur(n - 1);\n}\n</code></pre> <pre><code>int func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n/* \u5faa\u73af O(1) */\nvoid loop(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n/* \u9012\u5f52 O(n) */\nvoid recur(int n) {\n    if (n == 1) return;\n    return recur(n - 1);\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u51fd\u6570 <code>loop()</code> \u548c <code>recur()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u4f46\u7a7a\u95f4\u590d\u6742\u5ea6\u4e0d\u540c\u3002</p> <ul> <li>\u51fd\u6570 <code>loop()</code> \u5728\u5faa\u73af\u4e2d\u8c03\u7528\u4e86 \\(n\\) \u6b21 <code>function()</code> \uff0c\u6bcf\u8f6e\u4e2d\u7684 <code>function()</code> \u90fd\u8fd4\u56de\u5e76\u91ca\u653e\u4e86\u6808\u5e27\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \u3002</li> <li>\u9012\u5f52\u51fd\u6570 <code>recur()</code> \u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4f1a\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>recur()</code> \uff0c\u4ece\u800c\u5360\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_computational_complexity/space_complexity/#243","title":"2.4.3 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u56fe 2-16 \u5c55\u793a\u4e86\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\uff08\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-16 \u00a0 \u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/space_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u5e38\u89c1\u4e8e\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u7684\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5faa\u73af\u4e2d\u521d\u59cb\u5316\u53d8\u91cf\u6216\u8c03\u7528\u51fd\u6570\u800c\u5360\u7528\u7684\u5185\u5b58\uff0c\u5728\u8fdb\u5165\u4e0b\u4e00\u5faa\u73af\u540e\u5c31\u4f1a\u88ab\u91ca\u653e\uff0c\u56e0\u6b64\u4e0d\u4f1a\u7d2f\u79ef\u5360\u7528\u7a7a\u95f4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def function() -&gt; int:\n    \"\"\"\u51fd\u6570\"\"\"\n    # \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n\ndef constant(n: int):\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    # \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    a = 0\n    nums = [0] * 10000\n    node = ListNode(0)\n    # \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        c = 0\n    # \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in range(n):\n        function()\n</code></pre> space_complexity.cpp<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    vector&lt;int&gt; nums(10000);\n    ListNode node(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u51fd\u6570 */\nint function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    final int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        function();\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u51fd\u6570 */\nint Function() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid Constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    int a = 0;\n    int b = 0;\n    int[] nums = new int[10000];\n    ListNode node = new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        Function();\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u51fd\u6570 */\nfunc function() int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c...\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc spaceConstant(n int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0\n    b := 0\n    nums := make([]int, 10000)\n    ListNode := newNode(0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    var c int\n    for i := 0; i &lt; n; i++ {\n        c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i := 0; i &lt; n; i++ {\n        function()\n    }\n    fmt.Println(a, b, nums, c, ListNode)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u51fd\u6570 */\n@discardableResult\nfunc function() -&gt; Int {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0\n}\n\n/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    let a = 0\n    var b = 0\n    let nums = Array(repeating: 0, count: 10000)\n    let node = ListNode(x: 0)\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        let c = 0\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for _ in 0 ..&lt; n {\n        function()\n    }\n}\n</code></pre> space_complexity.js<pre><code>/* \u51fd\u6570 */\nfunction constFunc() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u51fd\u6570 */\nfunction constFunc(): number {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nfunction constant(n: number): void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a = 0;\n    const b = 0;\n    const nums = new Array(10000);\n    const node = new ListNode(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        const c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (let i = 0; i &lt; n; i++) {\n        constFunc();\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u51fd\u6570 */\nint function() {\n  // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n  return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n  // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n  final int a = 0;\n  int b = 0;\n  List&lt;int&gt; nums = List.filled(10000, 0);\n  ListNode node = ListNode(0);\n  // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    int c = 0;\n  }\n  // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n  for (var i = 0; i &lt; n; i++) {\n    function();\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u51fd\u6570 */\nfn function() -&gt;i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\n#[allow(unused)]\nfn constant(n: i32) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const A: i32 = 0;\n    let b = 0;\n    let nums = vec![0; 10000];\n    let node = ListNode::new(0);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        let c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for i in 0..n {\n        function();\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u51fd\u6570 */\nint func() {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n/* \u5e38\u6570\u9636 */\nvoid constant(int n) {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const int a = 0;\n    int b = 0;\n    int nums[1000];\n    ListNode *node = newListNode(0);\n    free(node);\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        int c = 0;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    for (int i = 0; i &lt; n; i++) {\n        func();\n    }\n}\n</code></pre> space_complexity.zig<pre><code>// \u51fd\u6570\nfn function() i32 {\n    // \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\n    return 0;\n}\n\n// \u5e38\u6570\u9636\nfn constant(n: i32) void {\n    // \u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5bf9\u8c61\u5360\u7528 O(1) \u7a7a\u95f4\n    const a: i32 = 0;\n    var b: i32 = 0;\n    var nums = [_]i32{0}**10000;\n    var node = inc.ListNode(i32){.val = 0};\n    var i: i32 = 0;\n    // \u5faa\u73af\u4e2d\u7684\u53d8\u91cf\u5360\u7528 O(1) \u7a7a\u95f4\n    while (i &lt; n) : (i += 1) {\n        var c: i32 = 0;\n        _ = c;\n    }\n    // \u5faa\u73af\u4e2d\u7684\u51fd\u6570\u5360\u7528 O(1) \u7a7a\u95f4\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        _ = function();\n    }\n    _ = a;\n    _ = b;\n    _ = nums;\n    _ = node;\n}\n</code></pre>"},{"location":"chapter_computational_complexity/space_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u5e38\u89c1\u4e8e\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u6b63\u6bd4\u7684\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u7b49\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def linear(n: int):\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    # \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    nums = [0] * n\n    # \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    hmap = dict[int, str]()\n    for i in range(n):\n        hmap[i] = str(i)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;int&gt; nums(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    vector&lt;ListNode&gt; nodes;\n    for (int i = 0; i &lt; n; i++) {\n        nodes.push_back(ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    unordered_map&lt;int, string&gt; map;\n    for (int i = 0; i &lt; n; i++) {\n        map[i] = to_string(i);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        nodes.add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Map&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        map.put(i, String.valueOf(i));\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nvoid Linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int[] nums = new int[n];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    List&lt;ListNode&gt; nodes = [];\n    for (int i = 0; i &lt; n; i++) {\n        nodes.Add(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    Dictionary&lt;int, string&gt; map = [];\n    for (int i = 0; i &lt; n; i++) {\n        map.Add(i, i.ToString());\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc spaceLinear(n int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    _ = make([]int, n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes []*node\n    for i := 0; i &lt; n; i++ {\n        nodes = append(nodes, newNode(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    m := make(map[int]string, n)\n    for i := 0; i &lt; n; i++ {\n        m[i] = strconv.Itoa(i)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let nums = Array(repeating: 0, count: n)\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let nodes = (0 ..&lt; n).map { ListNode(x: $0) }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let map = Dictionary(uniqueKeysWithValues: (0 ..&lt; n).map { ($0, \"\\($0)\") })\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    const nums = new Array(n);\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const nodes: ListNode[] = [];\n    for (let i = 0; i &lt; n; i++) {\n        nodes.push(new ListNode(i));\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    const map = new Map();\n    for (let i = 0; i &lt; n; i++) {\n        map.set(i, i.toString());\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n  // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;int&gt; nums = List.filled(n, 0);\n  // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  List&lt;ListNode&gt; nodes = [];\n  for (var i = 0; i &lt; n; i++) {\n    nodes.add(ListNode(i));\n  }\n  // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n  Map&lt;int, String&gt; map = HashMap();\n  for (var i = 0; i &lt; n; i++) {\n    map.putIfAbsent(i, () =&gt; i.toString());\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\n#[allow(unused)]\nfn linear(n: i32) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nums = vec![0; n as usize];\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut nodes = Vec::new();\n    for i in 0..n {\n        nodes.push(ListNode::new(i))\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    let mut map = HashMap::new();\n    for i in 0..n {\n        map.insert(i, i.to_string());\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u7ebf\u6027\u9636 */\nvoid linear(int n) {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    int *nums = malloc(sizeof(int) * n);\n    free(nums);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    ListNode **nodes = malloc(sizeof(ListNode *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        nodes[i] = newListNode(i);\n    }\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(nodes[i]);\n    }\n    free(nodes);\n\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    HashTable *h = NULL;\n    for (int i = 0; i &lt; n; i++) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = i;\n        tmp-&gt;val = i;\n        HASH_ADD_INT(h, key, tmp);\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    HashTable *curr, *tmp;\n    HASH_ITER(hh, h, curr, tmp) {\n        HASH_DEL(h, curr);\n        free(curr);\n    }\n}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(comptime n: i32) !void {\n    // \u957f\u5ea6\u4e3a n \u7684\u6570\u7ec4\u5360\u7528 O(n) \u7a7a\u95f4\n    var nums = [_]i32{0}**n;\n    // \u957f\u5ea6\u4e3a n \u7684\u5217\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var nodes = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        try nodes.append(i);\n    }\n    // \u957f\u5ea6\u4e3a n \u7684\u54c8\u5e0c\u8868\u5360\u7528 O(n) \u7a7a\u95f4\n    var map = std.AutoArrayHashMap(i32, []const u8).init(std.heap.page_allocator);\n    defer map.deinit();\n    var j: i32 = 0;\n    while (j &lt; n) : (j += 1) {\n        const string = try std.fmt.allocPrint(std.heap.page_allocator, \"{d}\", .{j});\n        defer std.heap.page_allocator.free(string);\n        try map.put(i, string);\n    }\n    _ = nums;\n}\n</code></pre> <p>\u5982\u56fe 2-17 \u6240\u793a\uff0c\u6b64\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5373\u540c\u65f6\u5b58\u5728 \\(n\\) \u4e2a\u672a\u8fd4\u56de\u7684 <code>linear_recur()</code> \u51fd\u6570\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def linear_recur(n: int):\n    \"\"\"\u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    print(\"\u9012\u5f52 n =\", n)\n    if n == 1:\n        return\n    linear_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; endl;\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    System.out.println(\"\u9012\u5f52 n = \" + n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid LinearRecur(int n) {\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n);\n    if (n == 1) return;\n    LinearRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceLinearRecur(n int) {\n    fmt.Println(\"\u9012\u5f52 n =\", n)\n    if n == 1 {\n        return\n    }\n    spaceLinearRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc linearRecur(n: Int) {\n    print(\"\u9012\u5f52 n = \\(n)\")\n    if n == 1 {\n        return\n    }\n    linearRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n) {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction linearRecur(n: number): void {\n    console.log(`\u9012\u5f52 n = ${n}`);\n    if (n === 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n  print('\u9012\u5f52 n = $n');\n  if (n == 1) return;\n  linearRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn linear_recur(n: i32) {\n    println!(\"\u9012\u5f52 n = {}\", n);\n    if n == 1 {return};\n    linear_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nvoid linearRecur(int n) {\n    printf(\"\u9012\u5f52 n = %d\\r\\n\", n);\n    if (n == 1)\n        return;\n    linearRecur(n - 1);\n}\n</code></pre> space_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn linearRecur(comptime n: i32) void {\n    std.debug.print(\"\u9012\u5f52 n = {}\\n\", .{n});\n    if (n == 1) return;\n    linearRecur(n - 1);\n}\n</code></pre> <p></p> <p> \u56fe 2-17 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u7ebf\u6027\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u5e38\u89c1\u4e8e\u77e9\u9635\u548c\u56fe\uff0c\u5143\u7d20\u6570\u91cf\u4e0e \\(n\\) \u6210\u5e73\u65b9\u5173\u7cfb\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def quadratic(n: int):\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    # \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    num_matrix = [[0] * n for _ in range(n)]\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    vector&lt;vector&lt;int&gt;&gt; numMatrix;\n    for (int i = 0; i &lt; n; i++) {\n        vector&lt;int&gt; tmp;\n        for (int j = 0; j &lt; n; j++) {\n            tmp.push_back(0);\n        }\n        numMatrix.push_back(tmp);\n    }\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[][] numMatrix = new int[n][n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;Integer&gt;&gt; numList = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n        for (int j = 0; j &lt; n; j++) {\n            tmp.add(0);\n        }\n        numList.add(tmp);\n    }\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nvoid Quadratic(int n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int[,] numMatrix = new int[n, n];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    List&lt;List&lt;int&gt;&gt; numList = [];\n    for (int i = 0; i &lt; n; i++) {\n        List&lt;int&gt; tmp = [];\n        for (int j = 0; j &lt; n; j++) {\n            tmp.Add(0);\n        }\n        numList.Add(tmp);\n    }\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc spaceQuadratic(n int) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    numMatrix := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        numMatrix[i] = make([]int, n)\n    }\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let numList = Array(repeating: Array(repeating: 0, count: n), count: n)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): void {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numMatrix = Array(n)\n        .fill(null)\n        .map(() =&gt; Array(n).fill(null));\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    const numList = [];\n    for (let i = 0; i &lt; n; i++) {\n        const tmp = [];\n        for (let j = 0; j &lt; n; j++) {\n            tmp.push(0);\n        }\n        numList.push(tmp);\n    }\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n  // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numMatrix = List.generate(n, (_) =&gt; List.filled(n, 0));\n  // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n  List&lt;List&lt;int&gt;&gt; numList = [];\n  for (var i = 0; i &lt; n; i++) {\n    List&lt;int&gt; tmp = [];\n    for (int j = 0; j &lt; n; j++) {\n      tmp.add(0);\n    }\n    numList.add(tmp);\n  }\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\n#[allow(unused)]\nfn quadratic(n: i32) {\n    // \u77e9\u9635\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let num_matrix = vec![vec![0; n as usize]; n as usize];\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    let mut num_list = Vec::new();\n    for i in 0..n {\n        let mut tmp = Vec::new();\n        for j in 0..n {\n            tmp.push(0);\n        }\n        num_list.push(tmp);\n    }\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nvoid quadratic(int n) {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    int **numMatrix = malloc(sizeof(int *) * n);\n    for (int i = 0; i &lt; n; i++) {\n        int *tmp = malloc(sizeof(int) * n);\n        for (int j = 0; j &lt; n; j++) {\n            tmp[j] = 0;\n        }\n        numMatrix[i] = tmp;\n    }\n\n    // \u5185\u5b58\u91ca\u653e\n    for (int i = 0; i &lt; n; i++) {\n        free(numMatrix[i]);\n    }\n    free(numMatrix);\n}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) !void {\n    // \u4e8c\u7ef4\u5217\u8868\u5360\u7528 O(n^2) \u7a7a\u95f4\n    var nodes = std.ArrayList(std.ArrayList(i32)).init(std.heap.page_allocator);\n    defer nodes.deinit();\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        var tmp = std.ArrayList(i32).init(std.heap.page_allocator);\n        defer tmp.deinit();\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            try tmp.append(0);\n        }\n        try nodes.append(tmp);\n    }\n}\n</code></pre> <p>\u5982\u56fe 2-18 \u6240\u793a\uff0c\u8be5\u51fd\u6570\u7684\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u5728\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u90fd\u521d\u59cb\u5316\u4e86\u4e00\u4e2a\u6570\u7ec4\uff0c\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\)\u3001\\(n-1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u5e73\u5747\u957f\u5ea6\u4e3a \\(n / 2\\) \uff0c\u56e0\u6b64\u603b\u4f53\u5360\u7528 \\(O(n^2)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def quadratic_recur(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 0:\n        return 0\n    # \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    nums = [0] * n\n    return quadratic_recur(n - 1)\n</code></pre> space_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    vector&lt;int&gt; nums(n);\n    cout &lt;&lt; \"\u9012\u5f52 n = \" &lt;&lt; n &lt;&lt; \" \u4e2d\u7684 nums \u957f\u5ea6 = \" &lt;&lt; nums.size() &lt;&lt; endl;\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    int[] nums = new int[n];\n    System.out.println(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.length);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint QuadraticRecur(int n) {\n    if (n &lt;= 0) return 0;\n    int[] nums = new int[n];\n    Console.WriteLine(\"\u9012\u5f52 n = \" + n + \" \u4e2d\u7684 nums \u957f\u5ea6 = \" + nums.Length);\n    return QuadraticRecur(n - 1);\n}\n</code></pre> space_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc spaceQuadraticRecur(n int) int {\n    if n &lt;= 0 {\n        return 0\n    }\n    nums := make([]int, n)\n    fmt.Printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d \\n\", n, len(nums))\n    return spaceQuadraticRecur(n - 1)\n}\n</code></pre> space_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\n@discardableResult\nfunc quadraticRecur(n: Int) -&gt; Int {\n    if n &lt;= 0 {\n        return 0\n    }\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = Array(repeating: 0, count: n)\n    print(\"\u9012\u5f52 n = \\(n) \u4e2d\u7684 nums \u957f\u5ea6 = \\(nums.count)\")\n    return quadraticRecur(n: n - 1)\n}\n</code></pre> space_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n) {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction quadraticRecur(n: number): number {\n    if (n &lt;= 0) return 0;\n    const nums = new Array(n);\n    console.log(`\u9012\u5f52 n = ${n} \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}`);\n    return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n  if (n &lt;= 0) return 0;\n  List&lt;int&gt; nums = List.filled(n, 0);\n  print('\u9012\u5f52 n = $n \u4e2d\u7684 nums \u957f\u5ea6 = ${nums.length}');\n  return quadraticRecur(n - 1);\n}\n</code></pre> space_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn quadratic_recur(n: i32) -&gt; i32 {\n    if n &lt;= 0 {return 0};\n    // \u6570\u7ec4 nums \u957f\u5ea6\u4e3a n, n-1, ..., 2, 1\n    let nums = vec![0; n as usize];\n    println!(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\", n, nums.len());\n    return quadratic_recur(n - 1);\n}\n</code></pre> space_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint quadraticRecur(int n) {\n    if (n &lt;= 0)\n        return 0;\n    int *nums = malloc(sizeof(int) * n);\n    printf(\"\u9012\u5f52 n = %d \u4e2d\u7684 nums \u957f\u5ea6 = %d\\r\\n\", n, n);\n    int res = quadraticRecur(n - 1);\n    free(nums);\n    return res;\n}\n</code></pre> space_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn quadraticRecur(comptime n: i32) i32 {\n    if (n &lt;= 0) return 0;\n    var nums = [_]i32{0}**n;\n    std.debug.print(\"\u9012\u5f52 n = {} \u4e2d\u7684 nums \u957f\u5ea6 = {}\\n\", .{n, nums.len});\n    return quadraticRecur(n - 1);\n}\n</code></pre> <p></p> <p> \u56fe 2-18 \u00a0 \u9012\u5f52\u51fd\u6570\u4ea7\u751f\u7684\u5e73\u65b9\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u6307\u6570\u9636\u5e38\u89c1\u4e8e\u4e8c\u53c9\u6811\u3002\u89c2\u5bdf\u56fe 2-19 \uff0c\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u201c\u6ee1\u4e8c\u53c9\u6811\u201d\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(2^n - 1\\) \uff0c\u5360\u7528 \\(O(2^n)\\) \u7a7a\u95f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig space_complexity.py<pre><code>def build_tree(n: int) -&gt; TreeNode | None:\n    \"\"\"\u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\"\"\"\n    if n == 0:\n        return None\n    root = TreeNode(0)\n    root.left = build_tree(n - 1)\n    root.right = build_tree(n - 1)\n    return root\n</code></pre> space_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return nullptr;\n    TreeNode *root = new TreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode buildTree(int n) {\n    if (n == 0)\n        return null;\n    TreeNode root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? BuildTree(int n) {\n    if (n == 0) return null;\n    TreeNode root = new(0) {\n        left = BuildTree(n - 1),\n        right = BuildTree(n - 1)\n    };\n    return root;\n}\n</code></pre> space_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n int) *treeNode {\n    if n == 0 {\n        return nil\n    }\n    root := newTreeNode(0)\n    root.left = buildTree(n - 1)\n    root.right = buildTree(n - 1)\n    return root\n}\n</code></pre> space_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunc buildTree(n: Int) -&gt; TreeNode? {\n    if n == 0 {\n        return nil\n    }\n    let root = TreeNode(x: 0)\n    root.left = buildTree(n: n - 1)\n    root.right = buildTree(n: n - 1)\n    return root\n}\n</code></pre> space_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n) {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfunction buildTree(n: number): TreeNode | null {\n    if (n === 0) return null;\n    const root = new TreeNode(0);\n    root.left = buildTree(n - 1);\n    root.right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode? buildTree(int n) {\n  if (n == 0) return null;\n  TreeNode root = TreeNode(0);\n  root.left = buildTree(n - 1);\n  root.right = buildTree(n - 1);\n  return root;\n}\n</code></pre> space_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nfn build_tree(n: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    if n == 0 {return None};\n    let root = TreeNode::new(0);\n    root.borrow_mut().left = build_tree(n - 1);\n    root.borrow_mut().right = build_tree(n - 1);\n    return Some(root);\n}\n</code></pre> space_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09 */\nTreeNode *buildTree(int n) {\n    if (n == 0)\n        return NULL;\n    TreeNode *root = newTreeNode(0);\n    root-&gt;left = buildTree(n - 1);\n    root-&gt;right = buildTree(n - 1);\n    return root;\n}\n</code></pre> space_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5efa\u7acb\u6ee1\u4e8c\u53c9\u6811\uff09\nfn buildTree(mem_allocator: std.mem.Allocator, n: i32) !?*inc.TreeNode(i32) {\n    if (n == 0) return null;\n    const root = try mem_allocator.create(inc.TreeNode(i32));\n    root.init(0);\n    root.left = try buildTree(mem_allocator, n - 1);\n    root.right = try buildTree(mem_allocator, n - 1);\n    return root;\n}\n</code></pre> <p></p> <p> \u56fe 2-19 \u00a0 \u6ee1\u4e8c\u53c9\u6811\u4ea7\u751f\u7684\u6307\u6570\u9636\u7a7a\u95f4\u590d\u6742\u5ea6 </p>"},{"location":"chapter_computational_complexity/space_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u5bf9\u6570\u9636\u5e38\u89c1\u4e8e\u5206\u6cbb\u7b97\u6cd5\u3002\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\uff0c\u8f93\u5165\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u6bcf\u8f6e\u9012\u5f52\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u534a\uff0c\u5f62\u6210\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u518d\u4f8b\u5982\u5c06\u6570\u5b57\u8f6c\u5316\u4e3a\u5b57\u7b26\u4e32\uff0c\u8f93\u5165\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5b83\u7684\u4f4d\u6570\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u5373\u5bf9\u5e94\u5b57\u7b26\u4e32\u957f\u5ea6\u4e3a \\(\\log_{10} n + 1\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_{10} n + 1) = O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/space_complexity/#244","title":"2.4.4 \u00a0 \u6743\u8861\u65f6\u95f4\u4e0e\u7a7a\u95f4","text":"<p>\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u80fd\u8fbe\u5230\u6700\u4f18\u3002\u7136\u800c\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u540c\u65f6\u4f18\u5316\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u975e\u5e38\u56f0\u96be\u3002</p> <p>\u964d\u4f4e\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u9700\u8981\u4ee5\u63d0\u5347\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a\u4ee3\u4ef7\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u6211\u4eec\u5c06\u727a\u7272\u5185\u5b58\u7a7a\u95f4\u6765\u63d0\u5347\u7b97\u6cd5\u8fd0\u884c\u901f\u5ea6\u7684\u601d\u8def\u79f0\u4e3a\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff1b\u53cd\u4e4b\uff0c\u5219\u79f0\u4e3a\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</p> <p>\u9009\u62e9\u54ea\u79cd\u601d\u8def\u53d6\u51b3\u4e8e\u6211\u4eec\u66f4\u770b\u91cd\u54ea\u4e2a\u65b9\u9762\u3002\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u65f6\u95f4\u6bd4\u7a7a\u95f4\u66f4\u5b9d\u8d35\uff0c\u56e0\u6b64\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u901a\u5e38\u662f\u66f4\u5e38\u7528\u7684\u7b56\u7565\u3002\u5f53\u7136\uff0c\u5728\u6570\u636e\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u63a7\u5236\u7a7a\u95f4\u590d\u6742\u5ea6\u4e5f\u975e\u5e38\u91cd\u8981\u3002</p>"},{"location":"chapter_computational_complexity/summary/","title":"2.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_computational_complexity/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<p>\u7b97\u6cd5\u6548\u7387\u8bc4\u4f30</p> <ul> <li>\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u662f\u8861\u91cf\u7b97\u6cd5\u4f18\u52a3\u7684\u4e24\u4e2a\u4e3b\u8981\u8bc4\u4ef7\u6307\u6807\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b9e\u9645\u6d4b\u8bd5\u6765\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u96be\u4ee5\u6d88\u9664\u6d4b\u8bd5\u73af\u5883\u7684\u5f71\u54cd\uff0c\u4e14\u4f1a\u8017\u8d39\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002</li> <li>\u590d\u6742\u5ea6\u5206\u6790\u53ef\u4ee5\u6d88\u9664\u5b9e\u9645\u6d4b\u8bd5\u7684\u5f0a\u7aef\uff0c\u5206\u6790\u7ed3\u679c\u9002\u7528\u4e8e\u6240\u6709\u8fd0\u884c\u5e73\u53f0\uff0c\u5e76\u4e14\u80fd\u591f\u63ed\u793a\u7b97\u6cd5\u5728\u4e0d\u540c\u6570\u636e\u89c4\u6a21\u4e0b\u7684\u6548\u7387\u3002</li> </ul> <p>\u65f6\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\uff0c\u53ef\u4ee5\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\uff0c\u4f46\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u53ef\u80fd\u5931\u6548\uff0c\u5982\u5728\u8f93\u5165\u7684\u6570\u636e\u91cf\u8f83\u5c0f\u6216\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\u65f6\uff0c\u65e0\u6cd5\u7cbe\u786e\u5bf9\u6bd4\u7b97\u6cd5\u6548\u7387\u7684\u4f18\u52a3\u3002</li> <li>\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f7f\u7528\u5927 \\(O\\) \u7b26\u53f7\u8868\u793a\uff0c\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u53cd\u6620\u5f53 \\(n\\) \u8d8b\u5411\u6b63\u65e0\u7a77\u65f6\uff0c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u7684\u589e\u957f\u7ea7\u522b\u3002</li> <li>\u63a8\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u4e24\u6b65\uff0c\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</li> <li>\u5e38\u89c1\u65f6\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n \\log n)\\)\u3001\\(O(n^2)\\)\u3001\\(O(2^n)\\) \u548c \\(O(n!)\\) \u7b49\u3002</li> <li>\u67d0\u4e9b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u975e\u56fa\u5b9a\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u4e3a\u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u51e0\u4e4e\u4e0d\u7528\uff0c\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u4e00\u822c\u9700\u8981\u6ee1\u8db3\u4e25\u683c\u6761\u4ef6\u624d\u80fd\u8fbe\u5230\u6700\u4f73\u60c5\u51b5\u3002</li> <li>\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53cd\u6620\u7b97\u6cd5\u5728\u968f\u673a\u6570\u636e\u8f93\u5165\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u6700\u63a5\u8fd1\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u7b97\u6cd5\u6027\u80fd\u3002\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u9700\u8981\u7edf\u8ba1\u8f93\u5165\u6570\u636e\u5206\u5e03\u4ee5\u53ca\u7efc\u5408\u540e\u7684\u6570\u5b66\u671f\u671b\u3002</li> </ul> <p>\u7a7a\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u4f5c\u7528\u7c7b\u4f3c\u4e8e\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u7528\u4e8e\u8861\u91cf\u7b97\u6cd5\u5360\u7528\u5185\u5b58\u7a7a\u95f4\u968f\u6570\u636e\u91cf\u589e\u957f\u7684\u8d8b\u52bf\u3002</li> <li>\u7b97\u6cd5\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u7684\u76f8\u5173\u5185\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u8f93\u5165\u7a7a\u95f4\u3001\u6682\u5b58\u7a7a\u95f4\u3001\u8f93\u51fa\u7a7a\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u7a7a\u95f4\u4e0d\u7eb3\u5165\u7a7a\u95f4\u590d\u6742\u5ea6\u8ba1\u7b97\u3002\u6682\u5b58\u7a7a\u95f4\u53ef\u5206\u4e3a\u6682\u5b58\u6570\u636e\u3001\u6808\u5e27\u7a7a\u95f4\u548c\u6307\u4ee4\u7a7a\u95f4\uff0c\u5176\u4e2d\u6808\u5e27\u7a7a\u95f4\u901a\u5e38\u4ec5\u5728\u9012\u5f52\u51fd\u6570\u4e2d\u5f71\u54cd\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u6211\u4eec\u901a\u5e38\u53ea\u5173\u6ce8\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5373\u7edf\u8ba1\u7b97\u6cd5\u5728\u6700\u5dee\u8f93\u5165\u6570\u636e\u548c\u6700\u5dee\u8fd0\u884c\u65f6\u523b\u4e0b\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5e38\u89c1\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece\u4f4e\u5230\u9ad8\u6392\u5217\u6709 \\(O(1)\\)\u3001\\(O(\\log n)\\)\u3001\\(O(n)\\)\u3001\\(O(n^2)\\) \u548c \\(O(2^n)\\) \u7b49\u3002</li> </ul>"},{"location":"chapter_computational_complexity/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u5c3e\u9012\u5f52\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u5417\uff1f</p> <p>\u7406\u8bba\u4e0a\uff0c\u5c3e\u9012\u5f52\u51fd\u6570\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002\u4e0d\u8fc7\u7edd\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\u3001Python\u3001C++\u3001Go\u3001C# \u7b49\uff09\u4e0d\u652f\u6301\u81ea\u52a8\u4f18\u5316\u5c3e\u9012\u5f52\uff0c\u56e0\u6b64\u901a\u5e38\u8ba4\u4e3a\u7a7a\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\) \u3002</p> <p>\u51fd\u6570\u548c\u65b9\u6cd5\u8fd9\u4e24\u4e2a\u672f\u8bed\u7684\u533a\u522b\u662f\u4ec0\u4e48\uff1f</p> <p>\u300c\u51fd\u6570 function\u300d\u53ef\u4ee5\u88ab\u72ec\u7acb\u6267\u884c\uff0c\u6240\u6709\u53c2\u6570\u90fd\u4ee5\u663e\u5f0f\u4f20\u9012\u3002\u300c\u65b9\u6cd5 method\u300d\u4e0e\u4e00\u4e2a\u5bf9\u8c61\u5173\u8054\uff0c\u88ab\u9690\u5f0f\u4f20\u9012\u7ed9\u8c03\u7528\u5b83\u7684\u5bf9\u8c61\uff0c\u80fd\u591f\u5bf9\u7c7b\u7684\u5b9e\u4f8b\u4e2d\u5305\u542b\u7684\u6570\u636e\u8fdb\u884c\u64cd\u4f5c\u3002</p> <p>\u4e0b\u9762\u4ee5\u51e0\u79cd\u5e38\u89c1\u7684\u7f16\u7a0b\u8bed\u8a00\u4e3a\u4f8b\u6765\u8bf4\u660e\u3002</p> <ul> <li>C \u8bed\u8a00\u662f\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\u8bed\u8a00\uff0c\u6ca1\u6709\u9762\u5411\u5bf9\u8c61\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u53ea\u6709\u51fd\u6570\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u521b\u5efa\u7ed3\u6784\u4f53\uff08struct\uff09\u6765\u6a21\u62df\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u4e0e\u7ed3\u6784\u4f53\u76f8\u5173\u8054\u7684\u51fd\u6570\u5c31\u76f8\u5f53\u4e8e\u5176\u4ed6\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u65b9\u6cd5\u3002</li> <li>Java \u548c C# \u662f\u9762\u5411\u5bf9\u8c61\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u4ee3\u7801\u5757\uff08\u65b9\u6cd5\uff09\u901a\u5e38\u4f5c\u4e3a\u67d0\u4e2a\u7c7b\u7684\u4e00\u90e8\u5206\u3002\u9759\u6001\u65b9\u6cd5\u7684\u884c\u4e3a\u7c7b\u4f3c\u4e8e\u51fd\u6570\uff0c\u56e0\u4e3a\u5b83\u88ab\u7ed1\u5b9a\u5728\u7c7b\u4e0a\uff0c\u4e0d\u80fd\u8bbf\u95ee\u7279\u5b9a\u7684\u5b9e\u4f8b\u53d8\u91cf\u3002</li> <li>C++ \u548c Python \u65e2\u652f\u6301\u8fc7\u7a0b\u5f0f\u7f16\u7a0b\uff08\u51fd\u6570\uff09\uff0c\u4e5f\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff08\u65b9\u6cd5\uff09\u3002</li> </ul> <p>\u56fe\u89e3\u201c\u5e38\u89c1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u201d\u53cd\u6620\u7684\u662f\u5426\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\uff1f</p> <p>\u4e0d\u662f\uff0c\u8be5\u56fe\u5c55\u793a\u7684\u662f\u7a7a\u95f4\u590d\u6742\u5ea6\uff0c\u5176\u53cd\u6620\u7684\u662f\u589e\u957f\u8d8b\u52bf\uff0c\u800c\u4e0d\u662f\u5360\u7528\u7a7a\u95f4\u7684\u7edd\u5bf9\u5927\u5c0f\u3002</p> <p>\u5047\u8bbe\u53d6 \\(n = 8\\) \uff0c\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u6bcf\u6761\u66f2\u7ebf\u7684\u503c\u4e0e\u51fd\u6570\u5bf9\u5e94\u4e0d\u4e0a\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6761\u66f2\u7ebf\u90fd\u5305\u542b\u4e00\u4e2a\u5e38\u6570\u9879\uff0c\u7528\u4e8e\u5c06\u53d6\u503c\u8303\u56f4\u538b\u7f29\u5230\u4e00\u4e2a\u89c6\u89c9\u8212\u9002\u7684\u8303\u56f4\u5185\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u56e0\u4e3a\u6211\u4eec\u901a\u5e38\u4e0d\u77e5\u9053\u6bcf\u4e2a\u65b9\u6cd5\u7684\u201c\u5e38\u6570\u9879\u201d\u590d\u6742\u5ea6\u662f\u591a\u5c11\uff0c\u6240\u4ee5\u4e00\u822c\u65e0\u6cd5\u4ec5\u51ed\u590d\u6742\u5ea6\u6765\u9009\u62e9 \\(n = 8\\) \u4e4b\u4e0b\u7684\u6700\u4f18\u89e3\u6cd5\u3002\u4f46\u5bf9\u4e8e \\(n = 8^5\\) \u5c31\u5f88\u597d\u9009\u4e86\uff0c\u8fd9\u65f6\u589e\u957f\u8d8b\u52bf\u5df2\u7ecf\u5360\u4e3b\u5bfc\u4e86\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/","title":"2.3 \u00a0 \u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u76f4\u89c2\u4e14\u51c6\u786e\u5730\u53cd\u6620\u7b97\u6cd5\u7684\u6548\u7387\u3002\u5982\u679c\u6211\u4eec\u60f3\u51c6\u786e\u9884\u4f30\u4e00\u6bb5\u4ee3\u7801\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u5e94\u8be5\u5982\u4f55\u64cd\u4f5c\u5462\uff1f</p> <ol> <li>\u786e\u5b9a\u8fd0\u884c\u5e73\u53f0\uff0c\u5305\u62ec\u786c\u4ef6\u914d\u7f6e\u3001\u7f16\u7a0b\u8bed\u8a00\u3001\u7cfb\u7edf\u73af\u5883\u7b49\uff0c\u8fd9\u4e9b\u56e0\u7d20\u90fd\u4f1a\u5f71\u54cd\u4ee3\u7801\u7684\u8fd0\u884c\u6548\u7387\u3002</li> <li>\u8bc4\u4f30\u5404\u79cd\u8ba1\u7b97\u64cd\u4f5c\u6240\u9700\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u4f8b\u5982\u52a0\u6cd5\u64cd\u4f5c <code>+</code> \u9700\u8981 1 ns \uff0c\u4e58\u6cd5\u64cd\u4f5c <code>*</code> \u9700\u8981 10 ns \uff0c\u6253\u5370\u64cd\u4f5c <code>print()</code> \u9700\u8981 5 ns \u7b49\u3002</li> <li>\u7edf\u8ba1\u4ee3\u7801\u4e2d\u6240\u6709\u7684\u8ba1\u7b97\u64cd\u4f5c\uff0c\u5e76\u5c06\u6240\u6709\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u6c42\u548c\uff0c\u4ece\u800c\u5f97\u5230\u8fd0\u884c\u65f6\u95f4\u3002</li> </ol> <p>\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\ndef algorithm(n: int):\n    a = 2      # 1 ns\n    a = a + 1  # 1 ns\n    a = a * 2  # 10 ns\n    # \u5faa\u73af n \u6b21\n    for _ in range(n):  # 1 ns\n        print(0)        # 5 ns\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        cout &lt;&lt; 0 &lt;&lt; endl;         // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        System.out.println(0);     // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid Algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {  // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        Console.WriteLine(0);      // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n int) {\n    a := 2     // 1 ns\n    a = a + 1  // 1 ns\n    a = a * 2  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for i := 0; i &lt; n; i++ {  // 1 ns\n        fmt.Println(a)        // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunc algorithm(n: Int) {\n    var a = 2 // 1 ns\n    a = a + 1 // 1 ns\n    a = a * 2 // 10 ns\n    // \u5faa\u73af n \u6b21\n    for _ in 0 ..&lt; n { // 1 ns\n        print(0) // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n) {\n    var a = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfunction algorithm(n: number): void {\n    var a: number = 2; // 1 ns\n    a = a + 1; // 1 ns\n    a = a * 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        console.log(0); // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n  int a = 2; // 1 ns\n  a = a + 1; // 1 ns\n  a = a * 2; // 10 ns\n  // \u5faa\u73af n \u6b21\n  for (int i = 0; i &lt; n; i++) { // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n    print(0); // 5 ns\n  }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: i32) {\n    let mut a = 2;      // 1 ns\n    a = a + 1;          // 1 ns\n    a = a * 2;          // 10 ns\n    // \u5faa\u73af n \u6b21\n    for _ in 0..n {     // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        println!(\"{}\", 0);  // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nvoid algorithm(int n) {\n    int a = 2;  // 1 ns\n    a = a + 1;  // 1 ns\n    a = a * 2;  // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // 1 ns \uff0c\u6bcf\u8f6e\u90fd\u8981\u6267\u884c i++\n        printf(\"%d\", 0);            // 5 ns\n    }\n}\n</code></pre> <pre><code>// \u5728\u67d0\u8fd0\u884c\u5e73\u53f0\u4e0b\nfn algorithm(n: usize) void {\n    var a: i32 = 2; // 1 ns\n    a += 1; // 1 ns\n    a *= 2; // 10 ns\n    // \u5faa\u73af n \u6b21\n    for (0..n) |_| { // 1 ns\n        std.debug.print(\"{}\\n\", .{0}); // 5 ns\n    }\n}\n</code></pre> <p>\u6839\u636e\u4ee5\u4e0a\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u4e3a \\((6n + 12)\\) ns \uff1a</p> \\[ 1 + 1 + 10 + (1 + 5) \\times n = 6n + 12 \\] <p>\u4f46\u5b9e\u9645\u4e0a\uff0c\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\u65e2\u4e0d\u5408\u7406\u4e5f\u4e0d\u73b0\u5b9e\u3002\u9996\u5148\uff0c\u6211\u4eec\u4e0d\u5e0c\u671b\u5c06\u9884\u4f30\u65f6\u95f4\u548c\u8fd0\u884c\u5e73\u53f0\u7ed1\u5b9a\uff0c\u56e0\u4e3a\u7b97\u6cd5\u9700\u8981\u5728\u5404\u79cd\u4e0d\u540c\u7684\u5e73\u53f0\u4e0a\u8fd0\u884c\u3002\u5176\u6b21\uff0c\u6211\u4eec\u5f88\u96be\u83b7\u77e5\u6bcf\u79cd\u64cd\u4f5c\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u8fd9\u7ed9\u9884\u4f30\u8fc7\u7a0b\u5e26\u6765\u4e86\u6781\u5927\u7684\u96be\u5ea6\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#231","title":"2.3.1 \u00a0 \u7edf\u8ba1\u65f6\u95f4\u589e\u957f\u8d8b\u52bf","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7edf\u8ba1\u7684\u4e0d\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\uff0c\u800c\u662f\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740\u6570\u636e\u91cf\u53d8\u5927\u65f6\u7684\u589e\u957f\u8d8b\u52bf\u3002</p> <p>\u201c\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u201d\u8fd9\u4e2a\u6982\u5ff5\u6bd4\u8f83\u62bd\u8c61\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a\u4f8b\u5b50\u6765\u52a0\u4ee5\u7406\u89e3\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7ed9\u5b9a\u4e09\u4e2a\u7b97\u6cd5 <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_A(n: int):\n    print(0)\n# \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\ndef algorithm_B(n: int):\n    for _ in range(n):\n        print(0)\n# \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\ndef algorithm_C(n: int):\n    for _ in range(1000000):\n        print(0)\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    cout &lt;&lt; 0 &lt;&lt; endl;\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    System.out.println(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        System.out.println(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        System.out.println(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmA(int n) {\n    Console.WriteLine(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid AlgorithmB(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        Console.WriteLine(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid AlgorithmC(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        Console.WriteLine(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_A(n int) {\n    fmt.Println(0)\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithm_B(n int) {\n    for i := 0; i &lt; n; i++ {\n        fmt.Println(0)\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithm_C(n int) {\n    for i := 0; i &lt; 1000000; i++ {\n        fmt.Println(0)\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmA(n: Int) {\n    print(0)\n}\n\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunc algorithmB(n: Int) {\n    for _ in 0 ..&lt; n {\n        print(0)\n    }\n}\n\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunc algorithmC(n: Int) {\n    for _ in 0 ..&lt; 1000000 {\n        print(0)\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n) {\n    console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n) {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n) {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_A(n: number): void {\n    console.log(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfunction algorithm_B(n: number): void {\n    for (let i = 0; i &lt; n; i++) {\n        console.log(0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfunction algorithm_C(n: number): void {\n    for (let i = 0; i &lt; 1000000; i++) {\n        console.log(0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmA(int n) {\n  print(0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithmB(int n) {\n  for (int i = 0; i &lt; n; i++) {\n    print(0);\n  }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithmC(int n) {\n  for (int i = 0; i &lt; 1000000; i++) {\n    print(0);\n  }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: i32) {\n    println!(\"{}\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) {\n    for _ in 0..n {\n        println!(\"{}\", 0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) {\n    for _ in 0..1000000 {\n        println!(\"{}\", 0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_A(int n) {\n    printf(\"%d\", 0);\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nvoid algorithm_B(int n) {\n    for (int i = 0; i &lt; n; i++) {\n        printf(\"%d\", 0);\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nvoid algorithm_C(int n) {\n    for (int i = 0; i &lt; 1000000; i++) {\n        printf(\"%d\", 0);\n    }\n}\n</code></pre> <pre><code>// \u7b97\u6cd5 A \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_A(n: usize) void {\n    _ = n;\n    std.debug.print(\"{}\\n\", .{0});\n}\n// \u7b97\u6cd5 B \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u7ebf\u6027\u9636\nfn algorithm_B(n: i32) void {\n    for (0..n) |_| {\n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n// \u7b97\u6cd5 C \u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u5e38\u6570\u9636\nfn algorithm_C(n: i32) void {\n    _ = n;\n    for (0..1000000) |_| { \n        std.debug.print(\"{}\\n\", .{0});\n    }\n}\n</code></pre> <p>\u56fe 2-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4e09\u4e2a\u7b97\u6cd5\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u7b97\u6cd5 <code>A</code> \u53ea\u6709 \\(1\\) \u4e2a\u6253\u5370\u64cd\u4f5c\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u4e0d\u968f\u7740 \\(n\\) \u589e\u5927\u800c\u589e\u957f\u3002\u6211\u4eec\u79f0\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>B</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(n\\) \u6b21\uff0c\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u7740 \\(n\\) \u589e\u5927\u5448\u7ebf\u6027\u589e\u957f\u3002\u6b64\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u88ab\u79f0\u4e3a\u201c\u7ebf\u6027\u9636\u201d\u3002</li> <li>\u7b97\u6cd5 <code>C</code> \u4e2d\u7684\u6253\u5370\u64cd\u4f5c\u9700\u8981\u5faa\u73af \\(1000000\\) \u6b21\uff0c\u867d\u7136\u8fd0\u884c\u65f6\u95f4\u5f88\u957f\uff0c\u4f46\u5b83\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\u3002\u56e0\u6b64 <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c <code>A</code> \u76f8\u540c\uff0c\u4ecd\u4e3a\u201c\u5e38\u6570\u9636\u201d\u3002</li> </ul> <p></p> <p> \u56fe 2-7 \u00a0 \u7b97\u6cd5 A\u3001B \u548c C \u7684\u65f6\u95f4\u589e\u957f\u8d8b\u52bf </p> <p>\u76f8\u8f83\u4e8e\u76f4\u63a5\u7edf\u8ba1\u7b97\u6cd5\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u6709\u54ea\u4e9b\u7279\u70b9\u5462\uff1f</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u80fd\u591f\u6709\u6548\u8bc4\u4f30\u7b97\u6cd5\u6548\u7387\u3002\u4f8b\u5982\uff0c\u7b97\u6cd5 <code>B</code> \u7684\u8fd0\u884c\u65f6\u95f4\u5448\u7ebf\u6027\u589e\u957f\uff0c\u5728 \\(n &gt; 1\\) \u65f6\u6bd4\u7b97\u6cd5 <code>A</code> \u66f4\u6162\uff0c\u5728 \\(n &gt; 1000000\\) \u65f6\u6bd4\u7b97\u6cd5 <code>C</code> \u66f4\u6162\u3002\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u8981\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8db3\u591f\u5927\uff0c\u590d\u6742\u5ea6\u4e3a\u201c\u5e38\u6570\u9636\u201d\u7684\u7b97\u6cd5\u4e00\u5b9a\u4f18\u4e8e\u201c\u7ebf\u6027\u9636\u201d\u7684\u7b97\u6cd5\uff0c\u8fd9\u6b63\u662f\u65f6\u95f4\u589e\u957f\u8d8b\u52bf\u7684\u542b\u4e49\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u66f4\u7b80\u4fbf\u3002\u663e\u7136\uff0c\u8fd0\u884c\u5e73\u53f0\u548c\u8ba1\u7b97\u64cd\u4f5c\u7c7b\u578b\u90fd\u4e0e\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u65e0\u5173\u3002\u56e0\u6b64\u5728\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5c06\u6240\u6709\u8ba1\u7b97\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u89c6\u4e3a\u76f8\u540c\u7684\u201c\u5355\u4f4d\u65f6\u95f4\u201d\uff0c\u4ece\u800c\u5c06\u201c\u8ba1\u7b97\u64cd\u4f5c\u8fd0\u884c\u65f6\u95f4\u7edf\u8ba1\u201d\u7b80\u5316\u4e3a\u201c\u8ba1\u7b97\u64cd\u4f5c\u6570\u91cf\u7edf\u8ba1\u201d\uff0c\u8fd9\u6837\u4e00\u6765\u4f30\u7b97\u96be\u5ea6\u5c31\u5927\u5927\u964d\u4f4e\u4e86\u3002</li> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002\u4f8b\u5982\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>A</code> \u548c <code>C</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u76f8\u540c\uff0c\u4f46\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u5dee\u522b\u5f88\u5927\u3002\u540c\u6837\uff0c\u5c3d\u7ba1\u7b97\u6cd5 <code>B</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6bd4 <code>C</code> \u9ad8\uff0c\u4f46\u5728\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7b97\u6cd5 <code>B</code> \u660e\u663e\u4f18\u4e8e\u7b97\u6cd5 <code>C</code> \u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5f88\u96be\u4ec5\u51ed\u65f6\u95f4\u590d\u6742\u5ea6\u5224\u65ad\u7b97\u6cd5\u6548\u7387\u7684\u9ad8\u4f4e\u3002\u5f53\u7136\uff0c\u5c3d\u7ba1\u5b58\u5728\u4e0a\u8ff0\u95ee\u9898\uff0c\u590d\u6742\u5ea6\u5206\u6790\u4ecd\u7136\u662f\u8bc4\u5224\u7b97\u6cd5\u6548\u7387\u6700\u6709\u6548\u4e14\u5e38\u7528\u7684\u65b9\u6cd5\u3002</li> </ul>"},{"location":"chapter_computational_complexity/time_complexity/#232","title":"2.3.2 \u00a0 \u51fd\u6570\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u8f93\u5165\u5927\u5c0f\u4e3a \\(n\\) \u7684\u51fd\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def algorithm(n: int):\n    a = 1      # +1\n    a = a + 1  # +1\n    a = a * 2  # +1\n    # \u5faa\u73af n \u6b21\n    for i in range(n):  # +1\n        print(0)        # +1\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        cout &lt;&lt; 0 &lt;&lt; endl;    // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        System.out.println(0);    // +1\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        Console.WriteLine(0);   // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1      // +1\n    a = a + 1   // +1\n    a = a * 2   // +1\n    // \u5faa\u73af n \u6b21\n    for i := 0; i &lt; n; i++ {   // +1\n        fmt.Println(a)         // +1\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +1\n    a = a + 1 // +1\n    a = a * 2 // +1\n    // \u5faa\u73af n \u6b21\n    for _ in 0 ..&lt; n { // +1\n        print(0) // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    var a = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void{\n    var a: number = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for(let i = 0; i &lt; n; i++){ // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        console.log(0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +1\n  a = a + 1; // +1\n  a = a * 2; // +1\n  // \u5faa\u73af n \u6b21\n  for (int i = 0; i &lt; n; i++) { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n    print(0); // +1\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;   // +1\n    a = a + 1;      // +1\n    a = a * 2;      // +1\n\n    // \u5faa\u73af n \u6b21\n    for _ in 0..n { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        println!(\"{}\", 0); // +1\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +1\n    a = a + 1;  // +1\n    a = a * 2;  // +1\n    // \u5faa\u73af n \u6b21\n    for (int i = 0; i &lt; n; i++) {   // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        printf(\"%d\", 0);            // +1\n    }\n}  \n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1; // +1\n    a += 1; // +1\n    a *= 2; // +1\n    // \u5faa\u73af n \u6b21\n    for (0..n) |_| { // +1\uff08\u6bcf\u8f6e\u90fd\u6267\u884c i ++\uff09\n        std.debug.print(\"{}\\n\", .{0}); // +1\n    }\n}\n</code></pre> <p>\u8bbe\u7b97\u6cd5\u7684\u64cd\u4f5c\u6570\u91cf\u662f\u4e00\u4e2a\u5173\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u7684\u51fd\u6570\uff0c\u8bb0\u4e3a \\(T(n)\\) \uff0c\u5219\u4ee5\u4e0a\u51fd\u6570\u7684\u64cd\u4f5c\u6570\u91cf\u4e3a\uff1a</p> \\[ T(n) = 3 + 2n \\] <p>\\(T(n)\\) \u662f\u4e00\u6b21\u51fd\u6570\uff0c\u8bf4\u660e\u5176\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u8d8b\u52bf\u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7ebf\u6027\u9636\u3002</p> <p>\u6211\u4eec\u5c06\u7ebf\u6027\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u8bb0\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e2a\u6570\u5b66\u7b26\u53f7\u79f0\u4e3a\u300c\u5927 \\(O\\) \u8bb0\u53f7 big-\\(O\\) notation\u300d\uff0c\u8868\u793a\u51fd\u6570 \\(T(n)\\) \u7684\u300c\u6e10\u8fd1\u4e0a\u754c asymptotic upper bound\u300d\u3002</p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u672c\u8d28\u4e0a\u662f\u8ba1\u7b97\u201c\u64cd\u4f5c\u6570\u91cf \\(T(n)\\)\u201d\u7684\u6e10\u8fd1\u4e0a\u754c\uff0c\u5b83\u5177\u6709\u660e\u786e\u7684\u6570\u5b66\u5b9a\u4e49\u3002</p> <p>\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c</p> <p>\u82e5\u5b58\u5728\u6b63\u5b9e\u6570 \\(c\\) \u548c\u5b9e\u6570 \\(n_0\\) \uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6240\u6709\u7684 \\(n &gt; n_0\\) \uff0c\u5747\u6709 \\(T(n) \\leq c \\cdot f(n)\\) \uff0c\u5219\u53ef\u8ba4\u4e3a \\(f(n)\\) \u7ed9\u51fa\u4e86 \\(T(n)\\) \u7684\u4e00\u4e2a\u6e10\u8fd1\u4e0a\u754c\uff0c\u8bb0\u4e3a \\(T(n) = O(f(n))\\) \u3002</p> <p>\u5982\u56fe 2-8 \u6240\u793a\uff0c\u8ba1\u7b97\u6e10\u8fd1\u4e0a\u754c\u5c31\u662f\u5bfb\u627e\u4e00\u4e2a\u51fd\u6570 \\(f(n)\\) \uff0c\u4f7f\u5f97\u5f53 \\(n\\) \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\\(T(n)\\) \u548c \\(f(n)\\) \u5904\u4e8e\u76f8\u540c\u7684\u589e\u957f\u7ea7\u522b\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570\u9879 \\(c\\) \u7684\u500d\u6570\u3002</p> <p></p> <p> \u56fe 2-8 \u00a0 \u51fd\u6570\u7684\u6e10\u8fd1\u4e0a\u754c </p>"},{"location":"chapter_computational_complexity/time_complexity/#233","title":"2.3.3 \u00a0 \u63a8\u7b97\u65b9\u6cd5","text":"<p>\u6e10\u8fd1\u4e0a\u754c\u7684\u6570\u5b66\u5473\u513f\u6709\u70b9\u91cd\uff0c\u5982\u679c\u4f60\u611f\u89c9\u6ca1\u6709\u5b8c\u5168\u7406\u89e3\uff0c\u4e5f\u65e0\u987b\u62c5\u5fc3\u3002\u6211\u4eec\u53ef\u4ee5\u5148\u638c\u63e1\u63a8\u7b97\u65b9\u6cd5\uff0c\u5728\u4e0d\u65ad\u7684\u5b9e\u8df5\u4e2d\uff0c\u5c31\u53ef\u4ee5\u9010\u6e10\u9886\u609f\u5176\u6570\u5b66\u610f\u4e49\u3002</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c\u786e\u5b9a \\(f(n)\\) \u4e4b\u540e\uff0c\u6211\u4eec\u4fbf\u53ef\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \u3002\u90a3\u4e48\u5982\u4f55\u786e\u5b9a\u6e10\u8fd1\u4e0a\u754c \\(f(n)\\) \u5462\uff1f\u603b\u4f53\u5206\u4e3a\u4e24\u6b65\uff1a\u9996\u5148\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff0c\u7136\u540e\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#1","title":"1. \u00a0 \u7b2c\u4e00\u6b65\uff1a\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf","text":"<p>\u9488\u5bf9\u4ee3\u7801\uff0c\u9010\u884c\u4ece\u4e0a\u5230\u4e0b\u8ba1\u7b97\u5373\u53ef\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4e0a\u8ff0 \\(c \\cdot f(n)\\) \u4e2d\u7684\u5e38\u6570\u9879 \\(c\\) \u53ef\u4ee5\u53d6\u4efb\u610f\u5927\u5c0f\uff0c\u56e0\u6b64\u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u4e2d\u7684\u5404\u79cd\u7cfb\u6570\u3001\u5e38\u6570\u9879\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002\u6839\u636e\u6b64\u539f\u5219\uff0c\u53ef\u4ee5\u603b\u7ed3\u51fa\u4ee5\u4e0b\u8ba1\u6570\u7b80\u5316\u6280\u5de7\u3002</p> <ol> <li>\u5ffd\u7565 \\(T(n)\\) \u4e2d\u7684\u5e38\u6570\u9879\u3002\u56e0\u4e3a\u5b83\u4eec\u90fd\u4e0e \\(n\\) \u65e0\u5173\uff0c\u6240\u4ee5\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4ea7\u751f\u5f71\u54cd\u3002</li> <li>\u7701\u7565\u6240\u6709\u7cfb\u6570\u3002\u4f8b\u5982\uff0c\u5faa\u73af \\(2n\\) \u6b21\u3001\\(5n + 1\\) \u6b21\u7b49\uff0c\u90fd\u53ef\u4ee5\u7b80\u5316\u8bb0\u4e3a \\(n\\) \u6b21\uff0c\u56e0\u4e3a \\(n\\) \u524d\u9762\u7684\u7cfb\u6570\u5bf9\u65f6\u95f4\u590d\u6742\u5ea6\u6ca1\u6709\u5f71\u54cd\u3002</li> <li>\u5faa\u73af\u5d4c\u5957\u65f6\u4f7f\u7528\u4e58\u6cd5\u3002\u603b\u64cd\u4f5c\u6570\u91cf\u7b49\u4e8e\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u64cd\u4f5c\u6570\u91cf\u4e4b\u79ef\uff0c\u6bcf\u4e00\u5c42\u5faa\u73af\u4f9d\u7136\u53ef\u4ee5\u5206\u522b\u5957\u7528\u7b2c <code>1.</code> \u70b9\u548c\u7b2c <code>2.</code> \u70b9\u7684\u6280\u5de7\u3002</li> </ol> <p>\u7ed9\u5b9a\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e0a\u8ff0\u6280\u5de7\u6765\u7edf\u8ba1\u64cd\u4f5c\u6570\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>def algorithm(n: int):\n    a = 1      # +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  # +0\uff08\u6280\u5de7 1\uff09\n    # +n\uff08\u6280\u5de7 2\uff09\n    for i in range(5 * n + 1):\n        print(0)\n    # +n*n\uff08\u6280\u5de7 3\uff09\n    for i in range(2 * n):\n        for j in range(n + 1):\n            print(0)\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        cout &lt;&lt; 0 &lt;&lt; endl;\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            cout &lt;&lt; 0 &lt;&lt; endl;\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        System.out.println(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            System.out.println(0);\n        }\n    }\n}\n</code></pre> <pre><code>void Algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        Console.WriteLine(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            Console.WriteLine(0);\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n int) {\n    a := 1     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i := 0; i &lt; 5 * n + 1; i++ {\n        fmt.Println(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i := 0; i &lt; 2 * n; i++ {\n        for j := 0; j &lt; n + 1; j++ {\n            fmt.Println(0)\n        }\n    }\n}\n</code></pre> <pre><code>func algorithm(n: Int) {\n    var a = 1 // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for _ in 0 ..&lt; (5 * n + 1) {\n        print(0)\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for _ in 0 ..&lt; (2 * n) {\n        for _ in 0 ..&lt; (n + 1) {\n            print(0)\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n) {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>function algorithm(n: number): void {\n    let a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (let i = 0; i &lt; 5 * n + 1; i++) {\n        console.log(0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (let i = 0; i &lt; 2 * n; i++) {\n        for (let j = 0; j &lt; n + 1; j++) {\n            console.log(0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n  int a = 1; // +0\uff08\u6280\u5de7 1\uff09\n  a = a + n; // +0\uff08\u6280\u5de7 1\uff09\n  // +n\uff08\u6280\u5de7 2\uff09\n  for (int i = 0; i &lt; 5 * n + 1; i++) {\n    print(0);\n  }\n  // +n*n\uff08\u6280\u5de7 3\uff09\n  for (int i = 0; i &lt; 2 * n; i++) {\n    for (int j = 0; j &lt; n + 1; j++) {\n      print(0);\n    }\n  }\n}\n</code></pre> <pre><code>fn algorithm(n: i32) {\n    let mut a = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for i in 0..(5 * n + 1) {\n        println!(\"{}\", 0);\n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for i in 0..(2 * n) {\n        for j in 0..(n + 1) {\n            println!(\"{}\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>void algorithm(int n) {\n    int a = 1;  // +0\uff08\u6280\u5de7 1\uff09\n    a = a + n;  // +0\uff08\u6280\u5de7 1\uff09\n    // +n\uff08\u6280\u5de7 2\uff09\n    for (int i = 0; i &lt; 5 * n + 1; i++) {\n        printf(\"%d\", 0);\n    }\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for (int i = 0; i &lt; 2 * n; i++) {\n        for (int j = 0; j &lt; n + 1; j++) {\n            printf(\"%d\", 0);\n        }\n    }\n}\n</code></pre> <pre><code>fn algorithm(n: usize) void {\n    var a: i32 = 1;     // +0\uff08\u6280\u5de7 1\uff09\n    a = a + @as(i32, @intCast(n));        // +0\uff08\u6280\u5de7 1\uff09\n\n    // +n\uff08\u6280\u5de7 2\uff09\n    for(0..(5 * n + 1)) |_| {\n        std.debug.print(\"{}\\n\", .{0}); \n    }\n\n    // +n*n\uff08\u6280\u5de7 3\uff09\n    for(0..(2 * n)) |_| {\n        for(0..(n + 1)) |_| {\n            std.debug.print(\"{}\\n\", .{0}); \n        }\n    }\n}\n</code></pre> <p>\u4ee5\u4e0b\u516c\u5f0f\u5c55\u793a\u4e86\u4f7f\u7528\u4e0a\u8ff0\u6280\u5de7\u524d\u540e\u7684\u7edf\u8ba1\u7ed3\u679c\uff0c\u4e24\u8005\u63a8\u7b97\u51fa\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002</p> \\[ \\begin{aligned} T(n) &amp; = 2n(n + 1) + (5n + 1) + 2 &amp; \\text{\u5b8c\u6574\u7edf\u8ba1 (-.-|||)} \\newline &amp; = 2n^2 + 7n + 3 \\newline T(n) &amp; = n^2 + n &amp; \\text{\u5077\u61d2\u7edf\u8ba1 (o.O)} \\end{aligned} \\]"},{"location":"chapter_computational_complexity/time_complexity/#2","title":"2. \u00a0 \u7b2c\u4e8c\u6b65\uff1a\u5224\u65ad\u6e10\u8fd1\u4e0a\u754c","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\u7531 \\(T(n)\\) \u4e2d\u6700\u9ad8\u9636\u7684\u9879\u6765\u51b3\u5b9a\u3002\u8fd9\u662f\u56e0\u4e3a\u5728 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u6700\u9ad8\u9636\u7684\u9879\u5c06\u53d1\u6325\u4e3b\u5bfc\u4f5c\u7528\uff0c\u5176\u4ed6\u9879\u7684\u5f71\u54cd\u90fd\u53ef\u4ee5\u5ffd\u7565\u3002</p> <p>\u8868 2-2 \u5c55\u793a\u4e86\u4e00\u4e9b\u4f8b\u5b50\uff0c\u5176\u4e2d\u4e00\u4e9b\u5938\u5f20\u7684\u503c\u662f\u4e3a\u4e86\u5f3a\u8c03\u201c\u7cfb\u6570\u65e0\u6cd5\u64bc\u52a8\u9636\u6570\u201d\u8fd9\u4e00\u7ed3\u8bba\u3002\u5f53 \\(n\\) \u8d8b\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u8fd9\u4e9b\u5e38\u6570\u53d8\u5f97\u65e0\u8db3\u8f7b\u91cd\u3002</p> <p> \u8868 2-2 \u00a0 \u4e0d\u540c\u64cd\u4f5c\u6570\u91cf\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> \u64cd\u4f5c\u6570\u91cf \\(T(n)\\) \u65f6\u95f4\u590d\u6742\u5ea6 \\(O(f(n))\\) \\(100000\\) \\(O(1)\\) \\(3n + 2\\) \\(O(n)\\) \\(2n^2 + 3n + 2\\) \\(O(n^2)\\) \\(n^3 + 10000n^2\\) \\(O(n^3)\\) \\(2^n + 10000n^{10000}\\) \\(O(2^n)\\)"},{"location":"chapter_computational_complexity/time_complexity/#234","title":"2.3.4 \u00a0 \u5e38\u89c1\u7c7b\u578b","text":"<p>\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b\u5982\u56fe 2-9 \u6240\u793a\uff08\u6309\u7167\u4ece\u4f4e\u5230\u9ad8\u7684\u987a\u5e8f\u6392\u5217\uff09\u3002</p> \\[ \\begin{aligned} O(1) &lt; O(\\log n) &lt; O(n) &lt; O(n \\log n) &lt; O(n^2) &lt; O(2^n) &lt; O(n!) \\newline \\text{\u5e38\u6570\u9636} &lt; \\text{\u5bf9\u6570\u9636} &lt; \\text{\u7ebf\u6027\u9636} &lt; \\text{\u7ebf\u6027\u5bf9\u6570\u9636} &lt; \\text{\u5e73\u65b9\u9636} &lt; \\text{\u6307\u6570\u9636} &lt; \\text{\u9636\u4e58\u9636} \\end{aligned} \\] <p></p> <p> \u56fe 2-9 \u00a0 \u5e38\u89c1\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7c7b\u578b </p>"},{"location":"chapter_computational_complexity/time_complexity/#1-o1","title":"1. \u00a0 \u5e38\u6570\u9636 \\(O(1)\\)","text":"<p>\u5e38\u6570\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u5373\u4e0d\u968f\u7740 \\(n\\) \u7684\u53d8\u5316\u800c\u53d8\u5316\u3002</p> <p>\u5728\u4ee5\u4e0b\u51fd\u6570\u4e2d\uff0c\u5c3d\u7ba1\u64cd\u4f5c\u6570\u91cf <code>size</code> \u53ef\u80fd\u5f88\u5927\uff0c\u4f46\u7531\u4e8e\u5176\u4e0e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u65e0\u5173\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def constant(n: int) -&gt; int:\n    \"\"\"\u5e38\u6570\u9636\"\"\"\n    count = 0\n    size = 100000\n    for _ in range(size):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e38\u6570\u9636 */\nint Constant(int n) {\n    int count = 0;\n    int size = 100000;\n    for (int i = 0; i &lt; size; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n int) int {\n    count := 0\n    size := 100000\n    for i := 0; i &lt; size; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e38\u6570\u9636 */\nfunc constant(n: Int) -&gt; Int {\n    var count = 0\n    let size = 100_000\n    for _ in 0 ..&lt; size {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n) {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e38\u6570\u9636 */\nfunction constant(n: number): number {\n    let count = 0;\n    const size = 100000;\n    for (let i = 0; i &lt; size; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n  int count = 0;\n  int size = 100000;\n  for (var i = 0; i &lt; size; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e38\u6570\u9636 */\nfn constant(n: i32) -&gt; i32 {\n    _ = n;\n    let mut count = 0;\n    let size = 100_000;\n    for _ in 0..size {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e38\u6570\u9636 */\nint constant(int n) {\n    int count = 0;\n    int size = 100000;\n    int i = 0;\n    for (int i = 0; i &lt; size; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5e38\u6570\u9636\nfn constant(n: i32) i32 {\n    _ = n;\n    var count: i32 = 0;\n    const size: i32 = 100_000;\n    var i: i32 = 0;\n    while(i&lt;size) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre>"},{"location":"chapter_computational_complexity/time_complexity/#2-on","title":"2. \u00a0 \u7ebf\u6027\u9636 \\(O(n)\\)","text":"<p>\u7ebf\u6027\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u7ebf\u6027\u7ea7\u522b\u589e\u957f\u3002\u7ebf\u6027\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5355\u5c42\u5faa\u73af\u4e2d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def linear(n: int) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\"\"\"\n    count = 0\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636 */\nint Linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++)\n        count++;\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n int) int {\n    count := 0\n    for i := 0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636 */\nfunc linear(n: Int) -&gt; Int {\n    var count = 0\n    for _ in 0 ..&lt; n {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n) {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636 */\nfunction linear(n: number): number {\n    let count = 0;\n    for (let i = 0; i &lt; n; i++) count++;\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n  int count = 0;\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636 */\nfn linear(n: i32) -&gt; i32 {\n    let mut count = 0;\n    for _ in 0..n {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636 */\nint linear(int n) {\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\nfn linear(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> <p>\u904d\u5386\u6570\u7ec4\u548c\u904d\u5386\u94fe\u8868\u7b49\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u6570\u7ec4\u6216\u94fe\u8868\u7684\u957f\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def array_traversal(nums: list[int]) -&gt; int:\n    \"\"\"\u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for num in nums:\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(vector&lt;int&gt; &amp;nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int num : nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint ArrayTraversal(int[] nums) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    foreach (int num in nums) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums []int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for range nums {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunc arrayTraversal(nums: [Int]) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfunction arrayTraversal(nums: number[]): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (let i = 0; i &lt; nums.length; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(List&lt;int&gt; nums) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n  for (var _num in nums) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nfn array_traversal(nums: &amp;[i32]) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for _ in nums {\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09 */\nint arrayTraversal(int *nums, int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u9636\uff08\u904d\u5386\u6570\u7ec4\uff09\nfn arrayTraversal(nums: []i32) i32 {\n    var count: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u6b63\u6bd4\n    for (nums) |_| {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u9700\u6839\u636e\u8f93\u5165\u6570\u636e\u7684\u7c7b\u578b\u6765\u5177\u4f53\u786e\u5b9a\u3002\u6bd4\u5982\u5728\u7b2c\u4e00\u4e2a\u793a\u4f8b\u4e2d\uff0c\u53d8\u91cf \\(n\\) \u4e3a\u8f93\u5165\u6570\u636e\u5927\u5c0f\uff1b\u5728\u7b2c\u4e8c\u4e2a\u793a\u4f8b\u4e2d\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u4e3a\u6570\u636e\u5927\u5c0f\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#3-on2","title":"3. \u00a0 \u5e73\u65b9\u9636 \\(O(n^2)\\)","text":"<p>\u5e73\u65b9\u9636\u7684\u64cd\u4f5c\u6570\u91cf\u76f8\u5bf9\u4e8e\u8f93\u5165\u6570\u636e\u5927\u5c0f \\(n\\) \u4ee5\u5e73\u65b9\u7ea7\u522b\u589e\u957f\u3002\u5e73\u65b9\u9636\u901a\u5e38\u51fa\u73b0\u5728\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u5916\u5c42\u5faa\u73af\u548c\u5185\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n)\\) \uff0c\u56e0\u6b64\u603b\u4f53\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def quadratic(n: int) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\"\"\"\n    count = 0\n    # \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for i in range(n):\n        for j in range(n):\n            count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636 */\nint Quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n int) int {\n    count := 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; n; j++ {\n            count++\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636 */\nfunc quadratic(n: Int) -&gt; Int {\n    var count = 0\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; n {\n            count += 1\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n) {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636 */\nfunction quadratic(n: number): number {\n    let count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n  int count = 0;\n  // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n  for (int i = 0; i &lt; n; i++) {\n    for (int j = 0; j &lt; n; j++) {\n      count++;\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636 */\nfn quadratic(n: i32) -&gt; i32 {\n    let mut count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for _ in 0..n {\n        for _ in 0..n {\n            count += 1;\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636 */\nint quadratic(int n) {\n    int count = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; n; j++) {\n            count++;\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\nfn quadratic(n: i32) i32 {\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u5faa\u73af\u6b21\u6570\u4e0e\u6570\u7ec4\u957f\u5ea6\u6210\u5e73\u65b9\u5173\u7cfb\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; n) : (j += 1) {\n            count += 1;\n        }\n    }\n    return count;\n}\n</code></pre> <p>\u56fe 2-10 \u5bf9\u6bd4\u4e86\u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u4e09\u79cd\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p></p> <p> \u56fe 2-10 \u00a0 \u5e38\u6570\u9636\u3001\u7ebf\u6027\u9636\u548c\u5e73\u65b9\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4ee5\u5192\u6ce1\u6392\u5e8f\u4e3a\u4f8b\uff0c\u5916\u5c42\u5faa\u73af\u6267\u884c \\(n - 1\\) \u6b21\uff0c\u5185\u5c42\u5faa\u73af\u6267\u884c \\(n-1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u5e73\u5747\u4e3a \\(n / 2\\) \u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O((n - 1) n / 2) = O(n^2)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def bubble_sort(nums: list[int]) -&gt; int:\n    \"\"\"\u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\"\"\"\n    count = 0  # \u8ba1\u6570\u5668\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(len(nums) - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp: int = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3  # \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(vector&lt;int&gt; &amp;nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int[] nums) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint BubbleSort(int[] nums) {\n    int count = 0;  // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums []int) int {\n    count := 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                tmp := nums[j]\n                nums[j] = nums[j+1]\n                nums[j+1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunc bubbleSort(nums: inout [Int]) -&gt; Int {\n    var count = 0 // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in stride(from: nums.count - 1, to: 0, by: -1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for j in 0 ..&lt; i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                count += 3 // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums) {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfunction bubbleSort(nums: number[]): number {\n    let count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(List&lt;int&gt; nums) {\n  int count = 0; // \u8ba1\u6570\u5668\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (var i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (var j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n      }\n    }\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nfn bubble_sort(nums: &amp;mut [i32]) -&gt; i32 {\n    let mut count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09 */\nint bubbleSort(int *nums, int n) {\n    int count = 0; // \u8ba1\u6570\u5668\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = n - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3; // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5e73\u65b9\u9636\uff08\u5192\u6ce1\u6392\u5e8f\uff09\nfn bubbleSort(nums: []i32) i32 {\n    var count: i32 = 0;  // \u8ba1\u6570\u5668 \n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: i32 = @as(i32, @intCast(nums.len)) - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                count += 3;  // \u5143\u7d20\u4ea4\u6362\u5305\u542b 3 \u4e2a\u5355\u5143\u64cd\u4f5c\n            }\n        }\n    }\n    return count;\n}\n</code></pre>"},{"location":"chapter_computational_complexity/time_complexity/#4-o2n","title":"4. \u00a0 \u6307\u6570\u9636 \\(O(2^n)\\)","text":"<p>\u751f\u7269\u5b66\u7684\u201c\u7ec6\u80de\u5206\u88c2\u201d\u662f\u6307\u6570\u9636\u589e\u957f\u7684\u5178\u578b\u4f8b\u5b50\uff1a\u521d\u59cb\u72b6\u6001\u4e3a \\(1\\) \u4e2a\u7ec6\u80de\uff0c\u5206\u88c2\u4e00\u8f6e\u540e\u53d8\u4e3a \\(2\\) \u4e2a\uff0c\u5206\u88c2\u4e24\u8f6e\u540e\u53d8\u4e3a \\(4\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5206\u88c2 \\(n\\) \u8f6e\u540e\u6709 \\(2^n\\) \u4e2a\u7ec6\u80de\u3002</p> <p>\u56fe 2-11 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u7ec6\u80de\u5206\u88c2\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def exponential(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    base = 1\n    # \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in range(n):\n        for _ in range(base):\n            count += 1\n        base *= 2\n    # count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0, base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Exponential(int n) {\n    int count = 0, bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc exponential(n int) int {\n    count, base := 0, 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for i := 0; i &lt; n; i++ {\n        for j := 0; j &lt; base; j++ {\n            count++\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc exponential(n: Int) -&gt; Int {\n    var count = 0\n    var base = 1\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0 ..&lt; n {\n        for _ in 0 ..&lt; base {\n            count += 1\n        }\n        base *= 2\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n) {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction exponential(n: number): number {\n    let count = 0,\n        base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = 0; j &lt; base; j++) {\n            count++;\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n  int count = 0, base = 1;\n  // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n  for (var i = 0; i &lt; n; i++) {\n    for (var j = 0; j &lt; base; j++) {\n      count++;\n    }\n    base *= 2;\n  }\n  // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn exponential(n: i32) -&gt; i32 {\n    let mut count = 0;\n    let mut base = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for _ in 0..n {\n        for _ in 0..base {\n            count += 1\n        }\n        base *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint exponential(int n) {\n    int count = 0;\n    int bas = 1;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    for (int i = 0; i &lt; n; i++) {\n        for (int j = 0; j &lt; bas; j++) {\n            count++;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn exponential(n: i32) i32 {\n    var count: i32 = 0;\n    var bas: i32 = 1;\n    var i: i32 = 0;\n    // \u7ec6\u80de\u6bcf\u8f6e\u4e00\u5206\u4e3a\u4e8c\uff0c\u5f62\u6210\u6570\u5217 1, 2, 4, 8, ..., 2^(n-1)\n    while (i &lt; n) : (i += 1) {\n        var j: i32 = 0;\n        while (j &lt; bas) : (j += 1) {\n            count += 1;\n        }\n        bas *= 2;\n    }\n    // count = 1 + 2 + 4 + 8 + .. + 2^(n-1) = 2^n - 1\n    return count;\n}\n</code></pre> <p></p> <p> \u56fe 2-11 \u00a0 \u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u5728\u5b9e\u9645\u7b97\u6cd5\u4e2d\uff0c\u6307\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4f8b\u5982\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u5176\u9012\u5f52\u5730\u4e00\u5206\u4e3a\u4e8c\uff0c\u7ecf\u8fc7 \\(n\\) \u6b21\u5206\u88c2\u540e\u505c\u6b62\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def exp_recur(n: int) -&gt; int:\n    \"\"\"\u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 1:\n        return 1\n    return exp_recur(n - 1) + exp_recur(n - 1) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint ExpRecur(int n) {\n    if (n == 1) return 1;\n    return ExpRecur(n - 1) + ExpRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc expRecur(n int) int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n-1) + expRecur(n-1) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc expRecur(n: Int) -&gt; Int {\n    if n == 1 {\n        return 1\n    }\n    return expRecur(n: n - 1) + expRecur(n: n - 1) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n) {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction expRecur(n: number): number {\n    if (n === 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n  if (n == 1) return 1;\n  return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn exp_recur(n: i32) -&gt; i32 {\n    if n == 1 {\n        return 1;\n    }\n    exp_recur(n - 1) + exp_recur(n - 1) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint expRecur(int n) {\n    if (n == 1)\n        return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> time_complexity.zig<pre><code>// \u6307\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn expRecur(n: i32) i32 {\n    if (n == 1) return 1;\n    return expRecur(n - 1) + expRecur(n - 1) + 1;\n}\n</code></pre> <p>\u6307\u6570\u9636\u589e\u957f\u975e\u5e38\u8fc5\u901f\uff0c\u5728\u7a77\u4e3e\u6cd5\uff08\u66b4\u529b\u641c\u7d22\u3001\u56de\u6eaf\u7b49\uff09\u4e2d\u6bd4\u8f83\u5e38\u89c1\u3002\u5bf9\u4e8e\u6570\u636e\u89c4\u6a21\u8f83\u5927\u7684\u95ee\u9898\uff0c\u6307\u6570\u9636\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff0c\u901a\u5e38\u9700\u8981\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u6216\u8d2a\u5fc3\u7b97\u6cd5\u7b49\u6765\u89e3\u51b3\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#5-olog-n","title":"5. \u00a0 \u5bf9\u6570\u9636 \\(O(\\log n)\\)","text":"<p>\u4e0e\u6307\u6570\u9636\u76f8\u53cd\uff0c\u5bf9\u6570\u9636\u53cd\u6620\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u60c5\u51b5\u3002\u8bbe\u8f93\u5165\u6570\u636e\u5927\u5c0f\u4e3a \\(n\\) \uff0c\u7531\u4e8e\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\uff0c\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u662f \\(\\log_2 n\\) \uff0c\u5373 \\(2^n\\) \u7684\u53cd\u51fd\u6570\u3002</p> <p>\u56fe 2-12 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6a21\u62df\u4e86\u201c\u6bcf\u8f6e\u7f29\u51cf\u5230\u4e00\u534a\u201d\u7684\u8fc7\u7a0b\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 n)\\) \uff0c\u7b80\u8bb0\u4e3a \\(O(\\log n)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def logarithmic(n: float) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\"\"\"\n    count = 0\n    while n &gt; 1:\n        n = n / 2\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint Logarithmic(float n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n /= 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09*/\nfunc logarithmic(n float64) int {\n    count := 0\n    for n &gt; 1 {\n        n = n / 2\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunc logarithmic(n: Double) -&gt; Int {\n    var count = 0\n    var n = n\n    while n &gt; 1 {\n        n = n / 2\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n) {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfunction logarithmic(n: number): number {\n    let count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(num n) {\n  int count = 0;\n  while (n &gt; 1) {\n    n = n / 2;\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nfn logarithmic(mut n: f32) -&gt; i32 {\n    let mut count = 0;\n    while n &gt; 1.0 {\n        n = n / 2.0;\n        count += 1;\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09 */\nint logarithmic(float n) {\n    int count = 0;\n    while (n &gt; 1) {\n        n = n / 2;\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u5faa\u73af\u5b9e\u73b0\uff09\nfn logarithmic(n: f32) i32 {\n    var count: i32 = 0;\n    var n_var = n;\n    while (n_var &gt; 1)\n    {\n        n_var = n_var / 2;\n        count +=1;\n    }\n    return count;\n}\n</code></pre> <p></p> <p> \u56fe 2-12 \u00a0 \u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e0e\u6307\u6570\u9636\u7c7b\u4f3c\uff0c\u5bf9\u6570\u9636\u4e5f\u5e38\u51fa\u73b0\u4e8e\u9012\u5f52\u51fd\u6570\u4e2d\u3002\u4ee5\u4e0b\u4ee3\u7801\u5f62\u6210\u4e86\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(\\log_2 n\\) \u7684\u9012\u5f52\u6811\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def log_recur(n: float) -&gt; int:\n    \"\"\"\u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n &lt;= 1:\n        return 0\n    return log_recur(n / 2) + 1\n</code></pre> time_complexity.cpp<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.java<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint LogRecur(float n) {\n    if (n &lt;= 1) return 0;\n    return LogRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.go<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09*/\nfunc logRecur(n float64) int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n/2) + 1\n}\n</code></pre> time_complexity.swift<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc logRecur(n: Double) -&gt; Int {\n    if n &lt;= 1 {\n        return 0\n    }\n    return logRecur(n: n / 2) + 1\n}\n</code></pre> time_complexity.js<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n) {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction logRecur(n: number): number {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(num n) {\n  if (n &lt;= 1) return 0;\n  return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn log_recur(n: f32) -&gt; i32 {\n    if n &lt;= 1.0 {\n        return 0;\n    }\n    log_recur(n / 2.0) + 1\n}\n</code></pre> time_complexity.c<pre><code>/* \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint logRecur(float n) {\n    if (n &lt;= 1)\n        return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> time_complexity.zig<pre><code>// \u5bf9\u6570\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn logRecur(n: f32) i32 {\n    if (n &lt;= 1) return 0;\n    return logRecur(n / 2) + 1;\n}\n</code></pre> <p>\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u7b97\u6cd5\u4e2d\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u591a\u201d\u548c\u201c\u5316\u7e41\u4e3a\u7b80\u201d\u7684\u7b97\u6cd5\u601d\u60f3\u3002\u5b83\u589e\u957f\u7f13\u6162\uff0c\u662f\u4ec5\u6b21\u4e8e\u5e38\u6570\u9636\u7684\u7406\u60f3\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <p>\\(O(\\log n)\\) \u7684\u5e95\u6570\u662f\u591a\u5c11\uff1f</p> <p>\u51c6\u786e\u6765\u8bf4\uff0c\u201c\u4e00\u5206\u4e3a \\(m\\)\u201d\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log_m n)\\) \u3002\u800c\u901a\u8fc7\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5177\u6709\u4e0d\u540c\u5e95\u6570\u3001\u76f8\u7b49\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff1a</p> \\[ O(\\log_m n) = O(\\log_k n / \\log_k m) = O(\\log_k n) \\] <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5e95\u6570 \\(m\\) \u53ef\u4ee5\u5728\u4e0d\u5f71\u54cd\u590d\u6742\u5ea6\u7684\u524d\u63d0\u4e0b\u8f6c\u6362\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u7701\u7565\u5e95\u6570 \\(m\\) \uff0c\u5c06\u5bf9\u6570\u9636\u76f4\u63a5\u8bb0\u4e3a \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#6-on-log-n","title":"6. \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636 \\(O(n \\log n)\\)","text":"<p>\u7ebf\u6027\u5bf9\u6570\u9636\u5e38\u51fa\u73b0\u4e8e\u5d4c\u5957\u5faa\u73af\u4e2d\uff0c\u4e24\u5c42\u5faa\u73af\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u522b\u4e3a \\(O(\\log n)\\) \u548c \\(O(n)\\) \u3002\u76f8\u5173\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def linear_log_recur(n: float) -&gt; int:\n    \"\"\"\u7ebf\u6027\u5bf9\u6570\u9636\"\"\"\n    if n &lt;= 1:\n        return 1\n    count: int = linear_log_recur(n // 2) + linear_log_recur(n // 2)\n    for _ in range(n):\n        count += 1\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint LinearLogRecur(float n) {\n    if (n &lt;= 1) return 1;\n    int count = LinearLogRecur(n / 2) + LinearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n float64) int {\n    if n &lt;= 1 {\n        return 1\n    }\n    count := linearLogRecur(n/2) + linearLogRecur(n/2)\n    for i := 0.0; i &lt; n; i++ {\n        count++\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunc linearLogRecur(n: Double) -&gt; Int {\n    if n &lt;= 1 {\n        return 1\n    }\n    var count = linearLogRecur(n: n / 2) + linearLogRecur(n: n / 2)\n    for _ in stride(from: 0, to: n, by: 1) {\n        count += 1\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n) {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfunction linearLogRecur(n: number): number {\n    if (n &lt;= 1) return 1;\n    let count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (let i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(num n) {\n  if (n &lt;= 1) return 1;\n  int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n  for (var i = 0; i &lt; n; i++) {\n    count++;\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nfn linear_log_recur(n: f32) -&gt; i32 {\n    if n &lt;= 1.0 {\n        return 1;\n    }\n    let mut count = linear_log_recur(n / 2.0) + linear_log_recur(n / 2.0);\n    for _ in 0 ..n as i32 {\n        count += 1;\n    }\n    return count\n}\n</code></pre> time_complexity.c<pre><code>/* \u7ebf\u6027\u5bf9\u6570\u9636 */\nint linearLogRecur(float n) {\n    if (n &lt;= 1)\n        return 1;\n    int count = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    for (int i = 0; i &lt; n; i++) {\n        count++;\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u7ebf\u6027\u5bf9\u6570\u9636\nfn linearLogRecur(n: f32) i32 {\n    if (n &lt;= 1) return 1;\n    var count: i32 = linearLogRecur(n / 2) + linearLogRecur(n / 2);\n    var i: f32 = 0;\n    while (i &lt; n) : (i += 1) {\n        count += 1;\n    }\n    return count;\n}\n</code></pre> <p>\u56fe 2-13 \u5c55\u793a\u4e86\u7ebf\u6027\u5bf9\u6570\u9636\u7684\u751f\u6210\u65b9\u5f0f\u3002\u4e8c\u53c9\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u603b\u6570\u90fd\u4e3a \\(n\\) \uff0c\u6811\u5171\u6709 \\(\\log_2 n + 1\\) \u5c42\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p></p> <p> \u56fe 2-13 \u00a0 \u7ebf\u6027\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u7b49\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#7-on","title":"7. \u00a0 \u9636\u4e58\u9636 \\(O(n!)\\)","text":"<p>\u9636\u4e58\u9636\u5bf9\u5e94\u6570\u5b66\u4e0a\u7684\u201c\u5168\u6392\u5217\u201d\u95ee\u9898\u3002\u7ed9\u5b9a \\(n\\) \u4e2a\u4e92\u4e0d\u91cd\u590d\u7684\u5143\u7d20\uff0c\u6c42\u5176\u6240\u6709\u53ef\u80fd\u7684\u6392\u5217\u65b9\u6848\uff0c\u65b9\u6848\u6570\u91cf\u4e3a\uff1a</p> \\[ n! = n \\times (n - 1) \\times (n - 2) \\times \\dots \\times 2 \\times 1 \\] <p>\u9636\u4e58\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u5b9e\u73b0\u3002\u5982\u56fe 2-14 \u548c\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u7b2c\u4e00\u5c42\u5206\u88c2\u51fa \\(n\\) \u4e2a\uff0c\u7b2c\u4e8c\u5c42\u5206\u88c2\u51fa \\(n - 1\\) \u4e2a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u7b2c \\(n\\) \u5c42\u65f6\u505c\u6b62\u5206\u88c2\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig time_complexity.py<pre><code>def factorial_recur(n: int) -&gt; int:\n    \"\"\"\u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\"\"\"\n    if n == 0:\n        return 1\n    count = 0\n    # \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in range(n):\n        count += factorial_recur(n - 1)\n    return count\n</code></pre> time_complexity.cpp<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.java<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.cs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint FactorialRecur(int n) {\n    if (n == 0) return 1;\n    int count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (int i = 0; i &lt; n; i++) {\n        count += FactorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.go<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n int) int {\n    if n == 0 {\n        return 1\n    }\n    count := 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for i := 0; i &lt; n; i++ {\n        count += factorialRecur(n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.swift<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunc factorialRecur(n: Int) -&gt; Int {\n    if n == 0 {\n        return 1\n    }\n    var count = 0\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0 ..&lt; n {\n        count += factorialRecur(n: n - 1)\n    }\n    return count\n}\n</code></pre> time_complexity.js<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n) {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.ts<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfunction factorialRecur(n: number): number {\n    if (n === 0) return 1;\n    let count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for (let i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.dart<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n  if (n == 0) return 1;\n  int count = 0;\n  // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n  for (var i = 0; i &lt; n; i++) {\n    count += factorialRecur(n - 1);\n  }\n  return count;\n}\n</code></pre> time_complexity.rs<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nfn factorial_recur(n: i32) -&gt; i32 {\n    if n == 0 {\n        return 1;\n    }\n    let mut count = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    for _ in 0..n {\n        count += factorial_recur(n - 1);\n    }\n    count\n}\n</code></pre> time_complexity.c<pre><code>/* \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09 */\nint factorialRecur(int n) {\n    if (n == 0)\n        return 1;\n    int count = 0;\n    for (int i = 0; i &lt; n; i++) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> time_complexity.zig<pre><code>// \u9636\u4e58\u9636\uff08\u9012\u5f52\u5b9e\u73b0\uff09\nfn factorialRecur(n: i32) i32 {\n    if (n == 0) return 1;\n    var count: i32 = 0;\n    var i: i32 = 0;\n    // \u4ece 1 \u4e2a\u5206\u88c2\u51fa n \u4e2a\n    while (i &lt; n) : (i += 1) {\n        count += factorialRecur(n - 1);\n    }\n    return count;\n}\n</code></pre> <p></p> <p> \u56fe 2-14 \u00a0 \u9636\u4e58\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6 </p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u5f53 \\(n \\geq 4\\) \u65f6\u6052\u6709 \\(n! &gt; 2^n\\) \uff0c\u6240\u4ee5\u9636\u4e58\u9636\u6bd4\u6307\u6570\u9636\u589e\u957f\u5f97\u66f4\u5feb\uff0c\u5728 \\(n\\) \u8f83\u5927\u65f6\u4e5f\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002</p>"},{"location":"chapter_computational_complexity/time_complexity/#235","title":"2.3.5 \u00a0 \u6700\u5dee\u3001\u6700\u4f73\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6","text":"<p>\u7b97\u6cd5\u7684\u65f6\u95f4\u6548\u7387\u5f80\u5f80\u4e0d\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u662f\u4e0e\u8f93\u5165\u6570\u636e\u7684\u5206\u5e03\u6709\u5173\u3002\u5047\u8bbe\u8f93\u5165\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d <code>nums</code> \u7531\u4ece \\(1\\) \u81f3 \\(n\\) \u7684\u6570\u5b57\u7ec4\u6210\uff0c\u6bcf\u4e2a\u6570\u5b57\u53ea\u51fa\u73b0\u4e00\u6b21\uff1b\u4f46\u5143\u7d20\u987a\u5e8f\u662f\u968f\u673a\u6253\u4e71\u7684\uff0c\u4efb\u52a1\u76ee\u6807\u662f\u8fd4\u56de\u5143\u7d20 \\(1\\) \u7684\u7d22\u5f15\u3002\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u5f53 <code>nums = [?, ?, ..., 1]</code> \uff0c\u5373\u5f53\u672b\u5c3e\u5143\u7d20\u662f \\(1\\) \u65f6\uff0c\u9700\u8981\u5b8c\u6574\u904d\u5386\u6570\u7ec4\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u5f53 <code>nums = [1, ?, ?, ...]</code> \uff0c\u5373\u5f53\u9996\u4e2a\u5143\u7d20\u4e3a \\(1\\) \u65f6\uff0c\u65e0\u8bba\u6570\u7ec4\u591a\u957f\u90fd\u4e0d\u9700\u8981\u7ee7\u7eed\u904d\u5386\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\Omega(1)\\) \u3002</li> </ul> <p>\u201c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0a\u754c\uff0c\u4f7f\u7528\u5927 \\(O\\) \u8bb0\u53f7\u8868\u793a\u3002\u76f8\u5e94\u5730\uff0c\u201c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u201d\u5bf9\u5e94\u51fd\u6570\u6e10\u8fd1\u4e0b\u754c\uff0c\u7528 \\(\\Omega\\) \u8bb0\u53f7\u8868\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig worst_best_time_complexity.py<pre><code>def random_numbers(n: int) -&gt; list[int]:\n    \"\"\"\u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a: 1, 2, ..., n \uff0c\u987a\u5e8f\u88ab\u6253\u4e71\"\"\"\n    # \u751f\u6210\u6570\u7ec4 nums =: 1, 2, 3, ..., n\n    nums = [i for i in range(1, n + 1)]\n    # \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    random.shuffle(nums)\n    return nums\n\ndef find_one(nums: list[int]) -&gt; int:\n    \"\"\"\u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\"\"\"\n    for i in range(len(nums)):\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        # \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1:\n            return i\n    return -1\n</code></pre> worst_best_time_complexity.cpp<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nvector&lt;int&gt; randomNumbers(int n) {\n    vector&lt;int&gt; nums(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u4f7f\u7528\u7cfb\u7edf\u65f6\u95f4\u751f\u6210\u968f\u673a\u79cd\u5b50\n    unsigned seed = chrono::system_clock::now().time_since_epoch().count();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    shuffle(nums.begin(), nums.end(), default_random_engine(seed));\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(vector&lt;int&gt; &amp;nums) {\n    for (int i = 0; i &lt; nums.size(); i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.java<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] randomNumbers(int n) {\n    Integer[] nums = new Integer[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    Collections.shuffle(Arrays.asList(nums));\n    // Integer[] -&gt; int[]\n    int[] res = new int[n];\n    for (int i = 0; i &lt; n; i++) {\n        res[i] = nums[i];\n    }\n    return res;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int[] nums) {\n    for (int i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.cs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint[] RandomNumbers(int n) {\n    int[] nums = new int[n];\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = 0; i &lt; nums.Length; i++) {\n        int index = new Random().Next(i, nums.Length);\n        (nums[i], nums[index]) = (nums[index], nums[i]);\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint FindOne(int[] nums) {\n    for (int i = 0; i &lt; nums.Length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.go<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n int) []int {\n    nums := make([]int, n)\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for i := 0; i &lt; n; i++ {\n        nums[i] = i + 1\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    rand.Shuffle(len(nums), func(i, j int) {\n        nums[i], nums[j] = nums[j], nums[i]\n    })\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums []int) int {\n    for i := 0; i &lt; len(nums); i++ {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.swift<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunc randomNumbers(n: Int) -&gt; [Int] {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    var nums = Array(1 ... n)\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle()\n    return nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunc findOne(nums: [Int]) -&gt; Int {\n    for i in nums.indices {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return i\n        }\n    }\n    return -1\n}\n</code></pre> worst_best_time_complexity.js<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n) {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums) {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.ts<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfunction randomNumbers(n: number): number[] {\n    const nums = Array(n);\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (let i = 0; i &lt; n; i++) {\n        const r = Math.floor(Math.random() * (i + 1));\n        const temp = nums[i];\n        nums[i] = nums[r];\n        nums[r] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfunction findOne(nums: number[]): number {\n    for (let i = 0; i &lt; nums.length; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] === 1) {\n            return i;\n        }\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.dart<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nList&lt;int&gt; randomNumbers(int n) {\n  final nums = List.filled(n, 0);\n  // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n  for (var i = 0; i &lt; n; i++) {\n    nums[i] = i + 1;\n  }\n  // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n  nums.shuffle();\n\n  return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(List&lt;int&gt; nums) {\n  for (var i = 0; i &lt; nums.length; i++) {\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n    // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n    if (nums[i] == 1) return i;\n  }\n\n  return -1;\n}\n</code></pre> worst_best_time_complexity.rs<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nfn random_numbers(n: i32) -&gt; Vec&lt;i32&gt; {\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    let mut nums = (1..=n).collect::&lt;Vec&lt;i32&gt;&gt;();\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    nums.shuffle(&amp;mut thread_rng());\n    nums\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nfn find_one(nums: &amp;[i32]) -&gt; Option&lt;usize&gt; {\n    for i in 0..nums.len() {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if nums[i] == 1 {\n            return Some(i);\n        }\n    }\n    None\n}\n</code></pre> worst_best_time_complexity.c<pre><code>/* \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71 */\nint *randomNumbers(int n) {\n    // \u5206\u914d\u5806\u533a\u5185\u5b58\uff08\u521b\u5efa\u4e00\u7ef4\u53ef\u53d8\u957f\u6570\u7ec4\uff1a\u6570\u7ec4\u4e2d\u5143\u7d20\u6570\u91cf\u4e3a n \uff0c\u5143\u7d20\u7c7b\u578b\u4e3a int \uff09\n    int *nums = (int *)malloc(n * sizeof(int));\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = i + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    for (int i = n - 1; i &gt; 0; i--) {\n        int j = rand() % (i + 1);\n        int temp = nums[i];\n        nums[i] = nums[j];\n        nums[j] = temp;\n    }\n    return nums;\n}\n\n/* \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15 */\nint findOne(int *nums, int n) {\n    for (int i = 0; i &lt; n; i++) {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (nums[i] == 1)\n            return i;\n    }\n    return -1;\n}\n</code></pre> worst_best_time_complexity.zig<pre><code>// \u751f\u6210\u4e00\u4e2a\u6570\u7ec4\uff0c\u5143\u7d20\u4e3a { 1, 2, ..., n }\uff0c\u987a\u5e8f\u88ab\u6253\u4e71\nfn randomNumbers(comptime n: usize) [n]i32 {\n    var nums: [n]i32 = undefined;\n    // \u751f\u6210\u6570\u7ec4 nums = { 1, 2, 3, ..., n }\n    for (&amp;nums, 0..) |*num, i| {\n        num.* = @as(i32, @intCast(i)) + 1;\n    }\n    // \u968f\u673a\u6253\u4e71\u6570\u7ec4\u5143\u7d20\n    const rand = std.crypto.random;\n    rand.shuffle(i32, &amp;nums);\n    return nums;\n}\n\n// \u67e5\u627e\u6570\u7ec4 nums \u4e2d\u6570\u5b57 1 \u6240\u5728\u7d22\u5f15\nfn findOne(nums: []i32) i32 {\n    for (nums, 0..) |num, i| {\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5934\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 O(1)\n        // \u5f53\u5143\u7d20 1 \u5728\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u8fbe\u5230\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6 O(n)\n        if (num == 1) return @intCast(i);\n    }\n    return -1;\n}\n</code></pre> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6211\u4eec\u5728\u5b9e\u9645\u4e2d\u5f88\u5c11\u4f7f\u7528\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u56e0\u4e3a\u901a\u5e38\u53ea\u6709\u5728\u5f88\u5c0f\u6982\u7387\u4e0b\u624d\u80fd\u8fbe\u5230\uff0c\u53ef\u80fd\u4f1a\u5e26\u6765\u4e00\u5b9a\u7684\u8bef\u5bfc\u6027\u3002\u800c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u4e3a\u5b9e\u7528\uff0c\u56e0\u4e3a\u5b83\u7ed9\u51fa\u4e86\u4e00\u4e2a\u6548\u7387\u5b89\u5168\u503c\uff0c\u8ba9\u6211\u4eec\u53ef\u4ee5\u653e\u5fc3\u5730\u4f7f\u7528\u7b97\u6cd5\u3002</p> <p>\u4ece\u4e0a\u8ff0\u793a\u4f8b\u53ef\u4ee5\u770b\u51fa\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ea\u51fa\u73b0\u4e8e\u201c\u7279\u6b8a\u7684\u6570\u636e\u5206\u5e03\u201d\uff0c\u8fd9\u4e9b\u60c5\u51b5\u7684\u51fa\u73b0\u6982\u7387\u53ef\u80fd\u5f88\u5c0f\uff0c\u5e76\u4e0d\u80fd\u771f\u5b9e\u5730\u53cd\u6620\u7b97\u6cd5\u8fd0\u884c\u6548\u7387\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f53\u73b0\u7b97\u6cd5\u5728\u968f\u673a\u8f93\u5165\u6570\u636e\u4e0b\u7684\u8fd0\u884c\u6548\u7387\uff0c\u7528 \\(\\Theta\\) \u8bb0\u53f7\u6765\u8868\u793a\u3002</p> <p>\u5bf9\u4e8e\u90e8\u5206\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u7b97\u51fa\u968f\u673a\u6570\u636e\u5206\u5e03\u4e0b\u7684\u5e73\u5747\u60c5\u51b5\u3002\u6bd4\u5982\u4e0a\u8ff0\u793a\u4f8b\uff0c\u7531\u4e8e\u8f93\u5165\u6570\u7ec4\u662f\u88ab\u6253\u4e71\u7684\uff0c\u56e0\u6b64\u5143\u7d20 \\(1\\) \u51fa\u73b0\u5728\u4efb\u610f\u7d22\u5f15\u7684\u6982\u7387\u90fd\u662f\u76f8\u7b49\u7684\uff0c\u90a3\u4e48\u7b97\u6cd5\u7684\u5e73\u5747\u5faa\u73af\u6b21\u6570\u5c31\u662f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a \\(n / 2\\) \uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(\\Theta(n / 2) = \\Theta(n)\\) \u3002</p> <p>\u4f46\u5bf9\u4e8e\u8f83\u4e3a\u590d\u6742\u7684\u7b97\u6cd5\uff0c\u8ba1\u7b97\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5f80\u5f80\u6bd4\u8f83\u56f0\u96be\uff0c\u56e0\u4e3a\u5f88\u96be\u5206\u6790\u51fa\u5728\u6570\u636e\u5206\u5e03\u4e0b\u7684\u6574\u4f53\u6570\u5b66\u671f\u671b\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4f5c\u4e3a\u7b97\u6cd5\u6548\u7387\u7684\u8bc4\u5224\u6807\u51c6\u3002</p> <p>\u4e3a\u4ec0\u4e48\u5f88\u5c11\u770b\u5230 \\(\\Theta\\) \u7b26\u53f7\uff1f</p> <p>\u53ef\u80fd\u7531\u4e8e \\(O\\) \u7b26\u53f7\u8fc7\u4e8e\u6717\u6717\u4e0a\u53e3\uff0c\u56e0\u6b64\u6211\u4eec\u5e38\u5e38\u4f7f\u7528\u5b83\u6765\u8868\u793a\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4f46\u4ece\u4e25\u683c\u610f\u4e49\u4e0a\u8bb2\uff0c\u8fd9\u79cd\u505a\u6cd5\u5e76\u4e0d\u89c4\u8303\u3002\u5728\u672c\u4e66\u548c\u5176\u4ed6\u8d44\u6599\u4e2d\uff0c\u82e5\u9047\u5230\u7c7b\u4f3c\u201c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u201d\u7684\u8868\u8ff0\uff0c\u8bf7\u5c06\u5176\u76f4\u63a5\u7406\u89e3\u4e3a \\(\\Theta(n)\\) \u3002</p>"},{"location":"chapter_data_structure/","title":"\u7b2c 3 \u7ae0 \u00a0 \u6570\u636e\u7ed3\u6784","text":"<p>Abstract</p> <p>\u6570\u636e\u7ed3\u6784\u5982\u540c\u4e00\u526f\u7a33\u56fa\u800c\u591a\u6837\u7684\u6846\u67b6\u3002</p> <p>\u5b83\u4e3a\u6570\u636e\u7684\u6709\u5e8f\u7ec4\u7ec7\u63d0\u4f9b\u4e86\u84dd\u56fe\uff0c\u7b97\u6cd5\u5f97\u4ee5\u5728\u6b64\u57fa\u7840\u4e0a\u751f\u52a8\u8d77\u6765\u3002</p>"},{"location":"chapter_data_structure/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b</li> <li>3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b</li> <li>3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *</li> <li>3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *</li> <li>3.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_data_structure/basic_data_types/","title":"3.2 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b","text":"<p>\u5f53\u8c08\u53ca\u8ba1\u7b97\u673a\u4e2d\u7684\u6570\u636e\u65f6\uff0c\u6211\u4eec\u4f1a\u60f3\u5230\u6587\u672c\u3001\u56fe\u7247\u3001\u89c6\u9891\u3001\u8bed\u97f3\u30013D \u6a21\u578b\u7b49\u5404\u79cd\u5f62\u5f0f\u3002\u5c3d\u7ba1\u8fd9\u4e9b\u6570\u636e\u7684\u7ec4\u7ec7\u5f62\u5f0f\u5404\u5f02\uff0c\u4f46\u5b83\u4eec\u90fd\u7531\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6784\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u662f CPU \u53ef\u4ee5\u76f4\u63a5\u8fdb\u884c\u8fd0\u7b97\u7684\u7c7b\u578b\uff0c\u5728\u7b97\u6cd5\u4e2d\u76f4\u63a5\u88ab\u4f7f\u7528\uff0c\u4e3b\u8981\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \u3002</li> <li>\u6d6e\u70b9\u6570\u7c7b\u578b <code>float</code>\u3001<code>double</code> \uff0c\u7528\u4e8e\u8868\u793a\u5c0f\u6570\u3002</li> <li>\u5b57\u7b26\u7c7b\u578b <code>char</code> \uff0c\u7528\u4e8e\u8868\u793a\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u6bcd\u3001\u6807\u70b9\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002</li> <li>\u5e03\u5c14\u7c7b\u578b <code>bool</code> \uff0c\u7528\u4e8e\u8868\u793a\u201c\u662f\u201d\u4e0e\u201c\u5426\u201d\u5224\u65ad\u3002</li> </ul> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4ee5\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u3002\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u5373\u4e3a \\(1\\) \u6bd4\u7279\u3002\u5728\u7edd\u5927\u591a\u6570\u73b0\u4ee3\u64cd\u4f5c\u7cfb\u7edf\u4e2d\uff0c\\(1\\) \u5b57\u8282\uff08byte\uff09\u7531 \\(8\\) \u6bd4\u7279\uff08bit\uff09\u7ec4\u6210\u3002</p> <p>\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5176\u5360\u7528\u7684\u7a7a\u95f4\u5927\u5c0f\u3002\u4e0b\u9762\u4ee5 Java \u4e3a\u4f8b\u3002</p> <ul> <li>\u6574\u6570\u7c7b\u578b <code>byte</code> \u5360\u7528 \\(1\\) byte = \\(8\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{8}\\) \u4e2a\u6570\u5b57\u3002</li> <li>\u6574\u6570\u7c7b\u578b <code>int</code> \u5360\u7528 \\(4\\) bytes = \\(32\\) bits \uff0c\u53ef\u4ee5\u8868\u793a \\(2^{32}\\) \u4e2a\u6570\u5b57\u3002</li> </ul> <p>\u8868 3-1 \u5217\u4e3e\u4e86 Java \u4e2d\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u3002\u6b64\u8868\u683c\u65e0\u987b\u6b7b\u8bb0\u786c\u80cc\uff0c\u5927\u81f4\u7406\u89e3\u5373\u53ef\uff0c\u9700\u8981\u65f6\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u6765\u56de\u5fc6\u3002</p> <p> \u8868 3-1 \u00a0 \u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u5360\u7528\u7a7a\u95f4\u548c\u53d6\u503c\u8303\u56f4 </p> \u7c7b\u578b \u7b26\u53f7 \u5360\u7528\u7a7a\u95f4 \u6700\u5c0f\u503c \u6700\u5927\u503c \u9ed8\u8ba4\u503c \u6574\u6570 <code>byte</code> 1 byte \\(-2^7\\) (\\(-128\\)) \\(2^7 - 1\\) (\\(127\\)) \\(0\\) <code>short</code> 2 bytes \\(-2^{15}\\) \\(2^{15} - 1\\) \\(0\\) <code>int</code> 4 bytes \\(-2^{31}\\) \\(2^{31} - 1\\) \\(0\\) <code>long</code> 8 bytes \\(-2^{63}\\) \\(2^{63} - 1\\) \\(0\\) \u6d6e\u70b9\u6570 <code>float</code> 4 bytes \\(1.175 \\times 10^{-38}\\) \\(3.403 \\times 10^{38}\\) \\(0.0\\text{f}\\) <code>double</code> 8 bytes \\(2.225 \\times 10^{-308}\\) \\(1.798 \\times 10^{308}\\) \\(0.0\\) \u5b57\u7b26 <code>char</code> 2 bytes \\(0\\) \\(2^{16} - 1\\) \\(0\\) \u5e03\u5c14 <code>bool</code> 1 byte \\(\\text{false}\\) \\(\\text{true}\\) \\(\\text{false}\\) <p>\u8bf7\u6ce8\u610f\uff0c\u8868 3-1 \u9488\u5bf9\u7684\u662f Java \u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u60c5\u51b5\u3002\u6bcf\u79cd\u7f16\u7a0b\u8bed\u8a00\u6709\u5404\u81ea\u7684\u6570\u636e\u7c7b\u578b\u5b9a\u4e49\uff0c\u5b83\u4eec\u7684\u5360\u7528\u7a7a\u95f4\u3001\u53d6\u503c\u8303\u56f4\u548c\u9ed8\u8ba4\u503c\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5728 Python \u4e2d\uff0c\u6574\u6570\u7c7b\u578b <code>int</code> \u53ef\u4ee5\u662f\u4efb\u610f\u5927\u5c0f\uff0c\u53ea\u53d7\u9650\u4e8e\u53ef\u7528\u5185\u5b58\uff1b\u6d6e\u70b9\u6570 <code>float</code> \u662f\u53cc\u7cbe\u5ea6 64 \u4f4d\uff1b\u6ca1\u6709 <code>char</code> \u7c7b\u578b\uff0c\u5355\u4e2a\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32 <code>str</code> \u3002</li> <li>C \u548c C++ \u672a\u660e\u786e\u89c4\u5b9a\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5927\u5c0f\uff0c\u800c\u56e0\u5b9e\u73b0\u548c\u5e73\u53f0\u5404\u5f02\u3002\u8868 3-1 \u9075\u5faa LP64 \u6570\u636e\u6a21\u578b\uff0c\u5176\u7528\u4e8e\u5305\u62ec Linux \u548c macOS \u5728\u5185\u7684 Unix 64 \u4f4d\u64cd\u4f5c\u7cfb\u7edf\u3002</li> <li>\u5b57\u7b26 <code>char</code> \u7684\u5927\u5c0f\u5728 C \u548c C++ \u4e2d\u4e3a 1 \u5b57\u8282\uff0c\u5728\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e2d\u53d6\u51b3\u4e8e\u7279\u5b9a\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\uff0c\u8be6\u89c1\u201c\u5b57\u7b26\u7f16\u7801\u201d\u7ae0\u8282\u3002</li> <li>\u5373\u4f7f\u8868\u793a\u5e03\u5c14\u91cf\u4ec5\u9700 1 \u4f4d\uff08\\(0\\) \u6216 \\(1\\)\uff09\uff0c\u5b83\u5728\u5185\u5b58\u4e2d\u901a\u5e38\u5b58\u50a8\u4e3a 1 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a CPU \u901a\u5e38\u5c06 1 \u5b57\u8282\u4f5c\u4e3a\u6700\u5c0f\u5bfb\u5740\u5185\u5b58\u5355\u5143\u3002</li> </ul> <p>\u90a3\u4e48\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u95f4\u6709\u4ec0\u4e48\u8054\u7cfb\u5462\uff1f\u6211\u4eec\u77e5\u9053\uff0c\u6570\u636e\u7ed3\u6784\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u4e0e\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002\u8fd9\u53e5\u8bdd\u7684\u4e3b\u8bed\u662f\u201c\u7ed3\u6784\u201d\u800c\u975e\u201c\u6570\u636e\u201d\u3002</p> <p>\u5982\u679c\u60f3\u8868\u793a\u201c\u4e00\u6392\u6570\u5b57\u201d\uff0c\u6211\u4eec\u81ea\u7136\u4f1a\u60f3\u5230\u4f7f\u7528\u6570\u7ec4\u3002\u8fd9\u662f\u56e0\u4e3a\u6570\u7ec4\u7684\u7ebf\u6027\u7ed3\u6784\u53ef\u4ee5\u8868\u793a\u6570\u5b57\u7684\u76f8\u90bb\u5173\u7cfb\u548c\u987a\u5e8f\u5173\u7cfb\uff0c\u4f46\u81f3\u4e8e\u5b58\u50a8\u7684\u5185\u5bb9\u662f\u6574\u6570 <code>int</code>\u3001\u5c0f\u6570 <code>float</code> \u6216\u662f\u5b57\u7b26 <code>char</code> \uff0c\u5219\u4e0e\u201c\u6570\u636e\u7ed3\u6784\u201d\u65e0\u5173\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u57fa\u672c\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u5185\u5bb9\u7c7b\u578b\u201d\uff0c\u800c\u6570\u636e\u7ed3\u6784\u63d0\u4f9b\u4e86\u6570\u636e\u7684\u201c\u7ec4\u7ec7\u65b9\u5f0f\u201d\u3002\u4f8b\u5982\u4ee5\u4e0b\u4ee3\u7801\uff0c\u6211\u4eec\u7528\u76f8\u540c\u7684\u6570\u636e\u7ed3\u6784\uff08\u6570\u7ec4\uff09\u6765\u5b58\u50a8\u4e0e\u8868\u793a\u4e0d\u540c\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u5305\u62ec <code>int</code>\u3001<code>float</code>\u3001<code>char</code>\u3001<code>bool</code> \u7b49\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nnumbers: list[int] = [0] * 5\ndecimals: list[float] = [0.0] * 5\n# Python \u7684\u5b57\u7b26\u5b9e\u9645\u4e0a\u662f\u957f\u5ea6\u4e3a 1 \u7684\u5b57\u7b26\u4e32\ncharacters: list[str] = ['0'] * 5\nbools: list[bool] = [False] * 5\n# Python \u7684\u5217\u8868\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\u5f15\u7528\ndata = [0, 0.0, 'a', False, ListNode(0)]\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[5];\nfloat decimals[5];\nchar characters[5];\nbool bools[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nboolean[] bools = new boolean[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint[] numbers = new int[5];\nfloat[] decimals = new float[5];\nchar[] characters = new char[5];\nbool[] bools = new bool[5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nvar numbers = [5]int{}\nvar decimals = [5]float64{}\nvar characters = [5]byte{}\nvar bools = [5]bool{}\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers = Array(repeating: Int(), count: 5)\nlet decimals = Array(repeating: Double(), count: 5)\nlet characters = Array(repeating: Character(\"a\"), count: 5)\nlet bools = Array(repeating: Bool(), count: 5)\n</code></pre> <pre><code>// JavaScript \u7684\u6570\u7ec4\u53ef\u4ee5\u81ea\u7531\u5b58\u50a8\u5404\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u548c\u5bf9\u8c61\nconst array = [0, 0.0, 'a', false];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nconst numbers: number[] = [];\nconst characters: string[] = [];\nconst bools: boolean[] = [];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nList&lt;int&gt; numbers = List.filled(5, 0);\nList&lt;double&gt; decimals = List.filled(5, 0.0);\nList&lt;String&gt; characters = List.filled(5, 'a');\nList&lt;bool&gt; bools = List.filled(5, false);\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nlet numbers: Vec&lt;i32&gt; = vec![0; 5];\nlet decimals: Vec&lt;f32&gt; = vec![0.0, 5];\nlet characters: Vec&lt;char&gt; = vec!['0'; 5];\nlet bools: Vec&lt;bool&gt; = vec![false; 5];\n</code></pre> <pre><code>// \u4f7f\u7528\u591a\u79cd\u57fa\u672c\u6570\u636e\u7c7b\u578b\u6765\u521d\u59cb\u5316\u6570\u7ec4\nint numbers[10];\nfloat decimals[10];\nchar characters[10];\nbool bools[10];\n</code></pre> <pre><code>\n</code></pre>"},{"location":"chapter_data_structure/character_encoding/","title":"3.4 \u00a0 \u5b57\u7b26\u7f16\u7801 *","text":"<p>\u5728\u8ba1\u7b97\u673a\u4e2d\uff0c\u6240\u6709\u6570\u636e\u90fd\u662f\u4ee5\u4e8c\u8fdb\u5236\u6570\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\uff0c\u5b57\u7b26 <code>char</code> \u4e5f\u4e0d\u4f8b\u5916\u3002\u4e3a\u4e86\u8868\u793a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u5efa\u7acb\u4e00\u5957\u201c\u5b57\u7b26\u96c6\u201d\uff0c\u89c4\u5b9a\u6bcf\u4e2a\u5b57\u7b26\u548c\u4e8c\u8fdb\u5236\u6570\u4e4b\u95f4\u7684\u4e00\u4e00\u5bf9\u5e94\u5173\u7cfb\u3002\u6709\u4e86\u5b57\u7b26\u96c6\u4e4b\u540e\uff0c\u8ba1\u7b97\u673a\u5c31\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5b8c\u6210\u4e8c\u8fdb\u5236\u6570\u5230\u5b57\u7b26\u7684\u8f6c\u6362\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#341-ascii","title":"3.4.1 \u00a0 ASCII \u5b57\u7b26\u96c6","text":"<p>\u300cASCII \u7801\u300d\u662f\u6700\u65e9\u51fa\u73b0\u7684\u5b57\u7b26\u96c6\uff0c\u5176\u5168\u79f0\u4e3a American Standard Code for Information Interchange\uff08\u7f8e\u56fd\u6807\u51c6\u4fe1\u606f\u4ea4\u6362\u4ee3\u7801\uff09\u3002\u5b83\u4f7f\u7528 7 \u4f4d\u4e8c\u8fdb\u5236\u6570\uff08\u4e00\u4e2a\u5b57\u8282\u7684\u4f4e 7 \u4f4d\uff09\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6700\u591a\u80fd\u591f\u8868\u793a 128 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002\u5982\u56fe 3-6 \u6240\u793a\uff0cASCII \u7801\u5305\u62ec\u82f1\u6587\u5b57\u6bcd\u7684\u5927\u5c0f\u5199\u3001\u6570\u5b57 0 ~ 9\u3001\u4e00\u4e9b\u6807\u70b9\u7b26\u53f7\uff0c\u4ee5\u53ca\u4e00\u4e9b\u63a7\u5236\u5b57\u7b26\uff08\u5982\u6362\u884c\u7b26\u548c\u5236\u8868\u7b26\uff09\u3002</p> <p></p> <p> \u56fe 3-6 \u00a0 ASCII \u7801 </p> <p>\u7136\u800c\uff0cASCII \u7801\u4ec5\u80fd\u591f\u8868\u793a\u82f1\u6587\u3002\u968f\u7740\u8ba1\u7b97\u673a\u7684\u5168\u7403\u5316\uff0c\u8bde\u751f\u4e86\u4e00\u79cd\u80fd\u591f\u8868\u793a\u66f4\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u96c6\u300cEASCII\u300d\u3002\u5b83\u5728 ASCII \u7684 7 \u4f4d\u57fa\u7840\u4e0a\u6269\u5c55\u5230 8 \u4f4d\uff0c\u80fd\u591f\u8868\u793a 256 \u4e2a\u4e0d\u540c\u7684\u5b57\u7b26\u3002</p> <p>\u5728\u4e16\u754c\u8303\u56f4\u5185\uff0c\u9646\u7eed\u51fa\u73b0\u4e86\u4e00\u6279\u9002\u7528\u4e8e\u4e0d\u540c\u5730\u533a\u7684 EASCII \u5b57\u7b26\u96c6\u3002\u8fd9\u4e9b\u5b57\u7b26\u96c6\u7684\u524d 128 \u4e2a\u5b57\u7b26\u7edf\u4e00\u4e3a ASCII \u7801\uff0c\u540e 128 \u4e2a\u5b57\u7b26\u5b9a\u4e49\u4e0d\u540c\uff0c\u4ee5\u9002\u5e94\u4e0d\u540c\u8bed\u8a00\u7684\u9700\u6c42\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#342-gbk","title":"3.4.2 \u00a0 GBK \u5b57\u7b26\u96c6","text":"<p>\u540e\u6765\u4eba\u4eec\u53d1\u73b0\uff0cEASCII \u7801\u4ecd\u7136\u65e0\u6cd5\u6ee1\u8db3\u8bb8\u591a\u8bed\u8a00\u7684\u5b57\u7b26\u6570\u91cf\u8981\u6c42\u3002\u6bd4\u5982\u6c49\u5b57\u6709\u8fd1\u5341\u4e07\u4e2a\uff0c\u5149\u65e5\u5e38\u4f7f\u7528\u7684\u5c31\u6709\u51e0\u5343\u4e2a\u3002\u4e2d\u56fd\u56fd\u5bb6\u6807\u51c6\u603b\u5c40\u4e8e 1980 \u5e74\u53d1\u5e03\u4e86\u300cGB2312\u300d\u5b57\u7b26\u96c6\uff0c\u5176\u6536\u5f55\u4e86 6763 \u4e2a\u6c49\u5b57\uff0c\u57fa\u672c\u6ee1\u8db3\u4e86\u6c49\u5b57\u7684\u8ba1\u7b97\u673a\u5904\u7406\u9700\u8981\u3002</p> <p>\u7136\u800c\uff0cGB2312 \u65e0\u6cd5\u5904\u7406\u90e8\u5206\u7f55\u89c1\u5b57\u548c\u7e41\u4f53\u5b57\u3002\u300cGBK\u300d\u5b57\u7b26\u96c6\u662f\u5728 GB2312 \u7684\u57fa\u7840\u4e0a\u6269\u5c55\u5f97\u5230\u7684\uff0c\u5b83\u5171\u6536\u5f55\u4e86 21886 \u4e2a\u6c49\u5b57\u3002\u5728 GBK \u7684\u7f16\u7801\u65b9\u6848\u4e2d\uff0cASCII \u5b57\u7b26\u4f7f\u7528\u4e00\u4e2a\u5b57\u8282\u8868\u793a\uff0c\u6c49\u5b57\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u8868\u793a\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#343-unicode","title":"3.4.3 \u00a0 Unicode \u5b57\u7b26\u96c6","text":"<p>\u968f\u7740\u8ba1\u7b97\u673a\u6280\u672f\u7684\u84ec\u52c3\u53d1\u5c55\uff0c\u5b57\u7b26\u96c6\u4e0e\u7f16\u7801\u6807\u51c6\u767e\u82b1\u9f50\u653e\uff0c\u800c\u8fd9\u5e26\u6765\u4e86\u8bb8\u591a\u95ee\u9898\u3002\u4e00\u65b9\u9762\uff0c\u8fd9\u4e9b\u5b57\u7b26\u96c6\u4e00\u822c\u53ea\u5b9a\u4e49\u4e86\u7279\u5b9a\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u65e0\u6cd5\u5728\u591a\u8bed\u8a00\u73af\u5883\u4e0b\u6b63\u5e38\u5de5\u4f5c\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u540c\u4e00\u79cd\u8bed\u8a00\u5b58\u5728\u591a\u79cd\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u5982\u679c\u4e24\u53f0\u8ba1\u7b97\u673a\u4f7f\u7528\u7684\u662f\u4e0d\u540c\u7684\u7f16\u7801\u6807\u51c6\uff0c\u5219\u5728\u4fe1\u606f\u4f20\u9012\u65f6\u5c31\u4f1a\u51fa\u73b0\u4e71\u7801\u3002</p> <p>\u90a3\u4e2a\u65f6\u4ee3\u7684\u7814\u7a76\u4eba\u5458\u5c31\u5728\u60f3\uff1a\u5982\u679c\u63a8\u51fa\u4e00\u4e2a\u8db3\u591f\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\uff0c\u5c06\u4e16\u754c\u8303\u56f4\u5185\u7684\u6240\u6709\u8bed\u8a00\u548c\u7b26\u53f7\u90fd\u6536\u5f55\u5176\u4e2d\uff0c\u4e0d\u5c31\u53ef\u4ee5\u89e3\u51b3\u8de8\u8bed\u8a00\u73af\u5883\u548c\u4e71\u7801\u95ee\u9898\u4e86\u5417\uff1f\u5728\u8fd9\u79cd\u60f3\u6cd5\u7684\u9a71\u52a8\u4e0b\uff0c\u4e00\u4e2a\u5927\u800c\u5168\u7684\u5b57\u7b26\u96c6 Unicode \u5e94\u8fd0\u800c\u751f\u3002</p> <p>\u300cUnicode\u300d\u7684\u4e2d\u6587\u540d\u79f0\u4e3a\u201c\u7edf\u4e00\u7801\u201d\uff0c\u7406\u8bba\u4e0a\u80fd\u5bb9\u7eb3 100 \u591a\u4e07\u4e2a\u5b57\u7b26\u3002\u5b83\u81f4\u529b\u4e8e\u5c06\u5168\u7403\u8303\u56f4\u5185\u7684\u5b57\u7b26\u7eb3\u5165\u7edf\u4e00\u7684\u5b57\u7b26\u96c6\u4e4b\u4e2d\uff0c\u63d0\u4f9b\u4e00\u79cd\u901a\u7528\u7684\u5b57\u7b26\u96c6\u6765\u5904\u7406\u548c\u663e\u793a\u5404\u79cd\u8bed\u8a00\u6587\u5b57\uff0c\u51cf\u5c11\u56e0\u4e3a\u7f16\u7801\u6807\u51c6\u4e0d\u540c\u800c\u4ea7\u751f\u7684\u4e71\u7801\u95ee\u9898\u3002</p> <p>\u81ea 1991 \u5e74\u53d1\u5e03\u4ee5\u6765\uff0cUnicode \u4e0d\u65ad\u6269\u5145\u65b0\u7684\u8bed\u8a00\u4e0e\u5b57\u7b26\u3002\u622a\u81f3 2022 \u5e74 9 \u6708\uff0cUnicode \u5df2\u7ecf\u5305\u542b 149186 \u4e2a\u5b57\u7b26\uff0c\u5305\u62ec\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\u3001\u7b26\u53f7\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u7b49\u3002\u5728\u5e9e\u5927\u7684 Unicode \u5b57\u7b26\u96c6\u4e2d\uff0c\u5e38\u7528\u7684\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\uff0c\u6709\u4e9b\u751f\u50fb\u7684\u5b57\u7b26\u5360\u7528 3 \u5b57\u8282\u751a\u81f3 4 \u5b57\u8282\u3002</p> <p>Unicode \u662f\u4e00\u79cd\u901a\u7528\u5b57\u7b26\u96c6\uff0c\u672c\u8d28\u4e0a\u662f\u7ed9\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u4e00\u4e2a\u7f16\u53f7\uff08\u79f0\u4e3a\u201c\u7801\u70b9\u201d\uff09\uff0c\u4f46\u5b83\u5e76\u6ca1\u6709\u89c4\u5b9a\u5728\u8ba1\u7b97\u673a\u4e2d\u5982\u4f55\u5b58\u50a8\u8fd9\u4e9b\u5b57\u7b26\u7801\u70b9\u3002\u6211\u4eec\u4e0d\u7981\u4f1a\u95ee\uff1a\u5f53\u591a\u79cd\u957f\u5ea6\u7684 Unicode \u7801\u70b9\u540c\u65f6\u51fa\u73b0\u5728\u4e00\u4e2a\u6587\u672c\u4e2d\u65f6\uff0c\u7cfb\u7edf\u5982\u4f55\u89e3\u6790\u5b57\u7b26\uff1f\u4f8b\u5982\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a 2 \u5b57\u8282\u7684\u7f16\u7801\uff0c\u7cfb\u7edf\u5982\u4f55\u786e\u8ba4\u5b83\u662f\u4e00\u4e2a 2 \u5b57\u8282\u7684\u5b57\u7b26\u8fd8\u662f\u4e24\u4e2a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff1f</p> <p>\u5bf9\u4e8e\u4ee5\u4e0a\u95ee\u9898\uff0c\u4e00\u79cd\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u662f\u5c06\u6240\u6709\u5b57\u7b26\u5b58\u50a8\u4e3a\u7b49\u957f\u7684\u7f16\u7801\u3002\u5982\u56fe 3-7 \u6240\u793a\uff0c\u201cHello\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u5b57\u8282\uff0c\u201c\u7b97\u6cd5\u201d\u4e2d\u7684\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9ad8\u4f4d\u586b 0 \u5c06\u201cHello \u7b97\u6cd5\u201d\u4e2d\u7684\u6240\u6709\u5b57\u7b26\u90fd\u7f16\u7801\u4e3a 2 \u5b57\u8282\u957f\u5ea6\u3002\u8fd9\u6837\u7cfb\u7edf\u5c31\u53ef\u4ee5\u6bcf\u9694 2 \u5b57\u8282\u89e3\u6790\u4e00\u4e2a\u5b57\u7b26\uff0c\u6062\u590d\u8fd9\u4e2a\u77ed\u8bed\u7684\u5185\u5bb9\u4e86\u3002</p> <p></p> <p> \u56fe 3-7 \u00a0 Unicode \u7f16\u7801\u793a\u4f8b </p> <p>\u7136\u800c ASCII \u7801\u5df2\u7ecf\u5411\u6211\u4eec\u8bc1\u660e\uff0c\u7f16\u7801\u82f1\u6587\u53ea\u9700 1 \u5b57\u8282\u3002\u82e5\u91c7\u7528\u4e0a\u8ff0\u65b9\u6848\uff0c\u82f1\u6587\u6587\u672c\u5360\u7528\u7a7a\u95f4\u7684\u5927\u5c0f\u5c06\u4f1a\u662f ASCII \u7f16\u7801\u4e0b\u7684\u4e24\u500d\uff0c\u975e\u5e38\u6d6a\u8d39\u5185\u5b58\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u66f4\u52a0\u9ad8\u6548\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002</p>"},{"location":"chapter_data_structure/character_encoding/#344-utf-8","title":"3.4.4 \u00a0 UTF-8 \u7f16\u7801","text":"<p>\u76ee\u524d\uff0cUTF-8 \u5df2\u6210\u4e3a\u56fd\u9645\u4e0a\u4f7f\u7528\u6700\u5e7f\u6cdb\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\u3002\u5b83\u662f\u4e00\u79cd\u53ef\u53d8\u957f\u5ea6\u7684\u7f16\u7801\uff0c\u4f7f\u7528 1 \u5230 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\uff0c\u6839\u636e\u5b57\u7b26\u7684\u590d\u6742\u6027\u800c\u53d8\u3002ASCII \u5b57\u7b26\u53ea\u9700 1 \u5b57\u8282\uff0c\u62c9\u4e01\u5b57\u6bcd\u548c\u5e0c\u814a\u5b57\u6bcd\u9700\u8981 2 \u5b57\u8282\uff0c\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u9700\u8981 3 \u5b57\u8282\uff0c\u5176\u4ed6\u7684\u4e00\u4e9b\u751f\u50fb\u5b57\u7b26\u9700\u8981 4 \u5b57\u8282\u3002</p> <p>UTF-8 \u7684\u7f16\u7801\u89c4\u5219\u5e76\u4e0d\u590d\u6742\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a 1 \u5b57\u8282\u7684\u5b57\u7b26\uff0c\u5c06\u6700\u9ad8\u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff0c\u5176\u4f59 7 \u4f4d\u8bbe\u7f6e\u4e3a Unicode \u7801\u70b9\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cASCII \u5b57\u7b26\u5728 Unicode \u5b57\u7b26\u96c6\u4e2d\u5360\u636e\u4e86\u524d 128 \u4e2a\u7801\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0cUTF-8 \u7f16\u7801\u53ef\u4ee5\u5411\u4e0b\u517c\u5bb9 ASCII \u7801\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 UTF-8 \u6765\u89e3\u6790\u5e74\u4ee3\u4e45\u8fdc\u7684 ASCII \u7801\u6587\u672c\u3002</li> <li>\u5bf9\u4e8e\u957f\u5ea6\u4e3a \\(n\\) \u5b57\u8282\u7684\u5b57\u7b26\uff08\u5176\u4e2d \\(n &gt; 1\\)\uff09\uff0c\u5c06\u9996\u4e2a\u5b57\u8282\u7684\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u7b2c \\(n + 1\\) \u4f4d\u8bbe\u7f6e\u4e3a \\(0\\) \uff1b\u4ece\u7b2c\u4e8c\u4e2a\u5b57\u8282\u5f00\u59cb\uff0c\u5c06\u6bcf\u4e2a\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \uff1b\u5176\u4f59\u6240\u6709\u4f4d\u7528\u4e8e\u586b\u5145\u5b57\u7b26\u7684 Unicode \u7801\u70b9\u3002</li> </ul> <p>\u56fe 3-8 \u5c55\u793a\u4e86\u201cHello\u7b97\u6cd5\u201d\u5bf9\u5e94\u7684 UTF-8 \u7f16\u7801\u3002\u89c2\u5bdf\u53d1\u73b0\uff0c\u7531\u4e8e\u6700\u9ad8 \\(n\\) \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(1\\) \uff0c\u56e0\u6b64\u7cfb\u7edf\u53ef\u4ee5\u901a\u8fc7\u8bfb\u53d6\u6700\u9ad8\u4f4d \\(1\\) \u7684\u4e2a\u6570\u6765\u89e3\u6790\u51fa\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(n\\) \u3002</p> <p>\u4f46\u4e3a\u4ec0\u4e48\u8981\u5c06\u5176\u4f59\u6240\u6709\u5b57\u8282\u7684\u9ad8 2 \u4f4d\u90fd\u8bbe\u7f6e\u4e3a \\(10\\) \u5462\uff1f\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a \\(10\\) \u80fd\u591f\u8d77\u5230\u6821\u9a8c\u7b26\u7684\u4f5c\u7528\u3002\u5047\u8bbe\u7cfb\u7edf\u4ece\u4e00\u4e2a\u9519\u8bef\u7684\u5b57\u8282\u5f00\u59cb\u89e3\u6790\u6587\u672c\uff0c\u5b57\u8282\u5934\u90e8\u7684 \\(10\\) \u80fd\u591f\u5e2e\u52a9\u7cfb\u7edf\u5feb\u901f\u5224\u65ad\u51fa\u5f02\u5e38\u3002</p> <p>\u4e4b\u6240\u4ee5\u5c06 \\(10\\) \u5f53\u4f5c\u6821\u9a8c\u7b26\uff0c\u662f\u56e0\u4e3a\u5728 UTF-8 \u7f16\u7801\u89c4\u5219\u4e0b\uff0c\u4e0d\u53ef\u80fd\u6709\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \u3002\u8fd9\u4e2a\u7ed3\u8bba\u53ef\u4ee5\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\uff1a\u5047\u8bbe\u4e00\u4e2a\u5b57\u7b26\u7684\u6700\u9ad8\u4e24\u4f4d\u662f \\(10\\) \uff0c\u8bf4\u660e\u8be5\u5b57\u7b26\u7684\u957f\u5ea6\u4e3a \\(1\\) \uff0c\u5bf9\u5e94 ASCII \u7801\u3002\u800c ASCII \u7801\u7684\u6700\u9ad8\u4f4d\u5e94\u8be5\u662f \\(0\\) \uff0c\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</p> <p></p> <p> \u56fe 3-8 \u00a0 UTF-8 \u7f16\u7801\u793a\u4f8b </p> <p>\u9664\u4e86 UTF-8 \u4e4b\u5916\uff0c\u5e38\u89c1\u7684\u7f16\u7801\u65b9\u5f0f\u8fd8\u5305\u62ec\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ul> <li>UTF-16 \u7f16\u7801\uff1a\u4f7f\u7528 2 \u6216 4 \u5b57\u8282\u6765\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u3002\u6240\u6709\u7684 ASCII \u5b57\u7b26\u548c\u5e38\u7528\u7684\u975e\u82f1\u6587\u5b57\u7b26\uff0c\u90fd\u7528 2 \u5b57\u8282\u8868\u793a\uff1b\u5c11\u6570\u5b57\u7b26\u9700\u8981\u7528\u5230 4 \u5b57\u8282\u8868\u793a\u3002\u5bf9\u4e8e 2 \u5b57\u8282\u7684\u5b57\u7b26\uff0cUTF-16 \u7f16\u7801\u4e0e Unicode \u7801\u70b9\u76f8\u7b49\u3002</li> <li>UTF-32 \u7f16\u7801\uff1a\u6bcf\u4e2a\u5b57\u7b26\u90fd\u4f7f\u7528 4 \u5b57\u8282\u3002\u8fd9\u610f\u5473\u7740 UTF-32 \u6bd4 UTF-8 \u548c UTF-16 \u66f4\u5360\u7528\u7a7a\u95f4\uff0c\u7279\u522b\u662f\u5bf9\u4e8e ASCII \u5b57\u7b26\u5360\u6bd4\u8f83\u9ad8\u7684\u6587\u672c\u3002</li> </ul> <p>\u4ece\u5b58\u50a8\u7a7a\u95f4\u5360\u7528\u7684\u89d2\u5ea6\u770b\uff0c\u4f7f\u7528 UTF-8 \u8868\u793a\u82f1\u6587\u5b57\u7b26\u975e\u5e38\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 1 \u5b57\u8282\uff1b\u4f7f\u7528 UTF-16 \u7f16\u7801\u67d0\u4e9b\u975e\u82f1\u6587\u5b57\u7b26\uff08\u4f8b\u5982\u4e2d\u6587\uff09\u4f1a\u66f4\u52a0\u9ad8\u6548\uff0c\u56e0\u4e3a\u5b83\u4ec5\u9700 2 \u5b57\u8282\uff0c\u800c UTF-8 \u53ef\u80fd\u9700\u8981 3 \u5b57\u8282\u3002</p> <p>\u4ece\u517c\u5bb9\u6027\u7684\u89d2\u5ea6\u770b\uff0cUTF-8 \u7684\u901a\u7528\u6027\u6700\u4f73\uff0c\u8bb8\u591a\u5de5\u5177\u548c\u5e93\u4f18\u5148\u652f\u6301 UTF-8 \u3002</p>"},{"location":"chapter_data_structure/character_encoding/#345","title":"3.4.5 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801","text":"<p>\u5bf9\u4e8e\u4ee5\u5f80\u7684\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff0c\u7a0b\u5e8f\u8fd0\u884c\u4e2d\u7684\u5b57\u7b26\u4e32\u90fd\u91c7\u7528 UTF-16 \u6216 UTF-32 \u8fd9\u7c7b\u7b49\u957f\u7684\u7f16\u7801\u3002\u5728\u7b49\u957f\u7f16\u7801\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b57\u7b26\u4e32\u770b\u4f5c\u6570\u7ec4\u6765\u5904\u7406\uff0c\u8fd9\u79cd\u505a\u6cd5\u5177\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u968f\u673a\u8bbf\u95ee\uff1aUTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\u3002UTF-8 \u662f\u4e00\u79cd\u53d8\u957f\u7f16\u7801\uff0c\u8981\u60f3\u627e\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u6211\u4eec\u9700\u8981\u4ece\u5b57\u7b26\u4e32\u7684\u5f00\u59cb\u5904\u904d\u5386\u5230\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\uff0c\u8fd9\u9700\u8981 \\(O(n)\\) \u7684\u65f6\u95f4\u3002</li> <li>\u5b57\u7b26\u8ba1\u6570\uff1a\u4e0e\u968f\u673a\u8bbf\u95ee\u7c7b\u4f3c\uff0c\u8ba1\u7b97 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u4e5f\u662f \\(O(1)\\) \u7684\u64cd\u4f5c\u3002\u4f46\u662f\uff0c\u8ba1\u7b97 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u7684\u957f\u5ea6\u9700\u8981\u904d\u5386\u6574\u4e2a\u5b57\u7b26\u4e32\u3002</li> <li>\u5b57\u7b26\u4e32\u64cd\u4f5c\uff1a\u5728 UTF-16 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u5f88\u591a\u5b57\u7b26\u4e32\u64cd\u4f5c\uff08\u5982\u5206\u5272\u3001\u8fde\u63a5\u3001\u63d2\u5165\u3001\u5220\u9664\u7b49\uff09\u66f4\u5bb9\u6613\u8fdb\u884c\u3002\u5728 UTF-8 \u7f16\u7801\u7684\u5b57\u7b26\u4e32\u4e0a\uff0c\u8fdb\u884c\u8fd9\u4e9b\u64cd\u4f5c\u901a\u5e38\u9700\u8981\u989d\u5916\u7684\u8ba1\u7b97\uff0c\u4ee5\u786e\u4fdd\u4e0d\u4f1a\u4ea7\u751f\u65e0\u6548\u7684 UTF-8 \u7f16\u7801\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u7f16\u7a0b\u8bed\u8a00\u7684\u5b57\u7b26\u7f16\u7801\u65b9\u6848\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5f88\u6709\u8da3\u7684\u8bdd\u9898\uff0c\u6d89\u53ca\u8bb8\u591a\u56e0\u7d20\u3002</p> <ul> <li>Java \u7684 <code>String</code> \u7c7b\u578b\u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u5b57\u8282\u3002\u8fd9\u662f\u56e0\u4e3a Java \u8bed\u8a00\u8bbe\u8ba1\u4e4b\u521d\uff0c\u4eba\u4eec\u8ba4\u4e3a 16 \u4f4d\u8db3\u4ee5\u8868\u793a\u6240\u6709\u53ef\u80fd\u7684\u5b57\u7b26\u3002\u7136\u800c\uff0c\u8fd9\u662f\u4e00\u4e2a\u4e0d\u6b63\u786e\u7684\u5224\u65ad\u3002\u540e\u6765 Unicode \u89c4\u8303\u6269\u5c55\u5230\u4e86\u8d85\u8fc7 16 \u4f4d\uff0c\u6240\u4ee5 Java \u4e2d\u7684\u5b57\u7b26\u73b0\u5728\u53ef\u80fd\u7531\u4e00\u5bf9 16 \u4f4d\u7684\u503c\uff08\u79f0\u4e3a\u201c\u4ee3\u7406\u5bf9\u201d\uff09\u8868\u793a\u3002</li> <li>JavaScript \u548c TypeScript \u7684\u5b57\u7b26\u4e32\u4f7f\u7528 UTF-16 \u7f16\u7801\u7684\u539f\u56e0\u4e0e Java \u7c7b\u4f3c\u3002\u5f53 1995 \u5e74 Netscape \u516c\u53f8\u9996\u6b21\u63a8\u51fa JavaScript \u8bed\u8a00\u65f6\uff0cUnicode \u8fd8\u5904\u4e8e\u53d1\u5c55\u65e9\u671f\uff0c\u90a3\u65f6\u5019\u4f7f\u7528 16 \u4f4d\u7684\u7f16\u7801\u5c31\u8db3\u4ee5\u8868\u793a\u6240\u6709\u7684 Unicode \u5b57\u7b26\u4e86\u3002</li> <li>C# \u4f7f\u7528 UTF-16 \u7f16\u7801\uff0c\u4e3b\u8981\u662f\u56e0\u4e3a .NET \u5e73\u53f0\u662f\u7531 Microsoft \u8bbe\u8ba1\u7684\uff0c\u800c Microsoft \u7684\u5f88\u591a\u6280\u672f\uff08\u5305\u62ec Windows \u64cd\u4f5c\u7cfb\u7edf\uff09\u90fd\u5e7f\u6cdb\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul> <p>\u7531\u4e8e\u4ee5\u4e0a\u7f16\u7a0b\u8bed\u8a00\u5bf9\u5b57\u7b26\u6570\u91cf\u7684\u4f4e\u4f30\uff0c\u5b83\u4eec\u4e0d\u5f97\u4e0d\u91c7\u53d6\u201c\u4ee3\u7406\u5bf9\u201d\u7684\u65b9\u5f0f\u6765\u8868\u793a\u8d85\u8fc7 16 \u4f4d\u957f\u5ea6\u7684 Unicode \u5b57\u7b26\u3002\u8fd9\u662f\u4e00\u4e2a\u4e0d\u5f97\u5df2\u4e3a\u4e4b\u7684\u65e0\u5948\u4e4b\u4e3e\u3002\u4e00\u65b9\u9762\uff0c\u5305\u542b\u4ee3\u7406\u5bf9\u7684\u5b57\u7b26\u4e32\u4e2d\uff0c\u4e00\u4e2a\u5b57\u7b26\u53ef\u80fd\u5360\u7528 2 \u5b57\u8282\u6216 4 \u5b57\u8282\uff0c\u4ece\u800c\u4e27\u5931\u4e86\u7b49\u957f\u7f16\u7801\u7684\u4f18\u52bf\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5904\u7406\u4ee3\u7406\u5bf9\u9700\u8981\u589e\u52a0\u989d\u5916\u4ee3\u7801\uff0c\u8fd9\u63d0\u9ad8\u4e86\u7f16\u7a0b\u7684\u590d\u6742\u6027\u548c\u8c03\u8bd5\u96be\u5ea6\u3002</p> <p>\u51fa\u4e8e\u4ee5\u4e0a\u539f\u56e0\uff0c\u90e8\u5206\u7f16\u7a0b\u8bed\u8a00\u63d0\u51fa\u4e86\u4e00\u4e9b\u4e0d\u540c\u7684\u7f16\u7801\u65b9\u6848\u3002</p> <ul> <li>Python \u4e2d\u7684 <code>str</code> \u4f7f\u7528 Unicode \u7f16\u7801\uff0c\u5e76\u91c7\u7528\u4e00\u79cd\u7075\u6d3b\u7684\u5b57\u7b26\u4e32\u8868\u793a\uff0c\u5b58\u50a8\u7684\u5b57\u7b26\u957f\u5ea6\u53d6\u51b3\u4e8e\u5b57\u7b26\u4e32\u4e2d\u6700\u5927\u7684 Unicode \u7801\u70b9\u3002\u82e5\u5b57\u7b26\u4e32\u4e2d\u5168\u90e8\u662f ASCII \u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 1 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u6709\u5b57\u7b26\u8d85\u51fa\u4e86 ASCII \u8303\u56f4\uff0c\u4f46\u5168\u90e8\u5728\u57fa\u672c\u591a\u8bed\u8a00\u5e73\u9762\uff08BMP\uff09\u5185\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 2 \u4e2a\u5b57\u8282\uff1b\u5982\u679c\u6709\u8d85\u51fa BMP \u7684\u5b57\u7b26\uff0c\u5219\u6bcf\u4e2a\u5b57\u7b26\u5360\u7528 4 \u4e2a\u5b57\u8282\u3002</li> <li>Go \u8bed\u8a00\u7684 <code>string</code> \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Go \u8bed\u8a00\u8fd8\u63d0\u4f9b\u4e86 <code>rune</code> \u7c7b\u578b\uff0c\u5b83\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> <li>Rust \u8bed\u8a00\u7684 str \u548c String \u7c7b\u578b\u5728\u5185\u90e8\u4f7f\u7528 UTF-8 \u7f16\u7801\u3002Rust \u4e5f\u63d0\u4f9b\u4e86 <code>char</code> \u7c7b\u578b\uff0c\u7528\u4e8e\u8868\u793a\u5355\u4e2a Unicode \u7801\u70b9\u3002</li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ee5\u4e0a\u8ba8\u8bba\u7684\u90fd\u662f\u5b57\u7b26\u4e32\u5728\u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u8fd9\u548c\u5b57\u7b26\u4e32\u5982\u4f55\u5728\u6587\u4ef6\u4e2d\u5b58\u50a8\u6216\u5728\u7f51\u7edc\u4e2d\u4f20\u8f93\u662f\u4e0d\u540c\u7684\u95ee\u9898\u3002\u5728\u6587\u4ef6\u5b58\u50a8\u6216\u7f51\u7edc\u4f20\u8f93\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u5b57\u7b26\u4e32\u7f16\u7801\u4e3a UTF-8 \u683c\u5f0f\uff0c\u4ee5\u8fbe\u5230\u6700\u4f18\u7684\u517c\u5bb9\u6027\u548c\u7a7a\u95f4\u6548\u7387\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/","title":"3.1 \u00a0 \u6570\u636e\u7ed3\u6784\u5206\u7c7b","text":"<p>\u5e38\u89c1\u7684\u6570\u636e\u7ed3\u6784\u5305\u62ec\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\uff0c\u5b83\u4eec\u53ef\u4ee5\u4ece\u201c\u903b\u8f91\u7ed3\u6784\u201d\u548c\u201c\u7269\u7406\u7ed3\u6784\u201d\u4e24\u4e2a\u7ef4\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002</p>"},{"location":"chapter_data_structure/classification_of_data_structure/#311","title":"3.1.1 \u00a0 \u903b\u8f91\u7ed3\u6784\uff1a\u7ebf\u6027\u4e0e\u975e\u7ebf\u6027","text":"<p>\u903b\u8f91\u7ed3\u6784\u63ed\u793a\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\u3002\u5728\u6570\u7ec4\u548c\u94fe\u8868\u4e2d\uff0c\u6570\u636e\u6309\u7167\u4e00\u5b9a\u987a\u5e8f\u6392\u5217\uff0c\u4f53\u73b0\u4e86\u6570\u636e\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff1b\u800c\u5728\u6811\u4e2d\uff0c\u6570\u636e\u4ece\u9876\u90e8\u5411\u4e0b\u6309\u5c42\u6b21\u6392\u5217\uff0c\u8868\u73b0\u51fa\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff1b\u56fe\u5219\u7531\u8282\u70b9\u548c\u8fb9\u6784\u6210\uff0c\u53cd\u6620\u4e86\u590d\u6742\u7684\u7f51\u7edc\u5173\u7cfb\u3002</p> <p>\u5982\u56fe 3-1 \u6240\u793a\uff0c\u903b\u8f91\u7ed3\u6784\u53ef\u5206\u4e3a\u201c\u7ebf\u6027\u201d\u548c\u201c\u975e\u7ebf\u6027\u201d\u4e24\u5927\u7c7b\u3002\u7ebf\u6027\u7ed3\u6784\u6bd4\u8f83\u76f4\u89c2\uff0c\u6307\u6570\u636e\u5728\u903b\u8f91\u5173\u7cfb\u4e0a\u5448\u7ebf\u6027\u6392\u5217\uff1b\u975e\u7ebf\u6027\u7ed3\u6784\u5219\u76f8\u53cd\uff0c\u5448\u975e\u7ebf\u6027\u6392\u5217\u3002</p> <ul> <li>\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3002</li> <li>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u56fe\u3001\u54c8\u5e0c\u8868\u3002</li> </ul> <p></p> <p> \u56fe 3-1 \u00a0 \u7ebf\u6027\u6570\u636e\u7ed3\u6784\u4e0e\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784 </p> <p>\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a\u6811\u5f62\u7ed3\u6784\u548c\u7f51\u72b6\u7ed3\u6784\u3002</p> <ul> <li>\u7ebf\u6027\u7ed3\u6784\uff1a\u6570\u7ec4\u3001\u94fe\u8868\u3001\u961f\u5217\u3001\u6808\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u4e00\u7684\u987a\u5e8f\u5173\u7cfb\u3002</li> <li>\u6811\u5f62\u7ed3\u6784\uff1a\u6811\u3001\u5806\u3001\u54c8\u5e0c\u8868\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u4e00\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> <li>\u7f51\u72b6\u7ed3\u6784\uff1a\u56fe\uff0c\u5143\u7d20\u4e4b\u95f4\u662f\u591a\u5bf9\u591a\u7684\u5173\u7cfb\u3002</li> </ul>"},{"location":"chapter_data_structure/classification_of_data_structure/#312","title":"3.1.2 \u00a0 \u7269\u7406\u7ed3\u6784\uff1a\u8fde\u7eed\u4e0e\u5206\u6563","text":"<p>\u5f53\u7b97\u6cd5\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6b63\u5728\u5904\u7406\u7684\u6570\u636e\u4e3b\u8981\u88ab\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\u3002\u56fe 3-2 \u5c55\u793a\u4e86\u4e00\u4e2a\u8ba1\u7b97\u673a\u5185\u5b58\u6761\uff0c\u5176\u4e2d\u6bcf\u4e2a\u9ed1\u8272\u65b9\u5757\u90fd\u5305\u542b\u4e00\u5757\u5185\u5b58\u7a7a\u95f4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5185\u5b58\u60f3\u8c61\u6210\u4e00\u4e2a\u5de8\u5927\u7684 Excel \u8868\u683c\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5355\u5143\u683c\u90fd\u53ef\u4ee5\u5b58\u50a8\u4e00\u5b9a\u5927\u5c0f\u7684\u6570\u636e\u3002</p> <p>\u7cfb\u7edf\u901a\u8fc7\u5185\u5b58\u5730\u5740\u6765\u8bbf\u95ee\u76ee\u6807\u4f4d\u7f6e\u7684\u6570\u636e\u3002\u5982\u56fe 3-2 \u6240\u793a\uff0c\u8ba1\u7b97\u673a\u6839\u636e\u7279\u5b9a\u89c4\u5219\u4e3a\u8868\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5206\u914d\u7f16\u53f7\uff0c\u786e\u4fdd\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u6709\u552f\u4e00\u7684\u5185\u5b58\u5730\u5740\u3002\u6709\u4e86\u8fd9\u4e9b\u5730\u5740\uff0c\u7a0b\u5e8f\u4fbf\u53ef\u4ee5\u8bbf\u95ee\u5185\u5b58\u4e2d\u7684\u6570\u636e\u3002</p> <p></p> <p> \u56fe 3-2 \u00a0 \u5185\u5b58\u6761\u3001\u5185\u5b58\u7a7a\u95f4\u3001\u5185\u5b58\u5730\u5740 </p> <p>Tip</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5c06\u5185\u5b58\u6bd4\u4f5c Excel \u8868\u683c\u662f\u4e00\u4e2a\u7b80\u5316\u7684\u7c7b\u6bd4\uff0c\u5b9e\u9645\u5185\u5b58\u7684\u5de5\u4f5c\u673a\u5236\u6bd4\u8f83\u590d\u6742\uff0c\u6d89\u53ca\u5230\u5730\u5740\u7a7a\u95f4\u3001\u5185\u5b58\u7ba1\u7406\u3001\u7f13\u5b58\u673a\u5236\u3001\u865a\u62df\u548c\u7269\u7406\u5185\u5b58\u7b49\u6982\u5ff5\u3002</p> <p>\u5185\u5b58\u662f\u6240\u6709\u7a0b\u5e8f\u7684\u5171\u4eab\u8d44\u6e90\uff0c\u5f53\u67d0\u5757\u5185\u5b58\u88ab\u67d0\u4e2a\u7a0b\u5e8f\u5360\u7528\u65f6\uff0c\u5219\u65e0\u6cd5\u88ab\u5176\u4ed6\u7a0b\u5e8f\u540c\u65f6\u4f7f\u7528\u4e86\u3002\u56e0\u6b64\u5728\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e2d\uff0c\u5185\u5b58\u8d44\u6e90\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u8003\u8651\u56e0\u7d20\u3002\u6bd4\u5982\uff0c\u7b97\u6cd5\u6240\u5360\u7528\u7684\u5185\u5b58\u5cf0\u503c\u4e0d\u5e94\u8d85\u8fc7\u7cfb\u7edf\u5269\u4f59\u7a7a\u95f2\u5185\u5b58\uff1b\u5982\u679c\u7f3a\u5c11\u8fde\u7eed\u5927\u5757\u7684\u5185\u5b58\u7a7a\u95f4\uff0c\u90a3\u4e48\u6240\u9009\u7528\u7684\u6570\u636e\u7ed3\u6784\u5fc5\u987b\u80fd\u591f\u5b58\u50a8\u5728\u5206\u6563\u7684\u5185\u5b58\u7a7a\u95f4\u5185\u3002</p> <p>\u5982\u56fe 3-3 \u6240\u793a\uff0c\u7269\u7406\u7ed3\u6784\u53cd\u6620\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u53ef\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u7269\u7406\u7ed3\u6784\u4ece\u5e95\u5c42\u51b3\u5b9a\u4e86\u6570\u636e\u7684\u8bbf\u95ee\u3001\u66f4\u65b0\u3001\u589e\u5220\u7b49\u64cd\u4f5c\u65b9\u6cd5\uff0c\u4e24\u79cd\u7269\u7406\u7ed3\u6784\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u65b9\u9762\u5448\u73b0\u51fa\u4e92\u8865\u7684\u7279\u70b9\u3002</p> <p></p> <p> \u56fe 3-3 \u00a0 \u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\u4e0e\u5206\u6563\u7a7a\u95f4\u5b58\u50a8 </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u57fa\u4e8e\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e8c\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002\u4f8b\u5982\uff0c\u6808\u548c\u961f\u5217\u65e2\u53ef\u4ee5\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\uff1b\u800c\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u6570\u7ec4\u548c\u94fe\u8868\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u3001\u77e9\u9635\u3001\u5f20\u91cf\uff08\u7ef4\u5ea6 \\(\\geq 3\\) \u7684\u6570\u7ec4\uff09\u7b49\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u53ef\u5b9e\u73b0\uff1a\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u3002</li> </ul> <p>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u79f0\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u610f\u5473\u7740\u6b64\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u76f8\u5bf9\u5e94\u5730\uff0c\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u79f0\u201c\u52a8\u6001\u6570\u636e\u7ed3\u6784\u201d\uff0c\u8fd9\u7c7b\u6570\u636e\u7ed3\u6784\u5728\u521d\u59cb\u5316\u540e\uff0c\u4ecd\u53ef\u4ee5\u5728\u7a0b\u5e8f\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5176\u957f\u5ea6\u8fdb\u884c\u8c03\u6574\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u611f\u89c9\u7269\u7406\u7ed3\u6784\u7406\u89e3\u8d77\u6765\u6709\u56f0\u96be\uff0c\u5efa\u8bae\u5148\u9605\u8bfb\u4e0b\u4e00\u7ae0\uff0c\u7136\u540e\u518d\u56de\u987e\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_data_structure/number_encoding/","title":"3.3 \u00a0 \u6570\u5b57\u7f16\u7801 *","text":"<p>Note</p> <p>\u5728\u672c\u4e66\u4e2d\uff0c\u6807\u9898\u5e26\u6709 * \u7b26\u53f7\u7684\u662f\u9009\u8bfb\u7ae0\u8282\u3002\u5982\u679c\u4f60\u65f6\u95f4\u6709\u9650\u6216\u611f\u5230\u7406\u89e3\u56f0\u96be\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\uff0c\u7b49\u5b66\u5b8c\u5fc5\u8bfb\u7ae0\u8282\u540e\u518d\u5355\u72ec\u653b\u514b\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#331","title":"3.3.1 \u00a0 \u6574\u6570\u7f16\u7801","text":"<p>\u5728\u4e0a\u4e00\u8282\u7684\u8868\u683c\u4e2d\u6211\u4eec\u53d1\u73b0\uff0c\u6240\u6709\u6574\u6570\u7c7b\u578b\u80fd\u591f\u8868\u793a\u7684\u8d1f\u6570\u90fd\u6bd4\u6b63\u6570\u591a\u4e00\u4e2a\uff0c\u4f8b\u5982 <code>byte</code> \u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \u3002\u8fd9\u4e2a\u73b0\u8c61\u6bd4\u8f83\u53cd\u76f4\u89c9\uff0c\u5b83\u7684\u5185\u5728\u539f\u56e0\u6d89\u53ca\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u76f8\u5173\u77e5\u8bc6\u3002</p> <p>\u9996\u5148\u9700\u8981\u6307\u51fa\uff0c\u6570\u5b57\u662f\u4ee5\u201c\u8865\u7801\u201d\u7684\u5f62\u5f0f\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u4e2d\u7684\u3002\u5728\u5206\u6790\u8fd9\u6837\u505a\u7684\u539f\u56e0\u4e4b\u524d\uff0c\u9996\u5148\u7ed9\u51fa\u4e09\u8005\u7684\u5b9a\u4e49\u3002</p> <ul> <li>\u539f\u7801\uff1a\u6211\u4eec\u5c06\u6570\u5b57\u7684\u4e8c\u8fdb\u5236\u8868\u793a\u7684\u6700\u9ad8\u4f4d\u89c6\u4e3a\u7b26\u53f7\u4f4d\uff0c\u5176\u4e2d \\(0\\) \u8868\u793a\u6b63\u6570\uff0c\\(1\\) \u8868\u793a\u8d1f\u6570\uff0c\u5176\u4f59\u4f4d\u8868\u793a\u6570\u5b57\u7684\u503c\u3002</li> <li>\u53cd\u7801\uff1a\u6b63\u6570\u7684\u53cd\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u53cd\u7801\u662f\u5bf9\u5176\u539f\u7801\u9664\u7b26\u53f7\u4f4d\u5916\u7684\u6240\u6709\u4f4d\u53d6\u53cd\u3002</li> <li>\u8865\u7801\uff1a\u6b63\u6570\u7684\u8865\u7801\u4e0e\u5176\u539f\u7801\u76f8\u540c\uff0c\u8d1f\u6570\u7684\u8865\u7801\u662f\u5728\u5176\u53cd\u7801\u7684\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u3002</li> </ul> <p>\u56fe 3-4 \u5c55\u793a\u4e86\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u4e4b\u95f4\u7684\u8f6c\u6362\u65b9\u6cd5\u3002</p> <p></p> <p> \u56fe 3-4 \u00a0 \u539f\u7801\u3001\u53cd\u7801\u4e0e\u8865\u7801\u4e4b\u95f4\u7684\u76f8\u4e92\u8f6c\u6362 </p> <p>\u300c\u539f\u7801 sign\u2013magnitude\u300d\u867d\u7136\u6700\u76f4\u89c2\uff0c\u4f46\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002\u4e00\u65b9\u9762\uff0c\u8d1f\u6570\u7684\u539f\u7801\u4e0d\u80fd\u76f4\u63a5\u7528\u4e8e\u8fd0\u7b97\u3002\u4f8b\u5982\u5728\u539f\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u5f97\u5230\u7684\u7ed3\u679c\u662f \\(-3\\) \uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 + 1000 \\; 0010 \\newline &amp; = 1000 \\; 0011 \\newline &amp; \\rightarrow -3 \\end{aligned} \\] <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u8ba1\u7b97\u673a\u5f15\u5165\u4e86\u300c\u53cd\u7801 1's complement\u300d\u3002\u5982\u679c\u6211\u4eec\u5148\u5c06\u539f\u7801\u8f6c\u6362\u4e3a\u53cd\u7801\uff0c\u5e76\u5728\u53cd\u7801\u4e0b\u8ba1\u7b97 \\(1 + (-2)\\) \uff0c\u6700\u540e\u5c06\u7ed3\u679c\u4ece\u53cd\u7801\u8f6c\u6362\u56de\u539f\u7801\uff0c\u5219\u53ef\u5f97\u5230\u6b63\u786e\u7ed3\u679c \\(-1\\) \u3002</p> \\[ \\begin{aligned} &amp; 1 + (-2) \\newline &amp; \\rightarrow 0000 \\; 0001 \\; \\text{(\u539f\u7801)} + 1000 \\; 0010 \\; \\text{(\u539f\u7801)} \\newline &amp; = 0000 \\; 0001 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1101 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1111 \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; \\rightarrow -1 \\end{aligned} \\] <p>\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57\u96f6\u7684\u539f\u7801\u6709 \\(+0\\) \u548c \\(-0\\) \u4e24\u79cd\u8868\u793a\u65b9\u5f0f\u3002\u8fd9\u610f\u5473\u7740\u6570\u5b57\u96f6\u5bf9\u5e94\u4e24\u4e2a\u4e0d\u540c\u7684\u4e8c\u8fdb\u5236\u7f16\u7801\uff0c\u8fd9\u53ef\u80fd\u4f1a\u5e26\u6765\u6b67\u4e49\u3002\u6bd4\u5982\u5728\u6761\u4ef6\u5224\u65ad\u4e2d\uff0c\u5982\u679c\u6ca1\u6709\u533a\u5206\u6b63\u96f6\u548c\u8d1f\u96f6\uff0c\u5219\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5224\u65ad\u7ed3\u679c\u51fa\u9519\u3002\u800c\u5982\u679c\u6211\u4eec\u60f3\u5904\u7406\u6b63\u96f6\u548c\u8d1f\u96f6\u6b67\u4e49\uff0c\u5219\u9700\u8981\u5f15\u5165\u989d\u5916\u7684\u5224\u65ad\u64cd\u4f5c\uff0c\u8fd9\u53ef\u80fd\u4f1a\u964d\u4f4e\u8ba1\u7b97\u673a\u7684\u8fd0\u7b97\u6548\u7387\u3002</p> \\[ \\begin{aligned} +0 &amp; \\rightarrow 0000 \\; 0000 \\newline -0 &amp; \\rightarrow 1000 \\; 0000 \\end{aligned} \\] <p>\u4e0e\u539f\u7801\u4e00\u6837\uff0c\u53cd\u7801\u4e5f\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u95ee\u9898\uff0c\u56e0\u6b64\u8ba1\u7b97\u673a\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u300c\u8865\u7801 2's complement\u300d\u3002\u6211\u4eec\u5148\u6765\u89c2\u5bdf\u4e00\u4e0b\u8d1f\u96f6\u7684\u539f\u7801\u3001\u53cd\u7801\u3001\u8865\u7801\u7684\u8f6c\u6362\u8fc7\u7a0b\uff1a</p> \\[ \\begin{aligned} -0 \\rightarrow \\; &amp; 1000 \\; 0000 \\; \\text{(\u539f\u7801)} \\newline = \\; &amp; 1111 \\; 1111 \\; \\text{(\u53cd\u7801)} \\newline = 1 \\; &amp; 0000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline \\end{aligned} \\] <p>\u5728\u8d1f\u96f6\u7684\u53cd\u7801\u57fa\u7840\u4e0a\u52a0 \\(1\\) \u4f1a\u4ea7\u751f\u8fdb\u4f4d\uff0c\u4f46 <code>byte</code> \u7c7b\u578b\u7684\u957f\u5ea6\u53ea\u6709 8 \u4f4d\uff0c\u56e0\u6b64\u6ea2\u51fa\u5230\u7b2c 9 \u4f4d\u7684 \\(1\\) \u4f1a\u88ab\u820d\u5f03\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8d1f\u96f6\u7684\u8865\u7801\u4e3a \\(0000 \\; 0000\\) \uff0c\u4e0e\u6b63\u96f6\u7684\u8865\u7801\u76f8\u540c\u3002\u8fd9\u610f\u5473\u7740\u5728\u8865\u7801\u8868\u793a\u4e2d\u53ea\u5b58\u5728\u4e00\u4e2a\u96f6\uff0c\u6b63\u8d1f\u96f6\u6b67\u4e49\u4ece\u800c\u5f97\u5230\u89e3\u51b3\u3002</p> <p>\u8fd8\u5269\u6700\u540e\u4e00\u4e2a\u7591\u60d1\uff1a<code>byte</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u662f \\([-128, 127]\\) \uff0c\u591a\u51fa\u6765\u7684\u4e00\u4e2a\u8d1f\u6570 \\(-128\\) \u662f\u5982\u4f55\u5f97\u5230\u7684\u5462\uff1f\u6211\u4eec\u6ce8\u610f\u5230\uff0c\u533a\u95f4 \\([-127, +127]\\) \u5185\u7684\u6240\u6709\u6574\u6570\u90fd\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\uff0c\u5e76\u4e14\u539f\u7801\u548c\u8865\u7801\u4e4b\u95f4\u53ef\u4ee5\u4e92\u76f8\u8f6c\u6362\u3002</p> <p>\u7136\u800c\uff0c\u8865\u7801 \\(1000 \\; 0000\\) \u662f\u4e00\u4e2a\u4f8b\u5916\uff0c\u5b83\u5e76\u6ca1\u6709\u5bf9\u5e94\u7684\u539f\u7801\u3002\u6839\u636e\u8f6c\u6362\u65b9\u6cd5\uff0c\u6211\u4eec\u5f97\u5230\u8be5\u8865\u7801\u7684\u539f\u7801\u4e3a \\(0000 \\; 0000\\) \u3002\u8fd9\u663e\u7136\u662f\u77db\u76fe\u7684\uff0c\u56e0\u4e3a\u8be5\u539f\u7801\u8868\u793a\u6570\u5b57 \\(0\\) \uff0c\u5b83\u7684\u8865\u7801\u5e94\u8be5\u662f\u81ea\u8eab\u3002\u8ba1\u7b97\u673a\u89c4\u5b9a\u8fd9\u4e2a\u7279\u6b8a\u7684\u8865\u7801 \\(1000 \\; 0000\\) \u4ee3\u8868 \\(-128\\) \u3002\u5b9e\u9645\u4e0a\uff0c\\((-1) + (-127)\\) \u5728\u8865\u7801\u4e0b\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u662f \\(-128\\) \u3002</p> \\[ \\begin{aligned} &amp; (-127) + (-1) \\newline &amp; \\rightarrow 1111 \\; 1111 \\; \\text{(\u539f\u7801)} + 1000 \\; 0001 \\; \\text{(\u539f\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u53cd\u7801)} + 1111  \\; 1110 \\; \\text{(\u53cd\u7801)} \\newline &amp; = 1000 \\; 0001 \\; \\text{(\u8865\u7801)} + 1111  \\; 1111 \\; \\text{(\u8865\u7801)} \\newline &amp; = 1000 \\; 0000 \\; \\text{(\u8865\u7801)} \\newline &amp; \\rightarrow -128 \\end{aligned} \\] <p>\u4f60\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0\u4e86\uff0c\u4e0a\u8ff0\u6240\u6709\u8ba1\u7b97\u90fd\u662f\u52a0\u6cd5\u8fd0\u7b97\u3002\u8fd9\u6697\u793a\u7740\u4e00\u4e2a\u91cd\u8981\u4e8b\u5b9e\uff1a\u8ba1\u7b97\u673a\u5185\u90e8\u7684\u786c\u4ef6\u7535\u8def\u4e3b\u8981\u662f\u57fa\u4e8e\u52a0\u6cd5\u8fd0\u7b97\u8bbe\u8ba1\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u52a0\u6cd5\u8fd0\u7b97\u76f8\u5bf9\u4e8e\u5176\u4ed6\u8fd0\u7b97\uff08\u6bd4\u5982\u4e58\u6cd5\u3001\u9664\u6cd5\u548c\u51cf\u6cd5\uff09\u6765\u8bf4\uff0c\u786c\u4ef6\u5b9e\u73b0\u8d77\u6765\u66f4\u7b80\u5355\uff0c\u66f4\u5bb9\u6613\u8fdb\u884c\u5e76\u884c\u5316\u5904\u7406\uff0c\u8fd0\u7b97\u901f\u5ea6\u66f4\u5feb\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u8ba1\u7b97\u673a\u53ea\u80fd\u505a\u52a0\u6cd5\u3002\u901a\u8fc7\u5c06\u52a0\u6cd5\u4e0e\u4e00\u4e9b\u57fa\u672c\u903b\u8f91\u8fd0\u7b97\u7ed3\u5408\uff0c\u8ba1\u7b97\u673a\u80fd\u591f\u5b9e\u73b0\u5404\u79cd\u5176\u4ed6\u7684\u6570\u5b66\u8fd0\u7b97\u3002\u4f8b\u5982\uff0c\u8ba1\u7b97\u51cf\u6cd5 \\(a - b\\) \u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u52a0\u6cd5 \\(a + (-b)\\) \uff1b\u8ba1\u7b97\u4e58\u6cd5\u548c\u9664\u6cd5\u53ef\u4ee5\u8f6c\u6362\u4e3a\u8ba1\u7b97\u591a\u6b21\u52a0\u6cd5\u6216\u51cf\u6cd5\u3002</p> <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u8ba1\u7b97\u673a\u4f7f\u7528\u8865\u7801\u7684\u539f\u56e0\uff1a\u57fa\u4e8e\u8865\u7801\u8868\u793a\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u7528\u540c\u6837\u7684\u7535\u8def\u548c\u64cd\u4f5c\u6765\u5904\u7406\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\uff0c\u4e0d\u9700\u8981\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\u6765\u5904\u7406\u51cf\u6cd5\uff0c\u5e76\u4e14\u65e0\u987b\u7279\u522b\u5904\u7406\u6b63\u8d1f\u96f6\u7684\u6b67\u4e49\u95ee\u9898\u3002\u8fd9\u5927\u5927\u7b80\u5316\u4e86\u786c\u4ef6\u8bbe\u8ba1\uff0c\u63d0\u9ad8\u4e86\u8fd0\u7b97\u6548\u7387\u3002</p> <p>\u8865\u7801\u7684\u8bbe\u8ba1\u975e\u5e38\u7cbe\u5999\uff0c\u56e0\u7bc7\u5e45\u5173\u7cfb\u6211\u4eec\u5c31\u5148\u4ecb\u7ecd\u5230\u8fd9\u91cc\uff0c\u5efa\u8bae\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8fdb\u4e00\u6b65\u6df1\u5165\u4e86\u89e3\u3002</p>"},{"location":"chapter_data_structure/number_encoding/#332","title":"3.3.2 \u00a0 \u6d6e\u70b9\u6570\u7f16\u7801","text":"<p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\uff1a<code>int</code> \u548c <code>float</code> \u957f\u5ea6\u76f8\u540c\uff0c\u90fd\u662f 4 bytes \uff0c\u4f46\u4e3a\u4ec0\u4e48 <code>float</code> \u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \uff1f\u8fd9\u975e\u5e38\u53cd\u76f4\u89c9\uff0c\u56e0\u4e3a\u6309\u7406\u8bf4 <code>float</code> \u9700\u8981\u8868\u793a\u5c0f\u6570\uff0c\u53d6\u503c\u8303\u56f4\u5e94\u8be5\u53d8\u5c0f\u624d\u5bf9\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8fd9\u662f\u56e0\u4e3a\u6d6e\u70b9\u6570 <code>float</code> \u91c7\u7528\u4e86\u4e0d\u540c\u7684\u8868\u793a\u65b9\u5f0f\u3002\u8bb0\u4e00\u4e2a 32-bit \u957f\u5ea6\u7684\u4e8c\u8fdb\u5236\u6570\u4e3a\uff1a</p> \\[ b_{31} b_{30} b_{29} \\ldots b_2 b_1 b_0 \\] <p>\u6839\u636e IEEE 754 \u6807\u51c6\uff0c32-bit \u957f\u5ea6\u7684 <code>float</code> \u7531\u4ee5\u4e0b\u4e09\u4e2a\u90e8\u5206\u6784\u6210\u3002</p> <ul> <li>\u7b26\u53f7\u4f4d \\(\\mathrm{S}\\) \uff1a\u5360 1 bit \uff0c\u5bf9\u5e94 \\(b_{31}\\) \u3002</li> <li>\u6307\u6570\u4f4d \\(\\mathrm{E}\\) \uff1a\u5360 8 bits \uff0c\u5bf9\u5e94 \\(b_{30} b_{29} \\ldots b_{23}\\) \u3002</li> <li>\u5206\u6570\u4f4d \\(\\mathrm{N}\\) \uff1a\u5360 23 bits \uff0c\u5bf9\u5e94 \\(b_{22} b_{21} \\ldots b_0\\) \u3002</li> </ul> <p>\u4e8c\u8fdb\u5236\u6570 <code>float</code> \u5bf9\u5e94\u503c\u7684\u8ba1\u7b97\u65b9\u6cd5\u4e3a\uff1a</p> \\[ \\text {val} = (-1)^{b_{31}} \\times 2^{\\left(b_{30} b_{29} \\ldots b_{23}\\right)_2-127} \\times\\left(1 . b_{22} b_{21} \\ldots b_0\\right)_2 \\] <p>\u8f6c\u5316\u5230\u5341\u8fdb\u5236\u4e0b\u7684\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a</p> \\[ \\text {val}=(-1)^{\\mathrm{S}} \\times 2^{\\mathrm{E} -127} \\times (1 + \\mathrm{N}) \\] <p>\u5176\u4e2d\u5404\u9879\u7684\u53d6\u503c\u8303\u56f4\u4e3a\uff1a</p> \\[ \\begin{aligned} \\mathrm{S} \\in &amp; \\{ 0, 1\\}, \\quad \\mathrm{E} \\in \\{ 1, 2, \\dots, 254 \\} \\newline (1 + \\mathrm{N}) = &amp; (1 + \\sum_{i=1}^{23} b_{23-i} 2^{-i}) \\subset [1, 2 - 2^{-23}] \\end{aligned} \\] <p></p> <p> \u56fe 3-5 \u00a0 IEEE 754 \u6807\u51c6\u4e0b\u7684 float \u7684\u8ba1\u7b97\u793a\u4f8b </p> <p>\u89c2\u5bdf\u56fe 3-5 \uff0c\u7ed9\u5b9a\u4e00\u4e2a\u793a\u4f8b\u6570\u636e \\(\\mathrm{S} = 0\\) \uff0c \\(\\mathrm{E} = 124\\) \uff0c\\(\\mathrm{N} = 2^{-2} + 2^{-3} = 0.375\\) \uff0c\u5219\u6709\uff1a</p> \\[ \\text { val } = (-1)^0 \\times 2^{124 - 127} \\times (1 + 0.375) = 0.171875 \\] <p>\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u56de\u7b54\u6700\u521d\u7684\u95ee\u9898\uff1a<code>float</code> \u7684\u8868\u793a\u65b9\u5f0f\u5305\u542b\u6307\u6570\u4f4d\uff0c\u5bfc\u81f4\u5176\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e <code>int</code> \u3002\u6839\u636e\u4ee5\u4e0a\u8ba1\u7b97\uff0c<code>float</code> \u53ef\u8868\u793a\u7684\u6700\u5927\u6b63\u6570\u4e3a \\(2^{254 - 127} \\times (2 - 2^{-23}) \\approx 3.4 \\times 10^{38}\\) \uff0c\u5207\u6362\u7b26\u53f7\u4f4d\u4fbf\u53ef\u5f97\u5230\u6700\u5c0f\u8d1f\u6570\u3002</p> <p>\u5c3d\u7ba1\u6d6e\u70b9\u6570 <code>float</code> \u6269\u5c55\u4e86\u53d6\u503c\u8303\u56f4\uff0c\u4f46\u5176\u526f\u4f5c\u7528\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002\u6574\u6570\u7c7b\u578b <code>int</code> \u5c06\u5168\u90e8 32 \u4f4d\u7528\u4e8e\u8868\u793a\u6570\u5b57\uff0c\u6570\u5b57\u662f\u5747\u5300\u5206\u5e03\u7684\uff1b\u800c\u7531\u4e8e\u6307\u6570\u4f4d\u7684\u5b58\u5728\uff0c\u6d6e\u70b9\u6570 <code>float</code> \u7684\u6570\u503c\u8d8a\u5927\uff0c\u76f8\u90bb\u4e24\u4e2a\u6570\u5b57\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u4f1a\u8d8b\u5411\u8d8a\u5927\u3002</p> <p>\u5982\u8868 3-2 \u6240\u793a\uff0c\u6307\u6570\u4f4d \\(E = 0\\) \u548c \\(E = 255\\) \u5177\u6709\u7279\u6b8a\u542b\u4e49\uff0c\u7528\u4e8e\u8868\u793a\u96f6\u3001\u65e0\u7a77\u5927\u3001\\(\\mathrm{NaN}\\) \u7b49\u3002</p> <p> \u8868 3-2 \u00a0 \u6307\u6570\u4f4d\u542b\u4e49 </p> \u6307\u6570\u4f4d E \u5206\u6570\u4f4d \\(\\mathrm{N} = 0\\) \u5206\u6570\u4f4d \\(\\mathrm{N} \\ne 0\\) \u8ba1\u7b97\u516c\u5f0f \\(0\\) \\(\\pm 0\\) \u6b21\u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{-126} \\times (0.\\mathrm{N})\\) \\(1, 2, \\dots, 254\\) \u6b63\u89c4\u6570 \u6b63\u89c4\u6570 \\((-1)^{\\mathrm{S}} \\times 2^{(\\mathrm{E} -127)} \\times (1.\\mathrm{N})\\) \\(255\\) \\(\\pm \\infty\\) \\(\\mathrm{NaN}\\) <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6b21\u6b63\u89c4\u6570\u663e\u8457\u63d0\u5347\u4e86\u6d6e\u70b9\u6570\u7684\u7cbe\u5ea6\u3002\u6700\u5c0f\u6b63\u6b63\u89c4\u6570\u4e3a \\(2^{-126}\\) \uff0c\u6700\u5c0f\u6b63\u6b21\u6b63\u89c4\u6570\u4e3a \\(2^{-126} \\times 2^{-23}\\) \u3002</p> <p>\u53cc\u7cbe\u5ea6 <code>double</code> \u4e5f\u91c7\u7528\u7c7b\u4f3c\u4e8e <code>float</code> \u7684\u8868\u793a\u65b9\u6cd5\uff0c\u5728\u6b64\u4e0d\u505a\u8d58\u8ff0\u3002</p>"},{"location":"chapter_data_structure/summary/","title":"3.5 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_data_structure/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u4ece\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u4e24\u4e2a\u89d2\u5ea6\u8fdb\u884c\u5206\u7c7b\u3002\u903b\u8f91\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5143\u7d20\u4e4b\u95f4\u7684\u903b\u8f91\u5173\u7cfb\uff0c\u800c\u7269\u7406\u7ed3\u6784\u63cf\u8ff0\u4e86\u6570\u636e\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u7684\u5b58\u50a8\u65b9\u5f0f\u3002</li> <li>\u5e38\u89c1\u7684\u903b\u8f91\u7ed3\u6784\u5305\u62ec\u7ebf\u6027\u3001\u6811\u72b6\u548c\u7f51\u72b6\u7b49\u3002\u901a\u5e38\u6211\u4eec\u6839\u636e\u903b\u8f91\u7ed3\u6784\u5c06\u6570\u636e\u7ed3\u6784\u5206\u4e3a\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\u3001\u56fe\u3001\u5806\uff09\u4e24\u79cd\u3002\u54c8\u5e0c\u8868\u7684\u5b9e\u73b0\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5f53\u7a0b\u5e8f\u8fd0\u884c\u65f6\uff0c\u6570\u636e\u88ab\u5b58\u50a8\u5728\u8ba1\u7b97\u673a\u5185\u5b58\u4e2d\u3002\u6bcf\u4e2a\u5185\u5b58\u7a7a\u95f4\u90fd\u62e5\u6709\u5bf9\u5e94\u7684\u5185\u5b58\u5730\u5740\uff0c\u7a0b\u5e8f\u901a\u8fc7\u8fd9\u4e9b\u5185\u5b58\u5730\u5740\u8bbf\u95ee\u6570\u636e\u3002</li> <li>\u7269\u7406\u7ed3\u6784\u4e3b\u8981\u5206\u4e3a\u8fde\u7eed\u7a7a\u95f4\u5b58\u50a8\uff08\u6570\u7ec4\uff09\u548c\u5206\u6563\u7a7a\u95f4\u5b58\u50a8\uff08\u94fe\u8868\uff09\u3002\u6240\u6709\u6570\u636e\u7ed3\u6784\u90fd\u662f\u7531\u6570\u7ec4\u3001\u94fe\u8868\u6216\u4e24\u8005\u7684\u7ec4\u5408\u5b9e\u73b0\u7684\u3002</li> <li>\u8ba1\u7b97\u673a\u4e2d\u7684\u57fa\u672c\u6570\u636e\u7c7b\u578b\u5305\u62ec\u6574\u6570 <code>byte</code>\u3001<code>short</code>\u3001<code>int</code>\u3001<code>long</code> \uff0c\u6d6e\u70b9\u6570 <code>float</code>\u3001<code>double</code> \uff0c\u5b57\u7b26 <code>char</code> \u548c\u5e03\u5c14 <code>boolean</code> \u3002\u5b83\u4eec\u7684\u53d6\u503c\u8303\u56f4\u53d6\u51b3\u4e8e\u5360\u7528\u7a7a\u95f4\u5927\u5c0f\u548c\u8868\u793a\u65b9\u5f0f\u3002</li> <li>\u539f\u7801\u3001\u53cd\u7801\u548c\u8865\u7801\u662f\u5728\u8ba1\u7b97\u673a\u4e2d\u7f16\u7801\u6570\u5b57\u7684\u4e09\u79cd\u65b9\u6cd5\uff0c\u5b83\u4eec\u4e4b\u95f4\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\u3002\u6574\u6570\u7684\u539f\u7801\u7684\u6700\u9ad8\u4f4d\u662f\u7b26\u53f7\u4f4d\uff0c\u5176\u4f59\u4f4d\u662f\u6570\u5b57\u7684\u503c\u3002</li> <li>\u6574\u6570\u5728\u8ba1\u7b97\u673a\u4e2d\u662f\u4ee5\u8865\u7801\u7684\u5f62\u5f0f\u5b58\u50a8\u7684\u3002\u5728\u8865\u7801\u8868\u793a\u4e0b\uff0c\u8ba1\u7b97\u673a\u53ef\u4ee5\u5bf9\u6b63\u6570\u548c\u8d1f\u6570\u7684\u52a0\u6cd5\u4e00\u89c6\u540c\u4ec1\uff0c\u4e0d\u9700\u8981\u4e3a\u51cf\u6cd5\u64cd\u4f5c\u5355\u72ec\u8bbe\u8ba1\u7279\u6b8a\u7684\u786c\u4ef6\u7535\u8def\uff0c\u5e76\u4e14\u4e0d\u5b58\u5728\u6b63\u8d1f\u96f6\u6b67\u4e49\u7684\u95ee\u9898\u3002</li> <li>\u6d6e\u70b9\u6570\u7684\u7f16\u7801\u7531 1 \u4f4d\u7b26\u53f7\u4f4d\u30018 \u4f4d\u6307\u6570\u4f4d\u548c 23 \u4f4d\u5206\u6570\u4f4d\u6784\u6210\u3002\u7531\u4e8e\u5b58\u5728\u6307\u6570\u4f4d\uff0c\u56e0\u6b64\u6d6e\u70b9\u6570\u7684\u53d6\u503c\u8303\u56f4\u8fdc\u5927\u4e8e\u6574\u6570\uff0c\u4ee3\u4ef7\u662f\u727a\u7272\u4e86\u7cbe\u5ea6\u3002</li> <li>ASCII \u7801\u662f\u6700\u65e9\u51fa\u73b0\u7684\u82f1\u6587\u5b57\u7b26\u96c6\uff0c\u957f\u5ea6\u4e3a 1 \u5b57\u8282\uff0c\u5171\u6536\u5f55 127 \u4e2a\u5b57\u7b26\u3002GBK \u5b57\u7b26\u96c6\u662f\u5e38\u7528\u7684\u4e2d\u6587\u5b57\u7b26\u96c6\uff0c\u5171\u6536\u5f55\u4e24\u4e07\u591a\u4e2a\u6c49\u5b57\u3002Unicode \u81f4\u529b\u4e8e\u63d0\u4f9b\u4e00\u4e2a\u5b8c\u6574\u7684\u5b57\u7b26\u96c6\u6807\u51c6\uff0c\u6536\u5f55\u4e16\u754c\u5185\u5404\u79cd\u8bed\u8a00\u7684\u5b57\u7b26\uff0c\u4ece\u800c\u89e3\u51b3\u7531\u4e8e\u5b57\u7b26\u7f16\u7801\u65b9\u6cd5\u4e0d\u4e00\u81f4\u800c\u5bfc\u81f4\u7684\u4e71\u7801\u95ee\u9898\u3002</li> <li>UTF-8 \u662f\u6700\u53d7\u6b22\u8fce\u7684 Unicode \u7f16\u7801\u65b9\u6cd5\uff0c\u901a\u7528\u6027\u975e\u5e38\u597d\u3002\u5b83\u662f\u4e00\u79cd\u53d8\u957f\u7684\u7f16\u7801\u65b9\u6cd5\uff0c\u5177\u6709\u5f88\u597d\u7684\u6269\u5c55\u6027\uff0c\u6709\u6548\u63d0\u5347\u4e86\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u6548\u7387\u3002UTF-16 \u548c UTF-32 \u662f\u7b49\u957f\u7684\u7f16\u7801\u65b9\u6cd5\u3002\u5728\u7f16\u7801\u4e2d\u6587\u65f6\uff0cUTF-16 \u6bd4 UTF-8 \u7684\u5360\u7528\u7a7a\u95f4\u66f4\u5c0f\u3002Java \u548c C# \u7b49\u7f16\u7a0b\u8bed\u8a00\u9ed8\u8ba4\u4f7f\u7528 UTF-16 \u7f16\u7801\u3002</li> </ul>"},{"location":"chapter_data_structure/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u540c\u65f6\u5305\u542b\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u548c\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff1f</p> <p>\u54c8\u5e0c\u8868\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u800c\u4e3a\u4e86\u89e3\u51b3\u54c8\u5e0c\u51b2\u7a81\uff0c\u6211\u4eec\u53ef\u80fd\u4f1a\u4f7f\u7528\u201c\u94fe\u5f0f\u5730\u5740\u201d\uff08\u540e\u7eed\u54c8\u5e0c\u8868\u7ae0\u8282\u4f1a\u8bb2\uff09\uff1a\u6570\u7ec4\u4e2d\u6bcf\u4e2a\u6876\u6307\u5411\u4e00\u4e2a\u94fe\u8868\uff0c\u5f53\u94fe\u8868\u957f\u5ea6\u8d85\u8fc7\u4e00\u5b9a\u9608\u503c\u65f6\uff0c\u53c8\u53ef\u80fd\u88ab\u8f6c\u5316\u4e3a\u6811\uff08\u901a\u5e38\u4e3a\u7ea2\u9ed1\u6811\uff09\u3002 \u4ece\u5b58\u50a8\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u54c8\u5e0c\u8868\u7684\u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u6876\u69fd\u4f4d\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u503c\uff0c\u4e5f\u53ef\u80fd\u5305\u542b\u4e00\u4e2a\u94fe\u8868\u6216\u6811\u3002\u56e0\u6b64\uff0c\u54c8\u5e0c\u8868\u53ef\u80fd\u540c\u65f6\u5305\u542b\u7ebf\u6027\uff08\u6570\u7ec4\u3001\u94fe\u8868\uff09\u548c\u975e\u7ebf\u6027\uff08\u6811\uff09\u6570\u636e\u7ed3\u6784\u3002</p> <p><code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u662f 1 byte \u5417\uff1f</p> <p><code>char</code> \u7c7b\u578b\u7684\u957f\u5ea6\u7531\u7f16\u7a0b\u8bed\u8a00\u91c7\u7528\u7684\u7f16\u7801\u65b9\u6cd5\u51b3\u5b9a\u3002\u4f8b\u5982\uff0cJava\u3001JavaScript\u3001TypeScript\u3001C# \u90fd\u91c7\u7528 UTF-16 \u7f16\u7801\uff08\u4fdd\u5b58 Unicode \u7801\u70b9\uff09\uff0c\u56e0\u6b64 char \u7c7b\u578b\u7684\u957f\u5ea6\u4e3a 2 bytes \u3002</p> <p>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u4e5f\u88ab\u79f0\u4e3a\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d \u662f\u5426\u6709\u6b67\u4e49\uff1f\u56e0\u4e3a\u6808\u4e5f\u53ef\u4ee5\u8fdb\u884c\u51fa\u6808\u548c\u5165\u6808\u7b49\u64cd\u4f5c\uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u662f\u201c\u52a8\u6001\u201d\u7684\u3002</p> <p>\u6808\u786e\u5b9e\u53ef\u4ee5\u5b9e\u73b0\u52a8\u6001\u7684\u6570\u636e\u64cd\u4f5c\uff0c\u4f46\u6570\u636e\u7ed3\u6784\u4ecd\u7136\u662f\u201c\u9759\u6001\u201d\uff08\u957f\u5ea6\u4e0d\u53ef\u53d8\uff09\u7684\u3002\u5c3d\u7ba1\u57fa\u4e8e\u6570\u7ec4\u7684\u6570\u636e\u7ed3\u6784\u53ef\u4ee5\u52a8\u6001\u5730\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\uff0c\u4f46\u5b83\u4eec\u7684\u5bb9\u91cf\u662f\u56fa\u5b9a\u7684\u3002\u5982\u679c\u6570\u636e\u91cf\u8d85\u51fa\u4e86\u9884\u5206\u914d\u7684\u5927\u5c0f\uff0c\u5c31\u9700\u8981\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u66f4\u5927\u7684\u6570\u7ec4\uff0c\u5e76\u5c06\u8001\u6570\u7ec4\u7684\u5185\u5bb9\u590d\u5236\u5230\u65b0\u6570\u7ec4\u4e2d\u3002</p> <p>\u5728\u6784\u5efa\u6808\uff08\u961f\u5217\uff09\u7684\u65f6\u5019\uff0c\u672a\u6307\u5b9a\u5b83\u7684\u5927\u5c0f\uff0c\u4e3a\u4ec0\u4e48\u5b83\u4eec\u662f\u201c\u9759\u6001\u6570\u636e\u7ed3\u6784\u201d\u5462\uff1f</p> <p>\u5728\u9ad8\u7ea7\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u65e0\u987b\u4eba\u5de5\u6307\u5b9a\u6808\uff08\u961f\u5217\uff09\u7684\u521d\u59cb\u5bb9\u91cf\uff0c\u8fd9\u4e2a\u5de5\u4f5c\u662f\u7531\u7c7b\u5185\u90e8\u81ea\u52a8\u5b8c\u6210\u7684\u3002\u4f8b\u5982\uff0cJava \u7684 ArrayList \u7684\u521d\u59cb\u5bb9\u91cf\u901a\u5e38\u4e3a 10 \u3002\u53e6\u5916\uff0c\u6269\u5bb9\u64cd\u4f5c\u4e5f\u662f\u81ea\u52a8\u5b9e\u73b0\u7684\u3002\u8be6\u89c1\u672c\u4e66\u7684\u201c\u5217\u8868\u201d\u7ae0\u8282\u3002</p>"},{"location":"chapter_divide_and_conquer/","title":"\u7b2c 12 \u7ae0 \u00a0 \u5206\u6cbb","text":"<p>Abstract</p> <p>\u96be\u9898\u88ab\u9010\u5c42\u62c6\u89e3\uff0c\u6bcf\u4e00\u6b21\u7684\u62c6\u89e3\u90fd\u4f7f\u5b83\u53d8\u5f97\u66f4\u4e3a\u7b80\u5355\u3002</p> <p>\u5206\u800c\u6cbb\u4e4b\u63ed\u793a\u4e86\u4e00\u4e2a\u91cd\u8981\u7684\u4e8b\u5b9e\uff1a\u4ece\u7b80\u5355\u505a\u8d77\uff0c\u4e00\u5207\u90fd\u4e0d\u518d\u590d\u6742\u3002</p>"},{"location":"chapter_divide_and_conquer/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5</li> <li>12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565</li> <li>12.3 \u00a0 \u6784\u5efa\u6811\u95ee\u9898</li> <li>12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898</li> <li>12.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_divide_and_conquer/binary_search_recur/","title":"12.2 \u00a0 \u5206\u6cbb\u641c\u7d22\u7b56\u7565","text":"<p>\u6211\u4eec\u5df2\u7ecf\u5b66\u8fc7\uff0c\u641c\u7d22\u7b97\u6cd5\u5206\u4e3a\u4e24\u5927\u7c7b\u3002</p> <ul> <li>\u66b4\u529b\u641c\u7d22\uff1a\u5b83\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</li> <li>\u81ea\u9002\u5e94\u641c\u7d22\uff1a\u5b83\u5229\u7528\u7279\u6709\u7684\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f\u6216\u5148\u9a8c\u4fe1\u606f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u548c\u6811\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u6bcf\u4e00\u6b65\u90fd\u5c06\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\u5206\u89e3\u4e3a\u4e00\u4e2a\u5c0f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff09\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u4e00\u76f4\u6301\u7eed\u5230\u6570\u7ec4\u4e3a\u7a7a\u6216\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u6811\u662f\u5206\u6cbb\u601d\u60f3\u7684\u4ee3\u8868\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u5806\u7b49\u6570\u636e\u7ed3\u6784\u4e2d\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7686\u4e3a \\(O(\\log n)\\) \u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u7b56\u7565\u5982\u4e0b\u6240\u793a\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4e8c\u5206\u67e5\u627e\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\uff08\u5728\u6570\u7ec4\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\u5206\u89e3\u4e3a\u5b50\u95ee\u9898\uff08\u5728\u6570\u7ec4\u7684\u4e00\u534a\u4e2d\u8fdb\u884c\u67e5\u627e\uff09\uff0c\u8fd9\u662f\u901a\u8fc7\u6bd4\u8f83\u4e2d\u95f4\u5143\u7d20\u548c\u76ee\u6807\u5143\u7d20\u6765\u5b9e\u73b0\u7684\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u5904\u7406\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5b83\u4e0d\u53d7\u5176\u4ed6\u5b50\u95ee\u9898\u7684\u5f71\u54cd\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u65e0\u987b\u5408\u5e76\uff1a\u4e8c\u5206\u67e5\u627e\u65e8\u5728\u67e5\u627e\u4e00\u4e2a\u7279\u5b9a\u5143\u7d20\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u3002\u5f53\u5b50\u95ee\u9898\u5f97\u5230\u89e3\u51b3\u65f6\uff0c\u539f\u95ee\u9898\u4e5f\u4f1a\u540c\u65f6\u5f97\u5230\u89e3\u51b3\u3002</li> </ul> <p>\u5206\u6cbb\u80fd\u591f\u63d0\u5347\u641c\u7d22\u6548\u7387\uff0c\u672c\u8d28\u4e0a\u662f\u56e0\u4e3a\u66b4\u529b\u641c\u7d22\u6bcf\u8f6e\u53ea\u80fd\u6392\u9664\u4e00\u4e2a\u9009\u9879\uff0c\u800c\u5206\u6cbb\u641c\u7d22\u6bcf\u8f6e\u53ef\u4ee5\u6392\u9664\u4e00\u534a\u9009\u9879\u3002</p>"},{"location":"chapter_divide_and_conquer/binary_search_recur/#1","title":"1. \u00a0 \u57fa\u4e8e\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e","text":"<p>\u5728\u4e4b\u524d\u7684\u7ae0\u8282\u4e2d\uff0c\u4e8c\u5206\u67e5\u627e\u662f\u57fa\u4e8e\u9012\u63a8\uff08\u8fed\u4ee3\uff09\u5b9e\u73b0\u7684\u3002\u73b0\u5728\u6211\u4eec\u57fa\u4e8e\u5206\u6cbb\uff08\u9012\u5f52\uff09\u6765\u5b9e\u73b0\u5b83\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u662f\u552f\u4e00\u7684\uff0c\u8bf7\u67e5\u627e\u5143\u7d20 <code>target</code> \u3002</p> <p>\u4ece\u5206\u6cbb\u89d2\u5ea6\uff0c\u6211\u4eec\u5c06\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u8bb0\u4e3a \\(f(i, j)\\) \u3002</p> <p>\u4ee5\u539f\u95ee\u9898 \\(f(0, n-1)\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u8fdb\u884c\u4e8c\u5206\u67e5\u627e\u3002</p> <ol> <li>\u8ba1\u7b97\u641c\u7d22\u533a\u95f4 \\([i, j]\\) \u7684\u4e2d\u70b9 \\(m\\) \uff0c\u6839\u636e\u5b83\u6392\u9664\u4e00\u534a\u641c\u7d22\u533a\u95f4\u3002</li> <li>\u9012\u5f52\u6c42\u89e3\u89c4\u6a21\u51cf\u5c0f\u4e00\u534a\u7684\u5b50\u95ee\u9898\uff0c\u53ef\u80fd\u4e3a \\(f(i, m-1)\\) \u6216 \\(f(m+1, j)\\) \u3002</li> <li>\u5faa\u73af\u7b2c <code>1.</code> \u6b65\u548c\u7b2c <code>2.</code> \u6b65\uff0c\u76f4\u81f3\u627e\u5230 <code>target</code> \u6216\u533a\u95f4\u4e3a\u7a7a\u65f6\u8fd4\u56de\u3002</li> </ol> <p>\u56fe 12-4 \u5c55\u793a\u4e86\u5728\u6570\u7ec4\u4e2d\u4e8c\u5206\u67e5\u627e\u5143\u7d20 \\(6\\) \u7684\u5206\u6cbb\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 12-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u7684\u5206\u6cbb\u8fc7\u7a0b </p> <p>\u5728\u5b9e\u73b0\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \u6765\u6c42\u89e3\u95ee\u9898 \\(f(i, j)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_recur.py<pre><code>def dfs(nums: list[int], target: int, i: int, j: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j)\"\"\"\n    # \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j:\n        return -1\n    # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    m = (i + j) // 2\n    if nums[m] &lt; target:\n        # \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j)\n    elif nums[m] &gt; target:\n        # \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1)\n    else:\n        # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n\ndef binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\"\"\"\n    n = len(nums)\n    # \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1)\n</code></pre> binary_search_recur.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(vector&lt;int&gt; &amp;nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    int n = nums.size();\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int[] nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int[] nums, int target) {\n    int n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint DFS(int[] nums, int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return DFS(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return DFS(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint BinarySearch(int[] nums, int target) {\n    int n = nums.Length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return DFS(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums []int, target, i, j int) int {\n    // \u5982\u679c\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u6ca1\u6709\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    //    \u8ba1\u7b97\u7d22\u5f15\u4e2d\u70b9\n    m := i + ((j - i) &gt;&gt; 1)\n    //\u5224\u65ad\u4e2d\u70b9\u4e0e\u76ee\u6807\u5143\u7d20\u5927\u5c0f\n    if nums[m] &lt; target {\n        // \u5c0f\u4e8e\u5219\u9012\u5f52\u53f3\u534a\u6570\u7ec4\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m+1, j)\n    } else if nums[m] &gt; target {\n        // \u5c0f\u4e8e\u5219\u9012\u5f52\u5de6\u534a\u6570\u7ec4\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m-1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums []int, target int) int {\n    n := len(nums)\n    return dfs(nums, target, 0, n-1)\n}\n</code></pre> binary_search_recur.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunc dfs(nums: [Int], target: Int, i: Int, j: Int) -&gt; Int {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j {\n        return -1\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    let m = (i + j) / 2\n    if nums[m] &lt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums: nums, target: target, i: m + 1, j: j)\n    } else if nums[m] &gt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums: nums, target: target, i: i, j: m - 1)\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    let n = nums.count\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums: nums, target: target, i: 0, j: n - 1)\n}\n</code></pre> binary_search_recur.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums, target, i, j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums, target) {\n    const n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfunction dfs(nums: number[], target: number, i: number, j: number): number {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    const m = i + ((j - i) &gt;&gt; 1);\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfunction binarySearch(nums: number[], target: number): number {\n    const n = nums.length;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(List&lt;int&gt; nums, int target, int i, int j) {\n  // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n  if (i &gt; j) {\n    return -1;\n  }\n  // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n  int m = (i + j) ~/ 2;\n  if (nums[m] &lt; target) {\n    // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n    return dfs(nums, target, m + 1, j);\n  } else if (nums[m] &gt; target) {\n    // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n    return dfs(nums, target, i, m - 1);\n  } else {\n    // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return m;\n  }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  int n = nums.length;\n  // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n  return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nfn dfs(nums: &amp;[i32], target: i32, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if i &gt; j { return -1; }\n    let m: i32 = (i + j) / 2;\n    if nums[m as usize] &lt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if nums[m as usize] &gt; target {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let n = nums.len() as i32;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    dfs(nums, target, 0, n - 1)\n}\n</code></pre> binary_search_recur.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff1a\u95ee\u9898 f(i, j) */\nint dfs(int nums[], int target, int i, int j) {\n    // \u82e5\u533a\u95f4\u4e3a\u7a7a\uff0c\u4ee3\u8868\u65e0\u76ee\u6807\u5143\u7d20\uff0c\u5219\u8fd4\u56de -1\n    if (i &gt; j) {\n        return -1;\n    }\n    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    int m = (i + j) / 2;\n    if (nums[m] &lt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(m+1, j)\n        return dfs(nums, target, m + 1, j);\n    } else if (nums[m] &gt; target) {\n        // \u9012\u5f52\u5b50\u95ee\u9898 f(i, m-1)\n        return dfs(nums, target, i, m - 1);\n    } else {\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        return m;\n    }\n}\n\n/* \u4e8c\u5206\u67e5\u627e */\nint binarySearch(int nums[], int target, int numsSize) {\n    int n = numsSize;\n    // \u6c42\u89e3\u95ee\u9898 f(0, n-1)\n    return dfs(nums, target, 0, n - 1);\n}\n</code></pre> binary_search_recur.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{binarySearch}\n</code></pre>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/","title":"12.3 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u68f5\u4e8c\u53c9\u6811\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \uff0c\u8bf7\u4ece\u4e2d\u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u8fd4\u56de\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u3002\u5047\u8bbe\u4e8c\u53c9\u6811\u4e2d\u6ca1\u6709\u503c\u91cd\u590d\u7684\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 12-5 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#1","title":"1. \u00a0 \u5224\u65ad\u662f\u5426\u4e3a\u5206\u6cbb\u95ee\u9898","text":"<p>\u539f\u95ee\u9898\u5b9a\u4e49\u4e3a\u4ece <code>preorder</code> \u548c <code>inorder</code> \u6784\u5efa\u4e8c\u53c9\u6811\uff0c\u662f\u4e00\u4e2a\u5178\u578b\u7684\u5206\u6cbb\u95ee\u9898\u3002</p> <ul> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u4ece\u5206\u6cbb\u7684\u89d2\u5ea6\u5207\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\u3001\u6784\u5efa\u53f3\u5b50\u6811\uff0c\u52a0\u4e0a\u4e00\u6b65\u64cd\u4f5c\uff1a\u521d\u59cb\u5316\u6839\u8282\u70b9\u3002\u800c\u5bf9\u4e8e\u6bcf\u68f5\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u590d\u7528\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u5c06\u5176\u5212\u5206\u4e3a\u66f4\u5c0f\u7684\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u7a7a\u5b50\u6811\uff09\u65f6\u7ec8\u6b62\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u5b83\u4eec\u4e4b\u95f4\u6ca1\u6709\u4ea4\u96c6\u3002\u5728\u6784\u5efa\u5de6\u5b50\u6811\u65f6\uff0c\u6211\u4eec\u53ea\u9700\u5173\u6ce8\u4e2d\u5e8f\u904d\u5386\u548c\u524d\u5e8f\u904d\u5386\u4e2d\u4e0e\u5de6\u5b50\u6811\u5bf9\u5e94\u7684\u90e8\u5206\u3002\u53f3\u5b50\u6811\u540c\u7406\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e00\u65e6\u5f97\u5230\u4e86\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u5b83\u4eec\u94fe\u63a5\u5230\u6839\u8282\u70b9\u4e0a\uff0c\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ul>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#2","title":"2. \u00a0 \u5982\u4f55\u5212\u5206\u5b50\u6811","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u8fd9\u9053\u9898\u53ef\u4ee5\u4f7f\u7528\u5206\u6cbb\u6765\u6c42\u89e3\uff0c\u4f46\u5982\u4f55\u901a\u8fc7\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u6765\u5212\u5206\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u5462\uff1f</p> <p>\u6839\u636e\u5b9a\u4e49\uff0c<code>preorder</code> \u548c <code>inorder</code> \u90fd\u53ef\u4ee5\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\u3002</p> <ul> <li>\u524d\u5e8f\u904d\u5386\uff1a<code>[ \u6839\u8282\u70b9 | \u5de6\u5b50\u6811 | \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> <li>\u4e2d\u5e8f\u904d\u5386\uff1a<code>[ \u5de6\u5b50\u6811 | \u6839\u8282\u70b9 \uff5c \u53f3\u5b50\u6811 ]</code> \uff0c\u4f8b\u5982\u56fe 12-5 \u7684\u6811\u5bf9\u5e94 <code>[ 9 | 3 | 1 2 7 ]</code> \u3002</li> </ul> <p>\u4ee5\u4e0a\u56fe\u6570\u636e\u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u56fe 12-6 \u6240\u793a\u7684\u6b65\u9aa4\u5f97\u5230\u5212\u5206\u7ed3\u679c\u3002</p> <ol> <li>\u524d\u5e8f\u904d\u5386\u7684\u9996\u5143\u7d20 3 \u662f\u6839\u8282\u70b9\u7684\u503c\u3002</li> <li>\u67e5\u627e\u6839\u8282\u70b9 3 \u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u5229\u7528\u8be5\u7d22\u5f15\u53ef\u5c06 <code>inorder</code> \u5212\u5206\u4e3a <code>[ 9 | 3 \uff5c 1 2 7 ]</code> \u3002</li> <li>\u6839\u636e <code>inorder</code> \u5212\u5206\u7ed3\u679c\uff0c\u6613\u5f97\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u5206\u522b\u4e3a 1 \u548c 3 \uff0c\u4ece\u800c\u53ef\u5c06 <code>preorder</code> \u5212\u5206\u4e3a <code>[ 3 | 9 | 2 1 7 ]</code> \u3002</li> </ol> <p></p> <p> \u56fe 12-6 \u00a0 \u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u5212\u5206\u5b50\u6811 </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#3","title":"3. \u00a0 \u57fa\u4e8e\u53d8\u91cf\u63cf\u8ff0\u5b50\u6811\u533a\u95f4","text":"<p>\u6839\u636e\u4ee5\u4e0a\u5212\u5206\u65b9\u6cd5\uff0c\u6211\u4eec\u5df2\u7ecf\u5f97\u5230\u6839\u8282\u70b9\u3001\u5de6\u5b50\u6811\u3001\u53f3\u5b50\u6811\u5728 <code>preorder</code> \u548c <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002\u800c\u4e3a\u4e86\u63cf\u8ff0\u8fd9\u4e9b\u7d22\u5f15\u533a\u95f4\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u3002</p> <ul> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(i\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u7684\u6839\u8282\u70b9\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u8bb0\u4e3a \\(m\\) \u3002</li> <li>\u5c06\u5f53\u524d\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u8bb0\u4e3a \\([l, r]\\) \u3002</li> </ul> <p>\u5982\u8868 12-1 \u6240\u793a\uff0c\u901a\u8fc7\u4ee5\u4e0a\u53d8\u91cf\u5373\u53ef\u8868\u793a\u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15\uff0c\u4ee5\u53ca\u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4\u3002</p> <p> \u8868 12-1 \u00a0 \u6839\u8282\u70b9\u548c\u5b50\u6811\u5728\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u4e2d\u7684\u7d22\u5f15 </p> \u6839\u8282\u70b9\u5728 <code>preorder</code> \u4e2d\u7684\u7d22\u5f15 \u5b50\u6811\u5728 <code>inorder</code> \u4e2d\u7684\u7d22\u5f15\u533a\u95f4 \u5f53\u524d\u6811 \\(i\\) \\([l, r]\\) \u5de6\u5b50\u6811 \\(i + 1\\) \\([l, m-1]\\) \u53f3\u5b50\u6811 \\(i + 1 + (m - l)\\) \\([m+1, r]\\) <p>\u8bf7\u6ce8\u610f\uff0c\u53f3\u5b50\u6811\u6839\u8282\u70b9\u7d22\u5f15\u4e2d\u7684 \\((m-l)\\) \u7684\u542b\u4e49\u662f\u201c\u5de6\u5b50\u6811\u7684\u8282\u70b9\u6570\u91cf\u201d\uff0c\u5efa\u8bae\u914d\u5408\u56fe 12-7 \u7406\u89e3\u3002</p> <p></p> <p> \u56fe 12-7 \u00a0 \u6839\u8282\u70b9\u548c\u5de6\u53f3\u5b50\u6811\u7684\u7d22\u5f15\u533a\u95f4\u8868\u793a </p>"},{"location":"chapter_divide_and_conquer/build_binary_tree_problem/#4","title":"4. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u63d0\u5347\u67e5\u8be2 \\(m\\) \u7684\u6548\u7387\uff0c\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>hmap</code> \u6765\u5b58\u50a8\u6570\u7ec4 <code>inorder</code> \u4e2d\u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig build_tree.py<pre><code>def dfs(\n    preorder: list[int],\n    inorder_map: dict[int, int],\n    i: int,\n    l: int,\n    r: int,\n) -&gt; TreeNode | None:\n    \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb\"\"\"\n    # \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0:\n        return None\n    # \u521d\u59cb\u5316\u6839\u8282\u70b9\n    root = TreeNode(preorder[i])\n    # \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    m = inorder_map[preorder[i]]\n    # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorder_map, i + 1, l, m - 1)\n    # \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r)\n    # \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n\ndef build_tree(preorder: list[int], inorder: list[int]) -&gt; TreeNode | None:\n    \"\"\"\u6784\u5efa\u4e8c\u53c9\u6811\"\"\"\n    # \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    inorder_map = {val: i for i, val in enumerate(inorder)}\n    root = dfs(preorder, inorder_map, 0, 0, len(inorder) - 1)\n    return root\n</code></pre> build_tree.cpp<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(vector&lt;int&gt; &amp;preorder, unordered_map&lt;int, int&gt; &amp;inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode *root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(vector&lt;int&gt; &amp;preorder, vector&lt;int&gt; &amp;inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    unordered_map&lt;int, int&gt; inorderMap;\n    for (int i = 0; i &lt; inorder.size(); i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorder.size() - 1);\n    return root;\n}\n</code></pre> build_tree.java<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode dfs(int[] preorder, Map&lt;Integer, Integer&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode buildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    Map&lt;Integer, Integer&gt; inorderMap = new HashMap&lt;&gt;();\n    for (int i = 0; i &lt; inorder.length; i++) {\n        inorderMap.put(inorder[i], i);\n    }\n    TreeNode root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.cs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? DFS(int[] preorder, Dictionary&lt;int, int&gt; inorderMap, int i, int l, int r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode root = new(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = DFS(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = DFS(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? BuildTree(int[] preorder, int[] inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    Dictionary&lt;int, int&gt; inorderMap = [];\n    for (int i = 0; i &lt; inorder.Length; i++) {\n        inorderMap.TryAdd(inorder[i], i);\n    }\n    TreeNode? root = DFS(preorder, inorderMap, 0, 0, inorder.Length - 1);\n    return root;\n}\n</code></pre> build_tree.go<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfsBuildTree(preorder []int, inorderMap map[int]int, i, l, r int) *TreeNode {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r-l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    root := NewTreeNode(preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    m := inorderMap[preorder[i]]\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.Left = dfsBuildTree(preorder, inorderMap, i+1, l, m-1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.Right = dfsBuildTree(preorder, inorderMap, i+1+m-l, m+1, r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder, inorder []int) *TreeNode {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    inorderMap := make(map[int]int, len(inorder))\n    for i := 0; i &lt; len(inorder); i++ {\n        inorderMap[inorder[i]] = i\n    }\n\n    root := dfsBuildTree(preorder, inorderMap, 0, 0, len(inorder)-1)\n    return root\n}\n</code></pre> build_tree.swift<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunc dfs(preorder: [Int], inorderMap: [Int: Int], i: Int, l: Int, r: Int) -&gt; TreeNode? {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0 {\n        return nil\n    }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    let root = TreeNode(x: preorder[i])\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    let m = inorderMap[preorder[i]]!\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1, l: l, r: m - 1)\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder: preorder, inorderMap: inorderMap, i: i + 1 + m - l, l: m + 1, r: r)\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunc buildTree(preorder: [Int], inorder: [Int]) -&gt; TreeNode? {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = inorder.enumerated().reduce(into: [:]) { $0[$1.element] = $1.offset }\n    return dfs(preorder: preorder, inorderMap: inorderMap, i: 0, l: 0, r: inorder.count - 1)\n}\n</code></pre> build_tree.js<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(preorder, inorderMap, i, l, r) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    const root = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder, inorder) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = new Map();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.ts<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfunction dfs(\n    preorder: number[],\n    inorderMap: Map&lt;number, number&gt;,\n    i: number,\n    l: number,\n    r: number\n): TreeNode | null {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0) return null;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    const root: TreeNode = new TreeNode(preorder[i]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    const m = inorderMap.get(preorder[i]);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfunction buildTree(preorder: number[], inorder: number[]): TreeNode | null {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let inorderMap = new Map&lt;number, number&gt;();\n    for (let i = 0; i &lt; inorder.length; i++) {\n        inorderMap.set(inorder[i], i);\n    }\n    const root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n    return root;\n}\n</code></pre> build_tree.dart<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode? dfs(\n  List&lt;int&gt; preorder,\n  Map&lt;int, int&gt; inorderMap,\n  int i,\n  int l,\n  int r,\n) {\n  // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n  if (r - l &lt; 0) {\n    return null;\n  }\n  // \u521d\u59cb\u5316\u6839\u8282\u70b9\n  TreeNode? root = TreeNode(preorder[i]);\n  // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n  int m = inorderMap[preorder[i]]!;\n  // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n  root.left = dfs(preorder, inorderMap, i + 1, l, m - 1);\n  // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n  root.right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r);\n  // \u8fd4\u56de\u6839\u8282\u70b9\n  return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode? buildTree(List&lt;int&gt; preorder, List&lt;int&gt; inorder) {\n  // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n  Map&lt;int, int&gt; inorderMap = {};\n  for (int i = 0; i &lt; inorder.length; i++) {\n    inorderMap[inorder[i]] = i;\n  }\n  TreeNode? root = dfs(preorder, inorderMap, 0, 0, inorder.length - 1);\n  return root;\n}\n</code></pre> build_tree.rs<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nfn dfs(preorder: &amp;[i32], inorder_map: &amp;HashMap&lt;i32, i32&gt;, i: i32, l: i32, r: i32) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if r - l &lt; 0 { return None; }\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    let root = TreeNode::new(preorder[i as usize]);\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    let m = inorder_map.get(&amp;preorder[i as usize]).unwrap();\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root.borrow_mut().left = dfs(preorder, inorder_map, i + 1, l, m - 1);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root.borrow_mut().right = dfs(preorder, inorder_map, i + 1 + m - l, m + 1, r);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    Some(root)\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nfn build_tree(preorder: &amp;[i32], inorder: &amp;[i32]) -&gt; Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt; {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    let mut inorder_map: HashMap&lt;i32, i32&gt; = HashMap::new();\n    for i in 0..inorder.len() {\n        inorder_map.insert(inorder[i], i as i32);\n    }\n    let root = dfs(preorder, &amp;inorder_map, 0, 0, inorder.len() as i32 - 1);\n    root\n}\n</code></pre> build_tree.c<pre><code>/* \u6784\u5efa\u4e8c\u53c9\u6811\uff1a\u5206\u6cbb */\nTreeNode *dfs(int *preorder, int *inorderMap, int i, int l, int r, int size) {\n    // \u5b50\u6811\u533a\u95f4\u4e3a\u7a7a\u65f6\u7ec8\u6b62\n    if (r - l &lt; 0)\n        return NULL;\n    // \u521d\u59cb\u5316\u6839\u8282\u70b9\n    TreeNode *root = (TreeNode *)malloc(sizeof(TreeNode));\n    root-&gt;val = preorder[i];\n    root-&gt;left = NULL;\n    root-&gt;right = NULL;\n    // \u67e5\u8be2 m \uff0c\u4ece\u800c\u5212\u5206\u5de6\u53f3\u5b50\u6811\n    int m = inorderMap[preorder[i]];\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u5de6\u5b50\u6811\n    root-&gt;left = dfs(preorder, inorderMap, i + 1, l, m - 1, size);\n    // \u5b50\u95ee\u9898\uff1a\u6784\u5efa\u53f3\u5b50\u6811\n    root-&gt;right = dfs(preorder, inorderMap, i + 1 + m - l, m + 1, r, size);\n    // \u8fd4\u56de\u6839\u8282\u70b9\n    return root;\n}\n\n/* \u6784\u5efa\u4e8c\u53c9\u6811 */\nTreeNode *buildTree(int *preorder, int preorderSize, int *inorder, int inorderSize) {\n    // \u521d\u59cb\u5316\u54c8\u5e0c\u8868\uff0c\u5b58\u50a8 inorder \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\n    int *inorderMap = (int *)malloc(sizeof(int) * MAX_SIZE);\n    for (int i = 0; i &lt; inorderSize; i++) {\n        inorderMap[inorder[i]] = i;\n    }\n    TreeNode *root = dfs(preorder, inorderMap, 0, 0, inorderSize - 1, inorderSize);\n    free(inorderMap);\n    return root;\n}\n</code></pre> build_tree.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{buildTree}\n</code></pre> <p>\u56fe 12-8 \u5c55\u793a\u4e86\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5404\u4e2a\u8282\u70b9\u662f\u5728\u5411\u4e0b\u201c\u9012\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\uff0c\u800c\u5404\u6761\u8fb9\uff08\u5f15\u7528\uff09\u662f\u5728\u5411\u4e0a\u201c\u5f52\u201d\u7684\u8fc7\u7a0b\u4e2d\u5efa\u7acb\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-8 \u00a0 \u6784\u5efa\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b </p> <p>\u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u5185\u7684\u524d\u5e8f\u904d\u5386 <code>preorder</code> \u548c\u4e2d\u5e8f\u904d\u5386 <code>inorder</code> \u7684\u5212\u5206\u7ed3\u679c\u5982\u56fe 12-9 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 12-9 \u00a0 \u6bcf\u4e2a\u9012\u5f52\u51fd\u6570\u4e2d\u7684\u5212\u5206\u7ed3\u679c </p> <p>\u8bbe\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u521d\u59cb\u5316\u6bcf\u4e00\u4e2a\u8282\u70b9\uff08\u6267\u884c\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs()</code> \uff09\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u54c8\u5e0c\u8868\u5b58\u50a8 <code>inorder</code> \u5143\u7d20\u5230\u7d22\u5f15\u7684\u6620\u5c04\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u7684\u6808\u5e27\u7a7a\u95f4\u3002\u56e0\u6b64\u603b\u4f53\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/","title":"12.1 \u00a0 \u5206\u6cbb\u7b97\u6cd5","text":"<p>\u300c\u5206\u6cbb divide and conquer\u300d\uff0c\u5168\u79f0\u5206\u800c\u6cbb\u4e4b\uff0c\u662f\u4e00\u79cd\u975e\u5e38\u91cd\u8981\u4e14\u5e38\u89c1\u7684\u7b97\u6cd5\u7b56\u7565\u3002\u5206\u6cbb\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff0c\u5305\u62ec\u201c\u5206\u201d\u548c\u201c\u6cbb\u201d\u4e24\u4e2a\u6b65\u9aa4\u3002</p> <ol> <li>\u5206\uff08\u5212\u5206\u9636\u6bb5\uff09\uff1a\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u6216\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898\u65f6\u7ec8\u6b62\u3002</li> <li>\u6cbb\uff08\u5408\u5e76\u9636\u6bb5\uff09\uff1a\u4ece\u5df2\u77e5\u89e3\u7684\u6700\u5c0f\u5b50\u95ee\u9898\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</li> </ol> <p>\u5982\u56fe 12-1 \u6240\u793a\uff0c\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\u4e4b\u4e00\u3002</p> <ol> <li>\u5206\uff1a\u9012\u5f52\u5730\u5c06\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\uff0c\u76f4\u5230\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\uff08\u6700\u5c0f\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u6cbb\uff1a\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u6709\u5e8f\u7684\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u8fdb\u884c\u5408\u5e76\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u7684\u539f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol> <p></p> <p> \u56fe 12-1 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1211","title":"12.1.1 \u00a0 \u5982\u4f55\u5224\u65ad\u5206\u6cbb\u95ee\u9898","text":"<p>\u4e00\u4e2a\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u5206\u6cbb\u89e3\u51b3\uff0c\u901a\u5e38\u53ef\u4ee5\u53c2\u8003\u4ee5\u4e0b\u51e0\u4e2a\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u539f\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u89c4\u6a21\u66f4\u5c0f\u3001\u7c7b\u4f3c\u7684\u5b50\u95ee\u9898\uff0c\u4ee5\u53ca\u80fd\u591f\u4ee5\u76f8\u540c\u65b9\u5f0f\u9012\u5f52\u5730\u8fdb\u884c\u5212\u5206\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u5b50\u95ee\u9898\u4e4b\u95f4\u6ca1\u6709\u91cd\u53e0\uff0c\u4e92\u4e0d\u4f9d\u8d56\uff0c\u53ef\u4ee5\u72ec\u7acb\u89e3\u51b3\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u539f\u95ee\u9898\u7684\u89e3\u901a\u8fc7\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\u5f97\u6765\u3002</li> </ol> <p>\u663e\u7136\uff0c\u5f52\u5e76\u6392\u5e8f\u6ee1\u8db3\u4ee5\u4e0a\u4e09\u6761\u5224\u65ad\u4f9d\u636e\u3002</p> <ol> <li>\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\uff1a\u9012\u5f52\u5730\u5c06\u6570\u7ec4\uff08\u539f\u95ee\u9898\uff09\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff1a\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u53ef\u4ee5\u72ec\u7acb\u5730\u8fdb\u884c\u6392\u5e8f\uff08\u5b50\u95ee\u9898\u53ef\u4ee5\u72ec\u7acb\u8fdb\u884c\u6c42\u89e3\uff09\u3002</li> <li>\u5b50\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u5408\u5e76\uff1a\u4e24\u4e2a\u6709\u5e8f\u5b50\u6570\u7ec4\uff08\u5b50\u95ee\u9898\u7684\u89e3\uff09\u53ef\u4ee5\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\uff08\u539f\u95ee\u9898\u7684\u89e3\uff09\u3002</li> </ol>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1212","title":"12.1.2 \u00a0 \u901a\u8fc7\u5206\u6cbb\u63d0\u5347\u6548\u7387","text":"<p>\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u7b97\u6cd5\u95ee\u9898\uff0c\u5f80\u5f80\u8fd8\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u5728\u6392\u5e8f\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u3001\u5f52\u5e76\u6392\u5e8f\u3001\u5806\u6392\u5e8f\u76f8\u8f83\u4e8e\u9009\u62e9\u3001\u5192\u6ce1\u3001\u63d2\u5165\u6392\u5e8f\u66f4\u5feb\uff0c\u5c31\u662f\u56e0\u4e3a\u5b83\u4eec\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u4e0d\u7981\u53d1\u95ee\uff1a\u4e3a\u4ec0\u4e48\u5206\u6cbb\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u5176\u5e95\u5c42\u903b\u8f91\u662f\u4ec0\u4e48\uff1f\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c06\u5927\u95ee\u9898\u5206\u89e3\u4e3a\u591a\u4e2a\u5b50\u95ee\u9898\u3001\u89e3\u51b3\u5b50\u95ee\u9898\u3001\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u5408\u5e76\u4e3a\u539f\u95ee\u9898\u7684\u89e3\uff0c\u8fd9\u51e0\u6b65\u7684\u6548\u7387\u4e3a\u4ec0\u4e48\u6bd4\u76f4\u63a5\u89e3\u51b3\u539f\u95ee\u9898\u7684\u6548\u7387\u66f4\u9ad8\uff1f\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u4ece\u64cd\u4f5c\u6570\u91cf\u548c\u5e76\u884c\u8ba1\u7b97\u4e24\u65b9\u9762\u6765\u8ba8\u8bba\u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1","title":"1. \u00a0 \u64cd\u4f5c\u6570\u91cf\u4f18\u5316","text":"<p>\u4ee5\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u4e3a\u4f8b\uff0c\u5176\u5904\u7406\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u9700\u8981 \\(O(n^2)\\) \u65f6\u95f4\u3002\u5047\u8bbe\u6211\u4eec\u6309\u7167\u56fe 12-2 \u6240\u793a\u7684\u65b9\u5f0f\uff0c\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u5219\u5212\u5206\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6bcf\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O((n / 2)^2)\\) \u65f6\u95f4\uff0c\u5408\u5e76\u4e24\u4e2a\u5b50\u6570\u7ec4\u9700\u8981 \\(O(n)\\) \u65f6\u95f4\uff0c\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ O(n + (\\frac{n}{2})^2 \\times 2 + n) = O(\\frac{n^2}{2} + 2n) \\] <p></p> <p> \u56fe 12-2 \u00a0 \u5212\u5206\u6570\u7ec4\u524d\u540e\u7684\u5192\u6ce1\u6392\u5e8f </p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u8ba1\u7b97\u4ee5\u4e0b\u4e0d\u7b49\u5f0f\uff0c\u5176\u5de6\u8fb9\u548c\u53f3\u8fb9\u5206\u522b\u4e3a\u5212\u5206\u524d\u548c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u603b\u6570\uff1a</p> \\[ \\begin{aligned} n^2 &amp; &gt; \\frac{n^2}{2} + 2n \\newline n^2 - \\frac{n^2}{2} - 2n &amp; &gt; 0 \\newline n(n - 4) &amp; &gt; 0 \\end{aligned} \\] <p>\u8fd9\u610f\u5473\u7740\u5f53 \\(n &gt; 4\\) \u65f6\uff0c\u5212\u5206\u540e\u7684\u64cd\u4f5c\u6570\u91cf\u66f4\u5c11\uff0c\u6392\u5e8f\u6548\u7387\u5e94\u8be5\u66f4\u9ad8\u3002\u8bf7\u6ce8\u610f\uff0c\u5212\u5206\u540e\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u7136\u662f\u5e73\u65b9\u9636 \\(O(n^2)\\) \uff0c\u53ea\u662f\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\u4e86\u3002</p> <p>\u8fdb\u4e00\u6b65\u60f3\uff0c\u5982\u679c\u6211\u4eec\u628a\u5b50\u6570\u7ec4\u4e0d\u65ad\u5730\u518d\u4ece\u4e2d\u70b9\u5904\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u505c\u6b62\u5212\u5206\u5462\uff1f\u8fd9\u79cd\u601d\u8def\u5b9e\u9645\u4e0a\u5c31\u662f\u201c\u5f52\u5e76\u6392\u5e8f\u201d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u518d\u601d\u8003\uff0c\u5982\u679c\u6211\u4eec\u591a\u8bbe\u7f6e\u51e0\u4e2a\u5212\u5206\u70b9\uff0c\u5c06\u539f\u6570\u7ec4\u5e73\u5747\u5212\u5206\u4e3a \\(k\\) \u4e2a\u5b50\u6570\u7ec4\u5462\uff1f\u8fd9\u79cd\u60c5\u51b5\u4e0e\u201c\u6876\u6392\u5e8f\u201d\u975e\u5e38\u7c7b\u4f3c\uff0c\u5b83\u975e\u5e38\u9002\u5408\u6392\u5e8f\u6d77\u91cf\u6570\u636e\uff0c\u7406\u8bba\u4e0a\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230 \\(O(n + k)\\) \u3002</p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#2","title":"2. \u00a0 \u5e76\u884c\u8ba1\u7b97\u4f18\u5316","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5206\u6cbb\u751f\u6210\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u56e0\u6b64\u901a\u5e38\u53ef\u4ee5\u5e76\u884c\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5206\u6cbb\u4e0d\u4ec5\u53ef\u4ee5\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8fd8\u6709\u5229\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</p> <p>\u5e76\u884c\u4f18\u5316\u5728\u591a\u6838\u6216\u591a\u5904\u7406\u5668\u7684\u73af\u5883\u4e2d\u5c24\u5176\u6709\u6548\uff0c\u56e0\u4e3a\u7cfb\u7edf\u53ef\u4ee5\u540c\u65f6\u5904\u7406\u591a\u4e2a\u5b50\u95ee\u9898\uff0c\u66f4\u52a0\u5145\u5206\u5730\u5229\u7528\u8ba1\u7b97\u8d44\u6e90\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u603b\u4f53\u7684\u8fd0\u884c\u65f6\u95f4\u3002</p> <p>\u6bd4\u5982\u5728\u56fe 12-3 \u6240\u793a\u7684\u201c\u6876\u6392\u5e8f\u201d\u4e2d\uff0c\u6211\u4eec\u5c06\u6d77\u91cf\u7684\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u5219\u53ef\u6240\u6709\u6876\u7684\u6392\u5e8f\u4efb\u52a1\u5206\u6563\u5230\u5404\u4e2a\u8ba1\u7b97\u5355\u5143\uff0c\u5b8c\u6210\u540e\u518d\u5408\u5e76\u7ed3\u679c\u3002</p> <p></p> <p> \u56fe 12-3 \u00a0 \u6876\u6392\u5e8f\u7684\u5e76\u884c\u8ba1\u7b97 </p>"},{"location":"chapter_divide_and_conquer/divide_and_conquer/#1213","title":"12.1.3 \u00a0 \u5206\u6cbb\u5e38\u89c1\u5e94\u7528","text":"<p>\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8bb8\u591a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff1a\u8be5\u7b97\u6cd5\u9996\u5148\u5c06\u70b9\u96c6\u5206\u6210\u4e24\u90e8\u5206\uff0c\u7136\u540e\u5206\u522b\u627e\u51fa\u4e24\u90e8\u5206\u4e2d\u7684\u6700\u8fd1\u70b9\u5bf9\uff0c\u6700\u540e\u627e\u51fa\u8de8\u8d8a\u4e24\u90e8\u5206\u7684\u6700\u8fd1\u70b9\u5bf9\u3002</li> <li>\u5927\u6574\u6570\u4e58\u6cd5\uff1a\u4f8b\u5982 Karatsuba \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u6574\u6570\u4e58\u6cd5\u5206\u89e3\u4e3a\u51e0\u4e2a\u8f83\u5c0f\u7684\u6574\u6570\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u77e9\u9635\u4e58\u6cd5\uff1a\u4f8b\u5982 Strassen \u7b97\u6cd5\uff0c\u5b83\u5c06\u5927\u77e9\u9635\u4e58\u6cd5\u5206\u89e3\u4e3a\u591a\u4e2a\u5c0f\u77e9\u9635\u7684\u4e58\u6cd5\u548c\u52a0\u6cd5\u3002</li> <li>\u6c49\u8bfa\u5854\u95ee\u9898\uff1a\u6c49\u8bfa\u5854\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u89e3\u51b3\uff0c\u8fd9\u662f\u5178\u578b\u7684\u5206\u6cbb\u7b56\u7565\u5e94\u7528\u3002</li> <li>\u6c42\u89e3\u9006\u5e8f\u5bf9\uff1a\u5728\u4e00\u4e2a\u5e8f\u5217\u4e2d\uff0c\u5982\u679c\u524d\u9762\u7684\u6570\u5b57\u5927\u4e8e\u540e\u9762\u7684\u6570\u5b57\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u6570\u5b57\u6784\u6210\u4e00\u4e2a\u9006\u5e8f\u5bf9\u3002\u6c42\u89e3\u9006\u5e8f\u5bf9\u95ee\u9898\u53ef\u4ee5\u5229\u7528\u5206\u6cbb\u7684\u601d\u60f3\uff0c\u501f\u52a9\u5f52\u5e76\u6392\u5e8f\u8fdb\u884c\u6c42\u89e3\u3002</li> </ul> <p>\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u5728\u7b97\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u7684\u8bbe\u8ba1\u4e2d\u5e94\u7528\u975e\u5e38\u5e7f\u6cdb\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\uff1a\u4e8c\u5206\u67e5\u627e\u662f\u5c06\u6709\u5e8f\u6570\u7ec4\u4ece\u4e2d\u70b9\u7d22\u5f15\u5904\u5206\u4e3a\u4e24\u90e8\u5206\uff0c\u7136\u540e\u6839\u636e\u76ee\u6807\u503c\u4e0e\u4e2d\u95f4\u5143\u7d20\u503c\u6bd4\u8f83\u7ed3\u679c\uff0c\u51b3\u5b9a\u6392\u9664\u54ea\u4e00\u534a\u533a\u95f4\uff0c\u5e76\u5728\u5269\u4f59\u533a\u95f4\u6267\u884c\u76f8\u540c\u7684\u4e8c\u5206\u64cd\u4f5c\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u672c\u8282\u5f00\u5934\u5df2\u4ecb\u7ecd\uff0c\u4e0d\u518d\u8d58\u8ff0\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\uff1a\u5feb\u901f\u6392\u5e8f\u662f\u9009\u53d6\u4e00\u4e2a\u57fa\u51c6\u503c\uff0c\u7136\u540e\u628a\u6570\u7ec4\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u4e00\u4e2a\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5c0f\uff0c\u53e6\u4e00\u5b50\u6570\u7ec4\u7684\u5143\u7d20\u6bd4\u57fa\u51c6\u503c\u5927\uff0c\u518d\u5bf9\u8fd9\u4e24\u90e8\u5206\u8fdb\u884c\u76f8\u540c\u7684\u5212\u5206\u64cd\u4f5c\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002</li> <li>\u6876\u6392\u5e8f\uff1a\u6876\u6392\u5e8f\u7684\u57fa\u672c\u601d\u60f3\u662f\u5c06\u6570\u636e\u5206\u6563\u5230\u591a\u4e2a\u6876\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u5404\u4e2a\u6876\u7684\u5143\u7d20\u4f9d\u6b21\u53d6\u51fa\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002</li> <li>\u6811\uff1a\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\u3001AVL \u6811\u3001\u7ea2\u9ed1\u6811\u3001B \u6811\u3001B+ \u6811\u7b49\uff0c\u5b83\u4eec\u7684\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u7b49\u64cd\u4f5c\u90fd\u53ef\u4ee5\u89c6\u4e3a\u5206\u6cbb\u7b56\u7565\u7684\u5e94\u7528\u3002</li> <li>\u5806\uff1a\u5806\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u5176\u5404\u79cd\u64cd\u4f5c\uff0c\u5982\u63d2\u5165\u3001\u5220\u9664\u548c\u5806\u5316\uff0c\u5b9e\u9645\u4e0a\u90fd\u9690\u542b\u4e86\u5206\u6cbb\u7684\u601d\u60f3\u3002</li> <li>\u54c8\u5e0c\u8868\uff1a\u867d\u7136\u54c8\u5e0c\u8868\u6765\u5e76\u4e0d\u76f4\u63a5\u5e94\u7528\u5206\u6cbb\uff0c\u4f46\u67d0\u4e9b\u54c8\u5e0c\u51b2\u7a81\u89e3\u51b3\u65b9\u6848\u95f4\u63a5\u5e94\u7528\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u4f8b\u5982\uff0c\u94fe\u5f0f\u5730\u5740\u4e2d\u7684\u957f\u94fe\u8868\u4f1a\u88ab\u8f6c\u5316\u4e3a\u7ea2\u9ed1\u6811\uff0c\u4ee5\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u51fa\uff0c\u5206\u6cbb\u662f\u4e00\u79cd\u201c\u6da6\u7269\u7ec6\u65e0\u58f0\u201d\u7684\u7b97\u6cd5\u601d\u60f3\uff0c\u9690\u542b\u5728\u5404\u79cd\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u4e4b\u4e2d\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/","title":"12.4 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898","text":"<p>\u5728\u5f52\u5e76\u6392\u5e8f\u548c\u6784\u5efa\u4e8c\u53c9\u6811\u4e2d\uff0c\u6211\u4eec\u90fd\u662f\u5c06\u539f\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a\u539f\u95ee\u9898\u4e00\u534a\u7684\u5b50\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u6c49\u8bfa\u5854\u95ee\u9898\uff0c\u6211\u4eec\u91c7\u7528\u4e0d\u540c\u7684\u5206\u89e3\u7b56\u7565\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e09\u6839\u67f1\u5b50\uff0c\u8bb0\u4e3a <code>A</code>\u3001<code>B</code> \u548c <code>C</code> \u3002\u8d77\u59cb\u72b6\u6001\u4e0b\uff0c\u67f1\u5b50 <code>A</code> \u4e0a\u5957\u7740 \\(n\\) \u4e2a\u5706\u76d8\uff0c\u5b83\u4eec\u4ece\u4e0a\u5230\u4e0b\u6309\u7167\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u3002\u6211\u4eec\u7684\u4efb\u52a1\u662f\u8981\u628a\u8fd9 \\(n\\) \u4e2a\u5706\u76d8\u79fb\u5230\u67f1\u5b50 <code>C</code> \u4e0a\uff0c\u5e76\u4fdd\u6301\u5b83\u4eec\u7684\u539f\u6709\u987a\u5e8f\u4e0d\u53d8\uff08\u5982\u56fe 12-10 \u6240\u793a\uff09\u3002\u5728\u79fb\u52a8\u5706\u76d8\u7684\u8fc7\u7a0b\u4e2d\uff0c\u9700\u8981\u9075\u5b88\u4ee5\u4e0b\u89c4\u5219\u3002</p> <ol> <li>\u5706\u76d8\u53ea\u80fd\u4ece\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u62ff\u51fa\uff0c\u4ece\u53e6\u4e00\u6839\u67f1\u5b50\u9876\u90e8\u653e\u5165\u3002</li> <li>\u6bcf\u6b21\u53ea\u80fd\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\u3002</li> <li>\u5c0f\u5706\u76d8\u5fc5\u987b\u65f6\u523b\u4f4d\u4e8e\u5927\u5706\u76d8\u4e4b\u4e0a\u3002</li> </ol> <p></p> <p> \u56fe 12-10 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u793a\u4f8b </p> <p>\u6211\u4eec\u5c06\u89c4\u6a21\u4e3a \\(i\\) \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u8bb0\u4f5c \\(f(i)\\) \u3002\u4f8b\u5982 \\(f(3)\\) \u4ee3\u8868\u5c06 \\(3\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u7684\u6c49\u8bfa\u5854\u95ee\u9898\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#1","title":"1. \u00a0 \u8003\u8651\u57fa\u672c\u60c5\u51b5","text":"<p>\u5982\u56fe 12-11 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(1)\\) \uff0c\u5373\u5f53\u53ea\u6709\u4e00\u4e2a\u5706\u76d8\u65f6\uff0c\u6211\u4eec\u5c06\u5b83\u76f4\u63a5\u4ece <code>A</code> \u79fb\u52a8\u81f3 <code>C</code> \u5373\u53ef\u3002</p> &lt;1&gt;&lt;2&gt; <p></p> <p></p> <p> \u56fe 12-11 \u00a0 \u89c4\u6a21\u4e3a 1 \u95ee\u9898\u7684\u89e3 </p> <p>\u5982\u56fe 12-12 \u6240\u793a\uff0c\u5bf9\u4e8e\u95ee\u9898 \\(f(2)\\) \uff0c\u5373\u5f53\u6709\u4e24\u4e2a\u5706\u76d8\u65f6\uff0c\u7531\u4e8e\u8981\u65f6\u523b\u6ee1\u8db3\u5c0f\u5706\u76d8\u5728\u5927\u5706\u76d8\u4e4b\u4e0a\uff0c\u56e0\u6b64\u9700\u8981\u501f\u52a9 <code>B</code> \u6765\u5b8c\u6210\u79fb\u52a8\u3002</p> <ol> <li>\u5148\u5c06\u4e0a\u9762\u7684\u5c0f\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u518d\u5c06\u5927\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002</li> <li>\u6700\u540e\u5c06\u5c0f\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-12 \u00a0 \u89c4\u6a21\u4e3a 2 \u95ee\u9898\u7684\u89e3 </p> <p>\u89e3\u51b3\u95ee\u9898 \\(f(2)\\) \u7684\u8fc7\u7a0b\u53ef\u603b\u7ed3\u4e3a\uff1a\u5c06\u4e24\u4e2a\u5706\u76d8\u501f\u52a9 <code>B</code> \u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u3002\u5176\u4e2d\uff0c<code>C</code> \u79f0\u4e3a\u76ee\u6807\u67f1\u3001<code>B</code> \u79f0\u4e3a\u7f13\u51b2\u67f1\u3002</p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#2","title":"2. \u00a0 \u5b50\u95ee\u9898\u5206\u89e3","text":"<p>\u5bf9\u4e8e\u95ee\u9898 \\(f(3)\\) \uff0c\u5373\u5f53\u6709\u4e09\u4e2a\u5706\u76d8\u65f6\uff0c\u60c5\u51b5\u53d8\u5f97\u7a0d\u5fae\u590d\u6742\u4e86\u4e00\u4e9b\u3002</p> <p>\u56e0\u4e3a\u5df2\u77e5 \\(f(1)\\) \u548c \\(f(2)\\) \u7684\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ece\u5206\u6cbb\u89d2\u5ea6\u601d\u8003\uff0c\u5c06 <code>A</code> \u9876\u90e8\u7684\u4e24\u4e2a\u5706\u76d8\u770b\u4f5c\u4e00\u4e2a\u6574\u4f53\uff0c\u6267\u884c\u56fe 12-13 \u6240\u793a\u7684\u6b65\u9aa4\u3002\u8fd9\u6837\u4e09\u4e2a\u5706\u76d8\u5c31\u88ab\u987a\u5229\u5730\u4ece <code>A</code> \u79fb\u81f3 <code>C</code> \u4e86\u3002</p> <ol> <li>\u4ee4 <code>B</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>C</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06 <code>A</code> \u4e2d\u5269\u4f59\u7684\u4e00\u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u52a8\u81f3 <code>C</code> \u3002</li> <li>\u4ee4 <code>C</code> \u4e3a\u76ee\u6807\u67f1\u3001<code>A</code> \u4e3a\u7f13\u51b2\u67f1\uff0c\u5c06\u4e24\u4e2a\u5706\u76d8\u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 12-13 \u00a0 \u89c4\u6a21\u4e3a 3 \u95ee\u9898\u7684\u89e3 </p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u6211\u4eec\u5c06\u95ee\u9898 \\(f(3)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(2)\\) \u548c\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u4e4b\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002\u8fd9\u8bf4\u660e\u5b50\u95ee\u9898\u662f\u72ec\u7acb\u7684\uff0c\u800c\u4e14\u89e3\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u53ef\u603b\u7ed3\u51fa\u56fe 12-14 \u6240\u793a\u7684\u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565\uff1a\u5c06\u539f\u95ee\u9898 \\(f(n)\\) \u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \u548c\u4e00\u4e2a\u5b50\u95ee\u9898 \\(f(1)\\) \uff0c\u5e76\u6309\u7167\u4ee5\u4e0b\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u3002</p> <ol> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>C</code> \u4ece <code>A</code> \u79fb\u81f3 <code>B</code> \u3002</li> <li>\u5c06\u5269\u4f59 \\(1\\) \u4e2a\u5706\u76d8\u4ece <code>A</code> \u76f4\u63a5\u79fb\u81f3 <code>C</code> \u3002</li> <li>\u5c06 \\(n-1\\) \u4e2a\u5706\u76d8\u501f\u52a9 <code>A</code> \u4ece <code>B</code> \u79fb\u81f3 <code>C</code> \u3002</li> </ol> <p>\u5bf9\u4e8e\u8fd9\u4e24\u4e2a\u5b50\u95ee\u9898 \\(f(n-1)\\) \uff0c\u53ef\u4ee5\u901a\u8fc7\u76f8\u540c\u7684\u65b9\u5f0f\u8fdb\u884c\u9012\u5f52\u5212\u5206\uff0c\u76f4\u81f3\u8fbe\u5230\u6700\u5c0f\u5b50\u95ee\u9898 \\(f(1)\\) \u3002\u800c \\(f(1)\\) \u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u53ea\u9700\u4e00\u6b21\u79fb\u52a8\u64cd\u4f5c\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 12-14 \u00a0 \u89e3\u51b3\u6c49\u8bfa\u5854\u95ee\u9898\u7684\u5206\u6cbb\u7b56\u7565 </p>"},{"location":"chapter_divide_and_conquer/hanota_problem/#3","title":"3. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u9012\u5f52\u51fd\u6570 <code>dfs(i, src, buf, tar)</code> \uff0c\u5b83\u7684\u4f5c\u7528\u662f\u5c06\u67f1 <code>src</code> \u9876\u90e8\u7684 \\(i\\) \u4e2a\u5706\u76d8\u501f\u52a9\u7f13\u51b2\u67f1 <code>buf</code> \u79fb\u52a8\u81f3\u76ee\u6807\u67f1 <code>tar</code> \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hanota.py<pre><code>def move(src: list[int], tar: list[int]):\n    \"\"\"\u79fb\u52a8\u4e00\u4e2a\u5706\u76d8\"\"\"\n    # \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    pan = src.pop()\n    # \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.append(pan)\n\ndef dfs(i: int, src: list[int], buf: list[int], tar: list[int]):\n    \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i)\"\"\"\n    # \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1:\n        move(src, tar)\n        return\n    # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf)\n    # \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    # \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar)\n\ndef solve_hanota(A: list[int], B: list[int], C: list[int]):\n    \"\"\"\u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898\"\"\"\n    n = len(A)\n    # \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C)\n</code></pre> hanota.cpp<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src.back();\n    src.pop_back();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push_back(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, vector&lt;int&gt; &amp;src, vector&lt;int&gt; &amp;buf, vector&lt;int&gt; &amp;tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(vector&lt;int&gt; &amp;A, vector&lt;int&gt; &amp;B, vector&lt;int&gt; &amp;C) {\n    int n = A.size();\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.java<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;Integer&gt; src, List&lt;Integer&gt; tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    Integer pan = src.remove(src.size() - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;Integer&gt; src, List&lt;Integer&gt; buf, List&lt;Integer&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;Integer&gt; A, List&lt;Integer&gt; B, List&lt;Integer&gt; C) {\n    int n = A.size();\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.cs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid Move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src[^1];\n    src.RemoveAt(src.Count - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.Add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid DFS(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        Move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    DFS(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    Move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    DFS(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid SolveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n    int n = A.Count;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    DFS(n, A, B, C);\n}\n</code></pre> hanota.go<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src, tar *list.List) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    pan := src.Back()\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.PushBack(pan.Value)\n    // \u79fb\u9664 src \u9876\u90e8\u5706\u76d8\n    src.Remove(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfsHanota(i int, src, buf, tar *list.List) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src, tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfsHanota(i-1, src, tar, buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfsHanota(i-1, buf, src, tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A, B, C *list.List) {\n    n := A.Len()\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfsHanota(n, A, B, C)\n}\n</code></pre> hanota.swift<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunc move(src: inout [Int], tar: inout [Int]) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    let pan = src.popLast()!\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.append(pan)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunc dfs(i: Int, src: inout [Int], buf: inout [Int], tar: inout [Int]) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move(src: &amp;src, tar: &amp;tar)\n        return\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i: i - 1, src: &amp;src, buf: &amp;tar, tar: &amp;buf)\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src: &amp;src, tar: &amp;tar)\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i: i - 1, src: &amp;buf, buf: &amp;src, tar: &amp;tar)\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunc solveHanota(A: inout [Int], B: inout [Int], C: inout [Int]) {\n    let n = A.count\n    // \u5217\u8868\u5c3e\u90e8\u662f\u67f1\u5b50\u9876\u90e8\n    // \u5c06 src \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(i: n, src: &amp;A, buf: &amp;B, tar: &amp;C)\n}\n</code></pre> hanota.js<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src, tar) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    const pan = src.pop();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i, src, buf, tar) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A, B, C) {\n    const n = A.length;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.ts<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfunction move(src: number[], tar: number[]): void {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    const pan = src.pop();\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfunction dfs(i: number, src: number[], buf: number[], tar: number[]): void {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i === 1) {\n        move(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfunction solveHanota(A: number[], B: number[], C: number[]): void {\n    const n = A.length;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.dart<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(List&lt;int&gt; src, List&lt;int&gt; tar) {\n  // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n  int pan = src.removeLast();\n  // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n  tar.add(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, List&lt;int&gt; src, List&lt;int&gt; buf, List&lt;int&gt; tar) {\n  // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n  if (i == 1) {\n    move(src, tar);\n    return;\n  }\n  // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n  dfs(i - 1, src, tar, buf);\n  // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n  move(src, tar);\n  // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n  dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(List&lt;int&gt; A, List&lt;int&gt; B, List&lt;int&gt; C) {\n  int n = A.length;\n  // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n  dfs(n, A, B, C);\n}\n</code></pre> hanota.rs<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nfn move_pan(src: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    let pan = src.remove(src.len() - 1);\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar.push(pan);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nfn dfs(i: i32, src: &amp;mut Vec&lt;i32&gt;, buf: &amp;mut Vec&lt;i32&gt;, tar: &amp;mut Vec&lt;i32&gt;) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if i == 1 {\n        move_pan(src, tar);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, tar, buf);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move_pan(src, tar);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, src, tar);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nfn solve_hanota(A: &amp;mut Vec&lt;i32&gt;, B: &amp;mut Vec&lt;i32&gt;, C: &amp;mut Vec&lt;i32&gt;) {\n    let n = A.len() as i32;\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(n, A, B, C);\n}\n</code></pre> hanota.c<pre><code>/* \u79fb\u52a8\u4e00\u4e2a\u5706\u76d8 */\nvoid move(int *src, int *srcSize, int *tar, int *tarSize) {\n    // \u4ece src \u9876\u90e8\u62ff\u51fa\u4e00\u4e2a\u5706\u76d8\n    int pan = src[*srcSize - 1];\n    src[*srcSize - 1] = 0;\n    (*srcSize)--;\n    // \u5c06\u5706\u76d8\u653e\u5165 tar \u9876\u90e8\n    tar[*tarSize] = pan;\n    (*tarSize)++;\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 f(i) */\nvoid dfs(int i, int *src, int *srcSize, int *buf, int *bufSize, int *tar, int *tarSize) {\n    // \u82e5 src \u53ea\u5269\u4e0b\u4e00\u4e2a\u5706\u76d8\uff0c\u5219\u76f4\u63a5\u5c06\u5176\u79fb\u5230 tar\n    if (i == 1) {\n        move(src, srcSize, tar, tarSize);\n        return;\n    }\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 src \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 tar \u79fb\u5230 buf\n    dfs(i - 1, src, srcSize, tar, tarSize, buf, bufSize);\n    // \u5b50\u95ee\u9898 f(1) \uff1a\u5c06 src \u5269\u4f59\u4e00\u4e2a\u5706\u76d8\u79fb\u5230 tar\n    move(src, srcSize, tar, tarSize);\n    // \u5b50\u95ee\u9898 f(i-1) \uff1a\u5c06 buf \u9876\u90e8 i-1 \u4e2a\u5706\u76d8\u501f\u52a9 src \u79fb\u5230 tar\n    dfs(i - 1, buf, bufSize, src, srcSize, tar, tarSize);\n}\n\n/* \u6c42\u89e3\u6c49\u8bfa\u5854\u95ee\u9898 */\nvoid solveHanota(int *A, int *ASize, int *B, int *BSize, int *C, int *CSize) {\n    // \u5c06 A \u9876\u90e8 n \u4e2a\u5706\u76d8\u501f\u52a9 B \u79fb\u5230 C\n    dfs(*ASize, A, ASize, B, BSize, C, CSize);\n}\n</code></pre> hanota.zig<pre><code>[class]{}-[func]{move}\n\n[class]{}-[func]{dfs}\n\n[class]{}-[func]{solveHanota}\n</code></pre> <p>\u5982\u56fe 12-15 \u6240\u793a\uff0c\u6c49\u8bfa\u5854\u95ee\u9898\u5f62\u6210\u4e00\u68f5\u9ad8\u5ea6\u4e3a \\(n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5f00\u542f\u7684 <code>dfs()</code> \u51fd\u6570\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 12-15 \u00a0 \u6c49\u8bfa\u5854\u95ee\u9898\u7684\u9012\u5f52\u6811 </p> <p>Quote</p> <p>\u6c49\u8bfa\u5854\u95ee\u9898\u6e90\u81ea\u4e00\u4e2a\u53e4\u8001\u7684\u4f20\u8bf4\u3002\u5728\u53e4\u5370\u5ea6\u7684\u4e00\u4e2a\u5bfa\u5e99\u91cc\uff0c\u50e7\u4fa3\u4eec\u6709\u4e09\u6839\u9ad8\u5927\u7684\u94bb\u77f3\u67f1\u5b50\uff0c\u4ee5\u53ca \\(64\\) \u4e2a\u5927\u5c0f\u4e0d\u4e00\u7684\u91d1\u5706\u76d8\u3002\u50e7\u4fa3\u4eec\u4e0d\u65ad\u5730\u79fb\u52a8\u5706\u76d8\uff0c\u4ed6\u4eec\u76f8\u4fe1\u5728\u6700\u540e\u4e00\u4e2a\u5706\u76d8\u88ab\u6b63\u786e\u653e\u7f6e\u7684\u90a3\u4e00\u523b\uff0c\u8fd9\u4e2a\u4e16\u754c\u5c31\u4f1a\u7ed3\u675f\u3002</p> <p>\u7136\u800c\uff0c\u5373\u4f7f\u50e7\u4fa3\u4eec\u6bcf\u79d2\u949f\u79fb\u52a8\u4e00\u6b21\uff0c\u603b\u5171\u9700\u8981\u5927\u7ea6 \\(2^{64} \\approx 1.84\u00d710^{19}\\) \u79d2\uff0c\u5408\u7ea6 \\(5850\\) \u4ebf\u5e74\uff0c\u8fdc\u8fdc\u8d85\u8fc7\u4e86\u73b0\u5728\u5bf9\u5b87\u5b99\u5e74\u9f84\u7684\u4f30\u8ba1\u3002\u6240\u4ee5\uff0c\u5018\u82e5\u8fd9\u4e2a\u4f20\u8bf4\u662f\u771f\u7684\uff0c\u6211\u4eec\u5e94\u8be5\u4e0d\u9700\u8981\u62c5\u5fc3\u4e16\u754c\u672b\u65e5\u7684\u5230\u6765\u3002</p>"},{"location":"chapter_divide_and_conquer/summary/","title":"12.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u5206\u6cbb\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565\uff0c\u5305\u62ec\u5206\uff08\u5212\u5206\uff09\u548c\u6cbb\uff08\u5408\u5e76\uff09\u4e24\u4e2a\u9636\u6bb5\uff0c\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\u3002</li> <li>\u5224\u65ad\u662f\u5426\u662f\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\u7684\u4f9d\u636e\u5305\u62ec\uff1a\u95ee\u9898\u80fd\u5426\u5206\u89e3\u3001\u5b50\u95ee\u9898\u662f\u5426\u72ec\u7acb\u3001\u5b50\u95ee\u9898\u80fd\u5426\u5408\u5e76\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u662f\u5206\u6cbb\u7b56\u7565\u7684\u5178\u578b\u5e94\u7528\uff0c\u5176\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u7b49\u957f\u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u76f4\u5230\u53ea\u5269\u4e00\u4e2a\u5143\u7d20\u65f6\u5f00\u59cb\u9010\u5c42\u5408\u5e76\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u5f15\u5165\u5206\u6cbb\u7b56\u7565\u5f80\u5f80\u53ef\u4ee5\u63d0\u5347\u7b97\u6cd5\u6548\u7387\u3002\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u7b56\u7565\u51cf\u5c11\u4e86\u64cd\u4f5c\u6570\u91cf\uff1b\u53e6\u4e00\u65b9\u9762\uff0c\u5206\u6cbb\u540e\u6709\u5229\u4e8e\u7cfb\u7edf\u7684\u5e76\u884c\u4f18\u5316\u3002</li> <li>\u5206\u6cbb\u65e2\u53ef\u4ee5\u89e3\u51b3\u8bb8\u591a\u7b97\u6cd5\u95ee\u9898\uff0c\u4e5f\u5e7f\u6cdb\u5e94\u7528\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\uff0c\u5904\u5904\u53ef\u89c1\u5176\u8eab\u5f71\u3002</li> <li>\u76f8\u8f83\u4e8e\u66b4\u529b\u641c\u7d22\uff0c\u81ea\u9002\u5e94\u641c\u7d22\u6548\u7387\u66f4\u9ad8\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u7684\u641c\u7d22\u7b97\u6cd5\u901a\u5e38\u662f\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u5b9e\u73b0\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u662f\u5206\u6cbb\u7b56\u7565\u7684\u53e6\u4e00\u4e2a\u5178\u578b\u5e94\u7528\uff0c\u5b83\u4e0d\u5305\u542b\u5c06\u5b50\u95ee\u9898\u7684\u89e3\u8fdb\u884c\u5408\u5e76\u7684\u6b65\u9aa4\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u5206\u6cbb\u5b9e\u73b0\u4e8c\u5206\u67e5\u627e\u3002</li> <li>\u5728\u6784\u5efa\u4e8c\u53c9\u6811\u7684\u95ee\u9898\u4e2d\uff0c\u6784\u5efa\u6811\uff08\u539f\u95ee\u9898\uff09\u53ef\u4ee5\u5212\u5206\u4e3a\u6784\u5efa\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\uff08\u5b50\u95ee\u9898\uff09\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u5212\u5206\u524d\u5e8f\u904d\u5386\u548c\u4e2d\u5e8f\u904d\u5386\u7684\u7d22\u5f15\u533a\u95f4\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u6c49\u8bfa\u5854\u95ee\u9898\u4e2d\uff0c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(n\\) \u7684\u95ee\u9898\u53ef\u4ee5\u5212\u5206\u4e3a\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(n-1\\) \u7684\u5b50\u95ee\u9898\u548c\u4e00\u4e2a\u89c4\u6a21\u4e3a \\(1\\) \u7684\u5b50\u95ee\u9898\u3002\u6309\u987a\u5e8f\u89e3\u51b3\u8fd9\u4e09\u4e2a\u5b50\u95ee\u9898\u540e\uff0c\u539f\u95ee\u9898\u968f\u4e4b\u5f97\u5230\u89e3\u51b3\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/","title":"\u7b2c 14 \u7ae0 \u00a0 \u52a8\u6001\u89c4\u5212","text":"<p>Abstract</p> <p>\u5c0f\u6eaa\u6c47\u5165\u6cb3\u6d41\uff0c\u6c5f\u6cb3\u6c47\u5165\u5927\u6d77\u3002</p> <p>\u52a8\u6001\u89c4\u5212\u5c06\u5c0f\u95ee\u9898\u7684\u89e3\u6c47\u96c6\u6210\u5927\u95ee\u9898\u7684\u7b54\u6848\uff0c\u4e00\u6b65\u6b65\u5f15\u9886\u6211\u4eec\u8d70\u5411\u89e3\u51b3\u95ee\u9898\u7684\u5f7c\u5cb8\u3002</p>"},{"location":"chapter_dynamic_programming/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212</li> <li>14.2 \u00a0 DP \u95ee\u9898\u7279\u6027</li> <li>14.3 \u00a0 DP \u89e3\u9898\u601d\u8def</li> <li>14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898</li> <li>14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898</li> <li>14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</li> <li>14.7 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_dynamic_programming/dp_problem_features/","title":"14.2 \u00a0 \u52a8\u6001\u89c4\u5212\u95ee\u9898\u7279\u6027","text":"<p>\u5728\u4e0a\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5b66\u4e60\u4e86\u52a8\u6001\u89c4\u5212\u662f\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u5206\u89e3\u6765\u6c42\u89e3\u539f\u95ee\u9898\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u7684\u4fa7\u91cd\u70b9\u4e0d\u540c\u3002</p> <ul> <li>\u5206\u6cbb\u7b97\u6cd5\u9012\u5f52\u5730\u5c06\u539f\u95ee\u9898\u5212\u5206\u4e3a\u591a\u4e2a\u76f8\u4e92\u72ec\u7acb\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u6700\u5c0f\u5b50\u95ee\u9898\uff0c\u5e76\u5728\u56de\u6eaf\u4e2d\u5408\u5e76\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u7ec8\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u4e5f\u5bf9\u95ee\u9898\u8fdb\u884c\u9012\u5f52\u5206\u89e3\uff0c\u4f46\u4e0e\u5206\u6cbb\u7b97\u6cd5\u7684\u4e3b\u8981\u533a\u522b\u662f\uff0c\u52a8\u6001\u89c4\u5212\u4e2d\u7684\u5b50\u95ee\u9898\u662f\u76f8\u4e92\u4f9d\u8d56\u7684\uff0c\u5728\u5206\u89e3\u8fc7\u7a0b\u4e2d\u4f1a\u51fa\u73b0\u8bb8\u591a\u91cd\u53e0\u5b50\u95ee\u9898\u3002</li> <li>\u56de\u6eaf\u7b97\u6cd5\u5728\u5c1d\u8bd5\u548c\u56de\u9000\u4e2d\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff0c\u5e76\u901a\u8fc7\u526a\u679d\u907f\u514d\u4e0d\u5fc5\u8981\u7684\u641c\u7d22\u5206\u652f\u3002\u539f\u95ee\u9898\u7684\u89e3\u7531\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\u6784\u6210\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u51b3\u7b56\u6b65\u9aa4\u4e4b\u524d\u7684\u5b50\u5e8f\u5217\u770b\u4f5c\u4e00\u4e2a\u5b50\u95ee\u9898\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5b83\u4eec\u4e0d\u4ec5\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u8fd8\u5177\u6709\u53e6\u5916\u4e24\u5927\u7279\u6027\uff1a\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</p>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1421","title":"14.2.1 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784","text":"<p>\u6211\u4eec\u5bf9\u722c\u697c\u68af\u95ee\u9898\u7a0d\u4f5c\u6539\u52a8\uff0c\u4f7f\u4e4b\u66f4\u52a0\u9002\u5408\u5c55\u793a\u6700\u4f18\u5b50\u7ed3\u6784\u6982\u5ff5\u3002</p> <p>\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u6bcf\u4e00\u9636\u697c\u68af\u4e0a\u90fd\u8d34\u6709\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u4f60\u5728\u8be5\u53f0\u9636\u6240\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\u3002\u7ed9\u5b9a\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\u6570\u7ec4 \\(cost\\) \uff0c\u5176\u4e2d \\(cost[i]\\) \u8868\u793a\u5728\u7b2c \\(i\\) \u4e2a\u53f0\u9636\u9700\u8981\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\\(cost[0]\\) \u4e3a\u5730\u9762\uff08\u8d77\u59cb\u70b9\uff09\u3002\u8bf7\u8ba1\u7b97\u6700\u5c11\u9700\u8981\u4ed8\u51fa\u591a\u5c11\u4ee3\u4ef7\u624d\u80fd\u5230\u8fbe\u9876\u90e8\uff1f</p> <p>\u5982\u56fe 14-6 \u6240\u793a\uff0c\u82e5\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u9636\u7684\u4ee3\u4ef7\u5206\u522b\u4e3a \\(1\\)\u3001\\(10\\)\u3001\\(1\\) \uff0c\u5219\u4ece\u5730\u9762\u722c\u5230\u7b2c \\(3\\) \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7\u4e3a \\(2\\) \u3002</p> <p></p> <p> \u56fe 14-6 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u6700\u5c0f\u4ee3\u4ef7 </p> <p>\u8bbe \\(dp[i]\\) \u4e3a\u722c\u5230\u7b2c \\(i\\) \u9636\u7d2f\u8ba1\u4ed8\u51fa\u7684\u4ee3\u4ef7\uff0c\u7531\u4e8e\u7b2c \\(i\\) \u9636\u53ea\u53ef\u80fd\u4ece \\(i - 1\\) \u9636\u6216 \\(i - 2\\) \u9636\u8d70\u6765\uff0c\u56e0\u6b64 \\(dp[i]\\) \u53ea\u53ef\u80fd\u7b49\u4e8e \\(dp[i - 1] + cost[i]\\) \u6216 \\(dp[i - 2] + cost[i]\\) \u3002\u4e3a\u4e86\u5c3d\u53ef\u80fd\u51cf\u5c11\u4ee3\u4ef7\uff0c\u6211\u4eec\u5e94\u8be5\u9009\u62e9\u4e24\u8005\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\uff1a</p> \\[ dp[i] = \\min(dp[i-1], dp[i-2]) + cost[i] \\] <p>\u8fd9\u4fbf\u53ef\u4ee5\u5f15\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u542b\u4e49\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u662f\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\u7684\u3002</p> <p>\u672c\u9898\u663e\u7136\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6211\u4eec\u4ece\u4e24\u4e2a\u5b50\u95ee\u9898\u6700\u4f18\u89e3 \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u4e2d\u6311\u9009\u51fa\u8f83\u4f18\u7684\u90a3\u4e00\u4e2a\uff0c\u5e76\u7528\u5b83\u6784\u5efa\u51fa\u539f\u95ee\u9898 \\(dp[i]\\) \u7684\u6700\u4f18\u89e3\u3002</p> <p>\u90a3\u4e48\uff0c\u4e0a\u4e00\u8282\u7684\u722c\u697c\u68af\u9898\u76ee\u6709\u6ca1\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u5462\uff1f\u5b83\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u770b\u4f3c\u662f\u4e00\u4e2a\u8ba1\u6570\u95ee\u9898\uff0c\u4f46\u5982\u679c\u6362\u4e00\u79cd\u95ee\u6cd5\uff1a\u201c\u6c42\u89e3\u6700\u5927\u65b9\u6848\u6570\u91cf\u201d\u3002\u6211\u4eec\u610f\u5916\u5730\u53d1\u73b0\uff0c\u867d\u7136\u9898\u76ee\u4fee\u6539\u524d\u540e\u662f\u7b49\u4ef7\u7684\uff0c\u4f46\u6700\u4f18\u5b50\u7ed3\u6784\u6d6e\u73b0\u51fa\u6765\u4e86\uff1a\u7b2c \\(n\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u7b49\u4e8e\u7b2c \\(n-1\\) \u9636\u548c\u7b2c \\(n-2\\) \u9636\u6700\u5927\u65b9\u6848\u6570\u91cf\u4e4b\u548c\u3002\u6240\u4ee5\u8bf4\uff0c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u89e3\u91ca\u65b9\u5f0f\u6bd4\u8f83\u7075\u6d3b\uff0c\u5728\u4e0d\u540c\u95ee\u9898\u4e2d\u4f1a\u6709\u4e0d\u540c\u7684\u542b\u4e49\u3002</p> <p>\u6839\u636e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u4ee5\u53ca\u521d\u59cb\u72b6\u6001 \\(dp[1] = cost[1]\\) \u548c \\(dp[2] = cost[2]\\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u52a8\u6001\u89c4\u5212\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1], dp[2] = cost[1], cost[2]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    return dp[n]\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDP(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = Math.Min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = min(dp[i-1], dp[i-2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDP(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1]\n    dp[2] = cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in stride(from: 3, through: n, by: 1) {\n        dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i]\n    }\n    return dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDP(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = Math.min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1] = cost[1];\n  dp[2] = cost[2];\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = min(dp[i - 1], dp[i - 2]) + cost[i];\n  }\n  return dp[n];\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 { return cost[n]; }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i] = cmp::min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    dp[n]\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDP(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int *dp = calloc(n + 1, sizeof(int));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = myMin(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    int res = dp[n];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDP(comptime cost: []i32) i32 {\n    comptime var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = cost[1];\n    dp[2] = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i] = @min(dp[i - 1], dp[i - 2]) + cost[i];\n    }\n    return dp[n];\n}\n</code></pre> <p>\u56fe 14-7 \u5c55\u793a\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-7 \u00a0 \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u672c\u9898\u4e5f\u53ef\u4ee5\u8fdb\u884c\u7a7a\u95f4\u4f18\u5316\uff0c\u5c06\u4e00\u7ef4\u538b\u7f29\u81f3\u96f6\u7ef4\uff0c\u4f7f\u5f97\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_cost_climbing_stairs_dp.py<pre><code>def min_cost_climbing_stairs_dp_comp(cost: list[int]) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(cost) - 1\n    if n == 1 or n == 2:\n        return cost[n]\n    a, b = cost[1], cost[2]\n    for i in range(3, n + 1):\n        a, b = b, min(a, b) + cost[i]\n    return b\n</code></pre> min_cost_climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(vector&lt;int&gt; &amp;cost) {\n    int n = cost.size() - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinCostClimbingStairsDPComp(int[] cost) {\n    int n = cost.Length - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = Math.Min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost []int) int {\n    n := len(cost) - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    min := func(a, b int) int {\n        if a &lt; b {\n            return a\n        }\n        return b\n    }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    a, b := cost[1], cost[2]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        tmp := b\n        b = min(a, tmp) + cost[i]\n        a = tmp\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minCostClimbingStairsDPComp(cost: [Int]) -&gt; Int {\n    let n = cost.count - 1\n    if n == 1 || n == 2 {\n        return cost[n]\n    }\n    var (a, b) = (cost[1], cost[2])\n    for i in stride(from: 3, through: n, by: 1) {\n        (a, b) = (b, min(a, b) + cost[i])\n    }\n    return b\n}\n</code></pre> min_cost_climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost) {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minCostClimbingStairsDPComp(cost: Array&lt;number&gt;): number {\n    const n = cost.length - 1;\n    if (n === 1 || n === 2) {\n        return cost[n];\n    }\n    let a = cost[1],\n        b = cost[2];\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = Math.min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(List&lt;int&gt; cost) {\n  int n = cost.length - 1;\n  if (n == 1 || n == 2) return cost[n];\n  int a = cost[1], b = cost[2];\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = min(a, tmp) + cost[i];\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_cost_climbing_stairs_dp_comp(cost: &amp;[i32]) -&gt; i32 {\n    let n = cost.len() - 1;\n    if n == 1 || n == 2 { return cost[n] };\n    let (mut a, mut b) = (cost[1], cost[2]);\n    for i in 3..=n {\n        let tmp = b;\n        b = cmp::min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    b\n}\n</code></pre> min_cost_climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minCostClimbingStairsDPComp(int cost[], int costSize) {\n    int n = costSize - 1;\n    if (n == 1 || n == 2)\n        return cost[n];\n    int a = cost[1], b = cost[2];\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = myMin(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> min_cost_climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\u6700\u5c0f\u4ee3\u4ef7\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minCostClimbingStairsDPComp(cost: []i32) i32 {\n    var n = cost.len - 1;\n    if (n == 1 or n == 2) {\n        return cost[n];\n    }\n    var a = cost[1];\n    var b = cost[2];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        var tmp = b;\n        b = @min(a, tmp) + cost[i];\n        a = tmp;\n    }\n    return b;\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/dp_problem_features/#1422","title":"14.2.2 \u00a0 \u65e0\u540e\u6548\u6027","text":"<p>\u65e0\u540e\u6548\u6027\u662f\u52a8\u6001\u89c4\u5212\u80fd\u591f\u6709\u6548\u89e3\u51b3\u95ee\u9898\u7684\u91cd\u8981\u7279\u6027\u4e4b\u4e00\uff0c\u5176\u5b9a\u4e49\u4e3a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u786e\u5b9a\u7684\u72b6\u6001\uff0c\u5b83\u7684\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u5f53\u524d\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002</p> <p>\u4ee5\u722c\u697c\u68af\u95ee\u9898\u4e3a\u4f8b\uff0c\u7ed9\u5b9a\u72b6\u6001 \\(i\\) \uff0c\u5b83\u4f1a\u53d1\u5c55\u51fa\u72b6\u6001 \\(i+1\\) \u548c\u72b6\u6001 \\(i+2\\) \uff0c\u5206\u522b\u5bf9\u5e94\u8df3 \\(1\\) \u6b65\u548c\u8df3 \\(2\\) \u6b65\u3002\u5728\u505a\u51fa\u8fd9\u4e24\u79cd\u9009\u62e9\u65f6\uff0c\u6211\u4eec\u65e0\u987b\u8003\u8651\u72b6\u6001 \\(i\\) \u4e4b\u524d\u7684\u72b6\u6001\uff0c\u5b83\u4eec\u5bf9\u72b6\u6001 \\(i\\) \u7684\u672a\u6765\u6ca1\u6709\u5f71\u54cd\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u7ed9\u722c\u697c\u68af\u95ee\u9898\u6dfb\u52a0\u4e00\u4e2a\u7ea6\u675f\uff0c\u60c5\u51b5\u5c31\u4e0d\u4e00\u6837\u4e86\u3002</p> <p>\u5e26\u7ea6\u675f\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u4f46\u4e0d\u80fd\u8fde\u7eed\u4e24\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-8 \u6240\u793a\uff0c\u722c\u4e0a\u7b2c \\(3\\) \u9636\u4ec5\u5269 \\(2\\) \u79cd\u53ef\u884c\u65b9\u6848\uff0c\u5176\u4e2d\u8fde\u7eed\u4e09\u6b21\u8df3 \\(1\\) \u9636\u7684\u65b9\u6848\u4e0d\u6ee1\u8db3\u7ea6\u675f\u6761\u4ef6\uff0c\u56e0\u6b64\u88ab\u820d\u5f03\u3002</p> <p></p> <p> \u56fe 14-8 \u00a0 \u5e26\u7ea6\u675f\u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u5728\u8be5\u95ee\u9898\u4e2d\uff0c\u5982\u679c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\uff0c\u90a3\u4e48\u4e0b\u4e00\u8f6e\u5c31\u5fc5\u987b\u8df3 \\(2\\) \u9636\u3002\u8fd9\u610f\u5473\u7740\uff0c\u4e0b\u4e00\u6b65\u9009\u62e9\u4e0d\u80fd\u7531\u5f53\u524d\u72b6\u6001\uff08\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u72ec\u7acb\u51b3\u5b9a\uff0c\u8fd8\u548c\u524d\u4e00\u4e2a\u72b6\u6001\uff08\u4e0a\u8f6e\u6240\u5728\u697c\u68af\u9636\u6570\uff09\u6709\u5173\u3002</p> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u6b64\u95ee\u9898\u5df2\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b \\(dp[i] = dp[i-1] + dp[i-2]\\) \u4e5f\u5931\u6548\u4e86\uff0c\u56e0\u4e3a \\(dp[i-1]\\) \u4ee3\u8868\u672c\u8f6e\u8df3 \\(1\\) \u9636\uff0c\u4f46\u5176\u4e2d\u5305\u542b\u4e86\u8bb8\u591a\u201c\u4e0a\u4e00\u8f6e\u662f\u8df3 \\(1\\) \u9636\u4e0a\u6765\u7684\u201d\u65b9\u6848\uff0c\u800c\u4e3a\u4e86\u6ee1\u8db3\u7ea6\u675f\uff0c\u6211\u4eec\u5c31\u4e0d\u80fd\u5c06 \\(dp[i-1]\\) \u76f4\u63a5\u8ba1\u5165 \\(dp[i]\\) \u4e2d\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u9700\u8981\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff1a\u72b6\u6001 \\([i, j]\\) \u8868\u793a\u5904\u5728\u7b2c \\(i\\) \u9636\u5e76\u4e14\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(j\\) \u9636\uff0c\u5176\u4e2d \\(j \\in \\{1, 2\\}\\) \u3002\u6b64\u72b6\u6001\u5b9a\u4e49\u6709\u6548\u5730\u533a\u5206\u4e86\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u8fd8\u662f \\(2\\) \u9636\uff0c\u6211\u4eec\u53ef\u4ee5\u636e\u6b64\u6765\u5224\u65ad\u5f53\u524d\u72b6\u6001\u662f\u4ece\u4f55\u800c\u6765\u7684\u3002</p> <ul> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(1\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ea\u80fd\u9009\u62e9\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 1]\\) \u53ea\u80fd\u4ece \\(dp[i-1, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> <li>\u5f53\u4e0a\u4e00\u8f6e\u8df3\u4e86 \\(2\\) \u9636\u65f6\uff0c\u4e0a\u4e0a\u4e00\u8f6e\u53ef\u9009\u62e9\u8df3 \\(1\\) \u9636\u6216\u8df3 \\(2\\) \u9636\uff0c\u5373 \\(dp[i, 2]\\) \u53ef\u4ee5\u4ece \\(dp[i-2, 1]\\) \u6216 \\(dp[i-2, 2]\\) \u8f6c\u79fb\u8fc7\u6765\u3002</li> </ul> <p>\u5982\u56fe 14-9 \u6240\u793a\uff0c\u5728\u8be5\u5b9a\u4e49\u4e0b\uff0c\\(dp[i, j]\\) \u8868\u793a\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u65b9\u6848\u6570\u3002\u6b64\u65f6\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ \\begin{cases} dp[i, 1] = dp[i-1, 2] \\\\ dp[i, 2] = dp[i-2, 1] + dp[i-2, 2] \\end{cases} \\] <p></p> <p> \u56fe 14-9 \u00a0 \u8003\u8651\u7ea6\u675f\u4e0b\u7684\u9012\u63a8\u5173\u7cfb </p> <p>\u6700\u7ec8\uff0c\u8fd4\u56de \\(dp[n, 1] + dp[n, 2]\\) \u5373\u53ef\uff0c\u4e24\u8005\u4e4b\u548c\u4ee3\u8868\u722c\u5230\u7b2c \\(n\\) \u9636\u7684\u65b9\u6848\u603b\u6570\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_constraint_dp.py<pre><code>def climbing_stairs_constraint_dp(n: int) -&gt; int:\n    \"\"\"\u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return 1\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [[0] * 3 for _ in range(n + 1)]\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1], dp[1][2] = 1, 0\n    dp[2][1], dp[2][2] = 0, 1\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    return dp[n][1] + dp[n][2]\n</code></pre> climbing_stairs_constraint_dp.cpp<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(3, 0));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.java<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[][] dp = new int[n + 1][3];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.cs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[,] dp = new int[n + 1, 3];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1, 1] = 1;\n    dp[1, 2] = 0;\n    dp[2, 1] = 0;\n    dp[2, 2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i, 1] = dp[i - 1, 2];\n        dp[i, 2] = dp[i - 2, 1] + dp[i - 2, 2];\n    }\n    return dp[n, 1] + dp[n, 2];\n}\n</code></pre> climbing_stairs_constraint_dp.go<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n int) int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([][3]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i][1] = dp[i-1][2]\n        dp[i][2] = dp[i-2][1] + dp[i-2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.swift<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsConstraintDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return 1\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: Array(repeating: 0, count: 3), count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1\n    dp[1][2] = 0\n    dp[2][1] = 0\n    dp[2][2] = 1\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in stride(from: 3, through: n, by: 1) {\n        dp[i][1] = dp[i - 1][2]\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2]\n    }\n    return dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.js<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n) {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = Array.from(new Array(n + 1), () =&gt; new Array(3));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.ts<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsConstraintDP(n: number): number {\n    if (n === 1 || n === 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(3));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.dart<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n  if (n == 1 || n == 2) {\n    return 1;\n  }\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(3, 0));\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1][1] = 1;\n  dp[1][2] = 0;\n  dp[2][1] = 0;\n  dp[2][2] = 1;\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i][1] = dp[i - 1][2];\n    dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n  }\n  return dp[n][1] + dp[n][2];\n}\n</code></pre> climbing_stairs_constraint_dp.rs<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_constraint_dp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 { return 1 };\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![vec![-1; 3]; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    dp[n][1] + dp[n][2]\n}\n</code></pre> climbing_stairs_constraint_dp.c<pre><code>/* \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsConstraintDP(int n) {\n    if (n == 1 || n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(3, sizeof(int));\n    }\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    int res = dp[n][1] + dp[n][2];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> climbing_stairs_constraint_dp.zig<pre><code>// \u5e26\u7ea6\u675f\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsConstraintDP(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return 1;\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_][3]i32{ [_]i32{ -1, -1, -1 } } ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1][1] = 1;\n    dp[1][2] = 0;\n    dp[2][1] = 0;\n    dp[2][2] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i][1] = dp[i - 1][2];\n        dp[i][2] = dp[i - 2][1] + dp[i - 2][2];\n    }\n    return dp[n][1] + dp[n][2];\n}\n</code></pre> <p>\u5728\u4e0a\u9762\u7684\u6848\u4f8b\u4e2d\uff0c\u7531\u4e8e\u4ec5\u9700\u591a\u8003\u8651\u524d\u9762\u4e00\u4e2a\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u72b6\u6001\u5b9a\u4e49\uff0c\u4f7f\u5f97\u95ee\u9898\u91cd\u65b0\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u95ee\u9898\u5177\u6709\u975e\u5e38\u4e25\u91cd\u7684\u201c\u6709\u540e\u6548\u6027\u201d\u3002</p> <p>\u722c\u697c\u68af\u4e0e\u969c\u788d\u751f\u6210</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\u3002\u89c4\u5b9a\u5f53\u722c\u5230\u7b2c \\(i\\) \u9636\u65f6\uff0c\u7cfb\u7edf\u81ea\u52a8\u4f1a\u5728\u7b2c \\(2i\\) \u9636\u4e0a\u653e\u4e0a\u969c\u788d\u7269\uff0c\u4e4b\u540e\u6240\u6709\u8f6e\u90fd\u4e0d\u5141\u8bb8\u8df3\u5230\u7b2c \\(2i\\) \u9636\u4e0a\u3002\u4f8b\u5982\uff0c\u524d\u4e24\u8f6e\u5206\u522b\u8df3\u5230\u4e86\u7b2c \\(2\\)\u3001\\(3\\) \u9636\u4e0a\uff0c\u5219\u4e4b\u540e\u5c31\u4e0d\u80fd\u8df3\u5230\u7b2c \\(4\\)\u3001\\(6\\) \u9636\u4e0a\u3002\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5728\u8fd9\u4e2a\u95ee\u9898\u4e2d\uff0c\u4e0b\u6b21\u8df3\u8dc3\u4f9d\u8d56\u8fc7\u53bb\u6240\u6709\u7684\u72b6\u6001\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b21\u8df3\u8dc3\u90fd\u4f1a\u5728\u66f4\u9ad8\u7684\u9636\u68af\u4e0a\u8bbe\u7f6e\u969c\u788d\uff0c\u5e76\u5f71\u54cd\u672a\u6765\u7684\u8df3\u8dc3\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u5f80\u5f80\u96be\u4ee5\u89e3\u51b3\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u590d\u6742\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff08\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898\uff09\u4e0d\u6ee1\u8db3\u65e0\u540e\u6548\u6027\u3002\u5bf9\u4e8e\u8fd9\u7c7b\u95ee\u9898\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u4f7f\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u4f8b\u5982\u542f\u53d1\u5f0f\u641c\u7d22\u3001\u9057\u4f20\u7b97\u6cd5\u3001\u5f3a\u5316\u5b66\u4e60\u7b49\uff0c\u4ece\u800c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f97\u5230\u53ef\u7528\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/","title":"14.3 \u00a0 \u52a8\u6001\u89c4\u5212\u89e3\u9898\u601d\u8def","text":"<p>\u4e0a\u4e24\u8282\u4ecb\u7ecd\u4e86\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4e3b\u8981\u7279\u5f81\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u4e00\u8d77\u63a2\u7a76\u4e24\u4e2a\u66f4\u52a0\u5b9e\u7528\u7684\u95ee\u9898\u3002</p> <ol> <li>\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u95ee\u9898\u662f\u4e0d\u662f\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff1f</li> <li>\u6c42\u89e3\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8be5\u4ece\u4f55\u5904\u5165\u624b\uff0c\u5b8c\u6574\u6b65\u9aa4\u662f\u4ec0\u4e48\uff1f</li> </ol>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1431","title":"14.3.1 \u00a0 \u95ee\u9898\u5224\u65ad","text":"<p>\u603b\u7684\u6765\u8bf4\uff0c\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u5305\u542b\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6ee1\u8db3\u65e0\u540e\u6548\u6027\uff0c\u90a3\u4e48\u5b83\u901a\u5e38\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u6c42\u89e3\u3002\u7136\u800c\uff0c\u6211\u4eec\u5f88\u96be\u4ece\u95ee\u9898\u63cf\u8ff0\u4e2d\u76f4\u63a5\u63d0\u53d6\u51fa\u8fd9\u4e9b\u7279\u6027\u3002\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f1a\u653e\u5bbd\u6761\u4ef6\uff0c\u5148\u89c2\u5bdf\u95ee\u9898\u662f\u5426\u9002\u5408\u4f7f\u7528\u56de\u6eaf\uff08\u7a77\u4e3e\uff09\u89e3\u51b3\u3002</p> <p>\u9002\u5408\u7528\u56de\u6eaf\u89e3\u51b3\u7684\u95ee\u9898\u901a\u5e38\u6ee1\u8db3\u201c\u51b3\u7b56\u6811\u6a21\u578b\u201d\uff0c\u8fd9\u79cd\u95ee\u9898\u53ef\u4ee5\u4f7f\u7528\u6811\u5f62\u7ed3\u6784\u6765\u63cf\u8ff0\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\uff0c\u6bcf\u4e00\u6761\u8def\u5f84\u4ee3\u8868\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\u3002</p> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u95ee\u9898\u5305\u542b\u660e\u786e\u7684\u51b3\u7b56\u6982\u5ff5\uff0c\u5e76\u4e14\u89e3\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u51b3\u7b56\u4ea7\u751f\u7684\uff0c\u90a3\u4e48\u5b83\u5c31\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u901a\u5e38\u53ef\u4ee5\u4f7f\u7528\u56de\u6eaf\u6765\u89e3\u51b3\u3002</p> <p>\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u52a8\u6001\u89c4\u5212\u95ee\u9898\u8fd8\u6709\u4e00\u4e9b\u5224\u65ad\u7684\u201c\u52a0\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u5305\u542b\u6700\u5927\uff08\u5c0f\uff09\u6216\u6700\u591a\uff08\u5c11\uff09\u7b49\u6700\u4f18\u5316\u63cf\u8ff0\u3002</li> <li>\u95ee\u9898\u7684\u72b6\u6001\u80fd\u591f\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u3001\u591a\u7ef4\u77e9\u9635\u6216\u6811\u6765\u8868\u793a\uff0c\u5e76\u4e14\u4e00\u4e2a\u72b6\u6001\u4e0e\u5176\u5468\u56f4\u7684\u72b6\u6001\u5b58\u5728\u9012\u63a8\u5173\u7cfb\u3002</li> </ul> <p>\u76f8\u5e94\u5730\uff0c\u4e5f\u5b58\u5728\u4e00\u4e9b\u201c\u51cf\u5206\u9879\u201d\u3002</p> <ul> <li>\u95ee\u9898\u7684\u76ee\u6807\u662f\u627e\u51fa\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e0d\u662f\u627e\u51fa\u6700\u4f18\u89e3\u3002</li> <li>\u95ee\u9898\u63cf\u8ff0\u4e2d\u6709\u660e\u663e\u7684\u6392\u5217\u7ec4\u5408\u7684\u7279\u5f81\uff0c\u9700\u8981\u8fd4\u56de\u5177\u4f53\u7684\u591a\u4e2a\u65b9\u6848\u3002</li> </ul> <p>\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\uff0c\u5e76\u5177\u6709\u8f83\u4e3a\u660e\u663e\u7684\u201c\u52a0\u5206\u9879\u201c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5047\u8bbe\u5b83\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u5e76\u5728\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u9a8c\u8bc1\u5b83\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1432","title":"14.3.2 \u00a0 \u95ee\u9898\u6c42\u89e3\u6b65\u9aa4","text":"<p>\u52a8\u6001\u89c4\u5212\u7684\u89e3\u9898\u6d41\u7a0b\u4f1a\u56e0\u95ee\u9898\u7684\u6027\u8d28\u548c\u96be\u5ea6\u800c\u6709\u6240\u4e0d\u540c\uff0c\u4f46\u901a\u5e38\u9075\u5faa\u4ee5\u4e0b\u6b65\u9aa4\uff1a\u63cf\u8ff0\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u5efa\u7acb \\(dp\\) \u8868\uff0c\u63a8\u5bfc\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u7b49\u3002</p> <p>\u4e3a\u4e86\u66f4\u5f62\u8c61\u5730\u5c55\u793a\u89e3\u9898\u6b65\u9aa4\uff0c\u6211\u4eec\u4f7f\u7528\u4e00\u4e2a\u7ecf\u5178\u95ee\u9898\u201c\u6700\u5c0f\u8def\u5f84\u548c\u201d\u6765\u4e3e\u4f8b\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a \\(n \\times m\\) \u7684\u4e8c\u7ef4\u7f51\u683c <code>grid</code> \uff0c\u7f51\u683c\u4e2d\u7684\u6bcf\u4e2a\u5355\u5143\u683c\u5305\u542b\u4e00\u4e2a\u975e\u8d1f\u6574\u6570\uff0c\u8868\u793a\u8be5\u5355\u5143\u683c\u7684\u4ee3\u4ef7\u3002\u673a\u5668\u4eba\u4ee5\u5de6\u4e0a\u89d2\u5355\u5143\u683c\u4e3a\u8d77\u59cb\u70b9\uff0c\u6bcf\u6b21\u53ea\u80fd\u5411\u4e0b\u6216\u8005\u5411\u53f3\u79fb\u52a8\u4e00\u6b65\uff0c\u76f4\u81f3\u5230\u8fbe\u53f3\u4e0b\u89d2\u5355\u5143\u683c\u3002\u8bf7\u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230\u53f3\u4e0b\u89d2\u7684\u6700\u5c0f\u8def\u5f84\u548c\u3002</p> <p>\u56fe 14-10 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u7f51\u683c\u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e3a \\(13\\) \u3002</p> <p></p> <p> \u56fe 14-10 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u793a\u4f8b\u6570\u636e </p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u672c\u9898\u7684\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u5c31\u662f\u4ece\u5f53\u524d\u683c\u5b50\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u3002\u8bbe\u5f53\u524d\u683c\u5b50\u7684\u884c\u5217\u7d22\u5f15\u4e3a \\([i, j]\\) \uff0c\u5219\u5411\u4e0b\u6216\u5411\u53f3\u8d70\u4e00\u6b65\u540e\uff0c\u7d22\u5f15\u53d8\u4e3a \\([i+1, j]\\) \u6216 \\([i, j+1]\\) \u3002\u56e0\u6b64\uff0c\u72b6\u6001\u5e94\u5305\u542b\u884c\u7d22\u5f15\u548c\u5217\u7d22\u5f15\u4e24\u4e2a\u53d8\u91cf\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u4ece\u8d77\u59cb\u70b9 \\([0, 0]\\) \u8d70\u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\uff0c\u89e3\u8bb0\u4e3a \\(dp[i, j]\\) \u3002</p> <p>\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u56fe 14-11 \u6240\u793a\u7684\u4e8c\u7ef4 \\(dp\\) \u77e9\u9635\uff0c\u5176\u5c3a\u5bf8\u4e0e\u8f93\u5165\u7f51\u683c \\(grid\\) \u76f8\u540c\u3002</p> <p></p> <p> \u56fe 14-11 \u00a0 \u72b6\u6001\u5b9a\u4e49\u4e0e dp \u8868 </p> <p>Note</p> <p>\u52a8\u6001\u89c4\u5212\u548c\u56de\u6eaf\u8fc7\u7a0b\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u4e00\u4e2a\u51b3\u7b56\u5e8f\u5217\uff0c\u800c\u72b6\u6001\u7531\u6240\u6709\u51b3\u7b56\u53d8\u91cf\u6784\u6210\u3002\u5b83\u5e94\u5f53\u5305\u542b\u63cf\u8ff0\u89e3\u9898\u8fdb\u5ea6\u7684\u6240\u6709\u53d8\u91cf\uff0c\u5176\u5305\u542b\u4e86\u8db3\u591f\u7684\u4fe1\u606f\uff0c\u80fd\u591f\u7528\u6765\u63a8\u5bfc\u51fa\u4e0b\u4e00\u4e2a\u72b6\u6001\u3002</p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u5b9a\u4e49\u4e00\u4e2a \\(dp\\) \u8868\u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u72b6\u6001\u7684\u6bcf\u4e2a\u72ec\u7acb\u53d8\u91cf\u90fd\u662f \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\u3002\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\\(dp\\) \u8868\u662f\u72b6\u6001\u548c\u5b50\u95ee\u9898\u7684\u89e3\u4e4b\u95f4\u7684\u6620\u5c04\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5bf9\u4e8e\u72b6\u6001 \\([i, j]\\) \uff0c\u5b83\u53ea\u80fd\u4ece\u4e0a\u8fb9\u683c\u5b50 \\([i-1, j]\\) \u548c\u5de6\u8fb9\u683c\u5b50 \\([i, j-1]\\) \u8f6c\u79fb\u800c\u6765\u3002\u56e0\u6b64\u6700\u4f18\u5b50\u7ed3\u6784\u4e3a\uff1a\u5230\u8fbe \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u7531 \\([i, j-1]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e0e \\([i-1, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c\u4e2d\u8f83\u5c0f\u7684\u90a3\u4e00\u4e2a\u51b3\u5b9a\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u63a8\u51fa\u56fe 14-12 \u6240\u793a\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, j] = \\min(dp[i-1, j], dp[i, j-1]) + grid[i, j] \\] <p></p> <p> \u56fe 14-12 \u00a0 \u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b </p> <p>Note</p> <p>\u6839\u636e\u5b9a\u4e49\u597d\u7684 \\(dp\\) \u8868\uff0c\u601d\u8003\u539f\u95ee\u9898\u548c\u5b50\u95ee\u9898\u7684\u5173\u7cfb\uff0c\u627e\u51fa\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u65b9\u6cd5\uff0c\u5373\u6700\u4f18\u5b50\u7ed3\u6784\u3002</p> <p>\u4e00\u65e6\u6211\u4eec\u627e\u5230\u4e86\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5728\u672c\u9898\u4e2d\uff0c\u5904\u5728\u9996\u884c\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u5de6\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u5904\u5728\u9996\u5217\u7684\u72b6\u6001\u53ea\u80fd\u4ece\u5176\u4e0a\u8fb9\u7684\u72b6\u6001\u5f97\u6765\uff0c\u56e0\u6b64\u9996\u884c \\(i = 0\\) \u548c\u9996\u5217 \\(j = 0\\) \u662f\u8fb9\u754c\u6761\u4ef6\u3002</p> <p>\u5982\u56fe 14-13 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u662f\u7531\u5176\u5de6\u65b9\u683c\u5b50\u548c\u4e0a\u65b9\u683c\u5b50\u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u5faa\u73af\u6765\u904d\u5386\u77e9\u9635\uff0c\u5916\u5faa\u73af\u904d\u5386\u5404\u884c\uff0c\u5185\u5faa\u73af\u904d\u5386\u5404\u5217\u3002</p> <p></p> <p> \u56fe 14-13 \u00a0 \u8fb9\u754c\u6761\u4ef6\u4e0e\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f </p> <p>Note</p> <p>\u8fb9\u754c\u6761\u4ef6\u5728\u52a8\u6001\u89c4\u5212\u4e2d\u7528\u4e8e\u521d\u59cb\u5316 \\(dp\\) \u8868\uff0c\u5728\u641c\u7d22\u4e2d\u7528\u4e8e\u526a\u679d\u3002</p> <p>\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f\u7684\u6838\u5fc3\u662f\u8981\u4fdd\u8bc1\u5728\u8ba1\u7b97\u5f53\u524d\u95ee\u9898\u7684\u89e3\u65f6\uff0c\u6240\u6709\u5b83\u4f9d\u8d56\u7684\u66f4\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u90fd\u5df2\u7ecf\u88ab\u6b63\u786e\u5730\u8ba1\u7b97\u51fa\u6765\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u5199\u51fa\u52a8\u6001\u89c4\u5212\u4ee3\u7801\u3002\u7136\u800c\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u601d\u60f3\uff0c\u56e0\u6b64\u6309\u7167\u201c\u66b4\u529b\u641c\u7d22 \\(\\rightarrow\\) \u8bb0\u5fc6\u5316\u641c\u7d22 \\(\\rightarrow\\) \u52a8\u6001\u89c4\u5212\u201d\u7684\u987a\u5e8f\u5b9e\u73b0\u66f4\u52a0\u7b26\u5408\u601d\u7ef4\u4e60\u60ef\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u4ece\u72b6\u6001 \\([i, j]\\) \u5f00\u59cb\u641c\u7d22\uff0c\u4e0d\u65ad\u5206\u89e3\u4e3a\u66f4\u5c0f\u7684\u72b6\u6001 \\([i-1, j]\\) \u548c \\([i, j-1]\\) \uff0c\u9012\u5f52\u51fd\u6570\u5305\u62ec\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, j]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u4ece \\([0, 0]\\) \u5230 \\([i, j]\\) \u7684\u6700\u5c0f\u8def\u5f84\u548c \\(dp[i, j]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53 \\(i = 0\\) \u4e14 \\(j = 0\\) \u65f6\uff0c\u8fd4\u56de\u4ee3\u4ef7 \\(grid[0, 0]\\) \u3002</li> <li>\u526a\u679d\uff1a\u5f53 \\(i &lt; 0\\) \u65f6\u6216 \\(j &lt; 0\\) \u65f6\u7d22\u5f15\u8d8a\u754c\uff0c\u6b64\u65f6\u8fd4\u56de\u4ee3\u4ef7 \\(+\\infty\\) \uff0c\u4ee3\u8868\u4e0d\u53ef\u884c\u3002</li> </ul> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dfs(grid: list[list[int]], i: int, j: int) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n    # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up = min_path_sum_dfs(grid, i - 1, j)\n    left = min_path_sum_dfs(grid, i, j - 1)\n    # \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) + grid[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(vector&lt;vector&lt;int&gt;&gt; &amp;grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint MinPathSumDFS(int[][] grid, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = MinPathSumDFS(grid, i - 1, j);\n    int left = MinPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.Min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid [][]int, i, j int) int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up := minPathSumDFS(grid, i-1, j)\n    left := minPathSumDFS(grid, i, j-1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return int(math.Min(float64(left), float64(up))) + grid[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc minPathSumDFS(grid: [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = minPathSumDFS(grid: grid, i: i - 1, j: j)\n    let left = minPathSumDFS(grid: grid, i: i, j: j - 1)\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return min(left, up) + grid[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(grid, i, j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction minPathSumDFS(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFS(grid, i - 1, j);\n    const left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return Math.min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(List&lt;List&lt;int&gt;&gt; grid, int i, int j) {\n  // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  int up = minPathSumDFS(grid, i - 1, j);\n  int left = minPathSumDFS(grid, i, j - 1);\n  // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  return min(left, up) + grid[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nfn min_path_sum_dfs(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = min_path_sum_dfs(grid, i - 1, j);\n    let left = min_path_sum_dfs(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    std::cmp::min(left, up) + grid[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22 */\nint minPathSumDFS(int grid[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFS(grid, i - 1, j);\n    int left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u66b4\u529b\u641c\u7d22\nfn minPathSumDFS(grid: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    var up = minPathSumDFS(grid, i - 1, j);\n    var left = minPathSumDFS(grid, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    return @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> <p>\u56fe 14-14 \u7ed9\u51fa\u4e86\u4ee5 \\(dp[2, 1]\\) \u4e3a\u6839\u8282\u70b9\u7684\u9012\u5f52\u6811\uff0c\u5176\u4e2d\u5305\u542b\u4e00\u4e9b\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5176\u6570\u91cf\u4f1a\u968f\u7740\u7f51\u683c <code>grid</code> \u7684\u5c3a\u5bf8\u53d8\u5927\u800c\u6025\u5267\u589e\u591a\u3002</p> <p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u9020\u6210\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u539f\u56e0\u4e3a\uff1a\u5b58\u5728\u591a\u6761\u8def\u5f84\u53ef\u4ee5\u4ece\u5de6\u4e0a\u89d2\u5230\u8fbe\u67d0\u4e00\u5355\u5143\u683c\u3002</p> <p></p> <p> \u56fe 14-14 \u00a0 \u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p> <p>\u6bcf\u4e2a\u72b6\u6001\u90fd\u6709\u5411\u4e0b\u548c\u5411\u53f3\u4e24\u79cd\u9009\u62e9\uff0c\u4ece\u5de6\u4e0a\u89d2\u8d70\u5230\u53f3\u4e0b\u89d2\u603b\u5171\u9700\u8981 \\(m + n - 2\\) \u6b65\uff0c\u6240\u4ee5\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^{m + n})\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u8fd9\u79cd\u8ba1\u7b97\u65b9\u5f0f\u672a\u8003\u8651\u4e34\u8fd1\u7f51\u683c\u8fb9\u754c\u7684\u60c5\u51b5\uff0c\u5f53\u5230\u8fbe\u7f51\u7edc\u8fb9\u754c\u65f6\u53ea\u5269\u4e0b\u4e00\u79cd\u9009\u62e9\uff0c\u56e0\u6b64\u5b9e\u9645\u7684\u8def\u5f84\u6570\u91cf\u4f1a\u5c11\u4e00\u4e9b\u3002</p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u6211\u4eec\u5f15\u5165\u4e00\u4e2a\u548c\u7f51\u683c <code>grid</code> \u76f8\u540c\u5c3a\u5bf8\u7684\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \uff0c\u7528\u4e8e\u8bb0\u5f55\u5404\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u8fdb\u884c\u526a\u679d\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dfs_mem(\n    grid: list[list[int]], mem: list[list[int]], i: int, j: int\n) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 and j == 0:\n        return grid[0][0]\n    # \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 or j &lt; 0:\n        return inf\n    # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1:\n        return mem[i][j]\n    # \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up = min_path_sum_dfs_mem(grid, mem, i - 1, j)\n    left = min_path_sum_dfs_mem(grid, mem, i, j - 1)\n    # \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(vector&lt;vector&lt;int&gt;&gt; &amp;grid, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Integer.MAX_VALUE;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint MinPathSumDFSMem(int[][] grid, int[][] mem, int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return int.MaxValue;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = MinPathSumDFSMem(grid, mem, i - 1, j);\n    int left = MinPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.Min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid, mem [][]int, i, j int) int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return math.MaxInt\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    up := minPathSumDFSMem(grid, mem, i-1, j)\n    left := minPathSumDFSMem(grid, mem, i, j-1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = int(math.Min(float64(left), float64(up))) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc minPathSumDFSMem(grid: [[Int]], mem: inout [[Int]], i: Int, j: Int) -&gt; Int {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0, j == 0 {\n        return grid[0][0]\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return .max\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][j] != -1 {\n        return mem[i][j]\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i - 1, j: j)\n    let left = minPathSumDFSMem(grid: grid, mem: &amp;mem, i: i, j: j - 1)\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = min(left, up) + grid[i][j]\n    return mem[i][j]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(grid, mem, i, j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] !== -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction minPathSumDFSMem(\n    grid: Array&lt;Array&lt;number&gt;&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    j: number\n): number {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i === 0 &amp;&amp; j === 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return Infinity;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    const up = minPathSumDFSMem(grid, mem, i - 1, j);\n    const left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = Math.min(left, up) + grid[i][j];\n    return mem[i][j];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(List&lt;List&lt;int&gt;&gt; grid, List&lt;List&lt;int&gt;&gt; mem, int i, int j) {\n  // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n  if (i == 0 &amp;&amp; j == 0) {\n    return grid[0][0];\n  }\n  // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n  if (i &lt; 0 || j &lt; 0) {\n    // \u5728 Dart \u4e2d\uff0cint \u7c7b\u578b\u662f\u56fa\u5b9a\u8303\u56f4\u7684\u6574\u6570\uff0c\u4e0d\u5b58\u5728\u8868\u793a\u201c\u65e0\u7a77\u5927\u201d\u7684\u503c\n    return BigInt.from(2).pow(31).toInt();\n  }\n  // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][j] != -1) {\n    return mem[i][j];\n  }\n  // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  int up = minPathSumDFSMem(grid, mem, i - 1, j);\n  int left = minPathSumDFSMem(grid, mem, i, j - 1);\n  // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n  mem[i][j] = min(left, up) + grid[i][j];\n  return mem[i][j];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn min_path_sum_dfs_mem(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;, mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: i32, j: i32) -&gt; i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if i == 0 &amp;&amp; j == 0 {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if i &lt; 0 || j &lt; 0 {\n        return i32::MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i as usize][j as usize] != -1 {\n        return mem[i as usize][j as usize];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    let up = min_path_sum_dfs_mem(grid, mem, i - 1, j);\n    let left = min_path_sum_dfs_mem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i as usize][j as usize] = std::cmp::min(left, up) + grid[i as usize][j as usize];\n    mem[i as usize][j as usize]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint minPathSumDFSMem(int grid[MAX_SIZE][MAX_SIZE], int mem[MAX_SIZE][MAX_SIZE], int i, int j) {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 &amp;&amp; j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 || j &lt; 0) {\n        return INT_MAX;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][j] != -1) {\n        return mem[i][j];\n    }\n    // \u5de6\u8fb9\u548c\u4e0a\u8fb9\u5355\u5143\u683c\u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    int up = minPathSumDFSMem(grid, mem, i - 1, j);\n    int left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[i][j] = myMin(left, up) != INT_MAX ? myMin(left, up) + grid[i][j] : INT_MAX;\n    return mem[i][j];\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn minPathSumDFSMem(grid: anytype, mem: anytype, i: i32, j: i32) i32 {\n    // \u82e5\u4e3a\u5de6\u4e0a\u89d2\u5355\u5143\u683c\uff0c\u5219\u7ec8\u6b62\u641c\u7d22\n    if (i == 0 and j == 0) {\n        return grid[0][0];\n    }\n    // \u82e5\u884c\u5217\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de +\u221e \u4ee3\u4ef7\n    if (i &lt; 0 or j &lt; 0) {\n        return std.math.maxInt(i32);\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] != -1) {\n        return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    }\n    // \u8ba1\u7b97\u4ece\u5de6\u4e0a\u89d2\u5230 (i-1, j) \u548c (i, j-1) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    var up = minPathSumDFSMem(grid, mem, i - 1, j);\n    var left = minPathSumDFSMem(grid, mem, i, j - 1);\n    // \u8fd4\u56de\u4ece\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u5de6\u4e0a\u89d2\u5230 (i, j) \u7684\u6700\u5c0f\u8def\u5f84\u4ee3\u4ef7\n    mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))] = @min(left, up) + grid[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n    return mem[@as(usize, @intCast(i))][@as(usize, @intCast(j))];\n}\n</code></pre> <p>\u5982\u56fe 14-15 \u6240\u793a\uff0c\u5728\u5f15\u5165\u8bb0\u5fc6\u5316\u540e\uff0c\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u72b6\u6001\u603b\u6570\uff0c\u5373\u7f51\u683c\u5c3a\u5bf8 \\(O(nm)\\) \u3002</p> <p></p> <p> \u56fe 14-15 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * m for _ in range(n)]\n    dp[0][0] = grid[0][0]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in range(1, m):\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in range(1, n):\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n):\n        for j in range(1, m):\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n    return dp[n - 1][m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n, vector&lt;int&gt;(m));\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n][m];\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDP(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n, m];\n    dp[0, 0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0, j] = dp[0, j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i, 0] = dp[i - 1, 0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i, j] = Math.Min(dp[i, j - 1], dp[i - 1, j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1, m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n)\n    for i := 0; i &lt; n; i++ {\n        dp[i] = make([]int, m)\n    }\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt; m; j++ {\n        dp[0][j] = dp[0][j-1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i := 1; i &lt; n; i++ {\n        dp[i][0] = dp[i-1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        for j := 1; j &lt; m; j++ {\n            dp[i][j] = int(math.Min(float64(dp[i][j-1]), float64(dp[i-1][j]))) + grid[i][j]\n        }\n    }\n    return dp[n-1][m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDP(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: m), count: n)\n    dp[0][0] = grid[0][0]\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in stride(from: 1, to: m, by: 1) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in stride(from: 1, to: n, by: 1) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in stride(from: 1, to: n, by: 1) {\n        for j in stride(from: 1, to: m, by: 1) {\n            dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j]\n        }\n    }\n    return dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDP(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n }, () =&gt;\n        Array.from({ length: m }, () =&gt; 0)\n    );\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (let i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt; n; i++) {\n        for (let j: number = 1; j &lt; m; j++) {\n            dp[i][j] = Math.min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n, (i) =&gt; List.filled(m, 0));\n  dp[0][0] = grid[0][0];\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt; m; j++) {\n    dp[0][j] = dp[0][j - 1] + grid[0][j];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n  for (int i = 1; i &lt; n; i++) {\n    dp[i][0] = dp[i - 1][0] + grid[i][0];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt; n; i++) {\n    for (int j = 1; j &lt; m; j++) {\n      dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n    }\n  }\n  return dp[n - 1][m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; m]; n];\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for i in 1..n {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..n {\n        for j in 1..m {\n            dp[i][j] = std::cmp::min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    dp[n - 1][m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212 */\nint minPathSumDP(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc(n * sizeof(int *));\n    for (int i = 0; i &lt; n; i++) {\n        dp[i] = calloc(m, sizeof(int));\n    }\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (int i = 1; i &lt; n; i++) {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt; n; i++) {\n        for (int j = 1; j &lt; m; j++) {\n            dp[i][j] = myMin(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    int res = dp[n - 1][m - 1];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt; n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u52a8\u6001\u89c4\u5212\nfn minPathSumDP(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][m]i32{[_]i32{0} ** m} ** n;\n    dp[0][0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (1..m) |j| {\n        dp[0][j] = dp[0][j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    for (1..n) |i| {\n        dp[i][0] = dp[i - 1][0] + grid[i][0];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n) |i| {\n        for (1..m) |j| {\n            dp[i][j] = @min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];\n        }\n    }\n    return dp[n - 1][m - 1];\n}\n</code></pre> <p>\u56fe 14-16 \u5c55\u793a\u4e86\u6700\u5c0f\u8def\u5f84\u548c\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\uff0c\u5176\u904d\u5386\u4e86\u6574\u4e2a\u7f51\u683c\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> <p>\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u4e3a \\(n \\times m\\) \uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nm)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-16 \u00a0 \u6700\u5c0f\u8def\u5f84\u548c\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/dp_solution_pipeline/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u683c\u5b50\u53ea\u4e0e\u5176\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u683c\u5b50\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u53ea\u7528\u4e00\u4e2a\u5355\u884c\u6570\u7ec4\u6765\u5b9e\u73b0 \\(dp\\) \u8868\u3002</p> <p>\u8bf7\u6ce8\u610f\uff0c\u56e0\u4e3a\u6570\u7ec4 <code>dp</code> \u53ea\u80fd\u8868\u793a\u4e00\u884c\u7684\u72b6\u6001\uff0c\u6240\u4ee5\u6211\u4eec\u65e0\u6cd5\u63d0\u524d\u521d\u59cb\u5316\u9996\u5217\u72b6\u6001\uff0c\u800c\u662f\u5728\u904d\u5386\u6bcf\u884c\u65f6\u66f4\u65b0\u5b83\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig min_path_sum.py<pre><code>def min_path_sum_dp_comp(grid: list[list[int]]) -&gt; int:\n    \"\"\"\u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(grid), len(grid[0])\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * m\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in range(1, m):\n        dp[j] = dp[j - 1] + grid[0][j]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in range(1, n):\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in range(1, m):\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n    return dp[m - 1]\n</code></pre> min_path_sum.cpp<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(vector&lt;vector&lt;int&gt;&gt; &amp;grid) {\n    int n = grid.size(), m = grid[0].size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.java<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int[][] grid) {\n    int n = grid.length, m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.cs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint MinPathSumDPComp(int[][] grid) {\n    int n = grid.Length, m = grid[0].Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[m];\n    dp[0] = grid[0][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = Math.Min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.go<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid [][]int) int {\n    n, m := len(grid), len(grid[0])\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j := 1; j &lt; m; j++ {\n        dp[j] = dp[j-1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt; n; i++ {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j := 1; j &lt; m; j++ {\n            dp[j] = int(math.Min(float64(dp[j-1]), float64(dp[j]))) + grid[i][j]\n        }\n    }\n    return dp[m-1]\n}\n</code></pre> min_path_sum.swift<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc minPathSumDPComp(grid: [[Int]]) -&gt; Int {\n    let n = grid.count\n    let m = grid[0].count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: m)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0]\n    for j in stride(from: 1, to: m, by: 1) {\n        dp[j] = dp[j - 1] + grid[0][j]\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in stride(from: 1, to: n, by: 1) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0]\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in stride(from: 1, to: m, by: 1) {\n            dp[j] = min(dp[j - 1], dp[j]) + grid[i][j]\n        }\n    }\n    return dp[m - 1]\n}\n</code></pre> min_path_sum.js<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid) {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.ts<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction minPathSumDPComp(grid: Array&lt;Array&lt;number&gt;&gt;): number {\n    const n = grid.length,\n        m = grid[0].length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = new Array(m);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (let j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt; m; j++) {\n            dp[j] = Math.min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre> min_path_sum.dart<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(List&lt;List&lt;int&gt;&gt; grid) {\n  int n = grid.length, m = grid[0].length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(m, 0);\n  dp[0] = grid[0][0];\n  for (int j = 1; j &lt; m; j++) {\n    dp[j] = dp[j - 1] + grid[0][j];\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n  for (int i = 1; i &lt; n; i++) {\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    dp[0] = dp[0] + grid[i][0];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n    for (int j = 1; j &lt; m; j++) {\n      dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];\n    }\n  }\n  return dp[m - 1];\n}\n</code></pre> min_path_sum.rs<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn min_path_sum_dp_comp(grid: &amp;Vec&lt;Vec&lt;i32&gt;&gt;) -&gt; i32 {\n    let (n, m) = (grid.len(), grid[0].len());\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; m];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for j in 1..m {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1..n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1..m {\n            dp[j] = std::cmp::min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    dp[m - 1]\n}\n</code></pre> min_path_sum.c<pre><code>/* \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint minPathSumDPComp(int grid[MAX_SIZE][MAX_SIZE], int n, int m) {\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(m, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (int j = 1; j &lt; m; j++) {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt; n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt; m; j++) {\n            dp[j] = myMin(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    int res = dp[m - 1];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> min_path_sum.zig<pre><code>// \u6700\u5c0f\u8def\u5f84\u548c\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn minPathSumDPComp(comptime grid: anytype) i32 {\n    comptime var n = grid.len;\n    comptime var m = grid[0].len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** m;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    dp[0] = grid[0][0];\n    for (1..m) |j| {\n        dp[j] = dp[j - 1] + grid[0][j];\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (1..n) |i| {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        dp[0] = dp[0] + grid[i][0];\n        for (1..m) |j| {\n            dp[j] = @min(dp[j - 1], dp[j]) + grid[i][j];\n        }\n    }\n    return dp[m - 1];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/edit_distance_problem/","title":"14.6 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u95ee\u9898","text":"<p>\u7f16\u8f91\u8ddd\u79bb\uff0c\u4e5f\u79f0 Levenshtein \u8ddd\u79bb\uff0c\u6307\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u4e92\u76f8\u8f6c\u6362\u7684\u6700\u5c11\u4fee\u6539\u6b21\u6570\uff0c\u901a\u5e38\u7528\u4e8e\u5728\u4fe1\u606f\u68c0\u7d22\u548c\u81ea\u7136\u8bed\u8a00\u5904\u7406\u4e2d\u5ea6\u91cf\u4e24\u4e2a\u5e8f\u5217\u7684\u76f8\u4f3c\u5ea6\u3002</p> <p>Question</p> <p>\u8f93\u5165\u4e24\u4e2a\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \uff0c\u8fd4\u56de\u5c06 \\(s\\) \u8f6c\u6362\u4e3a \\(t\\) \u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u4f60\u53ef\u4ee5\u5728\u4e00\u4e2a\u5b57\u7b26\u4e32\u4e2d\u8fdb\u884c\u4e09\u79cd\u7f16\u8f91\u64cd\u4f5c\uff1a\u63d2\u5165\u4e00\u4e2a\u5b57\u7b26\u3001\u5220\u9664\u4e00\u4e2a\u5b57\u7b26\u3001\u5c06\u5b57\u7b26\u66ff\u6362\u4e3a\u4efb\u610f\u4e00\u4e2a\u5b57\u7b26\u3002</p> <p>\u5982\u56fe 14-27 \u6240\u793a\uff0c\u5c06 <code>kitten</code> \u8f6c\u6362\u4e3a <code>sitting</code> \u9700\u8981\u7f16\u8f91 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u4e0e 1 \u6b21\u6dfb\u52a0\u64cd\u4f5c\uff1b\u5c06 <code>hello</code> \u8f6c\u6362\u4e3a <code>algo</code> \u9700\u8981 3 \u6b65\uff0c\u5305\u62ec 2 \u6b21\u66ff\u6362\u64cd\u4f5c\u548c 1 \u6b21\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 14-27 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u53ef\u4ee5\u5f88\u81ea\u7136\u5730\u7528\u51b3\u7b56\u6811\u6a21\u578b\u6765\u89e3\u91ca\u3002\u5b57\u7b26\u4e32\u5bf9\u5e94\u6811\u8282\u70b9\uff0c\u4e00\u8f6e\u51b3\u7b56\uff08\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\uff09\u5bf9\u5e94\u6811\u7684\u4e00\u6761\u8fb9\u3002</p> <p>\u5982\u56fe 14-28 \u6240\u793a\uff0c\u5728\u4e0d\u9650\u5236\u64cd\u4f5c\u7684\u60c5\u51b5\u4e0b\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u53ef\u4ee5\u6d3e\u751f\u51fa\u8bb8\u591a\u6761\u8fb9\uff0c\u6bcf\u6761\u8fb9\u5bf9\u5e94\u4e00\u79cd\u64cd\u4f5c\uff0c\u8fd9\u610f\u5473\u7740\u4ece <code>hello</code> \u8f6c\u6362\u5230 <code>algo</code> \u6709\u8bb8\u591a\u79cd\u53ef\u80fd\u7684\u8def\u5f84\u3002</p> <p>\u4ece\u51b3\u7b56\u6811\u7684\u89d2\u5ea6\u770b\uff0c\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u8282\u70b9 <code>hello</code> \u548c\u8282\u70b9 <code>algo</code> \u4e4b\u95f4\u7684\u6700\u77ed\u8def\u5f84\u3002</p> <p></p> <p> \u56fe 14-28 \u00a0 \u57fa\u4e8e\u51b3\u7b56\u6811\u6a21\u578b\u8868\u793a\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898 </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u6bcf\u4e00\u8f6e\u7684\u51b3\u7b56\u662f\u5bf9\u5b57\u7b26\u4e32 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\u64cd\u4f5c\u3002</p> <p>\u6211\u4eec\u5e0c\u671b\u5728\u7f16\u8f91\u64cd\u4f5c\u7684\u8fc7\u7a0b\u4e2d\uff0c\u95ee\u9898\u7684\u89c4\u6a21\u9010\u6e10\u7f29\u5c0f\uff0c\u8fd9\u6837\u624d\u80fd\u6784\u5efa\u5b50\u95ee\u9898\u3002\u8bbe\u5b57\u7b26\u4e32 \\(s\\) \u548c \\(t\\) \u7684\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \uff0c\u6211\u4eec\u5148\u8003\u8651\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u3002</p> <ul> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u76f8\u540c\uff0c\u6211\u4eec\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u76f4\u63a5\u8003\u8651 \\(s[n-2]\\) \u548c \\(t[m-2]\\) \u3002</li> <li>\u82e5 \\(s[n-1]\\) \u548c \\(t[m-1]\\) \u4e0d\u540c\uff0c\u6211\u4eec\u9700\u8981\u5bf9 \\(s\\) \u8fdb\u884c\u4e00\u6b21\u7f16\u8f91\uff08\u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\uff09\uff0c\u4f7f\u5f97\u4e24\u5b57\u7b26\u4e32\u5c3e\u90e8\u7684\u5b57\u7b26\u76f8\u540c\uff0c\u4ece\u800c\u53ef\u4ee5\u8df3\u8fc7\u5b83\u4eec\uff0c\u8003\u8651\u89c4\u6a21\u66f4\u5c0f\u7684\u95ee\u9898\u3002</li> </ul> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5728\u5b57\u7b26\u4e32 \\(s\\) \u4e2d\u8fdb\u884c\u7684\u6bcf\u4e00\u8f6e\u51b3\u7b56\uff08\u7f16\u8f91\u64cd\u4f5c\uff09\uff0c\u90fd\u4f1a\u4f7f\u5f97 \\(s\\) \u548c \\(t\\) \u4e2d\u5269\u4f59\u7684\u5f85\u5339\u914d\u5b57\u7b26\u53d1\u751f\u53d8\u5316\u3002\u56e0\u6b64\uff0c\u72b6\u6001\u4e3a\u5f53\u524d\u5728 \\(s\\) \u548c \\(t\\) \u4e2d\u8003\u8651\u7684\u7b2c \\(i\\) \u548c\u7b2c \\(j\\) \u4e2a\u5b57\u7b26\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u72b6\u6001 \\([i, j]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\uff1a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002</p> <p>\u81f3\u6b64\uff0c\u5f97\u5230\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((i+1) \\times (j+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u8003\u8651\u5b50\u95ee\u9898 \\(dp[i, j]\\) \uff0c\u5176\u5bf9\u5e94\u7684\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5c3e\u90e8\u5b57\u7b26\u4e3a \\(s[i-1]\\) \u548c \\(t[j-1]\\) \uff0c\u53ef\u6839\u636e\u4e0d\u540c\u7f16\u8f91\u64cd\u4f5c\u5206\u4e3a\u56fe 14-29 \u6240\u793a\u7684\u4e09\u79cd\u60c5\u51b5\u3002</p> <ol> <li>\u5728 \\(s[i-1]\\) \u4e4b\u540e\u6dfb\u52a0 \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i, j-1]\\) \u3002</li> <li>\u5220\u9664 \\(s[i-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j]\\) \u3002</li> <li>\u5c06 \\(s[i-1]\\) \u66ff\u6362\u4e3a \\(t[j-1]\\) \uff0c\u5219\u5269\u4f59\u5b50\u95ee\u9898 \\(dp[i-1, j-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 14-29 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u72b6\u6001\u8f6c\u79fb </p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u53ef\u5f97\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\\(dp[i, j]\\) \u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(dp[i, j-1]\\)\u3001\\(dp[i-1, j]\\)\u3001\\(dp[i-1, j-1]\\) \u4e09\u8005\u4e2d\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u518d\u52a0\u4e0a\u672c\u6b21\u7684\u7f16\u8f91\u6b65\u6570 \\(1\\) \u3002\u5bf9\u5e94\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = \\min(dp[i, j-1], dp[i-1, j], dp[i-1, j-1]) + 1 \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5f53 \\(s[i-1]\\) \u548c \\(t[j-1]\\) \u76f8\u540c\u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0b\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, j] = dp[i-1, j-1] \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u4e24\u5b57\u7b26\u4e32\u90fd\u4e3a\u7a7a\u65f6\uff0c\u7f16\u8f91\u6b65\u6570\u4e3a \\(0\\) \uff0c\u5373 \\(dp[0, 0] = 0\\) \u3002\u5f53 \\(s\\) \u4e3a\u7a7a\u4f46 \\(t\\) \u4e0d\u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(t\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u884c \\(dp[0, j] = j\\) \u3002\u5f53 \\(s\\) \u4e0d\u4e3a\u7a7a\u4f46 \\(t\\) \u4e3a\u7a7a\u65f6\uff0c\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u7b49\u4e8e \\(s\\) \u7684\u957f\u5ea6\uff0c\u5373\u9996\u5217 \\(dp[i, 0] = i\\) \u3002</p> <p>\u89c2\u5bdf\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff0c\u89e3 \\(dp[i, j]\\) \u4f9d\u8d56\u5de6\u65b9\u3001\u4e0a\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u89e3\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.py<pre><code>def edit_distance_dp(s: str, t: str) -&gt; int:\n    \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(s), len(t)\n    dp = [[0] * (m + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in range(1, n + 1):\n        dp[i][0] = i\n    for j in range(1, m + 1):\n        dp[0][j] = j\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for j in range(1, m + 1):\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            else:\n                # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1\n    return dp[n][m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(m + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[][] dp = new int[n + 1][m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = Math.min(Math.min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDP(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[,] dp = new int[n + 1, m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i, 0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0, j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i, j] = dp[i - 1, j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i, j] = Math.Min(Math.Min(dp[i, j - 1], dp[i - 1, j]), dp[i - 1, j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n, m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, m+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i := 1; i &lt;= n; i++ {\n        dp[i][0] = i\n    }\n    for j := 1; j &lt;= m; j++ {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for j := 1; j &lt;= m; j++ {\n            if s[i-1] == t[j-1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i-1][j-1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = MinInt(MinInt(dp[i][j-1], dp[i-1][j]), dp[i-1][j-1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDP(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: Array(repeating: 0, count: m + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in stride(from: 1, through: n, by: 1) {\n        dp[i][0] = i\n    }\n    for j in stride(from: 1, through: m, by: 1) {\n        dp[0][j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in stride(from: 1, through: n, by: 1) {\n        for j in stride(from: 1, through: m, by: 1) {\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1]\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1\n            }\n        }\n    }\n    return dp[n][m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt; new Array(m + 1).fill(0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDP(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: m + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (let j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let j = 1; j &lt;= m; j++) {\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] =\n                    Math.min(dp[i][j - 1], dp[i - 1][j], dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (_) =&gt; List.filled(m + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    dp[i][0] = i;\n  }\n  for (int j = 1; j &lt;= m; j++) {\n    dp[0][j] = j;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int j = 1; j &lt;= m; j++) {\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n        dp[i][j] = dp[i - 1][j - 1];\n      } else {\n        // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n        dp[i][j] = min(min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n      }\n    }\n  }\n  return dp[n][m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![vec![0; m + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for i in 1..= n {\n        dp[i][0] = i as i32;\n    }\n    for j in 1..m {\n        dp[0][j] = j as i32;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..=n {\n        for j in 1..=m {\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = std::cmp::min(std::cmp::min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    dp[n][m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212 */\nint editDistanceDP(char *s, char *t, int n, int m) {\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(m + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        dp[i][0] = i;\n    }\n    for (int j = 1; j &lt;= m; j++) {\n        dp[0][j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int j = 1; j &lt;= m; j++) {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = myMin(myMin(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    int res = dp[n][m];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u52a8\u6001\u89c4\u5212\nfn editDistanceDP(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_][m + 1]i32{[_]i32{0} ** (m + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..n + 1) |i| {\n        dp[i][0] = @intCast(i);\n    }\n    for (1..m + 1) |j| {\n        dp[0][j] = @intCast(j);\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..m + 1) |j| {\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[i][j] = dp[i - 1][j - 1];\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[i][j] = @min(@min(dp[i][j - 1], dp[i - 1][j]), dp[i - 1][j - 1]) + 1;\n            }\n        }\n    }\n    return dp[n][m];\n}\n</code></pre> <p>\u5982\u56fe 14-30 \u6240\u793a\uff0c\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u8fc7\u7a0b\u4e0e\u80cc\u5305\u95ee\u9898\u975e\u5e38\u7c7b\u4f3c\uff0c\u90fd\u53ef\u4ee5\u770b\u4f5c\u586b\u5199\u4e00\u4e2a\u4e8c\u7ef4\u7f51\u683c\u7684\u8fc7\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-30 \u00a0 \u7f16\u8f91\u8ddd\u79bb\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/edit_distance_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e \\(dp[i,j]\\) \u662f\u7531\u4e0a\u65b9 \\(dp[i-1, j]\\)\u3001\u5de6\u65b9 \\(dp[i, j-1]\\)\u3001\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \u8f6c\u79fb\u800c\u6765\u7684\uff0c\u800c\u6b63\u5e8f\u904d\u5386\u4f1a\u4e22\u5931\u5de6\u4e0a\u65b9 \\(dp[i-1, j-1]\\) \uff0c\u5012\u5e8f\u904d\u5386\u65e0\u6cd5\u63d0\u524d\u6784\u5efa \\(dp[i, j-1]\\) \uff0c\u56e0\u6b64\u4e24\u79cd\u904d\u5386\u987a\u5e8f\u90fd\u4e0d\u53ef\u53d6\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>leftup</code> \u6765\u6682\u5b58\u5de6\u4e0a\u65b9\u7684\u89e3 \\(dp[i-1, j-1]\\) \uff0c\u4ece\u800c\u53ea\u9700\u8003\u8651\u5de6\u65b9\u548c\u4e0a\u65b9\u7684\u89e3\u3002\u6b64\u65f6\u7684\u60c5\u51b5\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u53ef\u4f7f\u7528\u6b63\u5e8f\u904d\u5386\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig edit_distance.py<pre><code>def edit_distance_dp_comp(s: str, t: str) -&gt; int:\n    \"\"\"\u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n, m = len(s), len(t)\n    dp = [0] * (m + 1)\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in range(1, m + 1):\n        dp[j] = j\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in range(1, n + 1):\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        leftup = dp[0]  # \u6682\u5b58 dp[i-1, j-1]\n        dp[0] += 1\n        # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in range(1, m + 1):\n            temp = dp[j]\n            if s[i - 1] == t[j - 1]:\n                # \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            else:\n                # \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(dp[j - 1], dp[j], leftup) + 1\n            leftup = temp  # \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n    return dp[m]\n</code></pre> edit_distance.cpp<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(string s, string t) {\n    int n = s.length(), m = t.length();\n    vector&lt;int&gt; dp(m + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.java<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n    int n = s.length(), m = t.length();\n    int[] dp = new int[m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s.charAt(i - 1) == t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(Math.min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.cs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint EditDistanceDPComp(string s, string t) {\n    int n = s.Length, m = t.Length;\n    int[] dp = new int[m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.Min(Math.Min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.go<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s string, t string) int {\n    n := len(s)\n    m := len(t)\n    dp := make([]int, m+1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j := 1; j &lt;= m; j++ {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i := 1; i &lt;= n; i++ {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        leftUp := dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j := 1; j &lt;= m; j++ {\n            temp := dp[j]\n            if s[i-1] == t[j-1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftUp\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = MinInt(MinInt(dp[j-1], dp[j]), leftUp) + 1\n            }\n            leftUp = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.swift<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc editDistanceDPComp(s: String, t: String) -&gt; Int {\n    let n = s.utf8CString.count\n    let m = t.utf8CString.count\n    var dp = Array(repeating: 0, count: m + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in stride(from: 1, through: m, by: 1) {\n        dp[j] = j\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in stride(from: 1, through: n, by: 1) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0] // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in stride(from: 1, through: m, by: 1) {\n            let temp = dp[j]\n            if s.utf8CString[i - 1] == t.utf8CString[j - 1] {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1\n            }\n            leftup = temp // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m]\n}\n</code></pre> edit_distance.js<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s, t) {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.ts<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction editDistanceDPComp(s: string, t: string): number {\n    const n = s.length,\n        m = t.length;\n    const dp = new Array(m + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (let j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (let i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (let j = 1; j &lt;= m; j++) {\n            const temp = dp[j];\n            if (s.charAt(i - 1) === t.charAt(j - 1)) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = Math.min(dp[j - 1], dp[j], leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre> edit_distance.dart<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(String s, String t) {\n  int n = s.length, m = t.length;\n  List&lt;int&gt; dp = List.filled(m + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n  for (int j = 1; j &lt;= m; j++) {\n    dp[j] = j;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n  for (int i = 1; i &lt;= n; i++) {\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n    int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n    dp[0] = i;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n    for (int j = 1; j &lt;= m; j++) {\n      int temp = dp[j];\n      if (s[i - 1] == t[j - 1]) {\n        // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n        dp[j] = leftup;\n      } else {\n        // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n        dp[j] = min(min(dp[j - 1], dp[j]), leftup) + 1;\n      }\n      leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n    }\n  }\n  return dp[m];\n}\n</code></pre> edit_distance.rs<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn edit_distance_dp_comp(s: &amp;str, t: &amp;str) -&gt; i32 {\n    let (n, m) = (s.len(), t.len());\n    let mut dp = vec![0; m + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for j in 1..m {\n        dp[j] = j as i32;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for i in 1..=n {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        let mut leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i as i32;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for j in 1..=m {\n            let temp = dp[j];\n            if s.chars().nth(i - 1) == t.chars().nth(j - 1) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = std::cmp::min(std::cmp::min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    dp[m]\n}\n</code></pre> edit_distance.c<pre><code>/* \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint editDistanceDPComp(char *s, char *t, int n, int m) {\n    int *dp = calloc(m + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (int j = 1; j &lt;= m; j++) {\n        dp[j] = j;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (int i = 1; i &lt;= n; i++) {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        int leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = i;\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (int j = 1; j &lt;= m; j++) {\n            int temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = myMin(myMin(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    int res = dp[m];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> edit_distance.zig<pre><code>// \u7f16\u8f91\u8ddd\u79bb\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn editDistanceDPComp(comptime s: []const u8, comptime t: []const u8) i32 {\n    comptime var n = s.len;\n    comptime var m = t.len;\n    var dp = [_]i32{0} ** (m + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\n    for (1..m + 1) |j| {\n        dp[j] = @intCast(j);\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\n    for (1..n + 1) |i| {\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u5217\n        var leftup = dp[0]; // \u6682\u5b58 dp[i-1, j-1]\n        dp[0] = @intCast(i);\n        // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u5217\n        for (1..m + 1) |j| {\n            var temp = dp[j];\n            if (s[i - 1] == t[j - 1]) {\n                // \u82e5\u4e24\u5b57\u7b26\u76f8\u7b49\uff0c\u5219\u76f4\u63a5\u8df3\u8fc7\u6b64\u4e24\u5b57\u7b26\n                dp[j] = leftup;\n            } else {\n                // \u6700\u5c11\u7f16\u8f91\u6b65\u6570 = \u63d2\u5165\u3001\u5220\u9664\u3001\u66ff\u6362\u8fd9\u4e09\u79cd\u64cd\u4f5c\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570 + 1\n                dp[j] = @min(@min(dp[j - 1], dp[j]), leftup) + 1;\n            }\n            leftup = temp; // \u66f4\u65b0\u4e3a\u4e0b\u4e00\u8f6e\u7684 dp[i-1, j-1]\n        }\n    }\n    return dp[m];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/","title":"14.1 \u00a0 \u521d\u63a2\u52a8\u6001\u89c4\u5212","text":"<p>\u300c\u52a8\u6001\u89c4\u5212 dynamic programming\u300d\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u7b97\u6cd5\u8303\u5f0f\uff0c\u5b83\u5c06\u4e00\u4e2a\u95ee\u9898\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\uff0c\u4ece\u800c\u5927\u5e45\u63d0\u5347\u65f6\u95f4\u6548\u7387\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u4ece\u4e00\u4e2a\u7ecf\u5178\u4f8b\u9898\u5165\u624b\uff0c\u5148\u7ed9\u51fa\u5b83\u7684\u66b4\u529b\u56de\u6eaf\u89e3\u6cd5\uff0c\u89c2\u5bdf\u5176\u4e2d\u5305\u542b\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u518d\u9010\u6b65\u5bfc\u51fa\u66f4\u9ad8\u6548\u7684\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p> <p>\u722c\u697c\u68af</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u5171\u6709 \\(n\\) \u9636\u7684\u697c\u68af\uff0c\u4f60\u6bcf\u6b65\u53ef\u4ee5\u4e0a \\(1\\) \u9636\u6216\u8005 \\(2\\) \u9636\uff0c\u8bf7\u95ee\u6709\u591a\u5c11\u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\uff1f</p> <p>\u5982\u56fe 14-1 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e00\u4e2a \\(3\\) \u9636\u697c\u68af\uff0c\u5171\u6709 \\(3\\) \u79cd\u65b9\u6848\u53ef\u4ee5\u722c\u5230\u697c\u9876\u3002</p> <p></p> <p> \u56fe 14-1 \u00a0 \u722c\u5230\u7b2c 3 \u9636\u7684\u65b9\u6848\u6570\u91cf </p> <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u65b9\u6848\u6570\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u56de\u6eaf\u6765\u7a77\u4e3e\u6240\u6709\u53ef\u80fd\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u722c\u697c\u68af\u60f3\u8c61\u4e3a\u4e00\u4e2a\u591a\u8f6e\u9009\u62e9\u7684\u8fc7\u7a0b\uff1a\u4ece\u5730\u9762\u51fa\u53d1\uff0c\u6bcf\u8f6e\u9009\u62e9\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u6bcf\u5f53\u5230\u8fbe\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u65b9\u6848\u6570\u91cf\u52a0 \\(1\\) \uff0c\u5f53\u8d8a\u8fc7\u697c\u68af\u9876\u90e8\u65f6\u5c31\u5c06\u5176\u526a\u679d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_backtrack.py<pre><code>def backtrack(choices: list[int], state: int, n: int, res: list[int]) -&gt; int:\n    \"\"\"\u56de\u6eaf\"\"\"\n    # \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n:\n        res[0] += 1\n    # \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices:\n        # \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n:\n            continue\n        # \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res)\n        # \u56de\u9000\n\ndef climbing_stairs_backtrack(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u56de\u6eaf\"\"\"\n    choices = [1, 2]  # \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    state = 0  # \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    res = [0]  # \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res)\n    return res[0]\n</code></pre> climbing_stairs_backtrack.cpp<pre><code>/* \u56de\u6eaf */\nvoid backtrack(vector&lt;int&gt; &amp;choices, int state, int n, vector&lt;int&gt; &amp;res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (auto &amp;choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    vector&lt;int&gt; choices = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0;                // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    vector&lt;int&gt; res = {0};        // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.java<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;Integer&gt; choices, int state, int n, List&lt;Integer&gt; res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (Integer choice : choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    List&lt;Integer&gt; choices = Arrays.asList(1, 2); // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n    res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.cs<pre><code>/* \u56de\u6eaf */\nvoid Backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    foreach (int choice in choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        Backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint ClimbingStairsBacktrack(int n) {\n    List&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    List&lt;int&gt; res = [0]; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    Backtrack(choices, state, n, res);\n    return res[0];\n}\n</code></pre> climbing_stairs_backtrack.go<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices []int, state, n int, res []int) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] = res[0] + 1\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for _, choice := range choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state+choice &gt; n {\n            continue\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state+choice, n, res)\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n int) int {\n    // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    choices := []int{1, 2}\n    // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    state := 0\n    res := make([]int, 1)\n    // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    res[0] = 0\n    backtrack(choices, state, n, res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.swift<pre><code>/* \u56de\u6eaf */\nfunc backtrack(choices: [Int], state: Int, n: Int, res: inout [Int]) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n {\n        res[0] += 1\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n {\n            continue\n        }\n        backtrack(choices: choices, state: state + choice, n: n, res: &amp;res)\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunc climbingStairsBacktrack(n: Int) -&gt; Int {\n    let choices = [1, 2] // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    let state = 0 // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    var res: [Int] = []\n    res.append(0) // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices: choices, state: state, n: n, res: &amp;res)\n    return res[0]\n}\n</code></pre> climbing_stairs_backtrack.js<pre><code>/* \u56de\u6eaf */\nfunction backtrack(choices, state, n, res) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n) {\n    const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.ts<pre><code>/* \u56de\u6eaf */\nfunction backtrack(\n    choices: number[],\n    state: number,\n    n: number,\n    res: Map&lt;0, any&gt;\n): void {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state === n) res.set(0, res.get(0) + 1);\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (const choice of choices) {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfunction climbingStairsBacktrack(n: number): number {\n    const choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    const state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    const res = new Map();\n    res.set(0, 0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(choices, state, n, res);\n    return res.get(0);\n}\n</code></pre> climbing_stairs_backtrack.dart<pre><code>/* \u56de\u6eaf */\nvoid backtrack(List&lt;int&gt; choices, int state, int n, List&lt;int&gt; res) {\n  // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n  if (state == n) {\n    res[0]++;\n  }\n  // \u904d\u5386\u6240\u6709\u9009\u62e9\n  for (int choice in choices) {\n    // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n    if (state + choice &gt; n) continue;\n    // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n    backtrack(choices, state + choice, n, res);\n    // \u56de\u9000\n  }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n  List&lt;int&gt; choices = [1, 2]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n  int state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n  List&lt;int&gt; res = [];\n  res.add(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n  backtrack(choices, state, n, res);\n  return res[0];\n}\n</code></pre> climbing_stairs_backtrack.rs<pre><code>/* \u56de\u6eaf */\nfn backtrack(choices: &amp;[i32], state: i32, n: i32, res: &amp;mut [i32]) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if state == n { res[0] = res[0] + 1; }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for &amp;choice in choices {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if state + choice &gt; n { continue; }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nfn climbing_stairs_backtrack(n: usize) -&gt; i32 {\n    let choices = vec![ 1, 2 ]; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    let state = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    let mut res = Vec::new();\n    res.push(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(&amp;choices, state, n as i32, &amp;mut res);\n    res[0]\n}\n</code></pre> climbing_stairs_backtrack.c<pre><code>/* \u56de\u6eaf */\nvoid backtrack(int *choices, int state, int n, int *res, int len) {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n)\n        res[0]++;\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (int i = 0; i &lt; len; i++) {\n        int choice = choices[i];\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n)\n            continue;\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res, len);\n        // \u56de\u9000\n    }\n}\n\n/* \u722c\u697c\u68af\uff1a\u56de\u6eaf */\nint climbingStairsBacktrack(int n) {\n    int choices[2] = {1, 2}; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    int state = 0;           // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    int *res = (int *)malloc(sizeof(int));\n    *res = 0; // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    int len = sizeof(choices) / sizeof(int);\n    backtrack(choices, state, n, res, len);\n    int result = *res;\n    free(res);\n    return result;\n}\n</code></pre> climbing_stairs_backtrack.zig<pre><code>// \u56de\u6eaf\nfn backtrack(choices: []i32, state: i32, n: i32, res: std.ArrayList(i32)) void {\n    // \u5f53\u722c\u5230\u7b2c n \u9636\u65f6\uff0c\u65b9\u6848\u6570\u91cf\u52a0 1\n    if (state == n) {\n        res.items[0] = res.items[0] + 1;\n    }\n    // \u904d\u5386\u6240\u6709\u9009\u62e9\n    for (choices) |choice| {\n        // \u526a\u679d\uff1a\u4e0d\u5141\u8bb8\u8d8a\u8fc7\u7b2c n \u9636\n        if (state + choice &gt; n) {\n            continue;\n        }\n        // \u5c1d\u8bd5\uff1a\u505a\u51fa\u9009\u62e9\uff0c\u66f4\u65b0\u72b6\u6001\n        backtrack(choices, state + choice, n, res);\n        // \u56de\u9000\n    }\n}\n\n// \u722c\u697c\u68af\uff1a\u56de\u6eaf\nfn climbingStairsBacktrack(n: usize) !i32 {\n    var choices = [_]i32{ 1, 2 }; // \u53ef\u9009\u62e9\u5411\u4e0a\u722c 1 \u9636\u6216 2 \u9636\n    var state: i32 = 0; // \u4ece\u7b2c 0 \u9636\u5f00\u59cb\u722c\n    var res = std.ArrayList(i32).init(std.heap.page_allocator);\n    defer res.deinit();\n    try res.append(0); // \u4f7f\u7528 res[0] \u8bb0\u5f55\u65b9\u6848\u6570\u91cf\n    backtrack(&amp;choices, state, @intCast(n), res);\n    return res.items[0];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1411","title":"14.1.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u56de\u6eaf\u7b97\u6cd5\u901a\u5e38\u5e76\u4e0d\u663e\u5f0f\u5730\u5bf9\u95ee\u9898\u8fdb\u884c\u62c6\u89e3\uff0c\u800c\u662f\u5c06\u6c42\u89e3\u95ee\u9898\u770b\u4f5c\u4e00\u7cfb\u5217\u51b3\u7b56\u6b65\u9aa4\uff0c\u901a\u8fc7\u8bd5\u63a2\u548c\u526a\u679d\uff0c\u641c\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u4ece\u95ee\u9898\u5206\u89e3\u7684\u89d2\u5ea6\u5206\u6790\u8fd9\u9053\u9898\u3002\u8bbe\u722c\u5230\u7b2c \\(i\\) \u9636\u5171\u6709 \\(dp[i]\\) \u79cd\u65b9\u6848\uff0c\u90a3\u4e48 \\(dp[i]\\) \u5c31\u662f\u539f\u95ee\u9898\uff0c\u5176\u5b50\u95ee\u9898\u5305\u62ec\uff1a</p> \\[ dp[i-1], dp[i-2], \\dots, dp[2], dp[1] \\] <p>\u7531\u4e8e\u6bcf\u8f6e\u53ea\u80fd\u4e0a \\(1\\) \u9636\u6216 \\(2\\) \u9636\uff0c\u56e0\u6b64\u5f53\u6211\u4eec\u7ad9\u5728\u7b2c \\(i\\) \u9636\u697c\u68af\u4e0a\u65f6\uff0c\u4e0a\u4e00\u8f6e\u53ea\u53ef\u80fd\u7ad9\u5728\u7b2c \\(i - 1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u4e0a\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ea\u80fd\u4ece\u7b2c \\(i -1\\) \u9636\u6216\u7b2c \\(i - 2\\) \u9636\u8fc8\u5411\u7b2c \\(i\\) \u9636\u3002</p> <p>\u7531\u6b64\u4fbf\u53ef\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u63a8\u8bba\uff1a\u722c\u5230\u7b2c \\(i - 1\\) \u9636\u7684\u65b9\u6848\u6570\u52a0\u4e0a\u722c\u5230\u7b2c \\(i - 2\\) \u9636\u7684\u65b9\u6848\u6570\u5c31\u7b49\u4e8e\u722c\u5230\u7b2c \\(i\\) \u9636\u7684\u65b9\u6848\u6570\u3002\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ dp[i] = dp[i-1] + dp[i-2] \\] <p>\u8fd9\u610f\u5473\u7740\u5728\u722c\u697c\u68af\u95ee\u9898\u4e2d\uff0c\u5404\u4e2a\u5b50\u95ee\u9898\u4e4b\u95f4\u5b58\u5728\u9012\u63a8\u5173\u7cfb\uff0c\u539f\u95ee\u9898\u7684\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u7684\u89e3\u6784\u5efa\u5f97\u6765\u3002\u56fe 14-2 \u5c55\u793a\u4e86\u8be5\u9012\u63a8\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 14-2 \u00a0 \u65b9\u6848\u6570\u91cf\u9012\u63a8\u5173\u7cfb </p> <p>\u6211\u4eec\u53ef\u4ee5\u6839\u636e\u9012\u63a8\u516c\u5f0f\u5f97\u5230\u66b4\u529b\u641c\u7d22\u89e3\u6cd5\u3002\u4ee5 \\(dp[n]\\) \u4e3a\u8d77\u59cb\u70b9\uff0c\u9012\u5f52\u5730\u5c06\u4e00\u4e2a\u8f83\u5927\u95ee\u9898\u62c6\u89e3\u4e3a\u4e24\u4e2a\u8f83\u5c0f\u95ee\u9898\u7684\u548c\uff0c\u76f4\u81f3\u5230\u8fbe\u6700\u5c0f\u5b50\u95ee\u9898 \\(dp[1]\\) \u548c \\(dp[2]\\) \u65f6\u8fd4\u56de\u3002\u5176\u4e2d\uff0c\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u662f\u5df2\u77e5\u7684\uff0c\u5373 \\(dp[1] = 1\\)\u3001\\(dp[2] = 2\\) \uff0c\u8868\u793a\u722c\u5230\u7b2c \\(1\\)\u3001\\(2\\) \u9636\u5206\u522b\u6709 \\(1\\)\u3001\\(2\\) \u79cd\u65b9\u6848\u3002</p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5b83\u548c\u6807\u51c6\u56de\u6eaf\u4ee3\u7801\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u4f46\u66f4\u52a0\u7b80\u6d01\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs.py<pre><code>def dfs(i: int) -&gt; int:\n    \"\"\"\u641c\u7d22\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1) + dfs(i - 2)\n    return count\n\ndef climbing_stairs_dfs(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u641c\u7d22\"\"\"\n    return dfs(n)\n</code></pre> climbing_stairs_dfs.cpp<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.java<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.cs<pre><code>/* \u641c\u7d22 */\nint DFS(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1) + DFS(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint ClimbingStairsDFS(int n) {\n    return DFS(n);\n}\n</code></pre> climbing_stairs_dfs.go<pre><code>/* \u641c\u7d22 */\nfunc dfs(i int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfs(i-1) + dfs(i-2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n int) int {\n    return dfs(n)\n}\n</code></pre> climbing_stairs_dfs.swift<pre><code>/* \u641c\u7d22 */\nfunc dfs(i: Int) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1) + dfs(i: i - 2)\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunc climbingStairsDFS(n: Int) -&gt; Int {\n    dfs(i: n)\n}\n</code></pre> climbing_stairs_dfs.js<pre><code>/* \u641c\u7d22 */\nfunction dfs(i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.ts<pre><code>/* \u641c\u7d22 */\nfunction dfs(i: number): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfunction climbingStairsDFS(n: number): number {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.dart<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1) + dfs(i - 2);\n  return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n  return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.rs<pre><code>/* \u641c\u7d22 */\nfn dfs(i: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 { return i as i32; }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1) + dfs(i - 2);\n    count\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nfn climbing_stairs_dfs(n: usize) -&gt; i32 {\n    dfs(n) \n}\n</code></pre> climbing_stairs_dfs.c<pre><code>/* \u641c\u7d22 */\nint dfs(int i) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u641c\u7d22 */\nint climbingStairsDFS(int n) {\n    return dfs(n);\n}\n</code></pre> climbing_stairs_dfs.zig<pre><code>// \u641c\u7d22\nfn dfs(i: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1) + dfs(i - 2);\n    return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u641c\u7d22\nfn climbingStairsDFS(comptime n: usize) i32 {\n    return dfs(n);\n}\n</code></pre> <p>\u56fe 14-3 \u5c55\u793a\u4e86\u66b4\u529b\u641c\u7d22\u5f62\u6210\u7684\u9012\u5f52\u6811\u3002\u5bf9\u4e8e\u95ee\u9898 \\(dp[n]\\) \uff0c\u5176\u9012\u5f52\u6811\u7684\u6df1\u5ea6\u4e3a \\(n\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002\u6307\u6570\u9636\u5c5e\u4e8e\u7206\u70b8\u5f0f\u589e\u957f\uff0c\u5982\u679c\u6211\u4eec\u8f93\u5165\u4e00\u4e2a\u6bd4\u8f83\u5927\u7684 \\(n\\) \uff0c\u5219\u4f1a\u9677\u5165\u6f2b\u957f\u7684\u7b49\u5f85\u4e4b\u4e2d\u3002</p> <p></p> <p> \u56fe 14-3 \u00a0 \u722c\u697c\u68af\u5bf9\u5e94\u9012\u5f52\u6811 </p> <p>\u89c2\u5bdf\u56fe 14-3 \uff0c\u6307\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u201c\u91cd\u53e0\u5b50\u95ee\u9898\u201d\u5bfc\u81f4\u7684\u3002\u4f8b\u5982 \\(dp[9]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[8]\\) \u548c \\(dp[7]\\) \uff0c\\(dp[8]\\) \u88ab\u5206\u89e3\u4e3a \\(dp[7]\\) \u548c \\(dp[6]\\) \uff0c\u4e24\u8005\u90fd\u5305\u542b\u5b50\u95ee\u9898 \\(dp[7]\\) \u3002</p> <p>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5b50\u95ee\u9898\u4e2d\u5305\u542b\u66f4\u5c0f\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u5b50\u5b50\u5b59\u5b59\u65e0\u7a77\u5c3d\u4e5f\u3002\u7edd\u5927\u90e8\u5206\u8ba1\u7b97\u8d44\u6e90\u90fd\u6d6a\u8d39\u5728\u8fd9\u4e9b\u91cd\u53e0\u7684\u95ee\u9898\u4e0a\u3002</p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1412","title":"14.1.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u63d0\u5347\u7b97\u6cd5\u6548\u7387\uff0c\u6211\u4eec\u5e0c\u671b\u6240\u6709\u7684\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u6570\u7ec4 <code>mem</code> \u6765\u8bb0\u5f55\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5e76\u5728\u641c\u7d22\u8fc7\u7a0b\u4e2d\u5c06\u91cd\u53e0\u5b50\u95ee\u9898\u526a\u679d\u3002</p> <ol> <li>\u5f53\u9996\u6b21\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u5c06\u5176\u8bb0\u5f55\u81f3 <code>mem[i]</code> \uff0c\u4ee5\u4fbf\u4e4b\u540e\u4f7f\u7528\u3002</li> <li>\u5f53\u518d\u6b21\u9700\u8981\u8ba1\u7b97 \\(dp[i]\\) \u65f6\uff0c\u6211\u4eec\u4fbf\u53ef\u76f4\u63a5\u4ece <code>mem[i]</code> \u4e2d\u83b7\u53d6\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u8be5\u5b50\u95ee\u9898\u3002</li> </ol> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dfs_mem.py<pre><code>def dfs(i: int, mem: list[int]) -&gt; int:\n    \"\"\"\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 or i == 2:\n        return i\n    # \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1:\n        return mem[i]\n    # dp[i] = dp[i-1] + dp[i-2]\n    count = dfs(i - 1, mem) + dfs(i - 2, mem)\n    # \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n\ndef climbing_stairs_dfs_mem(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    mem = [-1] * (n + 1)\n    return dfs(n, mem)\n</code></pre> climbing_stairs_dfs_mem.cpp<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, vector&lt;int&gt; &amp;mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    vector&lt;int&gt; mem(n + 1, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.java<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int[] mem = new int[n + 1];\n    Arrays.fill(mem, -1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.cs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint DFS(int i, int[] mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = DFS(i - 1, mem) + DFS(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint ClimbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int[] mem = new int[n + 1];\n    Array.Fill(mem, -1);\n    return DFS(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.go<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfsMem(i int, mem []int) int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    count := dfsMem(i-1, mem) + dfsMem(i-2, mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n int) int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    mem := make([]int, n+1)\n    for i := range mem {\n        mem[i] = -1\n    }\n    return dfsMem(n, mem)\n}\n</code></pre> climbing_stairs_dfs_mem.swift<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc dfs(i: Int, mem: inout [Int]) -&gt; Int {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 {\n        return i\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 {\n        return mem[i]\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i: i - 1, mem: &amp;mem) + dfs(i: i - 2, mem: &amp;mem)\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count\n    return count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc climbingStairsDFSMem(n: Int) -&gt; Int {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    var mem = Array(repeating: -1, count: n + 1)\n    return dfs(i: n, mem: &amp;mem)\n}\n</code></pre> climbing_stairs_dfs_mem.js<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i, mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.ts<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction dfs(i: number, mem: number[]): number {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i === 1 || i === 2) return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    const count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction climbingStairsDFSMem(n: number): number {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    const mem = new Array(n + 1).fill(-1);\n    return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.dart<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, List&lt;int&gt; mem) {\n  // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n  if (i == 1 || i == 2) return i;\n  // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n  if (mem[i] != -1) return mem[i];\n  // dp[i] = dp[i-1] + dp[i-2]\n  int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n  // \u8bb0\u5f55 dp[i]\n  mem[i] = count;\n  return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n  // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n  List&lt;int&gt; mem = List.filled(n + 1, -1);\n  return dfs(n, mem);\n}\n</code></pre> climbing_stairs_dfs_mem.rs<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn dfs(i: usize, mem: &amp;mut [i32]) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if i == 1 || i == 2 { return i as i32; }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if mem[i] != -1 { return mem[i]; }\n    // dp[i] = dp[i-1] + dp[i-2]\n    let count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    count\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn climbing_stairs_dfs_mem(n: usize) -&gt; i32 {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    let mut mem = vec![-1; n + 1];\n    dfs(n, &amp;mut mem)\n}\n</code></pre> climbing_stairs_dfs_mem.c<pre><code>/* \u8bb0\u5fc6\u5316\u641c\u7d22 */\nint dfs(int i, int *mem) {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 || i == 2)\n        return i;\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1)\n        return mem[i];\n    // dp[i] = dp[i-1] + dp[i-2]\n    int count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n/* \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint climbingStairsDFSMem(int n) {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    int *mem = (int *)malloc((n + 1) * sizeof(int));\n    for (int i = 0; i &lt;= n; i++) {\n        mem[i] = -1;\n    }\n    int result = dfs(n, mem);\n    free(mem);\n    return result;\n}\n</code></pre> climbing_stairs_dfs_mem.zig<pre><code>// \u8bb0\u5fc6\u5316\u641c\u7d22\nfn dfs(i: usize, mem: []i32) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (i == 1 or i == 2) {\n        return @intCast(i);\n    }\n    // \u82e5\u5b58\u5728\u8bb0\u5f55 dp[i] \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u4e4b\n    if (mem[i] != -1) {\n        return mem[i];\n    }\n    // dp[i] = dp[i-1] + dp[i-2]\n    var count = dfs(i - 1, mem) + dfs(i - 2, mem);\n    // \u8bb0\u5f55 dp[i]\n    mem[i] = count;\n    return count;\n}\n\n// \u722c\u697c\u68af\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn climbingStairsDFSMem(comptime n: usize) i32 {\n    // mem[i] \u8bb0\u5f55\u722c\u5230\u7b2c i \u9636\u7684\u65b9\u6848\u603b\u6570\uff0c-1 \u4ee3\u8868\u65e0\u8bb0\u5f55\n    var mem = [_]i32{ -1 } ** (n + 1);\n    return dfs(n, &amp;mem);\n}\n</code></pre> <p>\u89c2\u5bdf\u56fe 14-4 \uff0c\u7ecf\u8fc7\u8bb0\u5fc6\u5316\u5904\u7406\u540e\uff0c\u6240\u6709\u91cd\u53e0\u5b50\u95ee\u9898\u90fd\u53ea\u9700\u8ba1\u7b97\u4e00\u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u8fd9\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u98de\u8dc3\u3002</p> <p></p> <p> \u56fe 14-4 \u00a0 \u8bb0\u5fc6\u5316\u641c\u7d22\u5bf9\u5e94\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1413","title":"14.1.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u8bb0\u5fc6\u5316\u641c\u7d22\u662f\u4e00\u79cd\u201c\u4ece\u9876\u81f3\u5e95\u201d\u7684\u65b9\u6cd5\uff1a\u6211\u4eec\u4ece\u539f\u95ee\u9898\uff08\u6839\u8282\u70b9\uff09\u5f00\u59cb\uff0c\u9012\u5f52\u5730\u5c06\u8f83\u5927\u5b50\u95ee\u9898\u5206\u89e3\u4e3a\u8f83\u5c0f\u5b50\u95ee\u9898\uff0c\u76f4\u81f3\u89e3\u5df2\u77e5\u7684\u6700\u5c0f\u5b50\u95ee\u9898\uff08\u53f6\u8282\u70b9\uff09\u3002\u4e4b\u540e\uff0c\u901a\u8fc7\u56de\u6eaf\u9010\u5c42\u6536\u96c6\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6784\u5efa\u51fa\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u4e0e\u4e4b\u76f8\u53cd\uff0c\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u201c\u4ece\u5e95\u81f3\u9876\u201d\u7684\u65b9\u6cd5\uff1a\u4ece\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\u5f00\u59cb\uff0c\u8fed\u4ee3\u5730\u6784\u5efa\u66f4\u5927\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u76f4\u81f3\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</p> <p>\u7531\u4e8e\u52a8\u6001\u89c4\u5212\u4e0d\u5305\u542b\u56de\u6eaf\u8fc7\u7a0b\uff0c\u56e0\u6b64\u53ea\u9700\u4f7f\u7528\u5faa\u73af\u8fed\u4ee3\u5b9e\u73b0\uff0c\u65e0\u987b\u4f7f\u7528\u9012\u5f52\u3002\u5728\u4ee5\u4e0b\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u521d\u59cb\u5316\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5b83\u8d77\u5230\u4e86\u4e0e\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u6570\u7ec4 <code>mem</code> \u76f8\u540c\u7684\u8bb0\u5f55\u4f5c\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return n\n    # \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp = [0] * (n + 1)\n    # \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1], dp[2] = 1, 2\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in range(3, n + 1):\n        dp[i] = dp[i - 1] + dp[i - 2]\n    return dp[n]\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    vector&lt;int&gt; dp(n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int[] dp = new int[n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    dp := make([]int, n+1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        dp[i] = dp[i-1] + dp[i-2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDP(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = Array(repeating: 0, count: n + 1)\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1\n    dp[2] = 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in stride(from: 3, through: n, by: 1) {\n        dp[i] = dp[i - 1] + dp[i - 2]\n    }\n    return dp[n]\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n) {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDP(n: number): number {\n    if (n === 1 || n === 2) return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    const dp = new Array(n + 1).fill(-1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (let i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n  if (n == 1 || n == 2) return n;\n  // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n  List&lt;int&gt; dp = List.filled(n + 1, 0);\n  // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n  dp[1] = 1;\n  dp[2] = 2;\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n  for (int i = 3; i &lt;= n; i++) {\n    dp[i] = dp[i - 1] + dp[i - 2];\n  }\n  return dp[n];\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp(n: usize) -&gt; i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if n == 1 || n == 2 { return n as i32; }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    let mut dp = vec![-1; n + 1];\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i in 3..=n {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    dp[n]\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDP(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    int *dp = (int *)malloc((n + 1) * sizeof(int));\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (int i = 3; i &lt;= n; i++) {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    int result = dp[n];\n    free(dp);\n    return result;\n}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u52a8\u6001\u89c4\u5212\nfn climbingStairsDP(comptime n: usize) i32 {\n    // \u5df2\u77e5 dp[1] \u548c dp[2] \uff0c\u8fd4\u56de\u4e4b\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    // \u521d\u59cb\u5316 dp \u8868\uff0c\u7528\u4e8e\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\n    var dp = [_]i32{-1} ** (n + 1);\n    // \u521d\u59cb\u72b6\u6001\uff1a\u9884\u8bbe\u6700\u5c0f\u5b50\u95ee\u9898\u7684\u89e3\n    dp[1] = 1;\n    dp[2] = 2;\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for (3..n + 1) |i| {\n        dp[i] = dp[i - 1] + dp[i - 2];\n    }\n    return dp[n];\n}\n</code></pre> <p>\u56fe 14-5 \u6a21\u62df\u4e86\u4ee5\u4e0a\u4ee3\u7801\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <p></p> <p> \u56fe 14-5 \u00a0 \u722c\u697c\u68af\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4e0e\u56de\u6eaf\u7b97\u6cd5\u4e00\u6837\uff0c\u52a8\u6001\u89c4\u5212\u4e5f\u4f7f\u7528\u201c\u72b6\u6001\u201d\u6982\u5ff5\u6765\u8868\u793a\u95ee\u9898\u6c42\u89e3\u7684\u7279\u5b9a\u9636\u6bb5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u5bf9\u5e94\u4e00\u4e2a\u5b50\u95ee\u9898\u4ee5\u53ca\u76f8\u5e94\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u4f8b\u5982\uff0c\u722c\u697c\u68af\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5f53\u524d\u6240\u5728\u697c\u68af\u9636\u6570 \\(i\\) \u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5185\u5bb9\uff0c\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u52a8\u6001\u89c4\u5212\u7684\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u5c06\u6570\u7ec4 <code>dp</code> \u79f0\u4e3a\u300c\\(dp\\) \u8868\u300d\uff0c\\(dp[i]\\) \u8868\u793a\u72b6\u6001 \\(i\\) \u5bf9\u5e94\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u5c06\u6700\u5c0f\u5b50\u95ee\u9898\u5bf9\u5e94\u7684\u72b6\u6001\uff08\u7b2c \\(1\\) \u9636\u548c\u7b2c \\(2\\) \u9636\u697c\u68af\uff09\u79f0\u4e3a\u300c\u521d\u59cb\u72b6\u6001\u300d\u3002</li> <li>\u5c06\u9012\u63a8\u516c\u5f0f \\(dp[i] = dp[i-1] + dp[i-2]\\) \u79f0\u4e3a\u300c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u300d\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/intro_to_dynamic_programming/#1414","title":"14.1.4 \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u7531\u4e8e \\(dp[i]\\) \u53ea\u4e0e \\(dp[i-1]\\) \u548c \\(dp[i-2]\\) \u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u987b\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4 <code>dp</code> \u6765\u5b58\u50a8\u6240\u6709\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u800c\u53ea\u9700\u4e24\u4e2a\u53d8\u91cf\u6eda\u52a8\u524d\u8fdb\u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig climbing_stairs_dp.py<pre><code>def climbing_stairs_dp_comp(n: int) -&gt; int:\n    \"\"\"\u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    if n == 1 or n == 2:\n        return n\n    a, b = 1, 2\n    for _ in range(3, n + 1):\n        a, b = b, a + b\n    return b\n</code></pre> climbing_stairs_dp.cpp<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.java<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.cs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint ClimbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.go<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n int) int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    a, b := 1, 2\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u4ece\u8f83\u5c0f\u5b50\u95ee\u9898\u9010\u6b65\u6c42\u89e3\u8f83\u5927\u5b50\u95ee\u9898\n    for i := 3; i &lt;= n; i++ {\n        a, b = b, a+b\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.swift<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc climbingStairsDPComp(n: Int) -&gt; Int {\n    if n == 1 || n == 2 {\n        return n\n    }\n    var a = 1\n    var b = 2\n    for _ in stride(from: 3, through: n, by: 1) {\n        (a, b) = (b, a + b)\n    }\n    return b\n}\n</code></pre> climbing_stairs_dp.js<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n) {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.ts<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction climbingStairsDPComp(n: number): number {\n    if (n === 1 || n === 2) return n;\n    let a = 1,\n        b = 2;\n    for (let i = 3; i &lt;= n; i++) {\n        const tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.dart<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n  if (n == 1 || n == 2) return n;\n  int a = 1, b = 2;\n  for (int i = 3; i &lt;= n; i++) {\n    int tmp = b;\n    b = a + b;\n    a = tmp;\n  }\n  return b;\n}\n</code></pre> climbing_stairs_dp.rs<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn climbing_stairs_dp_comp(n: usize) -&gt; i32 {\n    if n == 1 || n == 2 { return n as i32; }\n    let (mut a, mut b) = (1, 2);\n    for _ in 3..=n {\n        let tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    b\n}\n</code></pre> climbing_stairs_dp.c<pre><code>/* \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint climbingStairsDPComp(int n) {\n    if (n == 1 || n == 2)\n        return n;\n    int a = 1, b = 2;\n    for (int i = 3; i &lt;= n; i++) {\n        int tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> climbing_stairs_dp.zig<pre><code>// \u722c\u697c\u68af\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn climbingStairsDPComp(comptime n: usize) i32 {\n    if (n == 1 or n == 2) {\n        return @intCast(n);\n    }\n    var a: i32 = 1;\n    var b: i32 = 2;\n    for (3..n + 1) |_| {\n        var tmp = b;\n        b = a + b;\n        a = tmp;\n    }\n    return b;\n}\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u4ee3\u7801\uff0c\u7531\u4e8e\u7701\u53bb\u4e86\u6570\u7ec4 <code>dp</code> \u5360\u7528\u7684\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</p> <p>\u5728\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e2d\uff0c\u5f53\u524d\u72b6\u6001\u5f80\u5f80\u4ec5\u4e0e\u524d\u9762\u6709\u9650\u4e2a\u72b6\u6001\u6709\u5173\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u53ea\u4fdd\u7559\u5fc5\u8981\u7684\u72b6\u6001\uff0c\u901a\u8fc7\u201c\u964d\u7ef4\u201d\u6765\u8282\u7701\u5185\u5b58\u7a7a\u95f4\u3002\u8fd9\u79cd\u7a7a\u95f4\u4f18\u5316\u6280\u5de7\u88ab\u79f0\u4e3a\u201c\u6eda\u52a8\u53d8\u91cf\u201d\u6216\u201c\u6eda\u52a8\u6570\u7ec4\u201d\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/","title":"14.4 \u00a0 0-1 \u80cc\u5305\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u975e\u5e38\u597d\u7684\u52a8\u6001\u89c4\u5212\u5165\u95e8\u9898\u76ee\uff0c\u662f\u52a8\u6001\u89c4\u5212\u4e2d\u6700\u5e38\u89c1\u7684\u95ee\u9898\u5f62\u5f0f\u3002\u5176\u5177\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982 0-1 \u80cc\u5305\u95ee\u9898\u3001\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u3001\u591a\u91cd\u80cc\u5305\u95ee\u9898\u7b49\u3002</p> <p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6765\u6c42\u89e3\u6700\u5e38\u89c1\u7684 0-1 \u80cc\u5305\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u89c2\u5bdf\u56fe 14-17 \uff0c\u7531\u4e8e\u7269\u54c1\u7f16\u53f7 \\(i\\) \u4ece \\(1\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u6570\u7ec4\u7d22\u5f15\u4ece \\(0\\) \u5f00\u59cb\u8ba1\u6570\uff0c\u56e0\u6b64\u7269\u54c1 \\(i\\) \u5bf9\u5e94\u91cd\u91cf \\(wgt[i-1]\\) \u548c\u4ef7\u503c \\(val[i-1]\\) \u3002</p> <p></p> <p> \u56fe 14-17 \u00a0 0-1 \u80cc\u5305\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06 0-1 \u80cc\u5305\u95ee\u9898\u770b\u4f5c\u4e00\u4e2a\u7531 \\(n\\) \u8f6e\u51b3\u7b56\u7ec4\u6210\u7684\u8fc7\u7a0b\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u4f53\u90fd\u6709\u4e0d\u653e\u5165\u548c\u653e\u5165\u4e24\u79cd\u51b3\u7b56\uff0c\u56e0\u6b64\u8be5\u95ee\u9898\u6ee1\u8db3\u51b3\u7b56\u6811\u6a21\u578b\u3002</p> <p>\u8be5\u95ee\u9898\u7684\u76ee\u6807\u662f\u6c42\u89e3\u201c\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u201d\uff0c\u56e0\u6b64\u8f83\u5927\u6982\u7387\u662f\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u95ee\u9898\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\u6765\u8bf4\uff0c\u4e0d\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff1b\u653e\u5165\u80cc\u5305\uff0c\u80cc\u5305\u5bb9\u91cf\u51cf\u5c0f\u3002\u7531\u6b64\u53ef\u5f97\u72b6\u6001\u5b9a\u4e49\uff1a\u5f53\u524d\u7269\u54c1\u7f16\u53f7 \\(i\\) \u548c\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u8bb0\u4e3a \\([i, c]\\) \u3002</p> <p>\u72b6\u6001 \\([i, c]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\uff0c\u8bb0\u4e3a \\(dp[i, c]\\) \u3002</p> <p>\u5f85\u6c42\u89e3\u7684\u662f \\(dp[n, cap]\\) \uff0c\u56e0\u6b64\u9700\u8981\u4e00\u4e2a\u5c3a\u5bf8\u4e3a \\((n+1) \\times (cap+1)\\) \u7684\u4e8c\u7ef4 \\(dp\\) \u8868\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u5f53\u6211\u4eec\u505a\u51fa\u7269\u54c1 \\(i\\) \u7684\u51b3\u7b56\u540e\uff0c\u5269\u4f59\u7684\u662f\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u7684\u51b3\u7b56\uff0c\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u4e0d\u53d8\uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u80cc\u5305\u5bb9\u91cf\u51cf\u5c11 \\(wgt[i-1]\\) \uff0c\u4ef7\u503c\u589e\u52a0 \\(val[i-1]\\) \uff0c\u72b6\u6001\u53d8\u5316\u4e3a \\([i-1, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4e0a\u8ff0\u5206\u6790\u5411\u6211\u4eec\u63ed\u793a\u4e86\u672c\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u6700\u5927\u4ef7\u503c \\(dp[i, c]\\) \u7b49\u4e8e\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \u548c\u653e\u5165\u7269\u54c1 \\(i\\) \u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\u3002\u7531\u6b64\u53ef\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i-1, c - wgt[i-1]] + val[i-1]) \\] <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf \\(wgt[i - 1]\\) \u8d85\u51fa\u5269\u4f59\u80cc\u5305\u5bb9\u91cf \\(c\\) \uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u65e0\u7269\u54c1\u6216\u65e0\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u65f6\u6700\u5927\u4ef7\u503c\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217 \\(dp[i, 0]\\) \u548c\u9996\u884c \\(dp[0, c]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u524d\u72b6\u6001 \\([i, c]\\) \u4ece\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c]\\) \u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001 \\([i-1, c-wgt[i-1]]\\) \u8f6c\u79fb\u800c\u6765\uff0c\u56e0\u6b64\u901a\u8fc7\u4e24\u5c42\u5faa\u73af\u6b63\u5e8f\u904d\u5386\u6574\u4e2a \\(dp\\) \u8868\u5373\u53ef\u3002</p> <p>\u6839\u636e\u4ee5\u4e0a\u5206\u6790\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u6309\u987a\u5e8f\u5b9e\u73b0\u66b4\u529b\u641c\u7d22\u3001\u8bb0\u5fc6\u5316\u641c\u7d22\u3001\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u3002</p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#1","title":"1. \u00a0 \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u641c\u7d22","text":"<p>\u641c\u7d22\u4ee3\u7801\u5305\u542b\u4ee5\u4e0b\u8981\u7d20\u3002</p> <ul> <li>\u9012\u5f52\u53c2\u6570\uff1a\u72b6\u6001 \\([i, c]\\) \u3002</li> <li>\u8fd4\u56de\u503c\uff1a\u5b50\u95ee\u9898\u7684\u89e3 \\(dp[i, c]\\) \u3002</li> <li>\u7ec8\u6b62\u6761\u4ef6\uff1a\u5f53\u7269\u54c1\u7f16\u53f7\u8d8a\u754c \\(i = 0\\) \u6216\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\u4e3a \\(0\\) \u65f6\uff0c\u7ec8\u6b62\u9012\u5f52\u5e76\u8fd4\u56de\u4ef7\u503c \\(0\\) \u3002</li> <li>\u526a\u679d\uff1a\u82e5\u5f53\u524d\u7269\u54c1\u91cd\u91cf\u8d85\u51fa\u80cc\u5305\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dfs(wgt: list[int], val: list[int], i: int, c: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\"\"\"\n    # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs(wgt, val, i - 1, c)\n    # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no = knapsack_dfs(wgt, val, i - 1, c)\n    yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes)\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes);\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int[] wgt, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint KnapsackDFS(int[] weight, int[] val, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFS(weight, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = KnapsackDFS(weight, val, i - 1, c);\n    int yes = KnapsackDFS(weight, val, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.Max(no, yes);\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt, val []int, i, c int) int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFS(wgt, val, i-1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no := knapsackDFS(wgt, val, i-1, c)\n    yes := knapsackDFS(wgt, val, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return int(math.Max(float64(no), float64(yes)))\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunc knapsackDFS(wgt: [Int], val: [Int], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c)\n    let yes = knapsackDFS(wgt: wgt, val: val, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return max(no, yes)\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(wgt, val, i, c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfunction knapsackDFS(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFS(wgt, val, i - 1, c);\n    const yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return Math.max(no, yes);\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(List&lt;int&gt; wgt, List&lt;int&gt; val, int i, int c) {\n  // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFS(wgt, val, i - 1, c);\n  }\n  // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n  int no = knapsackDFS(wgt, val, i - 1, c);\n  int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n  return max(no, yes);\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nfn knapsack_dfs(wgt: &amp;[i32], val: &amp;[i32], i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsack_dfs(wgt, val, i - 1, c);\n    let yes = knapsack_dfs(wgt, val, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    std::cmp::max(no, yes)\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22 */\nint knapsackDFS(int wgt[], int val[], int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFS(wgt, val, i - 1, c);\n    int yes = knapsackDFS(wgt, val, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return myMax(no, yes);\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u66b4\u529b\u641c\u7d22\nfn knapsackDFS(wgt: []i32, val: []i32, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFS(wgt, val, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    var no = knapsackDFS(wgt, val, i - 1, c);\n    var yes = knapsackDFS(wgt, val, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    return @max(no, yes);\n}\n</code></pre> <p>\u5982\u56fe 14-18 \u6240\u793a\uff0c\u7531\u4e8e\u6bcf\u4e2a\u7269\u54c1\u90fd\u4f1a\u4ea7\u751f\u4e0d\u9009\u548c\u9009\u4e24\u6761\u641c\u7d22\u5206\u652f\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(2^n)\\) \u3002</p> <p>\u89c2\u5bdf\u9012\u5f52\u6811\uff0c\u5bb9\u6613\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u4f8b\u5982 \\(dp[1, 10]\\) \u7b49\u3002\u800c\u5f53\u7269\u54c1\u8f83\u591a\u3001\u80cc\u5305\u5bb9\u91cf\u8f83\u5927\uff0c\u5c24\u5176\u662f\u76f8\u540c\u91cd\u91cf\u7684\u7269\u54c1\u8f83\u591a\u65f6\uff0c\u91cd\u53e0\u5b50\u95ee\u9898\u7684\u6570\u91cf\u5c06\u4f1a\u5927\u5e45\u589e\u591a\u3002</p> <p></p> <p> \u56fe 14-18 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u66b4\u529b\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#2","title":"2. \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22","text":"<p>\u4e3a\u4e86\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\uff0c\u6211\u4eec\u501f\u52a9\u8bb0\u5fc6\u5217\u8868 <code>mem</code> \u6765\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u5176\u4e2d <code>mem[i][c]</code> \u5bf9\u5e94 \\(dp[i, c]\\) \u3002</p> <p>\u5f15\u5165\u8bb0\u5fc6\u5316\u4e4b\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5b50\u95ee\u9898\u6570\u91cf\uff0c\u4e5f\u5c31\u662f \\(O(n \\times cap)\\) \u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dfs_mem(\n    wgt: list[int], val: list[int], mem: list[list[int]], i: int, c: int\n) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\"\"\"\n    # \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 or c == 0:\n        return 0\n    # \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1:\n        return mem[i][c]\n    # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c:\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    # \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no = knapsack_dfs_mem(wgt, val, mem, i - 1, c)\n    yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1]\n    # \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, vector&lt;vector&lt;int&gt;&gt; &amp;mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int[] wgt, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint KnapsackDFSMem(int[] weight, int[] val, int[][] mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (weight[i - 1] &gt; c) {\n        return KnapsackDFSMem(weight, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = KnapsackDFSMem(weight, val, mem, i - 1, c);\n    int yes = KnapsackDFSMem(weight, val, mem, i - 1, c - weight[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.Max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt, val []int, mem [][]int, i, c int) int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i-1] &gt; c {\n        return knapsackDFSMem(wgt, val, mem, i-1, c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    no := knapsackDFSMem(wgt, val, mem, i-1, c)\n    yes := knapsackDFSMem(wgt, val, mem, i-1, c-wgt[i-1]) + val[i-1]\n    // \u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = int(math.Max(float64(no), float64(yes)))\n    return mem[i][c]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunc knapsackDFSMem(wgt: [Int], val: [Int], mem: inout [[Int]], i: Int, c: Int) -&gt; Int {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c]\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c {\n        return knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c)\n    let yes = knapsackDFSMem(wgt: wgt, val: val, mem: &amp;mem, i: i - 1, c: c - wgt[i - 1]) + val[i - 1]\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = max(no, yes)\n    return mem[i][c]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(wgt, val, mem, i, c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfunction knapsackDFSMem(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    mem: Array&lt;Array&lt;number&gt;&gt;,\n    i: number,\n    c: number\n): number {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i === 0 || c === 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] !== -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    const no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    const yes =\n        knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = Math.max(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(\n  List&lt;int&gt; wgt,\n  List&lt;int&gt; val,\n  List&lt;List&lt;int&gt;&gt; mem,\n  int i,\n  int c,\n) {\n  // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n  if (i == 0 || c == 0) {\n    return 0;\n  }\n  // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n  if (mem[i][c] != -1) {\n    return mem[i][c];\n  }\n  // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n  if (wgt[i - 1] &gt; c) {\n    return knapsackDFSMem(wgt, val, mem, i - 1, c);\n  }\n  // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n  int no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n  int yes = knapsackDFSMem(wgt, val, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n  // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n  mem[i][c] = max(no, yes);\n  return mem[i][c];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nfn knapsack_dfs_mem(wgt: &amp;[i32], val: &amp;[i32], mem: &amp;mut Vec&lt;Vec&lt;i32&gt;&gt;, i: usize, c: usize) -&gt; i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if i == 0 || c == 0 {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if mem[i][c] != -1 {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if wgt[i - 1] &gt; c as i32 {\n        return knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    let no = knapsack_dfs_mem(wgt, val, mem, i - 1, c);\n    let yes = knapsack_dfs_mem(wgt, val, mem, i - 1, c - wgt[i - 1] as usize) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = std::cmp::max(no, yes);\n    mem[i][c]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22 */\nint knapsackDFSMem(int wgt[], int val[], int memCols, int **mem, int i, int c) {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 || c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    int no = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c);\n    int yes = knapsackDFSMem(wgt, val, memCols, mem, i - 1, c - wgt[i - 1]) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = myMax(no, yes);\n    return mem[i][c];\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u8bb0\u5fc6\u5316\u641c\u7d22\nfn knapsackDFSMem(wgt: []i32, val: []i32, mem: anytype, i: usize, c: usize) i32 {\n    // \u82e5\u5df2\u9009\u5b8c\u6240\u6709\u7269\u54c1\u6216\u80cc\u5305\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u5219\u8fd4\u56de\u4ef7\u503c 0\n    if (i == 0 or c == 0) {\n        return 0;\n    }\n    // \u82e5\u5df2\u6709\u8bb0\u5f55\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (mem[i][c] != -1) {\n        return mem[i][c];\n    }\n    // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u53ea\u80fd\u9009\u62e9\u4e0d\u653e\u5165\u80cc\u5305\n    if (wgt[i - 1] &gt; c) {\n        return knapsackDFSMem(wgt, val, mem, i - 1, c);\n    }\n    // \u8ba1\u7b97\u4e0d\u653e\u5165\u548c\u653e\u5165\u7269\u54c1 i \u7684\u6700\u5927\u4ef7\u503c\n    var no = knapsackDFSMem(wgt, val, mem, i - 1, c);\n    var yes = knapsackDFSMem(wgt, val, mem, i - 1, c - @as(usize, @intCast(wgt[i - 1]))) + val[i - 1];\n    // \u8bb0\u5f55\u5e76\u8fd4\u56de\u4e24\u79cd\u65b9\u6848\u4e2d\u4ef7\u503c\u66f4\u5927\u7684\u90a3\u4e00\u4e2a\n    mem[i][c] = @max(no, yes);\n    return mem[i][c];\n}\n</code></pre> <p>\u56fe 14-19 \u5c55\u793a\u4e86\u5728\u8bb0\u5fc6\u5316\u641c\u7d22\u4e2d\u88ab\u526a\u6389\u7684\u641c\u7d22\u5206\u652f\u3002</p> <p></p> <p> \u56fe 14-19 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u8bb0\u5fc6\u5316\u641c\u7d22\u9012\u5f52\u6811 </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#3","title":"3. \u00a0 \u65b9\u6cd5\u4e09\uff1a\u52a8\u6001\u89c4\u5212","text":"<p>\u52a8\u6001\u89c4\u5212\u5b9e\u8d28\u4e0a\u5c31\u662f\u5728\u72b6\u6001\u8f6c\u79fb\u4e2d\u586b\u5145 \\(dp\\) \u8868\u7684\u8fc7\u7a0b\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint KnapsackDP(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c - weight[i - 1]] + val[i - 1], dp[i - 1, c]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i-1][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in stride(from: 1, through: n, by: 1) {\n        for c in stride(from: 1, through: cap, by: 1) {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(n + 1)\n        .fill(0)\n        .map(() =&gt; Array(cap + 1).fill(0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i - 1][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i - 1][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[n][cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint knapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i - 1][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn knapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i - 1][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> <p>\u5982\u56fe 14-20 \u6240\u793a\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u7531\u6570\u7ec4 <code>dp</code> \u5927\u5c0f\u51b3\u5b9a\uff0c\u5373 \\(O(n \\times cap)\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-20 \u00a0 0-1 \u80cc\u5305\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/knapsack_problem/#4","title":"4. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u90fd\u53ea\u4e0e\u5176\u4e0a\u4e00\u884c\u7684\u72b6\u6001\u6709\u5173\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6570\u7ec4\u6eda\u52a8\u524d\u8fdb\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \u3002</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\uff0c\u6211\u4eec\u80fd\u5426\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u5b9e\u73b0\u7a7a\u95f4\u4f18\u5316\u5462\uff1f\u89c2\u5bdf\u53ef\u77e5\uff0c\u6bcf\u4e2a\u72b6\u6001\u90fd\u662f\u7531\u6b63\u4e0a\u65b9\u6216\u5de6\u4e0a\u65b9\u7684\u683c\u5b50\u8f6c\u79fb\u8fc7\u6765\u7684\u3002\u5047\u8bbe\u53ea\u6709\u4e00\u4e2a\u6570\u7ec4\uff0c\u5f53\u5f00\u59cb\u904d\u5386\u7b2c \\(i\\) \u884c\u65f6\uff0c\u8be5\u6570\u7ec4\u5b58\u50a8\u7684\u4ecd\u7136\u662f\u7b2c \\(i-1\\) \u884c\u7684\u72b6\u6001\u3002</p> <ul> <li>\u5982\u679c\u91c7\u53d6\u6b63\u5e8f\u904d\u5386\uff0c\u90a3\u4e48\u904d\u5386\u5230 \\(dp[i, j]\\) \u65f6\uff0c\u5de6\u4e0a\u65b9 \\(dp[i-1, 1]\\) ~ \\(dp[i-1, j-1]\\) \u503c\u53ef\u80fd\u5df2\u7ecf\u88ab\u8986\u76d6\uff0c\u6b64\u65f6\u5c31\u65e0\u6cd5\u5f97\u5230\u6b63\u786e\u7684\u72b6\u6001\u8f6c\u79fb\u7ed3\u679c\u3002</li> <li>\u5982\u679c\u91c7\u53d6\u5012\u5e8f\u904d\u5386\uff0c\u5219\u4e0d\u4f1a\u53d1\u751f\u8986\u76d6\u95ee\u9898\uff0c\u72b6\u6001\u8f6c\u79fb\u53ef\u4ee5\u6b63\u786e\u8fdb\u884c\u3002</li> </ul> <p>\u56fe 14-21 \u5c55\u793a\u4e86\u5728\u5355\u4e2a\u6570\u7ec4\u4e0b\u4ece\u7b2c \\(i = 1\\) \u884c\u8f6c\u6362\u81f3\u7b2c \\(i = 2\\) \u884c\u7684\u8fc7\u7a0b\u3002\u8bf7\u601d\u8003\u6b63\u5e8f\u904d\u5386\u548c\u5012\u5e8f\u904d\u5386\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-21 \u00a0 0-1 \u80cc\u5305\u7684\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4 \\(i\\) \u76f4\u63a5\u5220\u9664\uff0c\u5e76\u4e14\u628a\u5185\u5faa\u73af\u66f4\u6539\u4e3a\u5012\u5e8f\u904d\u5386\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig knapsack.py<pre><code>def knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u5012\u5e8f\u904d\u5386\n        for c in range(cap, 0, -1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> knapsack.cpp<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.java<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.cs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint KnapsackDPComp(int[] weight, int[] val, int cap) {\n    int n = weight.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt; 0; c--) {\n            if (weight[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - weight[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.go<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for c := cap; c &gt;= 1; c-- {\n            if wgt[i-1] &lt;= c {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.swift<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc knapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in stride(from: 1, through: n, by: 1) {\n        // \u5012\u5e8f\u904d\u5386\n        for c in stride(from: cap, through: 1, by: -1) {\n            if wgt[i - 1] &lt;= c {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> knapsack.js<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.ts<pre><code>/* 0-1 \u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction knapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array(cap + 1).fill(0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (let c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> knapsack.dart<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    // \u5012\u5e8f\u904d\u5386\n    for (int c = cap; c &gt;= 1; c--) {\n      if (wgt[i - 1] &lt;= c) {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> knapsack.rs<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        // \u5012\u5e8f\u904d\u5386\n        for c in (1..=cap).rev() {\n            if wgt[i - 1] &lt;= c as i32 {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> knapsack.c<pre><code>/* 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint knapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        // \u5012\u5e8f\u904d\u5386\n        for (int c = cap; c &gt;= 1; c--) {\n            if (wgt[i - 1] &lt;= c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> knapsack.zig<pre><code>// 0-1 \u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn knapsackDPComp(wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        // \u5012\u5e8f\u904d\u5386\n        var c = cap;\n        while (c &gt; 0) : (c -= 1) {\n            if (wgt[i - 1] &lt; c) {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/summary/","title":"14.7 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u52a8\u6001\u89c4\u5212\u5bf9\u95ee\u9898\u8fdb\u884c\u5206\u89e3\uff0c\u5e76\u901a\u8fc7\u5b58\u50a8\u5b50\u95ee\u9898\u7684\u89e3\u6765\u89c4\u907f\u91cd\u590d\u8ba1\u7b97\uff0c\u63d0\u9ad8 \u8ba1\u7b97\u6548\u7387\u3002</li> <li>\u4e0d\u8003\u8651\u65f6\u95f4\u7684\u524d\u63d0\u4e0b\uff0c\u6240\u6709\u52a8\u6001\u89c4\u5212\u95ee\u9898\u90fd\u53ef\u4ee5\u7528\u56de\u6eaf\uff08\u66b4\u529b\u641c\u7d22\uff09\u8fdb\u884c\u6c42\u89e3\uff0c\u4f46\u9012\u5f52\u6811\u4e2d\u5b58\u5728\u5927\u91cf\u7684\u91cd\u53e0\u5b50\u95ee\u9898\uff0c\u6548\u7387\u6781\u4f4e\u3002\u901a\u8fc7\u5f15\u5165\u8bb0\u5fc6\u5316\u5217\u8868\uff0c\u53ef\u4ee5\u5b58\u50a8\u6240\u6709\u8ba1\u7b97\u8fc7\u7684\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u4ece\u800c\u4fdd\u8bc1\u91cd\u53e0\u5b50\u95ee\u9898\u53ea\u88ab\u8ba1\u7b97\u4e00\u6b21\u3002</li> <li>\u8bb0\u5fc6\u5316\u9012\u5f52\u662f\u4e00\u79cd\u4ece\u9876\u81f3\u5e95\u7684\u9012\u5f52\u5f0f\u89e3\u6cd5\uff0c\u800c\u4e0e\u4e4b\u5bf9\u5e94\u7684\u52a8\u6001\u89c4\u5212\u662f\u4e00\u79cd\u4ece\u5e95\u81f3\u9876\u7684\u9012\u63a8\u5f0f\u89e3\u6cd5\uff0c\u5176\u5982\u540c\u201c\u586b\u5199\u8868\u683c\u201d\u4e00\u6837\u3002\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u4ec5\u4f9d\u8d56\u67d0\u4e9b\u5c40\u90e8\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u6d88\u9664 \\(dp\\) \u8868\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff0c\u4ece\u800c\u964d\u4f4e\u7a7a\u95f4\u590d\u6742\u5ea6\u3002</li> <li>\u5b50\u95ee\u9898\u5206\u89e3\u662f\u4e00\u79cd\u901a\u7528\u7684\u7b97\u6cd5\u601d\u8def\uff0c\u5728\u5206\u6cbb\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u4e2d\u5177\u6709\u4e0d\u540c\u7684\u6027\u8d28\u3002</li> <li>\u52a8\u6001\u89c4\u5212\u95ee\u9898\u6709\u4e09\u5927\u7279\u6027\uff1a\u91cd\u53e0\u5b50\u95ee\u9898\u3001\u6700\u4f18\u5b50\u7ed3\u6784\u3001\u65e0\u540e\u6548\u6027\u3002</li> <li>\u5982\u679c\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u4ece\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u5efa\u5f97\u6765\uff0c\u5219\u5b83\u5c31\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u3002</li> <li>\u65e0\u540e\u6548\u6027\u6307\u5bf9\u4e8e\u4e00\u4e2a\u72b6\u6001\uff0c\u5176\u672a\u6765\u53d1\u5c55\u53ea\u4e0e\u8be5\u72b6\u6001\u6709\u5173\uff0c\u800c\u4e0e\u8fc7\u53bb\u7ecf\u5386\u7684\u6240\u6709\u72b6\u6001\u65e0\u5173\u3002\u8bb8\u591a\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u4e0d\u5177\u6709\u65e0\u540e\u6548\u6027\uff0c\u65e0\u6cd5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u5feb\u901f\u6c42\u89e3\u3002</li> </ul> <p>\u80cc\u5305\u95ee\u9898</p> <ul> <li>\u80cc\u5305\u95ee\u9898\u662f\u6700\u5178\u578b\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\u4e4b\u4e00\uff0c\u5177\u6709 0-1 \u80cc\u5305\u3001\u5b8c\u5168\u80cc\u5305\u3001\u591a\u91cd\u80cc\u5305\u7b49\u53d8\u79cd\u3002</li> <li>0-1 \u80cc\u5305\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u524d \\(i\\) \u4e2a\u7269\u54c1\u5728\u5269\u4f59\u5bb9\u91cf\u4e3a \\(c\\) \u7684\u80cc\u5305\u4e2d\u7684\u6700\u5927\u4ef7\u503c\u3002\u6839\u636e\u4e0d\u653e\u5165\u80cc\u5305\u548c\u653e\u5165\u80cc\u5305\u4e24\u79cd\u51b3\u7b56\uff0c\u53ef\u5f97\u5230\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u5e76\u6784\u5efa\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\uff0c\u7531\u4e8e\u6bcf\u4e2a\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u9700\u8981\u5012\u5e8f\u904d\u5386\u5217\u8868\uff0c\u907f\u514d\u5de6\u4e0a\u65b9\u72b6\u6001\u88ab\u8986\u76d6\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u6bcf\u79cd\u7269\u54c1\u7684\u9009\u53d6\u6570\u91cf\u65e0\u9650\u5236\uff0c\u56e0\u6b64\u9009\u62e9\u653e\u5165\u7269\u54c1\u7684\u72b6\u6001\u8f6c\u79fb\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\u3002\u7531\u4e8e\u72b6\u6001\u4f9d\u8d56\u6b63\u4e0a\u65b9\u548c\u6b63\u5de6\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u5728\u7a7a\u95f4\u4f18\u5316\u4e2d\u5e94\u5f53\u6b63\u5e8f\u904d\u5386\u3002</li> <li>\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u4e2a\u53d8\u79cd\u3002\u5b83\u4ece\u6c42\u201c\u6700\u5927\u201d\u4ef7\u503c\u53d8\u4e3a\u6c42\u201c\u6700\u5c0f\u201d\u786c\u5e01\u6570\u91cf\uff0c\u56e0\u6b64\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\max()\\) \u5e94\u6539\u4e3a \\(\\min()\\) \u3002\u4ece\u8ffd\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u5230\u8ffd\u6c42\u201c\u6070\u597d\u201d\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u4f7f\u7528 \\(amt + 1\\) \u6765\u8868\u793a\u201c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u201d\u7684\u65e0\u6548\u89e3\u3002</li> <li>\u96f6\u94b1\u5151\u6362 II \u95ee\u9898\u4ece\u6c42\u201c\u6700\u5c11\u786c\u5e01\u6570\u91cf\u201d\u6539\u4e3a\u6c42\u201c\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u201d\uff0c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u76f8\u5e94\u5730\u4ece \\(\\min()\\) \u6539\u4e3a\u6c42\u548c\u8fd0\u7b97\u7b26\u3002</li> </ul> <p>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898</p> <ul> <li>\u7f16\u8f91\u8ddd\u79bb\uff08Levenshtein \u8ddd\u79bb\uff09\u7528\u4e8e\u8861\u91cf\u4e24\u4e2a\u5b57\u7b26\u4e32\u4e4b\u95f4\u7684\u76f8\u4f3c\u5ea6\uff0c\u5176\u5b9a\u4e49\u4e3a\u4ece\u4e00\u4e2a\u5b57\u7b26\u4e32\u5230\u53e6\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\uff0c\u7f16\u8f91\u64cd\u4f5c\u5305\u62ec\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\u3002</li> <li>\u7f16\u8f91\u8ddd\u79bb\u95ee\u9898\u7684\u72b6\u6001\u5b9a\u4e49\u4e3a\u5c06 \\(s\\) \u7684\u524d \\(i\\) \u4e2a\u5b57\u7b26\u66f4\u6539\u4e3a \\(t\\) \u7684\u524d \\(j\\) \u4e2a\u5b57\u7b26\u6240\u9700\u7684\u6700\u5c11\u7f16\u8f91\u6b65\u6570\u3002\u5f53 \\(s[i] \\ne t[j]\\) \u65f6\uff0c\u5177\u6709\u4e09\u79cd\u51b3\u7b56\uff1a\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66ff\u6362\uff0c\u5b83\u4eec\u90fd\u6709\u76f8\u5e94\u7684\u5269\u4f59\u5b50\u95ee\u9898\u3002\u636e\u6b64\u4fbf\u53ef\u4ee5\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\u4e0e\u6784\u5efa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u3002\u800c\u5f53 \\(s[i] = t[j]\\) \u65f6\uff0c\u65e0\u987b\u7f16\u8f91\u5f53\u524d\u5b57\u7b26\u3002</li> <li>\u5728\u7f16\u8f91\u8ddd\u79bb\u4e2d\uff0c\u72b6\u6001\u4f9d\u8d56\u5176\u6b63\u4e0a\u65b9\u3001\u6b63\u5de6\u65b9\u3001\u5de6\u4e0a\u65b9\u7684\u72b6\u6001\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u6b63\u5e8f\u6216\u5012\u5e8f\u904d\u5386\u90fd\u65e0\u6cd5\u6b63\u786e\u5730\u8fdb\u884c\u72b6\u6001\u8f6c\u79fb\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u5229\u7528\u4e00\u4e2a\u53d8\u91cf\u6682\u5b58\u5de6\u4e0a\u65b9\u72b6\u6001\uff0c\u4ece\u800c\u8f6c\u5316\u5230\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7b49\u4ef7\u7684\u60c5\u51b5\uff0c\u53ef\u4ee5\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</li> </ul>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/","title":"14.5 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898","text":"<p>\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5148\u6c42\u89e3\u53e6\u4e00\u4e2a\u5e38\u89c1\u7684\u80cc\u5305\u95ee\u9898\uff1a\u5b8c\u5168\u80cc\u5305\uff0c\u518d\u4e86\u89e3\u5b83\u7684\u4e00\u79cd\u7279\u4f8b\uff1a\u96f6\u94b1\u5151\u6362\u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1451","title":"14.5.1 \u00a0 \u5b8c\u5168\u80cc\u5305","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80fd\u653e\u5165\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 14-22 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-22 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\uff0c\u533a\u522b\u4ec5\u5728\u4e8e\u4e0d\u9650\u5236\u7269\u54c1\u7684\u9009\u62e9\u6b21\u6570\u3002</p> <ul> <li>\u5728 0-1 \u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u53ea\u6709\u4e00\u4e2a\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u53ea\u80fd\u4ece\u524d \\(i-1\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> <li>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e2d\uff0c\u6bcf\u79cd\u7269\u54c1\u7684\u6570\u91cf\u662f\u65e0\u9650\u7684\uff0c\u56e0\u6b64\u5c06\u7269\u54c1 \\(i\\) \u653e\u5165\u80cc\u5305\u540e\uff0c\u4ecd\u53ef\u4ee5\u4ece\u524d \\(i\\) \u4e2a\u7269\u54c1\u4e2d\u9009\u62e9\u3002</li> </ul> <p>\u5728\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u89c4\u5b9a\u4e0b\uff0c\u72b6\u6001 \\([i, c]\\) \u7684\u53d8\u5316\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u4e0d\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u76f8\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i-1, c]\\) \u3002</li> <li>\u653e\u5165\u7269\u54c1 \\(i\\) \uff1a\u4e0e 0-1 \u80cc\u5305\u95ee\u9898\u4e0d\u540c\uff0c\u8f6c\u79fb\u81f3 \\([i, c-wgt[i-1]]\\) \u3002</li> </ul> <p>\u4ece\u800c\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u53d8\u4e3a\uff1a</p> \\[ dp[i, c] = \\max(dp[i-1, c], dp[i, c - wgt[i-1]] + val[i-1]) \\]"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5bf9\u6bd4\u4e24\u9053\u9898\u76ee\u7684\u4ee3\u7801\uff0c\u72b6\u6001\u8f6c\u79fb\u4e2d\u6709\u4e00\u5904\u4ece \\(i-1\\) \u53d8\u4e3a \\(i\\) \uff0c\u5176\u4f59\u5b8c\u5168\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (cap + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n    return dp[n][cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(cap + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDP(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i, c] = dp[i - 1, c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i, c] = Math.Max(dp[i - 1, c], dp[i, c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n, cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, cap+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i-1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = int(math.Max(float64(dp[i-1][c]), float64(dp[i][c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDP(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: cap + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in stride(from: 1, through: n, by: 1) {\n        for c in stride(from: 1, through: cap, by: 1) {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[n][cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDP(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: cap + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = Math.max(\n                    dp[i - 1][c],\n                    dp[i][c - wgt[i - 1]] + val[i - 1]\n                );\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(cap + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[i][c] = dp[i - 1][c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[i][c] = max(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; cap + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = std::cmp::max(dp[i - 1][c], dp[i][c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDP(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(cap + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = myMax(dp[i - 1][c], dp[i][c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[n][cap];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    return res;\n}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDP(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][cap + 1]i32{[_]i32{0} ** (cap + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[i][c] = dp[i - 1][c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[i][c] = @max(dp[i - 1][c], dp[i][c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[n][cap];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7531\u4e8e\u5f53\u524d\u72b6\u6001\u662f\u4ece\u5de6\u8fb9\u548c\u4e0a\u8fb9\u7684\u72b6\u6001\u8f6c\u79fb\u800c\u6765\u7684\uff0c\u56e0\u6b64\u7a7a\u95f4\u4f18\u5316\u540e\u5e94\u8be5\u5bf9 \\(dp\\) \u8868\u4e2d\u7684\u6bcf\u4e00\u884c\u8fdb\u884c\u6b63\u5e8f\u904d\u5386\u3002</p> <p>\u8fd9\u4e2a\u904d\u5386\u987a\u5e8f\u4e0e 0-1 \u80cc\u5305\u6b63\u597d\u76f8\u53cd\u3002\u8bf7\u501f\u52a9\u56fe 14-23 \u6765\u7406\u89e3\u4e24\u8005\u7684\u533a\u522b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-23 \u00a0 \u5b8c\u5168\u80cc\u5305\u95ee\u9898\u5728\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p> <p>\u4ee3\u7801\u5b9e\u73b0\u6bd4\u8f83\u7b80\u5355\uff0c\u4ec5\u9700\u5c06\u6570\u7ec4 <code>dp</code> \u7684\u7b2c\u4e00\u7ef4\u5220\u9664\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig unbounded_knapsack.py<pre><code>def unbounded_knapsack_dp_comp(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(wgt)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (cap + 1)\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for c in range(1, cap + 1):\n            if wgt[i - 1] &gt; c:\n                # \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n    return dp[cap]\n</code></pre> unbounded_knapsack.cpp<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    int n = wgt.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(cap + 1, 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.java<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.cs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint UnboundedKnapsackDPComp(int[] wgt, int[] val, int cap) {\n    int n = wgt.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.Max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.go<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt, val []int, cap int) int {\n    n := len(wgt)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, cap+1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        for c := 1; c &lt;= cap; c++ {\n            if wgt[i-1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = int(math.Max(float64(dp[c]), float64(dp[c-wgt[i-1]]+val[i-1])))\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.swift<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc unboundedKnapsackDPComp(wgt: [Int], val: [Int], cap: Int) -&gt; Int {\n    let n = wgt.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: cap + 1)\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in stride(from: 1, through: n, by: 1) {\n        for c in stride(from: 1, through: cap, by: 1) {\n            if wgt[i - 1] &gt; c {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1])\n            }\n        }\n    }\n    return dp[cap]\n}\n</code></pre> unbounded_knapsack.js<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(wgt, val, cap) {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.ts<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction unboundedKnapsackDPComp(\n    wgt: Array&lt;number&gt;,\n    val: Array&lt;number&gt;,\n    cap: number\n): number {\n    const n = wgt.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: cap + 1 }, () =&gt; 0);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = Math.max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre> unbounded_knapsack.dart<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  int n = wgt.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(cap + 1, 0);\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int c = 1; c &lt;= cap; c++) {\n      if (wgt[i - 1] &gt; c) {\n        // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n        dp[c] = dp[c];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n        dp[c] = max(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n      }\n    }\n  }\n  return dp[cap];\n}\n</code></pre> unbounded_knapsack.rs<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn unbounded_knapsack_dp_comp(wgt: &amp;[i32], val: &amp;[i32], cap: usize) -&gt; i32 {\n    let n = wgt.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; cap + 1];\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for c in 1..=cap {\n            if wgt[i - 1] &gt; c as i32 {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = std::cmp::max(dp[c], dp[c - wgt[i - 1] as usize] + val[i - 1]);\n            }\n        }\n    }\n    dp[cap]\n}\n</code></pre> unbounded_knapsack.c<pre><code>/* \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint unboundedKnapsackDPComp(int wgt[], int val[], int cap, int wgtSize) {\n    int n = wgtSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(cap + 1, sizeof(int));\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int c = 1; c &lt;= cap; c++) {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = myMax(dp[c], dp[c - wgt[i - 1]] + val[i - 1]);\n            }\n        }\n    }\n    int res = dp[cap];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> unbounded_knapsack.zig<pre><code>// \u5b8c\u5168\u80cc\u5305\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn unboundedKnapsackDPComp(comptime wgt: []i32, val: []i32, comptime cap: usize) i32 {\n    comptime var n = wgt.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (cap + 1);\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..cap + 1) |c| {\n            if (wgt[i - 1] &gt; c) {\n                // \u82e5\u8d85\u8fc7\u80cc\u5305\u5bb9\u91cf\uff0c\u5219\u4e0d\u9009\u7269\u54c1 i\n                dp[c] = dp[c];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u7269\u54c1 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5927\u503c\n                dp[c] = @max(dp[c], dp[c - @as(usize, @intCast(wgt[i - 1]))] + val[i - 1]);\n            }\n        }\n    }\n    return dp[cap];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1452","title":"14.5.2 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u5927\u7c7b\u52a8\u6001\u89c4\u5212\u95ee\u9898\u7684\u4ee3\u8868\uff0c\u5176\u62e5\u6709\u5f88\u591a\u53d8\u79cd\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 14-24 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-24 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_1","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u96f6\u94b1\u5151\u6362\u53ef\u4ee5\u770b\u4f5c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\uff0c\u4e24\u8005\u5177\u6709\u4ee5\u4e0b\u8054\u7cfb\u4e0e\u4e0d\u540c\u70b9\u3002</p> <ul> <li>\u4e24\u9053\u9898\u53ef\u4ee5\u76f8\u4e92\u8f6c\u6362\uff0c\u201c\u7269\u54c1\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u201d\u3001\u201c\u7269\u54c1\u91cd\u91cf\u201d\u5bf9\u5e94\u201c\u786c\u5e01\u9762\u503c\u201d\u3001\u201c\u80cc\u5305\u5bb9\u91cf\u201d\u5bf9\u5e94\u201c\u76ee\u6807\u91d1\u989d\u201d\u3002</li> <li>\u4f18\u5316\u76ee\u6807\u76f8\u53cd\uff0c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u8981\u6700\u5927\u5316\u7269\u54c1\u4ef7\u503c\uff0c\u96f6\u94b1\u5151\u6362\u95ee\u9898\u662f\u8981\u6700\u5c0f\u5316\u786c\u5e01\u6570\u91cf\u3002</li> <li>\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u662f\u6c42\u201c\u4e0d\u8d85\u8fc7\u201d\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u89e3\uff0c\u96f6\u94b1\u5151\u6362\u662f\u6c42\u201c\u6070\u597d\u201d\u51d1\u5230\u76ee\u6807\u91d1\u989d\u7684\u89e3\u3002</li> </ul> <p>\u7b2c\u4e00\u6b65\uff1a\u601d\u8003\u6bcf\u8f6e\u7684\u51b3\u7b56\uff0c\u5b9a\u4e49\u72b6\u6001\uff0c\u4ece\u800c\u5f97\u5230 \\(dp\\) \u8868</p> <p>\u72b6\u6001 \\([i, a]\\) \u5bf9\u5e94\u7684\u5b50\u95ee\u9898\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\uff0c\u8bb0\u4e3a \\(dp[i, a]\\) \u3002</p> <p>\u4e8c\u7ef4 \\(dp\\) \u8868\u7684\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt+1)\\) \u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u627e\u51fa\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u8fdb\u800c\u63a8\u5bfc\u51fa\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b</p> <p>\u672c\u9898\u4e0e\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u7684\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u5b58\u5728\u4ee5\u4e0b\u4e24\u70b9\u5dee\u5f02\u3002</p> <ul> <li>\u672c\u9898\u8981\u6c42\u6700\u5c0f\u503c\uff0c\u56e0\u6b64\u9700\u5c06\u8fd0\u7b97\u7b26 \\(\\max()\\) \u66f4\u6539\u4e3a \\(\\min()\\) \u3002</li> <li>\u4f18\u5316\u4e3b\u4f53\u662f\u786c\u5e01\u6570\u91cf\u800c\u975e\u5546\u54c1\u4ef7\u503c\uff0c\u56e0\u6b64\u5728\u9009\u4e2d\u786c\u5e01\u65f6\u6267\u884c \\(+1\\) \u5373\u53ef\u3002</li> </ul> \\[ dp[i, a] = \\min(dp[i-1, a], dp[i, a - coins[i-1]] + 1) \\] <p>\u7b2c\u4e09\u6b65\uff1a\u786e\u5b9a\u8fb9\u754c\u6761\u4ef6\u548c\u72b6\u6001\u8f6c\u79fb\u987a\u5e8f</p> <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u51d1\u51fa\u5b83\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u4e3a \\(0\\) \uff0c\u5373\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p> <p>\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u610f \\(&gt; 0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u5373\u662f\u65e0\u6548\u89e3\u3002\u4e3a\u4f7f\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(\\min()\\) \u51fd\u6570\u80fd\u591f\u8bc6\u522b\u5e76\u8fc7\u6ee4\u65e0\u6548\u89e3\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528 \\(+ \\infty\\) \u6765\u8868\u793a\u5b83\u4eec\uff0c\u5373\u4ee4\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(+ \\infty\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_1","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u5e76\u672a\u63d0\u4f9b \\(+ \\infty\\) \u53d8\u91cf\uff0c\u53ea\u80fd\u4f7f\u7528\u6574\u578b <code>int</code> \u7684\u6700\u5927\u503c\u6765\u4ee3\u66ff\u3002\u800c\u8fd9\u53c8\u4f1a\u5bfc\u81f4\u5927\u6570\u8d8a\u754c\uff1a\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e2d\u7684 \\(+ 1\\) \u64cd\u4f5c\u53ef\u80fd\u53d1\u751f\u6ea2\u51fa\u3002</p> <p>\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u6570\u5b57 \\(amt + 1\\) \u6765\u8868\u793a\u65e0\u6548\u89e3\uff0c\u56e0\u4e3a\u51d1\u51fa \\(amt\\) \u7684\u786c\u5e01\u6570\u91cf\u6700\u591a\u4e3a \\(amt\\) \u3002\u6700\u540e\u8fd4\u56de\u524d\uff0c\u5224\u65ad \\(dp[n, amt]\\) \u662f\u5426\u7b49\u4e8e \\(amt + 1\\) \uff0c\u82e5\u662f\u5219\u8fd4\u56de \\(-1\\) \uff0c\u4ee3\u8868\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py<pre><code>def coin_change_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in range(1, amt + 1):\n        dp[0][a] = MAX\n    # \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n    return dp[n][amt] if dp[n][amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDP(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0, a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i, a] = Math.Min(dp[i - 1, a], dp[i, a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n, amt] != MAX ? dp[n, amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a := 1; a &lt;= amt; a++ {\n        dp[0][a] = max\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = int(math.Min(float64(dp[i-1][a]), float64(dp[i][a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[n][amt] != max {\n        return dp[n][amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in stride(from: 1, through: amt, by: 1) {\n        dp[0][a] = MAX\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in stride(from: 1, through: n, by: 1) {\n        for a in stride(from: 1, through: amt, by: 1) {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[n][amt] != MAX ? dp[n][amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (let a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = Math.min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[n][amt] !== MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n  for (int a = 1; a &lt;= amt; a++) {\n    dp[0][a] = MAX;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n        dp[i][a] = min(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[n][amt] != MAX ? dp[n][amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for a in 1..= amt {\n        dp[0][a] = max;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = std::cmp::min(dp[i - 1][a], dp[i][a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[n][amt] != max { return dp[n][amt] as i32; } else { -1 }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (int a = 1; a &lt;= amt; a++) {\n        dp[0][a] = MAX;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = myMin(dp[i - 1][a], dp[i][a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[n][amt] != MAX ? dp[n][amt] : -1;\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u9996\u884c\u9996\u5217\n    for (1..amt + 1) |a| {\n        dp[0][a] = max;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = @min(dp[i - 1][a], dp[i][a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[n][amt] != max) {\n        return @intCast(dp[n][amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre> <p>\u56fe 14-25 \u5c55\u793a\u4e86\u96f6\u94b1\u5151\u6362\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b\uff0c\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u975e\u5e38\u76f8\u4f3c\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt;&lt;13&gt;&lt;14&gt;&lt;15&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 14-25 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898\u7684\u52a8\u6001\u89c4\u5212\u8fc7\u7a0b </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_1","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u96f6\u94b1\u5151\u6362\u7684\u7a7a\u95f4\u4f18\u5316\u7684\u5904\u7406\u65b9\u5f0f\u548c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u4e00\u81f4\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change.py<pre><code>def coin_change_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    MAX = amt + 1\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [MAX] * (amt + 1)\n    dp[0] = 0\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n    return dp[amt] if dp[amt] != MAX else -1\n</code></pre> coin_change.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Arrays.fill(dp, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    Array.Fill(dp, MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.Min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins []int, amt int) int {\n    n := len(coins)\n    max := amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    for i := 1; i &lt;= amt; i++ {\n        dp[i] = max\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = int(math.Min(float64(dp[a]), float64(dp[a-coins[i-1]]+1)))\n            }\n        }\n    }\n    if dp[amt] != max {\n        return dp[amt]\n    }\n    return -1\n}\n</code></pre> coin_change.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    let MAX = amt + 1\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: MAX, count: amt + 1)\n    dp[0] = 0\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in stride(from: 1, through: n, by: 1) {\n        for a in stride(from: 1, through: amt, by: 1) {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1)\n            }\n        }\n    }\n    return dp[amt] != MAX ? dp[amt] : -1\n}\n</code></pre> coin_change.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins, amt) {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    const MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; MAX);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = Math.min(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    return dp[amt] !== MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  int MAX = amt + 1;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, MAX);\n  dp[0] = 0;\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n        dp[a] = min(dp[a], dp[a - coins[i - 1]] + 1);\n      }\n    }\n  }\n  return dp[amt] != MAX ? dp[amt] : -1;\n}\n</code></pre> coin_change.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    let max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp.fill(max);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = std::cmp::min(dp[a], dp[a - coins[i - 1] as usize] + 1);\n            }\n        }\n    }\n    if dp[amt] != max { return dp[amt] as i32; } else { -1 }\n}\n</code></pre> coin_change.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    int MAX = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = myMin(dp[a], dp[a - coins[i - 1]] + 1);\n            }\n        }\n    }\n    int res = dp[amt] != MAX ? dp[amt] : -1;\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> coin_change.zig<pre><code>// \u96f6\u94b1\u5151\u6362\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    comptime var max = amt + 1;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    @memset(&amp;dp, max);\n    dp[0] = 0;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = @min(dp[a], dp[a - @as(usize, @intCast(coins[i - 1]))] + 1);\n            }\n        }\n    }\n    if (dp[amt] != max) {\n        return @intCast(dp[amt]);\n    } else {\n        return -1;\n    }\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1453-ii","title":"14.5.3 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u786c\u5e01\u7ec4\u5408\u6570\u91cf\u3002\u793a\u4f8b\u5982\u56fe 14-26 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 14-26 \u00a0 \u96f6\u94b1\u5151\u6362\u95ee\u9898 II \u7684\u793a\u4f8b\u6570\u636e </p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#1_2","title":"1. \u00a0 \u52a8\u6001\u89c4\u5212\u601d\u8def","text":"<p>\u76f8\u6bd4\u4e8e\u4e0a\u4e00\u9898\uff0c\u672c\u9898\u76ee\u6807\u662f\u6c42\u7ec4\u5408\u6570\u91cf\uff0c\u56e0\u6b64\u5b50\u95ee\u9898\u53d8\u4e3a\uff1a\u524d \\(i\\) \u79cd\u786c\u5e01\u80fd\u591f\u51d1\u51fa\u91d1\u989d \\(a\\) \u7684\u7ec4\u5408\u6570\u91cf\u3002\u800c \\(dp\\) \u8868\u4ecd\u7136\u662f\u5c3a\u5bf8\u4e3a \\((n+1) \\times (amt + 1)\\) \u7684\u4e8c\u7ef4\u77e9\u9635\u3002</p> <p>\u5f53\u524d\u72b6\u6001\u7684\u7ec4\u5408\u6570\u91cf\u7b49\u4e8e\u4e0d\u9009\u5f53\u524d\u786c\u5e01\u4e0e\u9009\u5f53\u524d\u786c\u5e01\u8fd9\u4e24\u79cd\u51b3\u7b56\u7684\u7ec4\u5408\u6570\u91cf\u4e4b\u548c\u3002\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> \\[ dp[i, a] = dp[i-1, a] + dp[i, a - coins[i-1]] \\] <p>\u5f53\u76ee\u6807\u91d1\u989d\u4e3a \\(0\\) \u65f6\uff0c\u65e0\u987b\u9009\u62e9\u4efb\u4f55\u786c\u5e01\u5373\u53ef\u51d1\u51fa\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u5e94\u5c06\u9996\u5217\u6240\u6709 \\(dp[i, 0]\\) \u90fd\u521d\u59cb\u5316\u4e3a \\(1\\) \u3002\u5f53\u65e0\u786c\u5e01\u65f6\uff0c\u65e0\u6cd5\u51d1\u51fa\u4efb\u4f55 \\(&gt;0\\) \u7684\u76ee\u6807\u91d1\u989d\uff0c\u56e0\u6b64\u9996\u884c\u6240\u6709 \\(dp[0, a]\\) \u90fd\u7b49\u4e8e \\(0\\) \u3002</p>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#2_2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py<pre><code>def coin_change_ii_dp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [[0] * (amt + 1) for _ in range(n + 1)]\n    # \u521d\u59cb\u5316\u9996\u5217\n    for i in range(n + 1):\n        dp[i][0] = 1\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n    return dp[n][amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;vector&lt;int&gt;&gt; dp(n + 1, vector&lt;int&gt;(amt + 1, 0));\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[][] dp = new int[n + 1][amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDP(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[,] dp = new int[n + 1, amt + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i, 0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i, a] = dp[i - 1, a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i, a] = dp[i - 1, a] + dp[i, a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n, amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([][]int, n+1)\n    for i := 0; i &lt;= n; i++ {\n        dp[i] = make([]int, amt+1)\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i := 0; i &lt;= n; i++ {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\uff1a\u5176\u4f59\u884c\u548c\u5217\n    for i := 1; i &lt;= n; i++ {\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i-1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i-1][a] + dp[i][a-coins[i-1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDP(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: Array(repeating: 0, count: amt + 1), count: n + 1)\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in stride(from: 0, through: n, by: 1) {\n        dp[i][0] = 1\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in stride(from: 1, through: n, by: 1) {\n        for a in stride(from: 1, through: amt, by: 1) {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[n][amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDP(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: n + 1 }, () =&gt;\n        Array.from({ length: amt + 1 }, () =&gt; 0)\n    );\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (let i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;List&lt;int&gt;&gt; dp = List.generate(n + 1, (index) =&gt; List.filled(amt + 1, 0));\n  // \u521d\u59cb\u5316\u9996\u5217\n  for (int i = 0; i &lt;= n; i++) {\n    dp[i][0] = 1;\n  }\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[i][a] = dp[i - 1][a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n        dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[n][amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![vec![0; amt + 1]; n + 1];\n    // \u521d\u59cb\u5316\u9996\u5217\n    for i in 0..= n {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[n][amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDP(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int **dp = malloc((n + 1) * sizeof(int *));\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i] = calloc(amt + 1, sizeof(int));\n    }\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (int i = 0; i &lt;= n; i++) {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[n][amt];\n    // \u91ca\u653e\u5185\u5b58\n    for (int i = 0; i &lt;= n; i++) {\n        free(dp[i]);\n    }\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDP(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_][amt + 1]i32{[_]i32{0} ** (amt + 1)} ** (n + 1);\n    // \u521d\u59cb\u5316\u9996\u5217\n    for (0..n + 1) |i| {\n        dp[i][0] = 1;\n    }\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[i][a] = dp[i - 1][a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[i][a] = dp[i - 1][a] + dp[i][a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[n][amt];\n}\n</code></pre>"},{"location":"chapter_dynamic_programming/unbounded_knapsack_problem/#3_2","title":"3. \u00a0 \u7a7a\u95f4\u4f18\u5316","text":"<p>\u7a7a\u95f4\u4f18\u5316\u5904\u7406\u65b9\u5f0f\u76f8\u540c\uff0c\u5220\u9664\u786c\u5e01\u7ef4\u5ea6\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_ii.py<pre><code>def coin_change_ii_dp_comp(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\"\"\"\n    n = len(coins)\n    # \u521d\u59cb\u5316 dp \u8868\n    dp = [0] * (amt + 1)\n    dp[0] = 1\n    # \u72b6\u6001\u8f6c\u79fb\n    for i in range(1, n + 1):\n        # \u6b63\u5e8f\u904d\u5386\n        for a in range(1, amt + 1):\n            if coins[i - 1] &gt; a:\n                # \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            else:\n                # \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n    return dp[amt]\n</code></pre> coin_change_ii.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(vector&lt;int&gt; &amp;coins, int amt) {\n    int n = coins.size();\n    // \u521d\u59cb\u5316 dp \u8868\n    vector&lt;int&gt; dp(amt + 1, 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.java<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.cs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint CoinChangeIIDPComp(int[] coins, int amt) {\n    int n = coins.Length;\n    // \u521d\u59cb\u5316 dp \u8868\n    int[] dp = new int[amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.go<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins []int, amt int) int {\n    n := len(coins)\n    // \u521d\u59cb\u5316 dp \u8868\n    dp := make([]int, amt+1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for i := 1; i &lt;= n; i++ {\n        // \u5012\u5e8f\u904d\u5386\n        for a := 1; a &lt;= amt; a++ {\n            if coins[i-1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a-coins[i-1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.swift<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunc coinChangeIIDPComp(coins: [Int], amt: Int) -&gt; Int {\n    let n = coins.count\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = Array(repeating: 0, count: amt + 1)\n    dp[0] = 1\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in stride(from: 1, through: n, by: 1) {\n        for a in stride(from: 1, through: amt, by: 1) {\n            if coins[i - 1] &gt; a {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a]\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]]\n            }\n        }\n    }\n    return dp[amt]\n}\n</code></pre> coin_change_ii.js<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins, amt) {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.ts<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u72b6\u6001\u538b\u7f29\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfunction coinChangeIIDPComp(coins: Array&lt;number&gt;, amt: number): number {\n    const n = coins.length;\n    // \u521d\u59cb\u5316 dp \u8868\n    const dp = Array.from({ length: amt + 1 }, () =&gt; 0);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (let i = 1; i &lt;= n; i++) {\n        for (let a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre> coin_change_ii.dart<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(List&lt;int&gt; coins, int amt) {\n  int n = coins.length;\n  // \u521d\u59cb\u5316 dp \u8868\n  List&lt;int&gt; dp = List.filled(amt + 1, 0);\n  dp[0] = 1;\n  // \u72b6\u6001\u8f6c\u79fb\n  for (int i = 1; i &lt;= n; i++) {\n    for (int a = 1; a &lt;= amt; a++) {\n      if (coins[i - 1] &gt; a) {\n        // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n        dp[a] = dp[a];\n      } else {\n        // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n        dp[a] = dp[a] + dp[a - coins[i - 1]];\n      }\n    }\n  }\n  return dp[amt];\n}\n</code></pre> coin_change_ii.rs<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nfn coin_change_ii_dp_comp(coins: &amp;[i32], amt: usize) -&gt; i32 {\n    let n = coins.len();\n    // \u521d\u59cb\u5316 dp \u8868\n    let mut dp = vec![0; amt + 1];\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for i in 1..=n {\n        for a in 1..=amt {\n            if coins[i - 1] &gt; a as i32 {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = dp[a] + dp[a - coins[i - 1] as usize];\n            }\n        }\n    }\n    dp[amt]\n}\n</code></pre> coin_change_ii.c<pre><code>/* \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212 */\nint coinChangeIIDPComp(int coins[], int amt, int coinsSize) {\n    int n = coinsSize;\n    // \u521d\u59cb\u5316 dp \u8868\n    int *dp = calloc(amt + 1, sizeof(int));\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (int i = 1; i &lt;= n; i++) {\n        for (int a = 1; a &lt;= amt; a++) {\n            if (coins[i - 1] &gt; a) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u4e4b\u548c\n                dp[a] = dp[a] + dp[a - coins[i - 1]];\n            }\n        }\n    }\n    int res = dp[amt];\n    // \u91ca\u653e\u5185\u5b58\n    free(dp);\n    return res;\n}\n</code></pre> coin_change_ii.zig<pre><code>// \u96f6\u94b1\u5151\u6362 II\uff1a\u7a7a\u95f4\u4f18\u5316\u540e\u7684\u52a8\u6001\u89c4\u5212\nfn coinChangeIIDPComp(comptime coins: []i32, comptime amt: usize) i32 {\n    comptime var n = coins.len;\n    // \u521d\u59cb\u5316 dp \u8868\n    var dp = [_]i32{0} ** (amt + 1);\n    dp[0] = 1;\n    // \u72b6\u6001\u8f6c\u79fb\n    for (1..n + 1) |i| {\n        for (1..amt + 1) |a| {\n            if (coins[i - 1] &gt; @as(i32, @intCast(a))) {\n                // \u82e5\u8d85\u8fc7\u76ee\u6807\u91d1\u989d\uff0c\u5219\u4e0d\u9009\u786c\u5e01 i\n                dp[a] = dp[a];\n            } else {\n                // \u4e0d\u9009\u548c\u9009\u786c\u5e01 i \u8fd9\u4e24\u79cd\u65b9\u6848\u7684\u8f83\u5c0f\u503c\n                dp[a] = dp[a] + dp[a - @as(usize, @intCast(coins[i - 1]))];\n            }\n        }\n    }\n    return dp[amt];\n}\n</code></pre>"},{"location":"chapter_graph/","title":"\u7b2c 9 \u7ae0 \u00a0 \u56fe","text":"<p>Abstract</p> <p>\u5728\u751f\u547d\u65c5\u9014\u4e2d\uff0c\u6211\u4eec\u5c31\u50cf\u662f\u4e00\u4e2a\u4e2a\u8282\u70b9\uff0c\u88ab\u65e0\u6570\u770b\u4e0d\u89c1\u7684\u8fb9\u76f8\u8fde\u3002</p> <p>\u6bcf\u4e00\u6b21\u7684\u76f8\u8bc6\u4e0e\u76f8\u79bb\uff0c\u90fd\u5728\u8fd9\u5f20\u5de8\u5927\u7684\u7f51\u7edc\u56fe\u4e2d\u7559\u4e0b\u72ec\u7279\u7684\u5370\u8bb0\u3002</p>"},{"location":"chapter_graph/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>9.1 \u00a0 \u56fe</li> <li>9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c</li> <li>9.3 \u00a0 \u56fe\u7684\u904d\u5386</li> <li>9.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_graph/graph/","title":"9.1 \u00a0 \u56fe","text":"<p>\u300c\u56fe graph\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u7531\u300c\u9876\u70b9 vertex\u300d\u548c\u300c\u8fb9 edge\u300d\u7ec4\u6210\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u56fe \\(G\\) \u62bd\u8c61\u5730\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9 \\(V\\) \u548c\u4e00\u7ec4\u8fb9 \\(E\\) \u7684\u96c6\u5408\u3002\u4ee5\u4e0b\u793a\u4f8b\u5c55\u793a\u4e86\u4e00\u4e2a\u5305\u542b 5 \u4e2a\u9876\u70b9\u548c 7 \u6761\u8fb9\u7684\u56fe\u3002</p> \\[ \\begin{aligned} V &amp; = \\{ 1, 2, 3, 4, 5 \\} \\newline E &amp; = \\{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \\} \\newline G &amp; = \\{ V, E \\} \\newline \\end{aligned} \\] <p>\u5982\u679c\u5c06\u9876\u70b9\u770b\u4f5c\u8282\u70b9\uff0c\u5c06\u8fb9\u770b\u4f5c\u8fde\u63a5\u5404\u4e2a\u8282\u70b9\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5c06\u56fe\u770b\u4f5c\u4e00\u79cd\u4ece\u94fe\u8868\u62d3\u5c55\u800c\u6765\u7684\u6570\u636e\u7ed3\u6784\u3002\u5982\u56fe 9-1 \u6240\u793a\uff0c\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u7684\u81ea\u7531\u5ea6\u66f4\u9ad8\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</p> <p></p> <p> \u56fe 9-1 \u00a0 \u94fe\u8868\u3001\u6811\u3001\u56fe\u4e4b\u95f4\u7684\u5173\u7cfb </p>"},{"location":"chapter_graph/graph/#911","title":"9.1.1 \u00a0 \u56fe\u5e38\u89c1\u7c7b\u578b\u4e0e\u672f\u8bed","text":"<p>\u6839\u636e\u8fb9\u662f\u5426\u5177\u6709\u65b9\u5411\uff0c\u53ef\u5206\u4e3a\u300c\u65e0\u5411\u56fe undirected graph\u300d\u548c\u300c\u6709\u5411\u56fe directed graph\u300d\uff0c\u5982\u56fe 9-2 \u6240\u793a\u3002</p> <ul> <li>\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u8fb9\u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u7684\u201c\u53cc\u5411\u201d\u8fde\u63a5\u5173\u7cfb\uff0c\u4f8b\u5982\u5fae\u4fe1\u6216 QQ \u4e2d\u7684\u201c\u597d\u53cb\u5173\u7cfb\u201d\u3002</li> <li>\u5728\u6709\u5411\u56fe\u4e2d\uff0c\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u5373 \\(A \\rightarrow B\\) \u548c \\(A \\leftarrow B\\) \u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4f8b\u5982\u5fae\u535a\u6216\u6296\u97f3\u4e0a\u7684\u201c\u5173\u6ce8\u201d\u4e0e\u201c\u88ab\u5173\u6ce8\u201d\u5173\u7cfb\u3002</li> </ul> <p></p> <p> \u56fe 9-2 \u00a0 \u6709\u5411\u56fe\u4e0e\u65e0\u5411\u56fe </p> <p>\u6839\u636e\u6240\u6709\u9876\u70b9\u662f\u5426\u8fde\u901a\uff0c\u53ef\u5206\u4e3a\u300c\u8fde\u901a\u56fe connected graph\u300d\u548c\u300c\u975e\u8fde\u901a\u56fe disconnected graph\u300d\uff0c\u5982\u56fe 9-3 \u6240\u793a\u3002</p> <ul> <li>\u5bf9\u4e8e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u53ef\u4ee5\u5230\u8fbe\u5176\u4f59\u4efb\u610f\u9876\u70b9\u3002</li> <li>\u5bf9\u4e8e\u975e\u8fde\u901a\u56fe\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002</li> </ul> <p></p> <p> \u56fe 9-3 \u00a0 \u8fde\u901a\u56fe\u4e0e\u975e\u8fde\u901a\u56fe </p> <p>\u6211\u4eec\u8fd8\u53ef\u4ee5\u4e3a\u8fb9\u6dfb\u52a0\u201c\u6743\u91cd\u201d\u53d8\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u5982\u56fe 9-4 \u6240\u793a\u7684\u300c\u6709\u6743\u56fe weighted graph\u300d\u3002\u4f8b\u5982\u5728\u201c\u738b\u8005\u8363\u8000\u201d\u7b49\u624b\u6e38\u4e2d\uff0c\u7cfb\u7edf\u4f1a\u6839\u636e\u5171\u540c\u6e38\u620f\u65f6\u95f4\u6765\u8ba1\u7b97\u73a9\u5bb6\u4e4b\u95f4\u7684\u201c\u4eb2\u5bc6\u5ea6\u201d\uff0c\u8fd9\u79cd\u4eb2\u5bc6\u5ea6\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6709\u6743\u56fe\u6765\u8868\u793a\u3002</p> <p></p> <p> \u56fe 9-4 \u00a0 \u6709\u6743\u56fe\u4e0e\u65e0\u6743\u56fe </p> <p>\u56fe\u6570\u636e\u7ed3\u6784\u5305\u542b\u4ee5\u4e0b\u5e38\u7528\u672f\u8bed\u3002</p> <ul> <li>\u300c\u90bb\u63a5 adjacency\u300d\uff1a\u5f53\u4e24\u9876\u70b9\u4e4b\u95f4\u5b58\u5728\u8fb9\u76f8\u8fde\u65f6\uff0c\u79f0\u8fd9\u4e24\u9876\u70b9\u201c\u90bb\u63a5\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u9876\u70b9 1 \u7684\u90bb\u63a5\u9876\u70b9\u4e3a\u9876\u70b9 2\u30013\u30015\u3002</li> <li>\u300c\u8def\u5f84 path\u300d\uff1a\u4ece\u9876\u70b9 A \u5230\u9876\u70b9 B \u7ecf\u8fc7\u7684\u8fb9\u6784\u6210\u7684\u5e8f\u5217\u88ab\u79f0\u4e3a\u4ece A \u5230 B \u7684\u201c\u8def\u5f84\u201d\u3002\u5728\u56fe 9-4 \u4e2d\uff0c\u8fb9\u5e8f\u5217 1-5-2-4 \u662f\u9876\u70b9 1 \u5230\u9876\u70b9 4 \u7684\u4e00\u6761\u8def\u5f84\u3002</li> <li>\u300c\u5ea6 degree\u300d\uff1a\u4e00\u4e2a\u9876\u70b9\u62e5\u6709\u7684\u8fb9\u6570\u3002\u5bf9\u4e8e\u6709\u5411\u56fe\uff0c\u300c\u5165\u5ea6 in-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u6307\u5411\u8be5\u9876\u70b9\uff0c\u300c\u51fa\u5ea6 out-degree\u300d\u8868\u793a\u6709\u591a\u5c11\u6761\u8fb9\u4ece\u8be5\u9876\u70b9\u6307\u51fa\u3002</li> </ul>"},{"location":"chapter_graph/graph/#912","title":"9.1.2 \u00a0 \u56fe\u7684\u8868\u793a","text":"<p>\u56fe\u7684\u5e38\u7528\u8868\u793a\u65b9\u5f0f\u5305\u62ec\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u3002\u4ee5\u4e0b\u4f7f\u7528\u65e0\u5411\u56fe\u8fdb\u884c\u4e3e\u4f8b\u3002</p>"},{"location":"chapter_graph/graph/#1","title":"1. \u00a0 \u90bb\u63a5\u77e9\u9635","text":"<p>\u8bbe\u56fe\u7684\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u300c\u90bb\u63a5\u77e9\u9635 adjacency matrix\u300d\u4f7f\u7528\u4e00\u4e2a \\(n \\times n\\) \u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u662f\u5426\u5b58\u5728\u8fb9\u3002</p> <p>\u5982\u56fe 9-5 \u6240\u793a\uff0c\u8bbe\u90bb\u63a5\u77e9\u9635\u4e3a \\(M\\)\u3001\u9876\u70b9\u5217\u8868\u4e3a \\(V\\) \uff0c\u90a3\u4e48\u77e9\u9635\u5143\u7d20 \\(M[i, j] = 1\\) \u8868\u793a\u9876\u70b9 \\(V[i]\\) \u5230\u9876\u70b9 \\(V[j]\\) \u4e4b\u95f4\u5b58\u5728\u8fb9\uff0c\u53cd\u4e4b \\(M[i, j] = 0\\) \u8868\u793a\u4e24\u9876\u70b9\u4e4b\u95f4\u65e0\u8fb9\u3002</p> <p></p> <p> \u56fe 9-5 \u00a0 \u56fe\u7684\u90bb\u63a5\u77e9\u9635\u8868\u793a </p> <p>\u90bb\u63a5\u77e9\u9635\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u9876\u70b9\u4e0d\u80fd\u4e0e\u81ea\u8eab\u76f8\u8fde\uff0c\u56e0\u6b64\u90bb\u63a5\u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u6ca1\u6709\u610f\u4e49\u3002</li> <li>\u5bf9\u4e8e\u65e0\u5411\u56fe\uff0c\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u7b49\u4ef7\uff0c\u6b64\u65f6\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\u3002</li> <li>\u5c06\u90bb\u63a5\u77e9\u9635\u7684\u5143\u7d20\u4ece \\(1\\) \u548c \\(0\\) \u66ff\u6362\u4e3a\u6743\u91cd\uff0c\u5219\u53ef\u8868\u793a\u6709\u6743\u56fe\u3002</li> </ul> <p>\u4f7f\u7528\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u8bbf\u95ee\u77e9\u9635\u5143\u7d20\u4ee5\u83b7\u53d6\u8fb9\uff0c\u56e0\u6b64\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002\u7136\u800c\uff0c\u77e9\u9635\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u5185\u5b58\u5360\u7528\u8f83\u591a\u3002</p>"},{"location":"chapter_graph/graph/#2","title":"2. \u00a0 \u90bb\u63a5\u8868","text":"<p>\u300c\u90bb\u63a5\u8868 adjacency list\u300d\u4f7f\u7528 \\(n\\) \u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u94fe\u8868\u8282\u70b9\u8868\u793a\u9876\u70b9\u3002\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff08\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u9876\u70b9\uff09\u3002\u56fe 9-6 \u5c55\u793a\u4e86\u4e00\u4e2a\u4f7f\u7528\u90bb\u63a5\u8868\u5b58\u50a8\u7684\u56fe\u7684\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 9-6 \u00a0 \u56fe\u7684\u90bb\u63a5\u8868\u8868\u793a </p> <p>\u90bb\u63a5\u8868\u4ec5\u5b58\u50a8\u5b9e\u9645\u5b58\u5728\u7684\u8fb9\uff0c\u800c\u8fb9\u7684\u603b\u6570\u901a\u5e38\u8fdc\u5c0f\u4e8e \\(n^2\\) \uff0c\u56e0\u6b64\u5b83\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002\u7136\u800c\uff0c\u5728\u90bb\u63a5\u8868\u4e2d\u9700\u8981\u901a\u8fc7\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u5176\u65f6\u95f4\u6548\u7387\u4e0d\u5982\u90bb\u63a5\u77e9\u9635\u3002</p> <p>\u89c2\u5bdf\u56fe 9-6 \uff0c\u90bb\u63a5\u8868\u7ed3\u6784\u4e0e\u54c8\u5e0c\u8868\u4e2d\u7684\u201c\u94fe\u5f0f\u5730\u5740\u201d\u975e\u5e38\u76f8\u4f3c\uff0c\u56e0\u6b64\u6211\u4eec\u4e5f\u53ef\u4ee5\u91c7\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u6765\u4f18\u5316\u6548\u7387\u3002\u6bd4\u5982\u5f53\u94fe\u8868\u8f83\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u5316\u4e3a AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u6548\u7387\u4ece \\(O(n)\\) \u4f18\u5316\u81f3 \\(O(\\log n)\\) \uff1b\u8fd8\u53ef\u4ee5\u628a\u94fe\u8868\u8f6c\u6362\u4e3a\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u81f3 \\(O(1)\\) \u3002</p>"},{"location":"chapter_graph/graph/#913","title":"9.1.3 \u00a0 \u56fe\u5e38\u89c1\u5e94\u7528","text":"<p>\u5982\u8868 9-1 \u6240\u793a\uff0c\u8bb8\u591a\u73b0\u5b9e\u7cfb\u7edf\u53ef\u4ee5\u7528\u56fe\u6765\u5efa\u6a21\uff0c\u76f8\u5e94\u7684\u95ee\u9898\u4e5f\u53ef\u4ee5\u7ea6\u5316\u4e3a\u56fe\u8ba1\u7b97\u95ee\u9898\u3002</p> <p> \u8868 9-1 \u00a0 \u73b0\u5b9e\u751f\u6d3b\u4e2d\u5e38\u89c1\u7684\u56fe </p> \u9876\u70b9 \u8fb9 \u56fe\u8ba1\u7b97\u95ee\u9898 \u793e\u4ea4\u7f51\u7edc \u7528\u6237 \u597d\u53cb\u5173\u7cfb \u6f5c\u5728\u597d\u53cb\u63a8\u8350 \u5730\u94c1\u7ebf\u8def \u7ad9\u70b9 \u7ad9\u70b9\u95f4\u7684\u8fde\u901a\u6027 \u6700\u77ed\u8def\u7ebf\u63a8\u8350 \u592a\u9633\u7cfb \u661f\u4f53 \u661f\u4f53\u95f4\u7684\u4e07\u6709\u5f15\u529b\u4f5c\u7528 \u884c\u661f\u8f68\u9053\u8ba1\u7b97"},{"location":"chapter_graph/graph_operations/","title":"9.2 \u00a0 \u56fe\u57fa\u7840\u64cd\u4f5c","text":"<p>\u56fe\u7684\u57fa\u7840\u64cd\u4f5c\u53ef\u5206\u4e3a\u5bf9\u201c\u8fb9\u201d\u7684\u64cd\u4f5c\u548c\u5bf9\u201c\u9876\u70b9\u201d\u7684\u64cd\u4f5c\u3002\u5728\u201c\u90bb\u63a5\u77e9\u9635\u201d\u548c\u201c\u90bb\u63a5\u8868\u201d\u4e24\u79cd\u8868\u793a\u65b9\u6cd5\u4e0b\uff0c\u5b9e\u73b0\u65b9\u5f0f\u6709\u6240\u4e0d\u540c\u3002</p>"},{"location":"chapter_graph/graph_operations/#921","title":"9.2.1 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u7684\u5b9e\u73b0","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u9876\u70b9\u6570\u91cf\u4e3a \\(n\\) \u7684\u65e0\u5411\u56fe\uff0c\u5219\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u5f0f\u5982\u56fe 9-7 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u6216\u5220\u9664\u8fb9\uff1a\u76f4\u63a5\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u4fee\u6539\u6307\u5b9a\u7684\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u800c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u9700\u8981\u540c\u65f6\u66f4\u65b0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u7684\u5c3e\u90e8\u6dfb\u52a0\u4e00\u884c\u4e00\u5217\uff0c\u5e76\u5168\u90e8\u586b \\(0\\) \u5373\u53ef\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u4e00\u884c\u4e00\u5217\u3002\u5f53\u5220\u9664\u9996\u884c\u9996\u5217\u65f6\u8fbe\u5230\u6700\u5dee\u60c5\u51b5\uff0c\u9700\u8981\u5c06 \\((n-1)^2\\) \u4e2a\u5143\u7d20\u201c\u5411\u5de6\u4e0a\u79fb\u52a8\u201d\uff0c\u4ece\u800c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u4f20\u5165 \\(n\\) \u4e2a\u9876\u70b9\uff0c\u521d\u59cb\u5316\u957f\u5ea6\u4e3a \\(n\\) \u7684\u9876\u70b9\u5217\u8868 <code>vertices</code> \uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff1b\u521d\u59cb\u5316 \\(n \\times n\\) \u5927\u5c0f\u7684\u90bb\u63a5\u77e9\u9635 <code>adjMat</code> \uff0c\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u77e9\u9635\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-7 \u00a0 \u90bb\u63a5\u77e9\u9635\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u8868\u793a\u56fe\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_matrix.py<pre><code>class GraphAdjMat:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, vertices: list[int], edges: list[list[int]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.vertices: list[int] = []\n        # \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n        self.adj_mat: list[list[int]] = []\n        # \u6dfb\u52a0\u9876\u70b9\n        for val in vertices:\n            self.add_vertex(val)\n        # \u6dfb\u52a0\u8fb9\n        # \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges:\n            self.add_edge(e[0], e[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.vertices)\n\n    def add_vertex(self, val: int):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        n = self.size()\n        # \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.append(val)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        new_row = [0] * n\n        self.adj_mat.append(new_row)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in self.adj_mat:\n            row.append(0)\n\n    def remove_vertex(self, index: int):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if index &gt;= self.size():\n            raise IndexError()\n        # \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.pop(index)\n        # \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in self.adj_mat:\n            row.pop(index)\n\n    def add_edge(self, i: int, j: int):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        # \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1\n        self.adj_mat[j][i] = 1\n\n    def remove_edge(self, i: int, j: int):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        # \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        # \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 or j &lt; 0 or i &gt;= self.size() or j &gt;= self.size() or i == j:\n            raise IndexError()\n        self.adj_mat[i][j] = 0\n        self.adj_mat[j][i] = 0\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u77e9\u9635\"\"\"\n        print(\"\u9876\u70b9\u5217\u8868 =\", self.vertices)\n        print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        print_matrix(self.adj_mat)\n</code></pre> graph_adjacency_matrix.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vector&lt;int&gt; vertices;       // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vector&lt;vector&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjMat(const vector&lt;int&gt; &amp;vertices, const vector&lt;vector&lt;int&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const vector&lt;int&gt; &amp;edge : edges) {\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() const {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.push_back(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        adjMat.emplace_back(vector&lt;int&gt;(n, 0));\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.push_back(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(int index) {\n        if (index &gt;= size()) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.erase(vertices.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.erase(adjMat.begin() + index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (vector&lt;int&gt; &amp;row : adjMat) {\n            row.erase(row.begin() + index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n            throw out_of_range(\"\u9876\u70b9\u4e0d\u5b58\u5728\");\n        }\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    void print() {\n        cout &lt;&lt; \"\u9876\u70b9\u5217\u8868 = \";\n        printVector(vertices);\n        cout &lt;&lt; \"\u90bb\u63a5\u77e9\u9635 =\" &lt;&lt; endl;\n        printVectorMatrix(adjMat);\n    }\n};\n</code></pre> graph_adjacency_matrix.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;Integer&gt; vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;Integer&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = new ArrayList&lt;&gt;();\n        this.adjMat = new ArrayList&lt;&gt;();\n        // \u6dfb\u52a0\u9876\u70b9\n        for (int val : vertices) {\n            addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (int[] e : edges) {\n            addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return vertices.size();\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(int val) {\n        int n = size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;Integer&gt; newRow = new ArrayList&lt;&gt;(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.add(0);\n        }\n        adjMat.add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(int index) {\n        if (index &gt;= size())\n            throw new IndexOutOfBoundsException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (List&lt;Integer&gt; row : adjMat) {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void addEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat.get(i).set(j, 1);\n        adjMat.get(j).set(i, 1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void removeEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j)\n            throw new IndexOutOfBoundsException();\n        adjMat.get(i).set(j, 0);\n        adjMat.get(j).set(i, 0);\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void print() {\n        System.out.print(\"\u9876\u70b9\u5217\u8868 = \");\n        System.out.println(vertices);\n        System.out.println(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.printMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    List&lt;int&gt; vertices;     // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    List&lt;List&lt;int&gt;&gt; adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjMat(int[] vertices, int[][] edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        foreach (int val in vertices) {\n            AddVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        foreach (int[] e in edges) {\n            AddEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return vertices.Count;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(int val) {\n        int n = Size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.Add(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        List&lt;int&gt; newRow = new(n);\n        for (int j = 0; j &lt; n; j++) {\n            newRow.Add(0);\n        }\n        adjMat.Add(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.Add(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(int index) {\n        if (index &gt;= Size())\n            throw new IndexOutOfRangeException();\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.RemoveAt(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        foreach (List&lt;int&gt; row in adjMat) {\n            row.RemoveAt(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void AddEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1;\n        adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    public void RemoveEdge(int i, int j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= Size() || j &gt;= Size() || i == j)\n            throw new IndexOutOfRangeException();\n        adjMat[i][j] = 0;\n        adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    public void Print() {\n        Console.Write(\"\u9876\u70b9\u5217\u8868 = \");\n        PrintUtil.PrintList(vertices);\n        Console.WriteLine(\"\u90bb\u63a5\u77e9\u9635 =\");\n        PrintUtil.PrintMatrix(adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjMat struct {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    vertices []int\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat [][]int\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjMat(vertices []int, edges [][]int) *graphAdjMat {\n    // \u6dfb\u52a0\u9876\u70b9\n    n := len(vertices)\n    adjMat := make([][]int, n)\n    for i := range adjMat {\n        adjMat[i] = make([]int, n)\n    }\n    // \u521d\u59cb\u5316\u56fe\n    g := &amp;graphAdjMat{\n        vertices: vertices,\n        adjMat:   adjMat,\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for i := range edges {\n        g.addEdge(edges[i][0], edges[i][1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjMat) size() int {\n    return len(g.vertices)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjMat) addVertex(val int) {\n    n := g.size()\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    g.vertices = append(g.vertices, val)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    newRow := make([]int, n)\n    g.adjMat = append(g.adjMat, newRow)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i], 0)\n    }\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjMat) removeVertex(index int) {\n    if index &gt;= g.size() {\n        return\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    g.vertices = append(g.vertices[:index], g.vertices[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    g.adjMat = append(g.adjMat[:index], g.adjMat[index+1:]...)\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for i := range g.adjMat {\n        g.adjMat[i] = append(g.adjMat[i][:index], g.adjMat[i][index+1:]...)\n    }\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) addEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    g.adjMat[i][j] = 1\n    g.adjMat[j][i] = 1\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nfunc (g *graphAdjMat) removeEdge(i, j int) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if i &lt; 0 || j &lt; 0 || i &gt;= g.size() || j &gt;= g.size() || i == j {\n        fmt.Errorf(\"%s\", \"Index Out Of Bounds Exception\")\n    }\n    g.adjMat[i][j] = 0\n    g.adjMat[j][i] = 0\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nfunc (g *graphAdjMat) print() {\n    fmt.Printf(\"\\t\u9876\u70b9\u5217\u8868 = %v\\n\", g.vertices)\n    fmt.Printf(\"\\t\u90bb\u63a5\u77e9\u9635 = \\n\")\n    for i := range g.adjMat {\n        fmt.Printf(\"\\t\\t\\t%v\\n\", g.adjMat[i])\n    }\n}\n</code></pre> graph_adjacency_matrix.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    private var vertices: [Int] // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    private var adjMat: [[Int]] // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(vertices: [Int], edges: [[Int]]) {\n        self.vertices = []\n        adjMat = []\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            addVertex(val: val)\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for e in edges {\n            addEdge(i: e[0], j: e[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    func size() -&gt; Int {\n        vertices.count\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    func addVertex(val: Int) {\n        let n = size()\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        vertices.append(val)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        let newRow = Array(repeating: 0, count: n)\n        adjMat.append(newRow)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for i in adjMat.indices {\n            adjMat[i].append(0)\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    func removeVertex(index: Int) {\n        if index &gt;= size() {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        vertices.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        adjMat.remove(at: index)\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for i in adjMat.indices {\n            adjMat[i].remove(at: index)\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func addEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        adjMat[i][j] = 1\n        adjMat[j][i] = 1\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    func removeEdge(i: Int, j: Int) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j {\n            fatalError(\"\u8d8a\u754c\")\n        }\n        adjMat[i][j] = 0\n        adjMat[j][i] = 0\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    func print() {\n        Swift.print(\"\u9876\u70b9\u5217\u8868 = \", terminator: \"\")\n        Swift.print(vertices)\n        Swift.print(\"\u90bb\u63a5\u77e9\u9635 =\")\n        PrintUtil.printMatrix(matrix: adjMat)\n    }\n}\n</code></pre> graph_adjacency_matrix.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices, edges) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val) {\n        const n = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow = [];\n        for (let j = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index) {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i, j) {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print() {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n    vertices: number[]; // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    adjMat: number[][]; // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n    /* \u6784\u9020\u51fd\u6570 */\n    constructor(vertices: number[], edges: number[][]) {\n        this.vertices = [];\n        this.adjMat = [];\n        // \u6dfb\u52a0\u9876\u70b9\n        for (const val of vertices) {\n            this.addVertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for (const e of edges) {\n            this.addEdge(e[0], e[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.vertices.length;\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(val: number): void {\n        const n: number = this.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        this.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        const newRow: number[] = [];\n        for (let j: number = 0; j &lt; n; j++) {\n            newRow.push(0);\n        }\n        this.adjMat.push(newRow);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for (const row of this.adjMat) {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(index: number): void {\n        if (index &gt;= this.size()) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        this.vertices.splice(index, 1);\n\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        this.adjMat.splice(index, 1);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for (const row of this.adjMat) {\n            row.splice(index, 1);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    addEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) === (j, i)\n        this.adjMat[i][j] = 1;\n        this.adjMat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    removeEdge(i: number, j: number): void {\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if (i &lt; 0 || j &lt; 0 || i &gt;= this.size() || j &gt;= this.size() || i === j) {\n            throw new RangeError('Index Out Of Bounds Exception');\n        }\n        this.adjMat[i][j] = 0;\n        this.adjMat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    print(): void {\n        console.log('\u9876\u70b9\u5217\u8868 = ', this.vertices);\n        console.log('\u90bb\u63a5\u77e9\u9635 =', this.adjMat);\n    }\n}\n</code></pre> graph_adjacency_matrix.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjMat {\n  List&lt;int&gt; vertices = []; // \u9876\u70b9\u5143\u7d20\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n  List&lt;List&lt;int&gt;&gt; adjMat = []; //\u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjMat(List&lt;int&gt; vertices, List&lt;List&lt;int&gt;&gt; edges) {\n    this.vertices = [];\n    this.adjMat = [];\n    // \u6dfb\u52a0\u9876\u70b9\n    for (int val in vertices) {\n      addVertex(val);\n    }\n    // \u6dfb\u52a0\u8fb9\n    // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    for (List&lt;int&gt; e in edges) {\n      addEdge(e[0], e[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return vertices.length;\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(int val) {\n    int n = size();\n    // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n    vertices.add(val);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n    List&lt;int&gt; newRow = List.filled(n, 0, growable: true);\n    adjMat.add(newRow);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.add(0);\n    }\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(int index) {\n    if (index &gt;= size()) {\n      throw IndexError;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    vertices.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    adjMat.removeAt(index);\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (List&lt;int&gt; row in adjMat) {\n      row.removeAt(index);\n    }\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void addEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n    adjMat[i][j] = 1;\n    adjMat[j][i] = 1;\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n  void removeEdge(int i, int j) {\n    // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n    if (i &lt; 0 || j &lt; 0 || i &gt;= size() || j &gt;= size() || i == j) {\n      throw IndexError;\n    }\n    adjMat[i][j] = 0;\n    adjMat[j][i] = 0;\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n  void printAdjMat() {\n    print(\"\u9876\u70b9\u5217\u8868 = $vertices\");\n    print(\"\u90bb\u63a5\u77e9\u9635 = \");\n    printMatrix(adjMat);\n  }\n}\n</code></pre> graph_adjacency_matrix.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjMat {\n    // \u9876\u70b9\u5217\u8868\uff0c\u5143\u7d20\u4ee3\u8868\u201c\u9876\u70b9\u503c\u201d\uff0c\u7d22\u5f15\u4ee3\u8868\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub vertices: Vec&lt;i32&gt;,\n    // \u90bb\u63a5\u77e9\u9635\uff0c\u884c\u5217\u7d22\u5f15\u5bf9\u5e94\u201c\u9876\u70b9\u7d22\u5f15\u201d\n    pub adj_mat: Vec&lt;Vec&lt;i32&gt;&gt;,\n}\n\nimpl GraphAdjMat {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(vertices: Vec&lt;i32&gt;, edges: Vec&lt;[usize; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjMat {\n            vertices: vec![],\n            adj_mat: vec![],\n        };\n        // \u6dfb\u52a0\u9876\u70b9\n        for val in vertices {\n            graph.add_vertex(val);\n        }\n        // \u6dfb\u52a0\u8fb9\n        // \u8bf7\u6ce8\u610f\uff0cedges \u5143\u7d20\u4ee3\u8868\u9876\u70b9\u7d22\u5f15\uff0c\u5373\u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        for edge in edges {\n            graph.add_edge(edge[0], edge[1])\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    pub fn size(&amp;self) -&gt; usize {\n        self.vertices.len()\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, val: i32) {\n        let n = self.size();\n        // \u5411\u9876\u70b9\u5217\u8868\u4e2d\u6dfb\u52a0\u65b0\u9876\u70b9\u7684\u503c\n        self.vertices.push(val);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u884c\n        self.adj_mat.push(vec![0; n]);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u6dfb\u52a0\u4e00\u5217\n        for row in &amp;mut self.adj_mat {\n            row.push(0);\n        }\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    pub fn remove_vertex(&amp;mut self, index: usize) {\n        if index &gt;= self.size() {\n            panic!(\"index error\")\n        }\n        // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n        self.vertices.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n        self.adj_mat.remove(index);\n        // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n        for row in &amp;mut self.adj_mat {\n            row.remove(index);\n        }\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        // \u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u90bb\u63a5\u77e9\u9635\u5173\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5bf9\u79f0\uff0c\u5373\u6ee1\u8db3 (i, j) == (j, i)\n        self.adj_mat[i][j] = 1;\n        self.adj_mat[j][i] = 1;\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n    pub fn remove_edge(&amp;mut self, i: usize, j: usize) {\n        // \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\n        // \u7d22\u5f15\u8d8a\u754c\u4e0e\u76f8\u7b49\u5904\u7406\n        if i &gt;= self.size() || j &gt;= self.size() || i == j {\n            panic!(\"index error\")\n        }\n        self.adj_mat[i][j] = 0;\n        self.adj_mat[j][i] = 0;\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\n    pub fn print(&amp;self) {\n        println!(\"\u9876\u70b9\u5217\u8868 = {:?}\", self.vertices);\n        println!(\"\u90bb\u63a5\u77e9\u9635 =\");\n        println!(\"[\");\n        for row in &amp;self.adj_mat {\n            println!(\"  {:?},\", row);\n        }\n        println!(\"]\")\n    }\n}\n</code></pre> graph_adjacency_matrix.c<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u77e9\u9635\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int vertices[MAX_SIZE];\n    int adjMat[MAX_SIZE][MAX_SIZE];\n    int size;\n} GraphAdjMat;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjMat *newGraphAdjMat() {\n    GraphAdjMat *graph = (GraphAdjMat *)malloc(sizeof(GraphAdjMat));\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        for (int j = 0; j &lt; MAX_SIZE; j++) {\n            graph-&gt;adjMat[i][j] = 0;\n        }\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjMat(GraphAdjMat *graph) {\n    free(graph);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjMat *graph, int val) {\n    if (graph-&gt;size == MAX_SIZE) {\n        fprintf(stderr, \"\u56fe\u7684\u9876\u70b9\u6570\u91cf\u5df2\u8fbe\u6700\u5927\u503c\\n\");\n        return;\n    }\n    // \u6dfb\u52a0\u7b2c n \u4e2a\u9876\u70b9\uff0c\u5e76\u5c06\u7b2c n \u884c\u548c\u5217\u7f6e\u96f6\n    int n = graph-&gt;size;\n    graph-&gt;vertices[n] = val;\n    for (int i = 0; i &lt;= n; i++) {\n        graph-&gt;adjMat[n][i] = graph-&gt;adjMat[i][n] = 0;\n    }\n    graph-&gt;size++;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjMat *graph, int index) {\n    if (index &lt; 0 || index &gt;= graph-&gt;size) {\n        fprintf(stderr, \"\u9876\u70b9\u7d22\u5f15\u8d8a\u754c\\n\");\n        return;\n    }\n    // \u5728\u9876\u70b9\u5217\u8868\u4e2d\u79fb\u9664\u7d22\u5f15 index \u7684\u9876\u70b9\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        graph-&gt;vertices[i] = graph-&gt;vertices[i + 1];\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u884c\n    for (int i = index; i &lt; graph-&gt;size - 1; i++) {\n        for (int j = 0; j &lt; graph-&gt;size; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i + 1][j];\n        }\n    }\n    // \u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u5220\u9664\u7d22\u5f15 index \u7684\u5217\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        for (int j = index; j &lt; graph-&gt;size - 1; j++) {\n            graph-&gt;adjMat[i][j] = graph-&gt;adjMat[i][j + 1];\n        }\n    }\n    graph-&gt;size--;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid addEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 1;\n    graph-&gt;adjMat[j][i] = 1;\n}\n\n/* \u5220\u9664\u8fb9 */\n// \u53c2\u6570 i, j \u5bf9\u5e94 vertices \u5143\u7d20\u7d22\u5f15\nvoid removeEdge(GraphAdjMat *graph, int i, int j) {\n    if (i &lt; 0 || j &lt; 0 || i &gt;= graph-&gt;size || j &gt;= graph-&gt;size || i == j) {\n        fprintf(stderr, \"\u8fb9\u7d22\u5f15\u8d8a\u754c\u6216\u76f8\u7b49\\n\");\n        return;\n    }\n    graph-&gt;adjMat[i][j] = 0;\n    graph-&gt;adjMat[j][i] = 0;\n}\n\n/* \u6253\u5370\u90bb\u63a5\u77e9\u9635 */\nvoid printGraphAdjMat(GraphAdjMat *graph) {\n    printf(\"\u9876\u70b9\u5217\u8868 = \");\n    printArray(graph-&gt;vertices, graph-&gt;size);\n    printf(\"\u90bb\u63a5\u77e9\u9635 =\\n\");\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        printArray(graph-&gt;adjMat[i], graph-&gt;size);\n    }\n}\n</code></pre> graph_adjacency_matrix.zig<pre><code>[class]{GraphAdjMat}-[func]{}\n</code></pre>"},{"location":"chapter_graph/graph_operations/#922","title":"9.2.2 \u00a0 \u57fa\u4e8e\u90bb\u63a5\u8868\u7684\u5b9e\u73b0","text":"<p>\u8bbe\u65e0\u5411\u56fe\u7684\u9876\u70b9\u603b\u6570\u4e3a \\(n\\)\u3001\u8fb9\u603b\u6570\u4e3a \\(m\\) \uff0c\u5219\u53ef\u6839\u636e\u56fe 9-8 \u6240\u793a\u7684\u65b9\u6cd5\u5b9e\u73b0\u5404\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u6dfb\u52a0\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u7684\u672b\u5c3e\u6dfb\u52a0\u8fb9\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002\u56e0\u4e3a\u662f\u65e0\u5411\u56fe\uff0c\u6240\u4ee5\u9700\u8981\u540c\u65f6\u6dfb\u52a0\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u5220\u9664\u8fb9\uff1a\u5728\u9876\u70b9\u5bf9\u5e94\u94fe\u8868\u4e2d\u67e5\u627e\u5e76\u5220\u9664\u6307\u5b9a\u8fb9\uff0c\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\u3002\u5728\u65e0\u5411\u56fe\u4e2d\uff0c\u9700\u8981\u540c\u65f6\u5220\u9664\u4e24\u4e2a\u65b9\u5411\u7684\u8fb9\u3002</li> <li>\u6dfb\u52a0\u9876\u70b9\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u94fe\u8868\uff0c\u5e76\u5c06\u65b0\u589e\u9876\u70b9\u4f5c\u4e3a\u94fe\u8868\u5934\u8282\u70b9\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u9876\u70b9\uff1a\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5220\u9664\u5305\u542b\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> <li>\u521d\u59cb\u5316\uff1a\u5728\u90bb\u63a5\u8868\u4e2d\u521b\u5efa \\(n\\) \u4e2a\u9876\u70b9\u548c \\(2m\\) \u6761\u8fb9\uff0c\u4f7f\u7528 \\(O(n + m)\\) \u65f6\u95f4\u3002</li> </ul> \u521d\u59cb\u5316\u90bb\u63a5\u8868\u6dfb\u52a0\u8fb9\u5220\u9664\u8fb9\u6dfb\u52a0\u9876\u70b9\u5220\u9664\u9876\u70b9 <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-8 \u00a0 \u90bb\u63a5\u8868\u7684\u521d\u59cb\u5316\u3001\u589e\u5220\u8fb9\u3001\u589e\u5220\u9876\u70b9 </p> <p>\u4ee5\u4e0b\u662f\u90bb\u63a5\u8868\u7684\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u6bd4\u56fe 9-8 \uff0c\u5b9e\u9645\u4ee3\u7801\u6709\u4ee5\u4e0b\u4e0d\u540c\u3002</p> <ul> <li>\u4e3a\u4e86\u65b9\u4fbf\u6dfb\u52a0\u4e0e\u5220\u9664\u9876\u70b9\uff0c\u4ee5\u53ca\u7b80\u5316\u4ee3\u7801\uff0c\u6211\u4eec\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u6765\u4ee3\u66ff\u94fe\u8868\u3002</li> <li>\u4f7f\u7528\u54c8\u5e0c\u8868\u6765\u5b58\u50a8\u90bb\u63a5\u8868\uff0c<code>key</code> \u4e3a\u9876\u70b9\u5b9e\u4f8b\uff0c<code>value</code> \u4e3a\u8be5\u9876\u70b9\u7684\u90bb\u63a5\u9876\u70b9\u5217\u8868\uff08\u94fe\u8868\uff09\u3002</li> </ul> <p>\u53e6\u5916\uff0c\u6211\u4eec\u5728\u90bb\u63a5\u8868\u4e2d\u4f7f\u7528 <code>Vertex</code> \u7c7b\u6765\u8868\u793a\u9876\u70b9\uff0c\u8fd9\u6837\u505a\u7684\u539f\u56e0\u662f\uff1a\u5982\u679c\u4e0e\u90bb\u63a5\u77e9\u9635\u4e00\u6837\uff0c\u7528\u5217\u8868\u7d22\u5f15\u6765\u533a\u5206\u4e0d\u540c\u9876\u70b9\uff0c\u90a3\u4e48\u5047\u8bbe\u8981\u5220\u9664\u7d22\u5f15\u4e3a \\(i\\) \u7684\u9876\u70b9\uff0c\u5219\u9700\u904d\u5386\u6574\u4e2a\u90bb\u63a5\u8868\uff0c\u5c06\u6240\u6709\u5927\u4e8e \\(i\\) \u7684\u7d22\u5f15\u5168\u90e8\u51cf \\(1\\) \uff0c\u6548\u7387\u5f88\u4f4e\u3002\u800c\u5982\u679c\u6bcf\u4e2a\u9876\u70b9\u90fd\u662f\u552f\u4e00\u7684 <code>Vertex</code> \u5b9e\u4f8b\uff0c\u5220\u9664\u67d0\u4e00\u9876\u70b9\u4e4b\u540e\u5c31\u65e0\u987b\u6539\u52a8\u5176\u4ed6\u9876\u70b9\u4e86\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_adjacency_list.py<pre><code>class GraphAdjList:\n    \"\"\"\u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\"\"\"\n\n    def __init__(self, edges: list[list[Vertex]]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        self.adj_list = dict[Vertex, list[Vertex]]()\n        # \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges:\n            self.add_vertex(edge[0])\n            self.add_vertex(edge[1])\n            self.add_edge(edge[0], edge[1])\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u9876\u70b9\u6570\u91cf\"\"\"\n        return len(self.adj_list)\n\n    def add_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u6dfb\u52a0\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list[vet1].append(vet2)\n        self.adj_list[vet2].append(vet1)\n\n    def remove_edge(self, vet1: Vertex, vet2: Vertex):\n        \"\"\"\u5220\u9664\u8fb9\"\"\"\n        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:\n            raise ValueError()\n        # \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list[vet1].remove(vet2)\n        self.adj_list[vet2].remove(vet1)\n\n    def add_vertex(self, vet: Vertex):\n        \"\"\"\u6dfb\u52a0\u9876\u70b9\"\"\"\n        if vet in self.adj_list:\n            return\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list[vet] = []\n\n    def remove_vertex(self, vet: Vertex):\n        \"\"\"\u5220\u9664\u9876\u70b9\"\"\"\n        if vet not in self.adj_list:\n            raise ValueError()\n        # \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.pop(vet)\n        # \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for vertex in self.adj_list:\n            if vet in self.adj_list[vertex]:\n                self.adj_list[vertex].remove(vet)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u90bb\u63a5\u8868\"\"\"\n        print(\"\u90bb\u63a5\u8868 =\")\n        for vertex in self.adj_list:\n            tmp = [v.val for v in self.adj_list[vertex]]\n            print(f\"{vertex.val}: {tmp},\")\n</code></pre> graph_adjacency_list.cpp<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  public:\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    unordered_map&lt;Vertex *, vector&lt;Vertex *&gt;&gt; adjList;\n\n    /* \u5728 vector \u4e2d\u5220\u9664\u6307\u5b9a\u8282\u70b9 */\n    void remove(vector&lt;Vertex *&gt; &amp;vec, Vertex *vet) {\n        for (int i = 0; i &lt; vec.size(); i++) {\n            if (vec[i] == vet) {\n                vec.erase(vec.begin() + i);\n                break;\n            }\n        }\n    }\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    GraphAdjList(const vector&lt;vector&lt;Vertex *&gt;&gt; &amp;edges) {\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const vector&lt;Vertex *&gt; &amp;edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    void addEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].push_back(vet2);\n        adjList[vet2].push_back(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    void removeEdge(Vertex *vet1, Vertex *vet2) {\n        if (!adjList.count(vet1) || !adjList.count(vet2) || vet1 == vet2)\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        remove(adjList[vet1], vet2);\n        remove(adjList[vet2], vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    void addVertex(Vertex *vet) {\n        if (adjList.count(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = vector&lt;Vertex *&gt;();\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    void removeVertex(Vertex *vet) {\n        if (!adjList.count(vet))\n            throw invalid_argument(\"\u4e0d\u5b58\u5728\u9876\u70b9\");\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.erase(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (auto &amp;adj : adjList) {\n            remove(adj.second, vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    void print() {\n        cout &lt;&lt; \"\u90bb\u63a5\u8868 =\" &lt;&lt; endl;\n        for (auto &amp;adj : adjList) {\n            const auto &amp;key = adj.first;\n            const auto &amp;vec = adj.second;\n            cout &lt;&lt; key-&gt;val &lt;&lt; \": \";\n            printVector(vetsToVals(vec));\n        }\n    }\n};\n</code></pre> graph_adjacency_list.java<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public GraphAdjList(Vertex[][] edges) {\n        this.adjList = new HashMap&lt;&gt;();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (Vertex[] edge : edges) {\n            addVertex(edge[0]);\n            addVertex(edge[1]);\n            addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public int size() {\n        return adjList.size();\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void addEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList.get(vet1).add(vet2);\n        adjList.get(vet2).add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void removeEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)\n            throw new IllegalArgumentException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList.get(vet1).remove(vet2);\n        adjList.get(vet2).remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void addVertex(Vertex vet) {\n        if (adjList.containsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.put(vet, new ArrayList&lt;&gt;());\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void removeVertex(Vertex vet) {\n        if (!adjList.containsKey(vet))\n            throw new IllegalArgumentException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (List&lt;Vertex&gt; list : adjList.values()) {\n            list.remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void print() {\n        System.out.println(\"\u90bb\u63a5\u8868 =\");\n        for (Map.Entry&lt;Vertex, List&lt;Vertex&gt;&gt; pair : adjList.entrySet()) {\n            List&lt;Integer&gt; tmp = new ArrayList&lt;&gt;();\n            for (Vertex vertex : pair.getValue())\n                tmp.add(vertex.val);\n            System.out.println(pair.getKey().val + \": \" + tmp + \",\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.cs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public Dictionary&lt;Vertex, List&lt;Vertex&gt;&gt; adjList;\n\n    /* \u6784\u9020\u51fd\u6570 */\n    public GraphAdjList(Vertex[][] edges) {\n        adjList = [];\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        foreach (Vertex[] edge in edges) {\n            AddVertex(edge[0]);\n            AddVertex(edge[1]);\n            AddEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    int Size() {\n        return adjList.Count;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public void AddEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1].Add(vet2);\n        adjList[vet2].Add(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public void RemoveEdge(Vertex vet1, Vertex vet2) {\n        if (!adjList.ContainsKey(vet1) || !adjList.ContainsKey(vet2) || vet1 == vet2)\n            throw new InvalidOperationException();\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1].Remove(vet2);\n        adjList[vet2].Remove(vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public void AddVertex(Vertex vet) {\n        if (adjList.ContainsKey(vet))\n            return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList.Add(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public void RemoveVertex(Vertex vet) {\n        if (!adjList.ContainsKey(vet))\n            throw new InvalidOperationException();\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.Remove(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        foreach (List&lt;Vertex&gt; list in adjList.Values) {\n            list.Remove(vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public void Print() {\n        Console.WriteLine(\"\u90bb\u63a5\u8868 =\");\n        foreach (KeyValuePair&lt;Vertex, List&lt;Vertex&gt;&gt; pair in adjList) {\n            List&lt;int&gt; tmp = [];\n            foreach (Vertex vertex in pair.Value)\n                tmp.Add(vertex.val);\n            Console.WriteLine(pair.Key.val + \": [\" + string.Join(\", \", tmp) + \"],\");\n        }\n    }\n}\n</code></pre> graph_adjacency_list.go<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntype graphAdjList struct {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList map[Vertex][]Vertex\n}\n\n/* \u6784\u9020\u51fd\u6570 */\nfunc newGraphAdjList(edges [][]Vertex) *graphAdjList {\n    g := &amp;graphAdjList{\n        adjList: make(map[Vertex][]Vertex),\n    }\n    // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n    for _, edge := range edges {\n        g.addVertex(edge[0])\n        g.addVertex(edge[1])\n        g.addEdge(edge[0], edge[1])\n    }\n    return g\n}\n\n/* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\nfunc (g *graphAdjList) size() int {\n    return len(g.adjList)\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nfunc (g *graphAdjList) addEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2, \u6dfb\u52a0\u533f\u540d struct{},\n    g.adjList[vet1] = append(g.adjList[vet1], vet2)\n    g.adjList[vet2] = append(g.adjList[vet2], vet1)\n}\n\n/* \u5220\u9664\u8fb9 */\nfunc (g *graphAdjList) removeEdge(vet1 Vertex, vet2 Vertex) {\n    _, ok1 := g.adjList[vet1]\n    _, ok2 := g.adjList[vet2]\n    if !ok1 || !ok2 || vet1 == vet2 {\n        panic(\"error\")\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    g.adjList[vet1] = DeleteSliceElms(g.adjList[vet1], vet2)\n    g.adjList[vet2] = DeleteSliceElms(g.adjList[vet2], vet1)\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nfunc (g *graphAdjList) addVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if ok {\n        return\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    g.adjList[vet] = make([]Vertex, 0)\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nfunc (g *graphAdjList) removeVertex(vet Vertex) {\n    _, ok := g.adjList[vet]\n    if !ok {\n        panic(\"error\")\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    delete(g.adjList, vet)\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for v, list := range g.adjList {\n        g.adjList[v] = DeleteSliceElms(list, vet)\n    }\n}\n\n/* \u6253\u5370\u90bb\u63a5\u8868 */\nfunc (g *graphAdjList) print() {\n    var builder strings.Builder\n    fmt.Printf(\"\u90bb\u63a5\u8868 = \\n\")\n    for k, v := range g.adjList {\n        builder.WriteString(\"\\t\\t\" + strconv.Itoa(k.Val) + \": \")\n        for _, vet := range v {\n            builder.WriteString(strconv.Itoa(vet.Val) + \" \")\n        }\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> graph_adjacency_list.swift<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    public private(set) var adjList: [Vertex: [Vertex]]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public init(edges: [[Vertex]]) {\n        adjList = [:]\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            addVertex(vet: edge[0])\n            addVertex(vet: edge[1])\n            addEdge(vet1: edge[0], vet2: edge[1])\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    public func size() -&gt; Int {\n        adjList.count\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    public func addEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        adjList[vet1]?.append(vet2)\n        adjList[vet2]?.append(vet1)\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    public func removeEdge(vet1: Vertex, vet2: Vertex) {\n        if adjList[vet1] == nil || adjList[vet2] == nil || vet1 == vet2 {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        adjList[vet1]?.removeAll(where: { $0 == vet2 })\n        adjList[vet2]?.removeAll(where: { $0 == vet1 })\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    public func addVertex(vet: Vertex) {\n        if adjList[vet] != nil {\n            return\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        adjList[vet] = []\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    public func removeVertex(vet: Vertex) {\n        if adjList[vet] == nil {\n            fatalError(\"\u53c2\u6570\u9519\u8bef\")\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        adjList.removeValue(forKey: vet)\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for key in adjList.keys {\n            adjList[key]?.removeAll(where: { $0 == vet })\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    public func print() {\n        Swift.print(\"\u90bb\u63a5\u8868 =\")\n        for pair in adjList {\n            var tmp: [Int] = []\n            for vertex in pair.value {\n                tmp.append(vertex.val)\n            }\n            Swift.print(\"\\(pair.key.val): \\(tmp),\")\n        }\n    }\n}\n</code></pre> graph_adjacency_list.js<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size() {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1, vet2) {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet) {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet) {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print() {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.ts<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    adjList: Map&lt;Vertex, Vertex[]&gt;;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(edges: Vertex[][]) {\n        this.adjList = new Map();\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for (const edge of edges) {\n            this.addVertex(edge[0]);\n            this.addVertex(edge[1]);\n            this.addEdge(edge[0], edge[1]);\n        }\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    size(): number {\n        return this.adjList.size;\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    addEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).push(vet2);\n        this.adjList.get(vet2).push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    removeEdge(vet1: Vertex, vet2: Vertex): void {\n        if (\n            !this.adjList.has(vet1) ||\n            !this.adjList.has(vet2) ||\n            vet1 === vet2\n        ) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        this.adjList.get(vet1).splice(this.adjList.get(vet1).indexOf(vet2), 1);\n        this.adjList.get(vet2).splice(this.adjList.get(vet2).indexOf(vet1), 1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    addVertex(vet: Vertex): void {\n        if (this.adjList.has(vet)) return;\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        this.adjList.set(vet, []);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    removeVertex(vet: Vertex): void {\n        if (!this.adjList.has(vet)) {\n            throw new Error('Illegal Argument Exception');\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        this.adjList.delete(vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for (const set of this.adjList.values()) {\n            const index: number = set.indexOf(vet);\n            if (index &gt; -1) {\n                set.splice(index, 1);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    print(): void {\n        console.log('\u90bb\u63a5\u8868 =');\n        for (const [key, value] of this.adjList.entries()) {\n            const tmp = [];\n            for (const vertex of value) {\n                tmp.push(vertex.val);\n            }\n            console.log(key.val + ': ' + tmp.join());\n        }\n    }\n}\n</code></pre> graph_adjacency_list.dart<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\nclass GraphAdjList {\n  // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  Map&lt;Vertex, List&lt;Vertex&gt;&gt; adjList = {};\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  GraphAdjList(List&lt;List&lt;Vertex&gt;&gt; edges) {\n    for (List&lt;Vertex&gt; edge in edges) {\n      addVertex(edge[0]);\n      addVertex(edge[1]);\n      addEdge(edge[0], edge[1]);\n    }\n  }\n\n  /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n  int size() {\n    return adjList.length;\n  }\n\n  /* \u6dfb\u52a0\u8fb9 */\n  void addEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    adjList[vet1]!.add(vet2);\n    adjList[vet2]!.add(vet1);\n  }\n\n  /* \u5220\u9664\u8fb9 */\n  void removeEdge(Vertex vet1, Vertex vet2) {\n    if (!adjList.containsKey(vet1) ||\n        !adjList.containsKey(vet2) ||\n        vet1 == vet2) {\n      throw ArgumentError;\n    }\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    adjList[vet1]!.remove(vet2);\n    adjList[vet2]!.remove(vet1);\n  }\n\n  /* \u6dfb\u52a0\u9876\u70b9 */\n  void addVertex(Vertex vet) {\n    if (adjList.containsKey(vet)) return;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    adjList[vet] = [];\n  }\n\n  /* \u5220\u9664\u9876\u70b9 */\n  void removeVertex(Vertex vet) {\n    if (!adjList.containsKey(vet)) {\n      throw ArgumentError;\n    }\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    adjList.remove(vet);\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    adjList.forEach((key, value) {\n      value.remove(vet);\n    });\n  }\n\n  /* \u6253\u5370\u90bb\u63a5\u8868 */\n  void printAdjList() {\n    print(\"\u90bb\u63a5\u8868 =\");\n    adjList.forEach((key, value) {\n      List&lt;int&gt; tmp = [];\n      for (Vertex vertex in value) {\n        tmp.add(vertex.val);\n      }\n      print(\"${key.val}: $tmp,\");\n    });\n  }\n}\n</code></pre> graph_adjacency_list.rs<pre><code>/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b\u578b */\npub struct GraphAdjList {\n    // \u90bb\u63a5\u8868\uff0ckey\uff1a\u9876\u70b9\uff0cvalue\uff1a\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    pub adj_list: HashMap&lt;Vertex, Vec&lt;Vertex&gt;&gt;,\n}\n\nimpl GraphAdjList {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(edges: Vec&lt;[Vertex; 2]&gt;) -&gt; Self {\n        let mut graph = GraphAdjList {\n            adj_list: HashMap::new(),\n        };\n        // \u6dfb\u52a0\u6240\u6709\u9876\u70b9\u548c\u8fb9\n        for edge in edges {\n            graph.add_vertex(edge[0]);\n            graph.add_vertex(edge[1]);\n            graph.add_edge(edge[0], edge[1]);\n        }\n\n        graph\n    }\n\n    /* \u83b7\u53d6\u9876\u70b9\u6570\u91cf */\n    #[allow(unused)]\n    pub fn size(&amp;self) -&gt; usize {\n        self.adj_list.len()\n    }\n\n    /* \u6dfb\u52a0\u8fb9 */\n    pub fn add_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u6dfb\u52a0\u8fb9 vet1 - vet2\n        self.adj_list.get_mut(&amp;vet1).unwrap().push(vet2);\n        self.adj_list.get_mut(&amp;vet2).unwrap().push(vet1);\n    }\n\n    /* \u5220\u9664\u8fb9 */\n    #[allow(unused)]\n    pub fn remove_edge(&amp;mut self, vet1: Vertex, vet2: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet1) || !self.adj_list.contains_key(&amp;vet2) || vet1 == vet2\n        {\n            panic!(\"value error\");\n        }\n        // \u5220\u9664\u8fb9 vet1 - vet2\n        self.adj_list\n            .get_mut(&amp;vet1)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet2);\n        self.adj_list\n            .get_mut(&amp;vet2)\n            .unwrap()\n            .retain(|&amp;vet| vet != vet1);\n    }\n\n    /* \u6dfb\u52a0\u9876\u70b9 */\n    pub fn add_vertex(&amp;mut self, vet: Vertex) {\n        if self.adj_list.contains_key(&amp;vet) {\n            return;\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n        self.adj_list.insert(vet, vec![]);\n    }\n\n    /* \u5220\u9664\u9876\u70b9 */\n    #[allow(unused)]\n    pub fn remove_vertex(&amp;mut self, vet: Vertex) {\n        if !self.adj_list.contains_key(&amp;vet) {\n            panic!(\"value error\");\n        }\n        // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n        self.adj_list.remove(&amp;vet);\n        // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n        for list in self.adj_list.values_mut() {\n            list.retain(|&amp;v| v != vet);\n        }\n    }\n\n    /* \u6253\u5370\u90bb\u63a5\u8868 */\n    pub fn print(&amp;self) {\n        println!(\"\u90bb\u63a5\u8868 =\");\n        for (vertex, list) in &amp;self.adj_list {\n            let list = list.iter().map(|vertex| vertex.val).collect::&lt;Vec&lt;i32&gt;&gt;();\n            println!(\"{}: {:?},\", vertex.val, list);\n        }\n    }\n}\n</code></pre> graph_adjacency_list.c<pre><code>/* \u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct AdjListNode {\n    Vertex *vertex;           // \u9876\u70b9\n    struct AdjListNode *next; // \u540e\u7ee7\u8282\u70b9\n} AdjListNode;\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid addEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *node = (AdjListNode *)malloc(sizeof(AdjListNode));\n    node-&gt;vertex = vet;\n    // \u5934\u63d2\u6cd5\n    node-&gt;next = head-&gt;next;\n    head-&gt;next = node;\n}\n\n/* \u5220\u9664\u8fb9\u8f85\u52a9\u51fd\u6570 */\nvoid removeEdgeHelper(AdjListNode *head, Vertex *vet) {\n    AdjListNode *pre = head;\n    AdjListNode *cur = head-&gt;next;\n    // \u5728\u94fe\u8868\u4e2d\u641c\u7d22 vet \u5bf9\u5e94\u8282\u70b9\n    while (cur != NULL &amp;&amp; cur-&gt;vertex != vet) {\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n    if (cur == NULL)\n        return;\n    // \u5c06 vet \u5bf9\u5e94\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\n    pre-&gt;next = cur-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(cur);\n}\n\n/* \u57fa\u4e8e\u90bb\u63a5\u8868\u5b9e\u73b0\u7684\u65e0\u5411\u56fe\u7c7b */\ntypedef struct {\n    AdjListNode *heads[MAX_SIZE]; // \u8282\u70b9\u6570\u7ec4\n    int size;                     // \u8282\u70b9\u6570\u91cf\n} GraphAdjList;\n\n/* \u6784\u9020\u51fd\u6570 */\nGraphAdjList *newGraphAdjList() {\n    GraphAdjList *graph = (GraphAdjList *)malloc(sizeof(GraphAdjList));\n    if (!graph) {\n        return NULL;\n    }\n    graph-&gt;size = 0;\n    for (int i = 0; i &lt; MAX_SIZE; i++) {\n        graph-&gt;heads[i] = NULL;\n    }\n    return graph;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delGraphAdjList(GraphAdjList *graph) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        AdjListNode *cur = graph-&gt;heads[i];\n        while (cur != NULL) {\n            AdjListNode *next = cur-&gt;next;\n            if (cur != graph-&gt;heads[i]) {\n                free(cur);\n            }\n            cur = next;\n        }\n        free(graph-&gt;heads[i]-&gt;vertex);\n        free(graph-&gt;heads[i]);\n    }\n    free(graph);\n}\n\n/* \u67e5\u627e\u9876\u70b9\u5bf9\u5e94\u7684\u8282\u70b9 */\nAdjListNode *findNode(GraphAdjList *graph, Vertex *vet) {\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i]-&gt;vertex == vet) {\n            return graph-&gt;heads[i];\n        }\n    }\n    return NULL;\n}\n\n/* \u6dfb\u52a0\u8fb9 */\nvoid addEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL &amp;&amp; head1 != head2);\n    // \u6dfb\u52a0\u8fb9 vet1 - vet2\n    addEdgeHelper(head1, vet2);\n    addEdgeHelper(head2, vet1);\n}\n\n/* \u5220\u9664\u8fb9 */\nvoid removeEdge(GraphAdjList *graph, Vertex *vet1, Vertex *vet2) {\n    AdjListNode *head1 = findNode(graph, vet1);\n    AdjListNode *head2 = findNode(graph, vet2);\n    assert(head1 != NULL &amp;&amp; head2 != NULL);\n    // \u5220\u9664\u8fb9 vet1 - vet2\n    removeEdgeHelper(head1, head2-&gt;vertex);\n    removeEdgeHelper(head2, head1-&gt;vertex);\n}\n\n/* \u6dfb\u52a0\u9876\u70b9 */\nvoid addVertex(GraphAdjList *graph, Vertex *vet) {\n    assert(graph != NULL &amp;&amp; graph-&gt;size &lt; MAX_SIZE);\n    AdjListNode *head = (AdjListNode *)malloc(sizeof(AdjListNode));\n    head-&gt;vertex = vet;\n    head-&gt;next = NULL;\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u6dfb\u52a0\u4e00\u4e2a\u65b0\u94fe\u8868\n    graph-&gt;heads[graph-&gt;size++] = head;\n}\n\n/* \u5220\u9664\u9876\u70b9 */\nvoid removeVertex(GraphAdjList *graph, Vertex *vet) {\n    AdjListNode *node = findNode(graph, vet);\n    assert(node != NULL);\n    // \u5728\u90bb\u63a5\u8868\u4e2d\u5220\u9664\u9876\u70b9 vet \u5bf9\u5e94\u7684\u94fe\u8868\n    AdjListNode *cur = node, *pre = NULL;\n    while (cur) {\n        pre = cur;\n        cur = cur-&gt;next;\n        free(pre);\n    }\n    // \u904d\u5386\u5176\u4ed6\u9876\u70b9\u7684\u94fe\u8868\uff0c\u5220\u9664\u6240\u6709\u5305\u542b vet \u7684\u8fb9\n    for (int i = 0; i &lt; graph-&gt;size; i++) {\n        cur = graph-&gt;heads[i];\n        pre = NULL;\n        while (cur) {\n            pre = cur;\n            cur = cur-&gt;next;\n            if (cur &amp;&amp; cur-&gt;vertex == vet) {\n                pre-&gt;next = cur-&gt;next;\n                free(cur);\n                break;\n            }\n        }\n    }\n    // \u5c06\u8be5\u9876\u70b9\u4e4b\u540e\u7684\u9876\u70b9\u5411\u524d\u79fb\u52a8\uff0c\u4ee5\u586b\u8865\u7a7a\u7f3a\n    int i;\n    for (i = 0; i &lt; graph-&gt;size; i++) {\n        if (graph-&gt;heads[i] == node)\n            break;\n    }\n    for (int j = i; j &lt; graph-&gt;size - 1; j++) {\n        graph-&gt;heads[j] = graph-&gt;heads[j + 1];\n    }\n    graph-&gt;size--;\n    free(vet);\n}\n</code></pre> graph_adjacency_list.zig<pre><code>[class]{GraphAdjList}-[func]{}\n</code></pre>"},{"location":"chapter_graph/graph_operations/#923","title":"9.2.3 \u00a0 \u6548\u7387\u5bf9\u6bd4","text":"<p>\u8bbe\u56fe\u4e2d\u5171\u6709 \\(n\\) \u4e2a\u9876\u70b9\u548c \\(m\\) \u6761\u8fb9\uff0c\u8868 9-2 \u5bf9\u6bd4\u4e86\u90bb\u63a5\u77e9\u9635\u548c\u90bb\u63a5\u8868\u7684\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u3002</p> <p> \u8868 9-2 \u00a0 \u90bb\u63a5\u77e9\u9635\u4e0e\u90bb\u63a5\u8868\u5bf9\u6bd4 </p> \u90bb\u63a5\u77e9\u9635 \u90bb\u63a5\u8868\uff08\u94fe\u8868\uff09 \u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09 \u5224\u65ad\u662f\u5426\u90bb\u63a5 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u8fb9 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u8fb9 \\(O(1)\\) \\(O(m)\\) \\(O(1)\\) \u6dfb\u52a0\u9876\u70b9 \\(O(n)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u9876\u70b9 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n)\\) \u5185\u5b58\u7a7a\u95f4\u5360\u7528 \\(O(n^2)\\) \\(O(n + m)\\) \\(O(n + m)\\) <p>\u89c2\u5bdf\u8868 9-2 \uff0c\u4f3c\u4e4e\u90bb\u63a5\u8868\uff08\u54c8\u5e0c\u8868\uff09\u7684\u65f6\u95f4\u6548\u7387\u4e0e\u7a7a\u95f4\u6548\u7387\u6700\u4f18\u3002\u4f46\u5b9e\u9645\u4e0a\uff0c\u5728\u90bb\u63a5\u77e9\u9635\u4e2d\u64cd\u4f5c\u8fb9\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u53ea\u9700\u4e00\u6b21\u6570\u7ec4\u8bbf\u95ee\u6216\u8d4b\u503c\u64cd\u4f5c\u5373\u53ef\u3002\u7efc\u5408\u6765\u770b\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\u7684\u539f\u5219\uff0c\u800c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u7684\u539f\u5219\u3002</p>"},{"location":"chapter_graph/graph_traversal/","title":"9.3 \u00a0 \u56fe\u7684\u904d\u5386","text":"<p>\u6811\u4ee3\u8868\u7684\u662f\u201c\u4e00\u5bf9\u591a\u201d\u7684\u5173\u7cfb\uff0c\u800c\u56fe\u5219\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u53ef\u4ee5\u8868\u793a\u4efb\u610f\u7684\u201c\u591a\u5bf9\u591a\u201d\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6811\u770b\u4f5c\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\u3002\u663e\u7136\uff0c\u6811\u7684\u904d\u5386\u64cd\u4f5c\u4e5f\u662f\u56fe\u7684\u904d\u5386\u64cd\u4f5c\u7684\u4e00\u79cd\u7279\u4f8b\u3002</p> <p>\u56fe\u548c\u6811\u90fd\u9700\u8981\u5e94\u7528\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u904d\u5386\u64cd\u4f5c\u3002\u56fe\u7684\u904d\u5386\u65b9\u5f0f\u53ef\u5206\u4e3a\u4e24\u79cd\uff1a\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal\u300d\u3002\u5b83\u4eec\u4e5f\u5e38\u88ab\u79f0\u4e3a\u300c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22 breadth-first search\u300d\u548c\u300c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22 depth-first search\u300d\uff0c\u7b80\u79f0 BFS \u548c DFS \u3002</p>"},{"location":"chapter_graph/graph_traversal/#931","title":"9.3.1 \u00a0 \u5e7f\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u4ece\u67d0\u4e2a\u8282\u70b9\u51fa\u53d1\uff0c\u59cb\u7ec8\u4f18\u5148\u8bbf\u95ee\u8ddd\u79bb\u6700\u8fd1\u7684\u9876\u70b9\uff0c\u5e76\u4e00\u5c42\u5c42\u5411\u5916\u6269\u5f20\u3002\u5982\u56fe 9-9 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u9996\u5148\u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u7136\u540e\u904d\u5386\u4e0b\u4e00\u4e2a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u8bbf\u95ee\u5b8c\u6bd5\u3002</p> <p></p> <p> \u56fe 9-9 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>BFS \u901a\u5e38\u501f\u52a9\u961f\u5217\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u961f\u5217\u5177\u6709\u201c\u5148\u5165\u5148\u51fa\u201d\u7684\u6027\u8d28\uff0c\u8fd9\u4e0e BFS \u7684\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u601d\u60f3\u5f02\u66f2\u540c\u5de5\u3002</p> <ol> <li>\u5c06\u904d\u5386\u8d77\u59cb\u9876\u70b9 <code>startVet</code> \u52a0\u5165\u961f\u5217\uff0c\u5e76\u5f00\u542f\u5faa\u73af\u3002</li> <li>\u5728\u5faa\u73af\u7684\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\uff0c\u5f39\u51fa\u961f\u9996\u9876\u70b9\u5e76\u8bb0\u5f55\u8bbf\u95ee\uff0c\u7136\u540e\u5c06\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u52a0\u5165\u5230\u961f\u5217\u5c3e\u90e8\u3002</li> <li>\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230\u6240\u6709\u9876\u70b9\u88ab\u8bbf\u95ee\u5b8c\u6bd5\u540e\u7ed3\u675f\u3002</li> </ol> <p>\u4e3a\u4e86\u9632\u6b62\u91cd\u590d\u904d\u5386\u9876\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u54ea\u4e9b\u8282\u70b9\u5df2\u88ab\u8bbf\u95ee\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_bfs.py<pre><code>def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]([start_vet])\n    # \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    que = deque[Vertex]([start_vet])\n    # \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while len(que) &gt; 0:\n        vet = que.popleft()  # \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adj_vet in graph.adj_list[vet]:\n            if adj_vet in visited:\n                continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.append(adj_vet)  # \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adj_vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n</code></pre> graph_bfs.cpp<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphBFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited = {startVet};\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    queue&lt;Vertex *&gt; que;\n    que.push(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.empty()) {\n        Vertex *vet = que.front();\n        que.pop();          // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push_back(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (auto adjVet : graph.adjList[vet]) {\n            if (visited.count(adjVet))\n                continue;            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.push(adjVet);        // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.emplace(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.java<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new LinkedList&lt;&gt;();\n    que.offer(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!que.isEmpty()) {\n        Vertex vet = que.poll(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.add(vet);            // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (Vertex adjVet : graph.adjList.get(vet)) {\n            if (visited.contains(adjVet))\n                continue;        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            que.offer(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.cs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphBFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [startVet];\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue&lt;Vertex&gt; que = new();\n    que.Enqueue(startVet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.Count &gt; 0) {\n        Vertex vet = que.Dequeue(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.Add(vet);               // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        foreach (Vertex adjVet in graph.adjList[vet]) {\n            if (visited.Contains(adjVet)) {\n                continue;          // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.Enqueue(adjVet);   // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.Add(adjVet);   // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.go<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    visited[startVet] = struct{}{}\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS, \u4f7f\u7528\u5207\u7247\u6a21\u62df\u961f\u5217\n    queue := make([]Vertex, 0)\n    queue = append(queue, startVet)\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    for len(queue) &gt; 0 {\n        // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        vet := queue[0]\n        queue = queue[1:]\n        // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        res = append(res, vet)\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for _, adjVet := range g.adjList[vet] {\n            _, isExist := visited[adjVet]\n            // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            if !isExist {\n                queue = append(queue, adjVet)\n                visited[adjVet] = struct{}{}\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.swift<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphBFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = [startVet]\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    var que: [Vertex] = [startVet]\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.isEmpty {\n        let vet = que.removeFirst() // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for adjVet in graph.adjList[vet] ?? [] {\n            if visited.contains(adjVet) {\n                continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.append(adjVet) // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.insert(adjVet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_bfs.js<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.ts<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    visited.add(startVet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    const que = [startVet];\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (que.length) {\n        const vet = que.shift(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        for (const adjVet of graph.adjList.get(vet) ?? []) {\n            if (visited.has(adjVet)) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            que.push(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\n            visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res;\n}\n</code></pre> graph_bfs.dart<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\nList&lt;Vertex&gt; graphBFS(GraphAdjList graph, Vertex startVet) {\n  // \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  visited.add(startVet);\n  // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n  Queue&lt;Vertex&gt; que = Queue();\n  que.add(startVet);\n  // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n  while (que.isNotEmpty) {\n    Vertex vet = que.removeFirst(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n    res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet in graph.adjList[vet]!) {\n      if (visited.contains(adjVet)) {\n        continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n      }\n      que.add(adjVet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n      visited.add(adjVet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    }\n  }\n  // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n  return res;\n}\n</code></pre> graph_bfs.rs<pre><code>/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    visited.insert(start_vet);\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    let mut que = VecDeque::new();\n    que.push_back(start_vet);\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while !que.is_empty() {\n        let vet = que.pop_front().unwrap(); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n                    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n            for &amp;adj_vet in adj_vets {\n                if visited.contains(&amp;adj_vet) {\n                    continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n                }\n                que.push_back(adj_vet); // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited.insert(adj_vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n        }\n    }\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res\n}\n</code></pre> graph_bfs.c<pre><code>/* \u8282\u70b9\u961f\u5217\u7ed3\u6784\u4f53 */\ntypedef struct {\n    Vertex *vertices[MAX_SIZE];\n    int front, rear, size;\n} Queue;\n\n/* \u6784\u9020\u51fd\u6570 */\nQueue *newQueue() {\n    Queue *q = (Queue *)malloc(sizeof(Queue));\n    q-&gt;front = q-&gt;rear = q-&gt;size = 0;\n    return q;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nint isEmpty(Queue *q) {\n    return q-&gt;size == 0;\n}\n\n/* \u5165\u961f\u64cd\u4f5c */\nvoid enqueue(Queue *q, Vertex *vet) {\n    q-&gt;vertices[q-&gt;rear] = vet;\n    q-&gt;rear = (q-&gt;rear + 1) % MAX_SIZE;\n    q-&gt;size++;\n}\n\n/* \u51fa\u961f\u64cd\u4f5c */\nVertex *dequeue(Queue *q) {\n    Vertex *vet = q-&gt;vertices[q-&gt;front];\n    q-&gt;front = (q-&gt;front + 1) % MAX_SIZE;\n    q-&gt;size--;\n    return vet;\n}\n\n/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **visited, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (visited[i] == vet)\n            return 1;\n    }\n    return 0;\n}\n\n/* \u5e7f\u5ea6\u4f18\u5148\u904d\u5386 BFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphBFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize, Vertex **visited, int *visitedSize) {\n    // \u961f\u5217\u7528\u4e8e\u5b9e\u73b0 BFS\n    Queue *queue = newQueue();\n    enqueue(queue, startVet);\n    visited[(*visitedSize)++] = startVet;\n    // \u4ee5\u9876\u70b9 vet \u4e3a\u8d77\u70b9\uff0c\u5faa\u73af\u76f4\u81f3\u8bbf\u95ee\u5b8c\u6240\u6709\u9876\u70b9\n    while (!isEmpty(queue)) {\n        Vertex *vet = dequeue(queue); // \u961f\u9996\u9876\u70b9\u51fa\u961f\n        res[(*resSize)++] = vet;      // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n        // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n        AdjListNode *node = findNode(graph, vet);\n        while (node != NULL) {\n            // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            if (!isVisited(visited, *visitedSize, node-&gt;vertex)) {\n                enqueue(queue, node-&gt;vertex);             // \u53ea\u5165\u961f\u672a\u8bbf\u95ee\u7684\u9876\u70b9\n                visited[(*visitedSize)++] = node-&gt;vertex; // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n            }\n            node = node-&gt;next;\n        }\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(queue);\n}\n</code></pre> graph_bfs.zig<pre><code>[class]{}-[func]{graphBFS}\n</code></pre> <p>\u4ee3\u7801\u76f8\u5bf9\u62bd\u8c61\uff0c\u5efa\u8bae\u5bf9\u7167\u56fe 9-10 \u6765\u52a0\u6df1\u7406\u89e3\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-10 \u00a0 \u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0d\u552f\u4e00\u3002\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u53ea\u8981\u6c42\u6309\u201c\u7531\u8fd1\u53ca\u8fdc\u201d\u7684\u987a\u5e8f\u904d\u5386\uff0c\u800c\u591a\u4e2a\u76f8\u540c\u8ddd\u79bb\u7684\u9876\u70b9\u7684\u904d\u5386\u987a\u5e8f\u5141\u8bb8\u88ab\u4efb\u610f\u6253\u4e71\u3002\u4ee5\u56fe 9-10 \u4e3a\u4f8b\uff0c\u9876\u70b9 \\(1\\)\u3001\\(3\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\uff0c\u9876\u70b9 \\(2\\)\u3001\\(4\\)\u3001\\(6\\) \u7684\u8bbf\u95ee\u987a\u5e8f\u4e5f\u53ef\u4ee5\u4efb\u610f\u4ea4\u6362\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u5165\u961f\u5e76\u51fa\u961f\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u5728\u904d\u5386\u90bb\u63a5\u9876\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u7531\u4e8e\u662f\u65e0\u5411\u56fe\uff0c\u56e0\u6b64\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \uff0c\u961f\u5217 <code>que</code> \u4e2d\u7684\u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/graph_traversal/#932","title":"9.3.2 \u00a0 \u6df1\u5ea6\u4f18\u5148\u904d\u5386","text":"<p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u518d\u56de\u5934\u7684\u904d\u5386\u65b9\u5f0f\u3002\u5982\u56fe 9-11 \u6240\u793a\uff0c\u4ece\u5de6\u4e0a\u89d2\u9876\u70b9\u51fa\u53d1\uff0c\u8bbf\u95ee\u5f53\u524d\u9876\u70b9\u7684\u67d0\u4e2a\u90bb\u63a5\u9876\u70b9\uff0c\u76f4\u5230\u8d70\u5230\u5c3d\u5934\u65f6\u8fd4\u56de\uff0c\u518d\u7ee7\u7eed\u8d70\u5230\u5c3d\u5934\u5e76\u8fd4\u56de\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u81f3\u6240\u6709\u9876\u70b9\u904d\u5386\u5b8c\u6210\u3002</p> <p></p> <p> \u56fe 9-11 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386 </p>"},{"location":"chapter_graph/graph_traversal/#1_1","title":"1. \u00a0 \u7b97\u6cd5\u5b9e\u73b0","text":"<p>\u8fd9\u79cd\u201c\u8d70\u5230\u5c3d\u5934\u518d\u8fd4\u56de\u201d\u7684\u7b97\u6cd5\u8303\u5f0f\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u5728\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u4e2d\uff0c\u6211\u4eec\u4e5f\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868 <code>visited</code> \u6765\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\uff0c\u4ee5\u907f\u514d\u91cd\u590d\u8bbf\u95ee\u9876\u70b9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig graph_dfs.py<pre><code>def dfs(graph: GraphAdjList, visited: set[Vertex], res: list[Vertex], vet: Vertex):\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570\"\"\"\n    res.append(vet)  # \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet)  # \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    # \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adj_list[vet]:\n        if adjVet in visited:\n            continue  # \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        # \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet)\n\ndef graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; list[Vertex]:\n    \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS\"\"\"\n    # \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    # \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res = []\n    # \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited = set[Vertex]()\n    dfs(graph, visited, res, start_vet)\n    return res\n</code></pre> graph_dfs.cpp<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList &amp;graph, unordered_set&lt;Vertex *&gt; &amp;visited, vector&lt;Vertex *&gt; &amp;res, Vertex *vet) {\n    res.push_back(vet);   // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.emplace(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex *adjVet : graph.adjList[vet]) {\n        if (visited.count(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvector&lt;Vertex *&gt; graphDFS(GraphAdjList &amp;graph, Vertex *startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    vector&lt;Vertex *&gt; res;\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    unordered_set&lt;Vertex *&gt; visited;\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.java<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList graph, Set&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (Vertex adjVet : graph.adjList.get(vet)) {\n        if (visited.contains(adjVet))\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = new ArrayList&lt;&gt;();\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    Set&lt;Vertex&gt; visited = new HashSet&lt;&gt;();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.cs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid DFS(GraphAdjList graph, HashSet&lt;Vertex&gt; visited, List&lt;Vertex&gt; res, Vertex vet) {\n    res.Add(vet);     // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.Add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    foreach (Vertex adjVet in graph.adjList[vet]) {\n        if (visited.Contains(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9                             \n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        DFS(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nList&lt;Vertex&gt; GraphDFS(GraphAdjList graph, Vertex startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    List&lt;Vertex&gt; res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    HashSet&lt;Vertex&gt; visited = [];\n    DFS(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.go<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex) {\n    // append \u64cd\u4f5c\u4f1a\u8fd4\u56de\u65b0\u7684\u7684\u5f15\u7528\uff0c\u5fc5\u987b\u8ba9\u539f\u5f15\u7528\u91cd\u65b0\u8d4b\u503c\u4e3a\u65b0slice\u7684\u5f15\u7528\n    *res = append(*res, vet)\n    visited[vet] = struct{}{}\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for _, adjVet := range g.adjList[vet] {\n        _, isExist := visited[adjVet]\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        if !isExist {\n            dfs(g, visited, res, adjVet)\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(g *graphAdjList, startVet Vertex) []Vertex {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    res := make([]Vertex, 0)\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    visited := make(map[Vertex]struct{})\n    dfs(g, visited, &amp;res, startVet)\n    // \u8fd4\u56de\u9876\u70b9\u904d\u5386\u5e8f\u5217\n    return res\n}\n</code></pre> graph_dfs.swift<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunc dfs(graph: GraphAdjList, visited: inout Set&lt;Vertex&gt;, res: inout [Vertex], vet: Vertex) {\n    res.append(vet) // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet) // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for adjVet in graph.adjList[vet] ?? [] {\n        if visited.contains(adjVet) {\n            continue // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: adjVet)\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunc graphDFS(graph: GraphAdjList, startVet: Vertex) -&gt; [Vertex] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    var res: [Vertex] = []\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    var visited: Set&lt;Vertex&gt; = []\n    dfs(graph: graph, visited: &amp;visited, res: &amp;res, vet: startVet)\n    return res\n}\n</code></pre> graph_dfs.js<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction dfs(graph, visited, res, vet) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph, startVet) {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.ts<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfunction dfs(\n    graph: GraphAdjList,\n    visited: Set&lt;Vertex&gt;,\n    res: Vertex[],\n    vet: Vertex\n): void {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    for (const adjVet of graph.adjList.get(vet)) {\n        if (visited.has(adjVet)) {\n            continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        }\n        // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n        dfs(graph, visited, res, adjVet);\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfunction graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    const res: Vertex[] = [];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    const visited: Set&lt;Vertex&gt; = new Set();\n    dfs(graph, visited, res, startVet);\n    return res;\n}\n</code></pre> graph_dfs.dart<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(\n  GraphAdjList graph,\n  Set&lt;Vertex&gt; visited,\n  List&lt;Vertex&gt; res,\n  Vertex vet,\n) {\n  res.add(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n  visited.add(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n  // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n  for (Vertex adjVet in graph.adjList[vet]!) {\n    if (visited.contains(adjVet)) {\n      continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n    }\n    // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n    dfs(graph, visited, res, adjVet);\n  }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\nList&lt;Vertex&gt; graphDFS(GraphAdjList graph, Vertex startVet) {\n  // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n  List&lt;Vertex&gt; res = [];\n  // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n  Set&lt;Vertex&gt; visited = {};\n  dfs(graph, visited, res, startVet);\n  return res;\n}\n</code></pre> graph_dfs.rs<pre><code>/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nfn dfs(graph: &amp;GraphAdjList, visited: &amp;mut HashSet&lt;Vertex&gt;, res: &amp;mut Vec&lt;Vertex&gt;, vet: Vertex) {\n    res.push(vet); // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    visited.insert(vet); // \u6807\u8bb0\u8be5\u9876\u70b9\u5df2\u88ab\u8bbf\u95ee\n                         // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    if let Some(adj_vets) = graph.adj_list.get(&amp;vet) {\n        for &amp;adj_vet in adj_vets {\n            if visited.contains(&amp;adj_vet) {\n                continue; // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n            }\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, visited, res, adj_vet);\n        }\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nfn graph_dfs(graph: GraphAdjList, start_vet: Vertex) -&gt; Vec&lt;Vertex&gt; {\n    // \u9876\u70b9\u904d\u5386\u5e8f\u5217\n    let mut res = vec![];\n    // \u54c8\u5e0c\u8868\uff0c\u7528\u4e8e\u8bb0\u5f55\u5df2\u88ab\u8bbf\u95ee\u8fc7\u7684\u9876\u70b9\n    let mut visited = HashSet::new();\n    dfs(&amp;graph, &amp;mut visited, &amp;mut res, start_vet);\n\n    res\n}\n</code></pre> graph_dfs.c<pre><code>/* \u68c0\u67e5\u9876\u70b9\u662f\u5426\u5df2\u88ab\u8bbf\u95ee */\nint isVisited(Vertex **res, int size, Vertex *vet) {\n    // \u904d\u5386\u67e5\u627e\u8282\u70b9\uff0c\u4f7f\u7528 O(n) \u65f6\u95f4\n    for (int i = 0; i &lt; size; i++) {\n        if (res[i] == vet) {\n            return 1;\n        }\n    }\n    return 0;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS \u8f85\u52a9\u51fd\u6570 */\nvoid dfs(GraphAdjList *graph, Vertex **res, int *resSize, Vertex *vet) {\n    // \u8bb0\u5f55\u8bbf\u95ee\u9876\u70b9\n    res[(*resSize)++] = vet;\n    // \u904d\u5386\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\n    AdjListNode *node = findNode(graph, vet);\n    while (node != NULL) {\n        // \u8df3\u8fc7\u5df2\u88ab\u8bbf\u95ee\u7684\u9876\u70b9\n        if (!isVisited(res, *resSize, node-&gt;vertex)) {\n            // \u9012\u5f52\u8bbf\u95ee\u90bb\u63a5\u9876\u70b9\n            dfs(graph, res, resSize, node-&gt;vertex);\n        }\n        node = node-&gt;next;\n    }\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 DFS */\n// \u4f7f\u7528\u90bb\u63a5\u8868\u6765\u8868\u793a\u56fe\uff0c\u4ee5\u4fbf\u83b7\u53d6\u6307\u5b9a\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\nvoid graphDFS(GraphAdjList *graph, Vertex *startVet, Vertex **res, int *resSize) {\n    dfs(graph, res, resSize, startVet);\n}\n</code></pre> graph_dfs.zig<pre><code>[class]{}-[func]{dfs}\n\n[class]{}-[func]{graphDFS}\n</code></pre> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 9-12 \u6240\u793a\u3002</p> <ul> <li>\u76f4\u865a\u7ebf\u4ee3\u8868\u5411\u4e0b\u9012\u63a8\uff0c\u8868\u793a\u5f00\u542f\u4e86\u4e00\u4e2a\u65b0\u7684\u9012\u5f52\u65b9\u6cd5\u6765\u8bbf\u95ee\u65b0\u9876\u70b9\u3002</li> <li>\u66f2\u865a\u7ebf\u4ee3\u8868\u5411\u4e0a\u56de\u6eaf\uff0c\u8868\u793a\u6b64\u9012\u5f52\u65b9\u6cd5\u5df2\u7ecf\u8fd4\u56de\uff0c\u56de\u6eaf\u5230\u4e86\u5f00\u542f\u6b64\u65b9\u6cd5\u7684\u4f4d\u7f6e\u3002</li> </ul> <p>\u4e3a\u4e86\u52a0\u6df1\u7406\u89e3\uff0c\u5efa\u8bae\u5c06\u56fe 9-12 \u4e0e\u4ee3\u7801\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u8111\u4e2d\u6a21\u62df\uff08\u6216\u8005\u7528\u7b14\u753b\u4e0b\u6765\uff09\u6574\u4e2a DFS \u8fc7\u7a0b\uff0c\u5305\u62ec\u6bcf\u4e2a\u9012\u5f52\u65b9\u6cd5\u4f55\u65f6\u5f00\u542f\u3001\u4f55\u65f6\u8fd4\u56de\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 9-12 \u00a0 \u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u6b65\u9aa4 </p> <p>\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5e8f\u5217\u662f\u5426\u552f\u4e00\uff1f</p> <p>\u4e0e\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u7c7b\u4f3c\uff0c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5e8f\u5217\u7684\u987a\u5e8f\u4e5f\u4e0d\u662f\u552f\u4e00\u7684\u3002\u7ed9\u5b9a\u67d0\u9876\u70b9\uff0c\u5148\u5f80\u54ea\u4e2a\u65b9\u5411\u63a2\u7d22\u90fd\u53ef\u4ee5\uff0c\u5373\u90bb\u63a5\u9876\u70b9\u7684\u987a\u5e8f\u53ef\u4ee5\u4efb\u610f\u6253\u4e71\uff0c\u90fd\u662f\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p> <p>\u4ee5\u6811\u7684\u904d\u5386\u4e3a\u4f8b\uff0c\u201c\u6839 \\(\\rightarrow\\) \u5de6 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u201c\u5de6 \\(\\rightarrow\\) \u53f3 \\(\\rightarrow\\) \u6839\u201d\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\uff0c\u5b83\u4eec\u5c55\u793a\u4e86\u4e09\u79cd\u904d\u5386\u4f18\u5148\u7ea7\uff0c\u7136\u800c\u8fd9\u4e09\u8005\u90fd\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u3002</p>"},{"location":"chapter_graph/graph_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\u6240\u6709\u9876\u70b9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(1\\) \u6b21\uff0c\u4f7f\u7528 \\(O(|V|)\\) \u65f6\u95f4\uff1b\u6240\u6709\u8fb9\u90fd\u4f1a\u88ab\u8bbf\u95ee \\(2\\) \u6b21\uff0c\u4f7f\u7528 \\(O(2|E|)\\) \u65f6\u95f4\uff1b\u603b\u4f53\u4f7f\u7528 \\(O(|V| + |E|)\\) \u65f6\u95f4\u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6\uff1a\u5217\u8868 <code>res</code> \uff0c\u54c8\u5e0c\u8868 <code>visited</code> \u9876\u70b9\u6570\u91cf\u6700\u591a\u4e3a \\(|V|\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u6700\u5927\u4e3a \\(|V|\\) \uff0c\u56e0\u6b64\u4f7f\u7528 \\(O(|V|)\\) \u7a7a\u95f4\u3002</p>"},{"location":"chapter_graph/summary/","title":"9.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_graph/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u56fe\u7531\u9876\u70b9\u548c\u8fb9\u7ec4\u6210\uff0c\u53ef\u4ee5\u8868\u793a\u4e3a\u4e00\u7ec4\u9876\u70b9\u548c\u4e00\u7ec4\u8fb9\u6784\u6210\u7684\u96c6\u5408\u3002</li> <li>\u76f8\u8f83\u4e8e\u7ebf\u6027\u5173\u7cfb\uff08\u94fe\u8868\uff09\u548c\u5206\u6cbb\u5173\u7cfb\uff08\u6811\uff09\uff0c\u7f51\u7edc\u5173\u7cfb\uff08\u56fe\uff09\u5177\u6709\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\uff0c\u56e0\u800c\u66f4\u4e3a\u590d\u6742\u3002</li> <li>\u6709\u5411\u56fe\u7684\u8fb9\u5177\u6709\u65b9\u5411\u6027\uff0c\u8fde\u901a\u56fe\u4e2d\u7684\u4efb\u610f\u9876\u70b9\u5747\u53ef\u8fbe\uff0c\u6709\u6743\u56fe\u7684\u6bcf\u6761\u8fb9\u90fd\u5305\u542b\u6743\u91cd\u53d8\u91cf\u3002</li> <li>\u90bb\u63a5\u77e9\u9635\u5229\u7528\u77e9\u9635\u6765\u8868\u793a\u56fe\uff0c\u6bcf\u4e00\u884c\uff08\u5217\uff09\u4ee3\u8868\u4e00\u4e2a\u9876\u70b9\uff0c\u77e9\u9635\u5143\u7d20\u4ee3\u8868\u8fb9\uff0c\u7528 \\(1\\) \u6216 \\(0\\) \u8868\u793a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u6709\u8fb9\u6216\u65e0\u8fb9\u3002\u90bb\u63a5\u77e9\u9635\u5728\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u4e0a\u6548\u7387\u5f88\u9ad8\uff0c\u4f46\u7a7a\u95f4\u5360\u7528\u8f83\u591a\u3002</li> <li>\u90bb\u63a5\u8868\u4f7f\u7528\u591a\u4e2a\u94fe\u8868\u6765\u8868\u793a\u56fe\uff0c\u7b2c \\(i\\) \u4e2a\u94fe\u8868\u5bf9\u5e94\u9876\u70b9 \\(i\\) \uff0c\u5176\u4e2d\u5b58\u50a8\u4e86\u8be5\u9876\u70b9\u7684\u6240\u6709\u90bb\u63a5\u9876\u70b9\u3002\u90bb\u63a5\u8868\u76f8\u5bf9\u4e8e\u90bb\u63a5\u77e9\u9635\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\uff0c\u4f46\u7531\u4e8e\u9700\u8981\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u8fb9\uff0c\u56e0\u6b64\u65f6\u95f4\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u90bb\u63a5\u8868\u4e2d\u7684\u94fe\u8868\u8fc7\u957f\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6216\u54c8\u5e0c\u8868\uff0c\u4ece\u800c\u63d0\u5347\u67e5\u8be2\u6548\u7387\u3002</li> <li>\u4ece\u7b97\u6cd5\u601d\u60f3\u7684\u89d2\u5ea6\u5206\u6790\uff0c\u90bb\u63a5\u77e9\u9635\u4f53\u73b0\u4e86\u201c\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4\u201d\uff0c\u90bb\u63a5\u8868\u4f53\u73b0\u4e86\u201c\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4\u201d\u3002</li> <li>\u56fe\u53ef\u7528\u4e8e\u5efa\u6a21\u5404\u7c7b\u73b0\u5b9e\u7cfb\u7edf\uff0c\u5982\u793e\u4ea4\u7f51\u7edc\u3001\u5730\u94c1\u7ebf\u8def\u7b49\u3002</li> <li>\u6811\u662f\u56fe\u7684\u4e00\u79cd\u7279\u4f8b\uff0c\u6811\u7684\u904d\u5386\u4e5f\u662f\u56fe\u7684\u904d\u5386\u7684\u4e00\u79cd\u7279\u4f8b\u3002</li> <li>\u56fe\u7684\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u7531\u8fd1\u53ca\u8fdc\u3001\u5c42\u5c42\u6269\u5f20\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u901a\u5e38\u501f\u52a9\u961f\u5217\u5b9e\u73b0\u3002</li> <li>\u56fe\u7684\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u662f\u4e00\u79cd\u4f18\u5148\u8d70\u5230\u5e95\u3001\u65e0\u8def\u53ef\u8d70\u65f6\u518d\u56de\u6eaf\u7684\u641c\u7d22\u65b9\u5f0f\uff0c\u5e38\u57fa\u4e8e\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> </ul>"},{"location":"chapter_graph/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u8def\u5f84\u7684\u5b9a\u4e49\u662f\u9876\u70b9\u5e8f\u5217\u8fd8\u662f\u8fb9\u5e8f\u5217\uff1f</p> <p>\u7ef4\u57fa\u767e\u79d1\u4e0a\u4e0d\u540c\u8bed\u8a00\u7248\u672c\u7684\u5b9a\u4e49\u4e0d\u4e00\u81f4\uff1a\u82f1\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u8fb9\u5e8f\u5217\u201d\uff0c\u800c\u4e2d\u6587\u7248\u662f\u201c\u8def\u5f84\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u201d\u3002\u4ee5\u4e0b\u662f\u82f1\u6587\u7248\u539f\u6587\uff1aIn graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices. \u5728\u672c\u6587\u4e2d\uff0c\u8def\u5f84\u88ab\u89c6\u4e3a\u4e00\u4e2a\u8fb9\u5e8f\u5217\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u9876\u70b9\u5e8f\u5217\u3002\u8fd9\u662f\u56e0\u4e3a\u4e24\u4e2a\u9876\u70b9\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u591a\u6761\u8fb9\u8fde\u63a5\uff0c\u6b64\u65f6\u6bcf\u6761\u8fb9\u90fd\u5bf9\u5e94\u4e00\u6761\u8def\u5f84\u3002</p> <p>\u975e\u8fde\u901a\u56fe\u4e2d\u662f\u5426\u4f1a\u6709\u65e0\u6cd5\u904d\u5386\u5230\u7684\u70b9\uff1f</p> <p>\u5728\u975e\u8fde\u901a\u56fe\u4e2d\uff0c\u4ece\u67d0\u4e2a\u9876\u70b9\u51fa\u53d1\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u9876\u70b9\u65e0\u6cd5\u5230\u8fbe\u3002\u904d\u5386\u975e\u8fde\u901a\u56fe\u9700\u8981\u8bbe\u7f6e\u591a\u4e2a\u8d77\u70b9\uff0c\u4ee5\u904d\u5386\u5230\u56fe\u7684\u6240\u6709\u8fde\u901a\u5206\u91cf\u3002</p> <p>\u5728\u90bb\u63a5\u8868\u4e2d\uff0c\u201c\u4e0e\u8be5\u9876\u70b9\u76f8\u8fde\u7684\u6240\u6709\u9876\u70b9\u201d\u7684\u9876\u70b9\u987a\u5e8f\u662f\u5426\u6709\u8981\u6c42\uff1f</p> <p>\u53ef\u4ee5\u662f\u4efb\u610f\u987a\u5e8f\u3002\u4f46\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u6309\u7167\u6307\u5b9a\u89c4\u5219\u6765\u6392\u5e8f\uff0c\u6bd4\u5982\u6309\u7167\u9876\u70b9\u6dfb\u52a0\u7684\u6b21\u5e8f\uff0c\u6216\u8005\u6309\u7167\u9876\u70b9\u503c\u5927\u5c0f\u7684\u987a\u5e8f\u7b49\uff0c\u8fd9\u6837\u6709\u52a9\u4e8e\u5feb\u901f\u67e5\u627e\u201c\u5e26\u6709\u67d0\u79cd\u6781\u503c\u201d\u7684\u9876\u70b9\u3002</p>"},{"location":"chapter_greedy/","title":"\u7b2c 15 \u7ae0 \u00a0 \u8d2a\u5fc3","text":"<p>Abstract</p> <p>\u5411\u65e5\u8475\u671d\u7740\u592a\u9633\u8f6c\u52a8\uff0c\u65f6\u523b\u8ffd\u6c42\u81ea\u8eab\u6210\u957f\u7684\u6700\u5927\u53ef\u80fd\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u5728\u4e00\u8f6e\u8f6e\u7684\u7b80\u5355\u9009\u62e9\u4e2d\uff0c\u9010\u6b65\u5bfc\u5411\u6700\u4f73\u7b54\u6848\u3002</p>"},{"location":"chapter_greedy/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5</li> <li>15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898</li> <li>15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898</li> <li>15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898</li> <li>15.5 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_greedy/fractional_knapsack_problem/","title":"15.2 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u4e2a\u7269\u54c1\uff0c\u7b2c \\(i\\) \u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(wgt[i-1]\\)\u3001\u4ef7\u503c\u4e3a \\(val[i-1]\\) \uff0c\u548c\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(cap\\) \u7684\u80cc\u5305\u3002\u6bcf\u4e2a\u7269\u54c1\u53ea\u80fd\u9009\u62e9\u4e00\u6b21\uff0c\u4f46\u53ef\u4ee5\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u4ef7\u503c\u6839\u636e\u9009\u62e9\u7684\u91cd\u91cf\u6bd4\u4f8b\u8ba1\u7b97\uff0c\u95ee\u5728\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u6700\u5927\u4ef7\u503c\u3002\u793a\u4f8b\u5982\u56fe 15-3 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-3 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5206\u6570\u80cc\u5305\u95ee\u9898\u548c 0-1 \u80cc\u5305\u95ee\u9898\u6574\u4f53\u4e0a\u975e\u5e38\u76f8\u4f3c\uff0c\u72b6\u6001\u5305\u542b\u5f53\u524d\u7269\u54c1 \\(i\\) \u548c\u5bb9\u91cf \\(c\\) \uff0c\u76ee\u6807\u662f\u6c42\u9650\u5b9a\u80cc\u5305\u5bb9\u91cf\u4e0b\u7684\u6700\u5927\u4ef7\u503c\u3002</p> <p>\u4e0d\u540c\u70b9\u5728\u4e8e\uff0c\u672c\u9898\u5141\u8bb8\u53ea\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\u3002\u5982\u56fe 15-4 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u7269\u54c1\u4efb\u610f\u5730\u8fdb\u884c\u5207\u5206\uff0c\u5e76\u6309\u7167\u91cd\u91cf\u6bd4\u4f8b\u6765\u8ba1\u7b97\u76f8\u5e94\u4ef7\u503c\u3002</p> <ol> <li>\u5bf9\u4e8e\u7269\u54c1 \\(i\\) \uff0c\u5b83\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c\u4e3a \\(val[i-1] / wgt[i-1]\\) \uff0c\u7b80\u79f0\u5355\u4f4d\u4ef7\u503c\u3002</li> <li>\u5047\u8bbe\u653e\u5165\u4e00\u90e8\u5206\u7269\u54c1 \\(i\\) \uff0c\u91cd\u91cf\u4e3a \\(w\\) \uff0c\u5219\u80cc\u5305\u589e\u52a0\u7684\u4ef7\u503c\u4e3a \\(w \\times val[i-1] / wgt[i-1]\\) \u3002</li> </ol> <p></p> <p> \u56fe 15-4 \u00a0 \u7269\u54c1\u5728\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u4ef7\u503c </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6700\u5927\u5316\u80cc\u5305\u5185\u7269\u54c1\u603b\u4ef7\u503c\uff0c\u672c\u8d28\u4e0a\u662f\u6700\u5927\u5316\u5355\u4f4d\u91cd\u91cf\u4e0b\u7684\u7269\u54c1\u4ef7\u503c\u3002\u7531\u6b64\u4fbf\u53ef\u63a8\u7406\u51fa\u56fe 15-5 \u6240\u793a\u7684\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\u3002</li> <li>\u904d\u5386\u6240\u6709\u7269\u54c1\uff0c\u6bcf\u8f6e\u8d2a\u5fc3\u5730\u9009\u62e9\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\u3002</li> <li>\u82e5\u5269\u4f59\u80cc\u5305\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u4f7f\u7528\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u586b\u6ee1\u80cc\u5305\u3002</li> </ol> <p></p> <p> \u56fe 15-5 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565 </p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5efa\u7acb\u4e86\u4e00\u4e2a\u7269\u54c1\u7c7b <code>Item</code> \uff0c\u4ee5\u4fbf\u5c06\u7269\u54c1\u6309\u7167\u5355\u4f4d\u4ef7\u503c\u8fdb\u884c\u6392\u5e8f\u3002\u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u5f53\u80cc\u5305\u5df2\u6ee1\u65f6\u8df3\u51fa\u5e76\u8fd4\u56de\u89e3\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig fractional_knapsack.py<pre><code>class Item:\n    \"\"\"\u7269\u54c1\"\"\"\n\n    def __init__(self, w: int, v: int):\n        self.w = w  # \u7269\u54c1\u91cd\u91cf\n        self.v = v  # \u7269\u54c1\u4ef7\u503c\n\ndef fractional_knapsack(wgt: list[int], val: list[int], cap: int) -&gt; int:\n    \"\"\"\u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3\"\"\"\n    # \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    items = [Item(w, v) for w, v in zip(wgt, val)]\n    # \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort(key=lambda item: item.v / item.w, reverse=True)\n    # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    res = 0\n    for item in items:\n        if item.w &lt;= cap:\n            # \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v\n            cap -= item.w\n        else:\n            # \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap\n            # \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n    return res\n</code></pre> fractional_knapsack.cpp<pre><code>/* \u7269\u54c1 */\nclass Item {\n  public:\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n\n    Item(int w, int v) : w(w), v(v) {\n    }\n};\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(vector&lt;int&gt; &amp;wgt, vector&lt;int&gt; &amp;val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    vector&lt;Item&gt; items;\n    for (int i = 0; i &lt; wgt.size(); i++) {\n        items.push_back(Item(wgt[i], val[i]));\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    sort(items.begin(), items.end(), [](Item &amp;a, Item &amp;b) { return (double)a.v / a.w &gt; (double)b.v / b.w; });\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    for (auto &amp;item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.java<pre><code>/* \u7269\u54c1 */\nclass Item {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n\n    public Item(int w, int v) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item[] items = new Item[wgt.length];\n    for (int i = 0; i &lt; wgt.length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Arrays.sort(items, Comparator.comparingDouble(item -&gt; -((double) item.v / item.w)));\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    for (Item item : items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double) item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.cs<pre><code>/* \u7269\u54c1 */\nclass Item(int w, int v) {\n    public int w = w; // \u7269\u54c1\u91cd\u91cf\n    public int v = v; // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble FractionalKnapsack(int[] wgt, int[] val, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item[] items = new Item[wgt.Length];\n    for (int i = 0; i &lt; wgt.Length; i++) {\n        items[i] = new Item(wgt[i], val[i]);\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    Array.Sort(items, (x, y) =&gt; (y.v / y.w).CompareTo(x.v / x.w));\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    double res = 0;\n    foreach (Item item in items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (double)item.v / item.w * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.go<pre><code>/* \u7269\u54c1 */\ntype Item struct {\n    w int // \u7269\u54c1\u91cd\u91cf\n    v int // \u7269\u54c1\u4ef7\u503c\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt []int, val []int, cap int) float64 {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    items := make([]Item, len(wgt))\n    for i := 0; i &lt; len(wgt); i++ {\n        items[i] = Item{wgt[i], val[i]}\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    sort.Slice(items, func(i, j int) bool {\n        return float64(items[i].v)/float64(items[i].w) &gt; float64(items[j].v)/float64(items[j].w)\n    })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    res := 0.0\n    for _, item := range items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += float64(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += float64(item.v) / float64(item.w) * float64(cap)\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.swift<pre><code>/* \u7269\u54c1 */\nclass Item {\n    var w: Int // \u7269\u54c1\u91cd\u91cf\n    var v: Int // \u7269\u54c1\u4ef7\u503c\n\n    init(w: Int, v: Int) {\n        self.w = w\n        self.v = v\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunc fractionalKnapsack(wgt: [Int], val: [Int], cap: Int) -&gt; Double {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    var items = zip(wgt, val).map { Item(w: $0, v: $1) }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort(by: { -(Double($0.v) / Double($0.w)) &lt; -(Double($1.v) / Double($1.w)) })\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    var res = 0.0\n    var cap = cap\n    for item in items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += Double(item.v)\n            cap -= item.w\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += Double(item.v) / Double(item.w) * Double(cap)\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    return res\n}\n</code></pre> fractional_knapsack.js<pre><code>/* \u7269\u54c1 */\nclass Item {\n    constructor(w, v) {\n        this.w = w; // \u7269\u54c1\u91cd\u91cf\n        this.v = v; // \u7269\u54c1\u4ef7\u503c\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt, val, cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    const items = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.ts<pre><code>/* \u7269\u54c1 */\nclass Item {\n    w: number; // \u7269\u54c1\u91cd\u91cf\n    v: number; // \u7269\u54c1\u4ef7\u503c\n\n    constructor(w: number, v: number) {\n        this.w = w;\n        this.v = v;\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfunction fractionalKnapsack(wgt: number[], val: number[], cap: number): number {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    const items: Item[] = wgt.map((w, i) =&gt; new Item(w, val[i]));\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort((a, b) =&gt; b.v / b.w - a.v / a.w);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let res = 0;\n    for (const item of items) {\n        if (item.w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (item.v / item.w) * cap;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    return res;\n}\n</code></pre> fractional_knapsack.dart<pre><code>/* \u7269\u54c1 */\nclass Item {\n  int w; // \u7269\u54c1\u91cd\u91cf\n  int v; // \u7269\u54c1\u4ef7\u503c\n\n  Item(this.w, this.v);\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\ndouble fractionalKnapsack(List&lt;int&gt; wgt, List&lt;int&gt; val, int cap) {\n  // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n  List&lt;Item&gt; items = List.generate(wgt.length, (i) =&gt; Item(wgt[i], val[i]));\n  // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n  items.sort((a, b) =&gt; (b.v / b.w).compareTo(a.v / a.w));\n  // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n  double res = 0;\n  for (Item item in items) {\n    if (item.w &lt;= cap) {\n      // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n      res += item.v;\n      cap -= item.w;\n    } else {\n      // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n      res += item.v / item.w * cap;\n      // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n      break;\n    }\n  }\n  return res;\n}\n</code></pre> fractional_knapsack.rs<pre><code>/* \u7269\u54c1 */\nstruct Item {\n    w: i32, // \u7269\u54c1\u91cd\u91cf\n    v: i32, // \u7269\u54c1\u4ef7\u503c\n}\n\nimpl Item {\n    fn new(w: i32, v: i32) -&gt; Self {\n        Self { w, v }\n    }\n}\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfn fractional_knapsack(wgt: &amp;[i32], val: &amp;[i32], mut cap: i32) -&gt; f64 {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    let mut items = wgt\n        .iter()\n        .zip(val.iter())\n        .map(|(&amp;w, &amp;v)| Item::new(w, v))\n        .collect::&lt;Vec&lt;Item&gt;&gt;();\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    items.sort_by(|a, b| {\n        (b.v as f64 / b.w as f64)\n            .partial_cmp(&amp;(a.v as f64 / a.w as f64))\n            .unwrap()\n    });\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    let mut res = 0.0;\n    for item in &amp;items {\n        if item.w &lt;= cap {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += item.v as f64;\n            cap -= item.w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += item.v as f64 / item.w as f64 * cap as f64;\n            // \u5df2\u65e0\u5269\u4f59\u5bb9\u91cf\uff0c\u56e0\u6b64\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    res\n}\n</code></pre> fractional_knapsack.c<pre><code>/* \u7269\u54c1 */\ntypedef struct {\n    int w; // \u7269\u54c1\u91cd\u91cf\n    int v; // \u7269\u54c1\u4ef7\u503c\n} Item;\n\n/* \u5206\u6570\u80cc\u5305\uff1a\u8d2a\u5fc3 */\nfloat fractionalKnapsack(int wgt[], int val[], int itemCount, int cap) {\n    // \u521b\u5efa\u7269\u54c1\u5217\u8868\uff0c\u5305\u542b\u4e24\u4e2a\u5c5e\u6027\uff1a\u91cd\u91cf\u3001\u4ef7\u503c\n    Item *items = malloc(sizeof(Item) * itemCount);\n    for (int i = 0; i &lt; itemCount; i++) {\n        items[i] = (Item){.w = wgt[i], .v = val[i]};\n    }\n    // \u6309\u7167\u5355\u4f4d\u4ef7\u503c item.v / item.w \u4ece\u9ad8\u5230\u4f4e\u8fdb\u884c\u6392\u5e8f\n    qsort(items, (size_t)itemCount, sizeof(Item), sortByValueDensity);\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\n    float res = 0.0;\n    for (int i = 0; i &lt; itemCount; i++) {\n        if (items[i].w &lt;= cap) {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u5145\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u6574\u4e2a\u88c5\u8fdb\u80cc\u5305\n            res += items[i].v;\n            cap -= items[i].w;\n        } else {\n            // \u82e5\u5269\u4f59\u5bb9\u91cf\u4e0d\u8db3\uff0c\u5219\u5c06\u5f53\u524d\u7269\u54c1\u7684\u4e00\u90e8\u5206\u88c5\u8fdb\u80cc\u5305\n            res += (float)cap / items[i].w * items[i].v;\n            cap = 0;\n            break;\n        }\n    }\n    free(items);\n    return res;\n}\n</code></pre> fractional_knapsack.zig<pre><code>[class]{Item}-[func]{}\n\n[class]{}-[func]{fractionalKnapsack}\n</code></pre> <p>\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u9700\u8981\u904d\u5386\u6574\u4e2a\u7269\u54c1\u5217\u8868\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u7269\u54c1\u6570\u91cf\u3002</p> <p>\u7531\u4e8e\u521d\u59cb\u5316\u4e86\u4e00\u4e2a <code>Item</code> \u5bf9\u8c61\u5217\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p>"},{"location":"chapter_greedy/fractional_knapsack_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u91c7\u7528\u53cd\u8bc1\u6cd5\u3002\u5047\u8bbe\u7269\u54c1 \\(x\\) \u662f\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\u7684\u7269\u54c1\uff0c\u4f7f\u7528\u67d0\u7b97\u6cd5\u6c42\u5f97\u6700\u5927\u4ef7\u503c\u4e3a <code>res</code> \uff0c\u4f46\u8be5\u89e3\u4e2d\u4e0d\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u73b0\u5728\u4ece\u80cc\u5305\u4e2d\u62ff\u51fa\u5355\u4f4d\u91cd\u91cf\u7684\u4efb\u610f\u7269\u54c1\uff0c\u5e76\u66ff\u6362\u4e3a\u5355\u4f4d\u91cd\u91cf\u7684\u7269\u54c1 \\(x\\) \u3002\u7531\u4e8e\u7269\u54c1 \\(x\\) \u7684\u5355\u4f4d\u4ef7\u503c\u6700\u9ad8\uff0c\u56e0\u6b64\u66ff\u6362\u540e\u7684\u603b\u4ef7\u503c\u4e00\u5b9a\u5927\u4e8e <code>res</code> \u3002\u8fd9\u4e0e <code>res</code> \u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u8bf4\u660e\u6700\u4f18\u89e3\u4e2d\u5fc5\u987b\u5305\u542b\u7269\u54c1 \\(x\\) \u3002</p> <p>\u5bf9\u4e8e\u8be5\u89e3\u4e2d\u7684\u5176\u4ed6\u7269\u54c1\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u6784\u5efa\u51fa\u4e0a\u8ff0\u77db\u76fe\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u5355\u4f4d\u4ef7\u503c\u66f4\u5927\u7684\u7269\u54c1\u603b\u662f\u66f4\u4f18\u9009\u62e9\uff0c\u8fd9\u8bf4\u660e\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p> <p>\u5982\u56fe 15-6 \u6240\u793a\uff0c\u5982\u679c\u5c06\u7269\u54c1\u91cd\u91cf\u548c\u7269\u54c1\u5355\u4f4d\u4ef7\u503c\u5206\u522b\u770b\u4f5c\u4e00\u5f20\u4e8c\u7ef4\u56fe\u8868\u7684\u6a2a\u8f74\u548c\u7eb5\u8f74\uff0c\u5219\u5206\u6570\u80cc\u5305\u95ee\u9898\u53ef\u8f6c\u5316\u4e3a\u201c\u6c42\u5728\u6709\u9650\u6a2a\u8f74\u533a\u95f4\u4e0b\u7684\u6700\u5927\u56f4\u6210\u9762\u79ef\u201d\u3002\u8fd9\u4e2a\u7c7b\u6bd4\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</p> <p></p> <p> \u56fe 15-6 \u00a0 \u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u51e0\u4f55\u8868\u793a </p>"},{"location":"chapter_greedy/greedy_algorithm/","title":"15.1 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5","text":"<p>\u300c\u8d2a\u5fc3\u7b97\u6cd5 greedy algorithm\u300d\u662f\u4e00\u79cd\u5e38\u89c1\u7684\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u5176\u57fa\u672c\u601d\u60f3\u662f\u5728\u95ee\u9898\u7684\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\uff0c\u90fd\u9009\u62e9\u5f53\u524d\u770b\u8d77\u6765\u6700\u4f18\u7684\u9009\u62e9\uff0c\u5373\u8d2a\u5fc3\u5730\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002\u8d2a\u5fc3\u7b97\u6cd5\u7b80\u6d01\u4e14\u9ad8\u6548\uff0c\u5728\u8bb8\u591a\u5b9e\u9645\u95ee\u9898\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002</p> <p>\u8d2a\u5fc3\u7b97\u6cd5\u548c\u52a8\u6001\u89c4\u5212\u90fd\u5e38\u7528\u4e8e\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u3002\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u4e00\u4e9b\u76f8\u4f3c\u4e4b\u5904\uff0c\u6bd4\u5982\u90fd\u4f9d\u8d56\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff0c\u4f46\u5de5\u4f5c\u539f\u7406\u4e0d\u540c\u3002</p> <ul> <li>\u52a8\u6001\u89c4\u5212\u4f1a\u6839\u636e\u4e4b\u524d\u9636\u6bb5\u7684\u6240\u6709\u51b3\u7b56\u6765\u8003\u8651\u5f53\u524d\u51b3\u7b56\uff0c\u5e76\u4f7f\u7528\u8fc7\u53bb\u5b50\u95ee\u9898\u7684\u89e3\u6765\u6784\u5efa\u5f53\u524d\u5b50\u95ee\u9898\u7684\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4f1a\u8003\u8651\u8fc7\u53bb\u7684\u51b3\u7b56\uff0c\u800c\u662f\u4e00\u8def\u5411\u524d\u5730\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u4e0d\u65ad\u7f29\u5c0f\u95ee\u9898\u8303\u56f4\uff0c\u76f4\u81f3\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> </ul> <p>\u6211\u4eec\u5148\u901a\u8fc7\u4f8b\u9898\u201c\u96f6\u94b1\u5151\u6362\u201d\u4e86\u89e3\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u3002\u8fd9\u9053\u9898\u5df2\u7ecf\u5728\u201c\u5b8c\u5168\u80cc\u5305\u95ee\u9898\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u76f8\u4fe1\u4f60\u5bf9\u5b83\u5e76\u4e0d\u964c\u751f\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a \\(n\\) \u79cd\u786c\u5e01\uff0c\u7b2c \\(i\\) \u79cd\u786c\u5e01\u7684\u9762\u503c\u4e3a \\(coins[i - 1]\\) \uff0c\u76ee\u6807\u91d1\u989d\u4e3a \\(amt\\) \uff0c\u6bcf\u79cd\u786c\u5e01\u53ef\u4ee5\u91cd\u590d\u9009\u53d6\uff0c\u95ee\u80fd\u591f\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u7684\u6700\u5c11\u786c\u5e01\u6570\u91cf\u3002\u5982\u679c\u65e0\u6cd5\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u672c\u9898\u91c7\u53d6\u7684\u8d2a\u5fc3\u7b56\u7565\u5982\u56fe 15-1 \u6240\u793a\u3002\u7ed9\u5b9a\u76ee\u6807\u91d1\u989d\uff0c\u6211\u4eec\u8d2a\u5fc3\u5730\u9009\u62e9\u4e0d\u5927\u4e8e\u4e14\u6700\u63a5\u8fd1\u5b83\u7684\u786c\u5e01\uff0c\u4e0d\u65ad\u5faa\u73af\u8be5\u6b65\u9aa4\uff0c\u76f4\u81f3\u51d1\u51fa\u76ee\u6807\u91d1\u989d\u4e3a\u6b62\u3002</p> <p></p> <p> \u56fe 15-1 \u00a0 \u96f6\u94b1\u5151\u6362\u7684\u8d2a\u5fc3\u7b56\u7565 </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\u3002\u4f60\u53ef\u80fd\u4f1a\u4e0d\u7531\u5730\u53d1\u51fa\u611f\u53f9\uff1aSo clean \uff01\u8d2a\u5fc3\u7b97\u6cd5\u4ec5\u7528\u7ea6\u5341\u884c\u4ee3\u7801\u5c31\u89e3\u51b3\u4e86\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig coin_change_greedy.py<pre><code>def coin_change_greedy(coins: list[int], amt: int) -&gt; int:\n    \"\"\"\u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3\"\"\"\n    # \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    i = len(coins) - 1\n    count = 0\n    # \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0:\n        # \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 and coins[i] &gt; amt:\n            i -= 1\n        # \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count += 1\n    # \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return count if amt == 0 else -1\n</code></pre> coin_change_greedy.cpp<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(vector&lt;int&gt; &amp;coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.size() - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.java<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.length - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.cs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint CoinChangeGreedy(int[] coins, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = coins.Length - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.go<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins []int, amt int) int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    i := len(coins) - 1\n    count := 0\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    for amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        for i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i--\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count++\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    if amt != 0 {\n        return -1\n    }\n    return count\n}\n</code></pre> coin_change_greedy.swift<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunc coinChangeGreedy(coins: [Int], amt: Int) -&gt; Int {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    var i = coins.count - 1\n    var count = 0\n    var amt = amt\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i]\n        count += 1\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1\n}\n</code></pre> coin_change_greedy.js<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins, amt) {\n    // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.ts<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfunction coinChangeGreedy(coins: number[], amt: number): number {\n    // \u5047\u8bbe coins \u6570\u7ec4\u6709\u5e8f\n    let i = coins.length - 1;\n    let count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt === 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.dart<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(List&lt;int&gt; coins, int amt) {\n  // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n  int i = coins.length - 1;\n  int count = 0;\n  // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n  while (amt &gt; 0) {\n    // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n    while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n      i--;\n    }\n    // \u9009\u62e9 coins[i]\n    amt -= coins[i];\n    count++;\n  }\n  // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n  return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.rs<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nfn coin_change_greedy(coins: &amp;[i32], mut amt: i32) -&gt; i32 {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    let mut i = coins.len() - 1;\n    let mut count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while amt &gt; 0 {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while i &gt; 0 &amp;&amp; coins[i] &gt; amt {\n            i -= 1;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count += 1;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    if amt == 0 {\n        count\n    } else {\n        -1\n    }\n}\n</code></pre> coin_change_greedy.c<pre><code>/* \u96f6\u94b1\u5151\u6362\uff1a\u8d2a\u5fc3 */\nint coinChangeGreedy(int *coins, int size, int amt) {\n    // \u5047\u8bbe coins \u5217\u8868\u6709\u5e8f\n    int i = size - 1;\n    int count = 0;\n    // \u5faa\u73af\u8fdb\u884c\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u5230\u65e0\u5269\u4f59\u91d1\u989d\n    while (amt &gt; 0) {\n        // \u627e\u5230\u5c0f\u4e8e\u4e14\u6700\u63a5\u8fd1\u5269\u4f59\u91d1\u989d\u7684\u786c\u5e01\n        while (i &gt; 0 &amp;&amp; coins[i] &gt; amt) {\n            i--;\n        }\n        // \u9009\u62e9 coins[i]\n        amt -= coins[i];\n        count++;\n    }\n    // \u82e5\u672a\u627e\u5230\u53ef\u884c\u65b9\u6848\uff0c\u5219\u8fd4\u56de -1\n    return amt == 0 ? count : -1;\n}\n</code></pre> coin_change_greedy.zig<pre><code>[class]{}-[func]{coinChangeGreedy}\n</code></pre>"},{"location":"chapter_greedy/greedy_algorithm/#1511","title":"15.1.1 \u00a0 \u8d2a\u5fc3\u7684\u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u64cd\u4f5c\u76f4\u63a5\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u800c\u4e14\u901a\u5e38\u6548\u7387\u4e5f\u5f88\u9ad8\u3002\u5728\u4ee5\u4e0a\u4ee3\u7801\u4e2d\uff0c\u8bb0\u786c\u5e01\u6700\u5c0f\u9762\u503c\u4e3a \\(\\min(coins)\\) \uff0c\u5219\u8d2a\u5fc3\u9009\u62e9\u6700\u591a\u5faa\u73af \\(amt / \\min(coins)\\) \u6b21\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(amt / \\min(coins))\\) \u3002\u8fd9\u6bd4\u52a8\u6001\u89c4\u5212\u89e3\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\times amt)\\) \u63d0\u5347\u4e86\u4e00\u4e2a\u6570\u91cf\u7ea7\u3002</p> <p>\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u9762\u503c\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u80fd\u627e\u5230\u6700\u4f18\u89e3\u3002\u56fe 15-2 \u7ed9\u51fa\u4e86\u4e24\u4e2a\u793a\u4f8b\u3002</p> <ul> <li>\u6b63\u4f8b \\(coins = [1, 5, 10, 20, 50, 100]\\)\uff1a\u5728\u8be5\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u7ed9\u5b9a\u4efb\u610f \\(amt\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u90fd\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 20, 50]\\)\uff1a\u5047\u8bbe \\(amt = 60\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 10\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(11\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(20 + 20 + 20\\) \uff0c\u4ec5\u9700 \\(3\\) \u679a\u786c\u5e01\u3002</li> <li>\u53cd\u4f8b \\(coins = [1, 49, 50]\\)\uff1a\u5047\u8bbe \\(amt = 98\\) \uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ea\u80fd\u627e\u5230 \\(50 + 1 \\times 48\\) \u7684\u5151\u6362\u7ec4\u5408\uff0c\u5171\u8ba1 \\(49\\) \u679a\u786c\u5e01\uff0c\u4f46\u52a8\u6001\u89c4\u5212\u53ef\u4ee5\u627e\u5230\u6700\u4f18\u89e3 \\(49 + 49\\) \uff0c\u4ec5\u9700 \\(2\\) \u679a\u786c\u5e01\u3002</li> </ul> <p></p> <p> \u56fe 15-2 \u00a0 \u8d2a\u5fc3\u65e0\u6cd5\u627e\u51fa\u6700\u4f18\u89e3\u7684\u793a\u4f8b </p> <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u65e0\u6cd5\u4fdd\u8bc1\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3\uff0c\u5e76\u4e14\u6709\u53ef\u80fd\u627e\u5230\u975e\u5e38\u5dee\u7684\u89e3\u3002\u5b83\u66f4\u9002\u5408\u7528\u52a8\u6001\u89c4\u5212\u89e3\u51b3\u3002</p> <p>\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9002\u7528\u60c5\u51b5\u5206\u4ee5\u4e0b\u4e24\u79cd\u3002</p> <ol> <li>\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u5f80\u5f80\u662f\u6700\u4f18\u9009\u62e9\uff0c\u56e0\u4e3a\u5b83\u5f80\u5f80\u6bd4\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u66f4\u9ad8\u6548\u3002</li> <li>\u53ef\u4ee5\u627e\u5230\u8fd1\u4f3c\u6700\u4f18\u89e3\uff1a\u8d2a\u5fc3\u7b97\u6cd5\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u4e5f\u662f\u53ef\u7528\u7684\u3002\u5bf9\u4e8e\u5f88\u591a\u590d\u6742\u95ee\u9898\u6765\u8bf4\uff0c\u5bfb\u627e\u5168\u5c40\u6700\u4f18\u89e3\u975e\u5e38\u56f0\u96be\uff0c\u80fd\u4ee5\u8f83\u9ad8\u6548\u7387\u627e\u5230\u6b21\u4f18\u89e3\u4e5f\u662f\u975e\u5e38\u4e0d\u9519\u7684\u3002</li> </ol>"},{"location":"chapter_greedy/greedy_algorithm/#1512","title":"15.1.2 \u00a0 \u8d2a\u5fc3\u7b97\u6cd5\u7279\u6027","text":"<p>\u90a3\u4e48\u95ee\u9898\u6765\u4e86\uff0c\u4ec0\u4e48\u6837\u7684\u95ee\u9898\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u5462\uff1f\u6216\u8005\u8bf4\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1f</p> <p>\u76f8\u8f83\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u4f7f\u7528\u6761\u4ef6\u66f4\u52a0\u82db\u523b\uff0c\u5176\u4e3b\u8981\u5173\u6ce8\u95ee\u9898\u7684\u4e24\u4e2a\u6027\u8d28\u3002</p> <ul> <li>\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff1a\u53ea\u6709\u5f53\u5c40\u90e8\u6700\u4f18\u9009\u62e9\u59cb\u7ec8\u53ef\u4ee5\u5bfc\u81f4\u5168\u5c40\u6700\u4f18\u89e3\u65f6\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u624d\u80fd\u4fdd\u8bc1\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u6700\u4f18\u5b50\u7ed3\u6784\u5df2\u7ecf\u5728\u201c\u52a8\u6001\u89c4\u5212\u201d\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u8fd9\u91cc\u4e0d\u518d\u8d58\u8ff0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4e00\u4e9b\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\u5e76\u4e0d\u660e\u663e\uff0c\u4f46\u4ecd\u7136\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u3002</p> <p>\u6211\u4eec\u4e3b\u8981\u63a2\u7a76\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u5224\u65ad\u65b9\u6cd5\u3002\u867d\u7136\u5b83\u7684\u63cf\u8ff0\u770b\u4e0a\u53bb\u6bd4\u8f83\u7b80\u5355\uff0c\u4f46\u5b9e\u9645\u4e0a\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\uff0c\u8bc1\u660e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u5e76\u975e\u6613\u4e8b\u3002</p> <p>\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\uff0c\u6211\u4eec\u867d\u7136\u80fd\u591f\u5bb9\u6613\u5730\u4e3e\u51fa\u53cd\u4f8b\uff0c\u5bf9\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u8fdb\u884c\u8bc1\u4f2a\uff0c\u4f46\u8bc1\u5b9e\u7684\u96be\u5ea6\u8f83\u5927\u3002\u5982\u679c\u95ee\uff1a\u6ee1\u8db3\u4ec0\u4e48\u6761\u4ef6\u7684\u786c\u5e01\u7ec4\u5408\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\uff1f\u6211\u4eec\u5f80\u5f80\u53ea\u80fd\u51ed\u501f\u76f4\u89c9\u6216\u4e3e\u4f8b\u5b50\u6765\u7ed9\u51fa\u4e00\u4e2a\u6a21\u68f1\u4e24\u53ef\u7684\u7b54\u6848\uff0c\u800c\u96be\u4ee5\u7ed9\u51fa\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\u3002</p> <p>Quote</p> <p>\u6709\u4e00\u7bc7\u8bba\u6587\u7ed9\u51fa\u4e86\u4e00\u4e2a \\(O(n^3)\\) \u65f6\u95f4\u590d\u6742\u5ea6\u7684\u7b97\u6cd5\uff0c\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u786c\u5e01\u7ec4\u5408\u80fd\u5426\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u627e\u51fa\u4efb\u610f\u91d1\u989d\u7684\u6700\u4f18\u89e3\u3002</p> <p>Pearson, David. A polynomial-time algorithm for the change-making problem. Operations Research Letters 33.3 (2005): 231-234.</p>"},{"location":"chapter_greedy/greedy_algorithm/#1513","title":"15.1.3 \u00a0 \u8d2a\u5fc3\u89e3\u9898\u6b65\u9aa4","text":"<p>\u8d2a\u5fc3\u95ee\u9898\u7684\u89e3\u51b3\u6d41\u7a0b\u5927\u4f53\u53ef\u5206\u4e3a\u4ee5\u4e0b\u4e09\u6b65\u3002</p> <ol> <li>\u95ee\u9898\u5206\u6790\uff1a\u68b3\u7406\u4e0e\u7406\u89e3\u95ee\u9898\u7279\u6027\uff0c\u5305\u62ec\u72b6\u6001\u5b9a\u4e49\u3001\u4f18\u5316\u76ee\u6807\u548c\u7ea6\u675f\u6761\u4ef6\u7b49\u3002\u8fd9\u4e00\u6b65\u5728\u56de\u6eaf\u548c\u52a8\u6001\u89c4\u5212\u4e2d\u90fd\u6709\u6d89\u53ca\u3002</li> <li>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\uff1a\u786e\u5b9a\u5982\u4f55\u5728\u6bcf\u4e00\u6b65\u4e2d\u505a\u51fa\u8d2a\u5fc3\u9009\u62e9\u3002\u8fd9\u4e2a\u7b56\u7565\u80fd\u591f\u5728\u6bcf\u4e00\u6b65\u51cf\u5c0f\u95ee\u9898\u7684\u89c4\u6a21\uff0c\u5e76\u6700\u7ec8\u89e3\u51b3\u6574\u4e2a\u95ee\u9898\u3002</li> <li>\u6b63\u786e\u6027\u8bc1\u660e\uff1a\u901a\u5e38\u9700\u8981\u8bc1\u660e\u95ee\u9898\u5177\u6709\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8fd9\u4e2a\u6b65\u9aa4\u53ef\u80fd\u9700\u8981\u7528\u5230\u6570\u5b66\u8bc1\u660e\uff0c\u4f8b\u5982\u5f52\u7eb3\u6cd5\u6216\u53cd\u8bc1\u6cd5\u7b49\u3002</li> </ol> <p>\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6c42\u89e3\u95ee\u9898\u7684\u6838\u5fc3\u6b65\u9aa4\uff0c\u4f46\u5b9e\u65bd\u8d77\u6765\u53ef\u80fd\u5e76\u4e0d\u5bb9\u6613\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e0d\u540c\u95ee\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\u7684\u5dee\u5f02\u8f83\u5927\u3002\u5bf9\u4e8e\u8bb8\u591a\u95ee\u9898\u6765\u8bf4\uff0c\u8d2a\u5fc3\u7b56\u7565\u6bd4\u8f83\u6d45\u663e\uff0c\u6211\u4eec\u901a\u8fc7\u4e00\u4e9b\u5927\u6982\u7684\u601d\u8003\u4e0e\u5c1d\u8bd5\u5c31\u80fd\u5f97\u51fa\u3002\u800c\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u7b56\u7565\u53ef\u80fd\u975e\u5e38\u9690\u853d\uff0c\u8fd9\u79cd\u60c5\u51b5\u5c31\u975e\u5e38\u8003\u9a8c\u4e2a\u4eba\u7684\u89e3\u9898\u7ecf\u9a8c\u4e0e\u7b97\u6cd5\u80fd\u529b\u4e86\u3002</li> <li>\u67d0\u4e9b\u8d2a\u5fc3\u7b56\u7565\u5177\u6709\u8f83\u5f3a\u7684\u8ff7\u60d1\u6027\u3002\u5f53\u6211\u4eec\u6ee1\u6000\u4fe1\u5fc3\u8bbe\u8ba1\u597d\u8d2a\u5fc3\u7b56\u7565\uff0c\u5199\u51fa\u89e3\u9898\u4ee3\u7801\u5e76\u63d0\u4ea4\u8fd0\u884c\uff0c\u5f88\u53ef\u80fd\u53d1\u73b0\u90e8\u5206\u6d4b\u8bd5\u6837\u4f8b\u65e0\u6cd5\u901a\u8fc7\u3002\u8fd9\u662f\u56e0\u4e3a\u8bbe\u8ba1\u7684\u8d2a\u5fc3\u7b56\u7565\u53ea\u662f\u201c\u90e8\u5206\u6b63\u786e\u201d\u7684\uff0c\u4e0a\u6587\u4ecb\u7ecd\u7684\u96f6\u94b1\u5151\u6362\u5c31\u662f\u4e00\u4e2a\u5178\u578b\u6848\u4f8b\u3002</li> </ul> <p>\u4e3a\u4e86\u4fdd\u8bc1\u6b63\u786e\u6027\uff0c\u6211\u4eec\u5e94\u8be5\u5bf9\u8d2a\u5fc3\u7b56\u7565\u8fdb\u884c\u4e25\u8c28\u7684\u6570\u5b66\u8bc1\u660e\uff0c\u901a\u5e38\u9700\u8981\u7528\u5230\u53cd\u8bc1\u6cd5\u6216\u6570\u5b66\u5f52\u7eb3\u6cd5\u3002</p> <p>\u7136\u800c\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u4e5f\u5f88\u53ef\u80fd\u4e0d\u662f\u4e00\u4ef6\u6613\u4e8b\u3002\u5982\u82e5\u6ca1\u6709\u5934\u7eea\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9009\u62e9\u9762\u5411\u6d4b\u8bd5\u7528\u4f8b\u8fdb\u884c\u4ee3\u7801\u8c03\u8bd5\uff0c\u4e00\u6b65\u6b65\u4fee\u6539\u4e0e\u9a8c\u8bc1\u8d2a\u5fc3\u7b56\u7565\u3002</p>"},{"location":"chapter_greedy/greedy_algorithm/#1514","title":"15.1.4 \u00a0 \u8d2a\u5fc3\u5178\u578b\u4f8b\u9898","text":"<p>\u8d2a\u5fc3\u7b97\u6cd5\u5e38\u5e38\u5e94\u7528\u5728\u6ee1\u8db3\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u7684\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u4ee5\u4e0b\u5217\u4e3e\u4e86\u4e00\u4e9b\u5178\u578b\u7684\u8d2a\u5fc3\u7b97\u6cd5\u95ee\u9898\u3002</p> <ul> <li>\u786c\u5e01\u627e\u96f6\u95ee\u9898\uff1a\u5728\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\u4e0b\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u603b\u662f\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u5047\u8bbe\u4f60\u6709\u4e00\u4e9b\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u5728\u4e00\u6bb5\u65f6\u95f4\u5185\u8fdb\u884c\uff0c\u4f60\u7684\u76ee\u6807\u662f\u5b8c\u6210\u5c3d\u53ef\u80fd\u591a\u7684\u4efb\u52a1\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4efb\u52a1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u7269\u54c1\u548c\u4e00\u4e2a\u8f7d\u91cd\u91cf\uff0c\u4f60\u7684\u76ee\u6807\u662f\u9009\u62e9\u4e00\u7ec4\u7269\u54c1\uff0c\u4f7f\u5f97\u603b\u91cd\u91cf\u4e0d\u8d85\u8fc7\u8f7d\u91cd\u91cf\uff0c\u4e14\u603b\u4ef7\u503c\u6700\u5927\u3002\u5982\u679c\u6bcf\u6b21\u90fd\u9009\u62e9\u6027\u4ef7\u6bd4\u6700\u9ad8\uff08\u4ef7\u503c / \u91cd\u91cf\uff09\u7684\u7269\u54c1\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5728\u4e00\u4e9b\u60c5\u51b5\u4e0b\u53ef\u4ee5\u5f97\u5230\u6700\u4f18\u89e3\u3002</li> <li>\u80a1\u7968\u4e70\u5356\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u80a1\u7968\u7684\u5386\u53f2\u4ef7\u683c\uff0c\u4f60\u53ef\u4ee5\u8fdb\u884c\u591a\u6b21\u4e70\u5356\uff0c\u4f46\u5982\u679c\u4f60\u5df2\u7ecf\u6301\u6709\u80a1\u7968\uff0c\u90a3\u4e48\u5728\u5356\u51fa\u4e4b\u524d\u4e0d\u80fd\u518d\u4e70\uff0c\u76ee\u6807\u662f\u83b7\u53d6\u6700\u5927\u5229\u6da6\u3002</li> <li>\u970d\u592b\u66fc\u7f16\u7801\uff1a\u970d\u592b\u66fc\u7f16\u7801\u662f\u4e00\u79cd\u7528\u4e8e\u65e0\u635f\u6570\u636e\u538b\u7f29\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002\u901a\u8fc7\u6784\u5efa\u970d\u592b\u66fc\u6811\uff0c\u6bcf\u6b21\u9009\u62e9\u51fa\u73b0\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u8282\u70b9\u5408\u5e76\uff0c\u6700\u540e\u5f97\u5230\u7684\u970d\u592b\u66fc\u6811\u7684\u5e26\u6743\u8def\u5f84\u957f\u5ea6\uff08\u7f16\u7801\u957f\u5ea6\uff09\u6700\u5c0f\u3002</li> <li>Dijkstra \u7b97\u6cd5\uff1a\u5b83\u662f\u4e00\u79cd\u89e3\u51b3\u7ed9\u5b9a\u6e90\u9876\u70b9\u5230\u5176\u4f59\u5404\u9876\u70b9\u7684\u6700\u77ed\u8def\u5f84\u95ee\u9898\u7684\u8d2a\u5fc3\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_greedy/max_capacity_problem/","title":"15.3 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898","text":"<p>Question</p> <p>\u8f93\u5165\u4e00\u4e2a\u6570\u7ec4 \\(ht\\) \uff0c\u5176\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u4ee3\u8868\u4e00\u4e2a\u5782\u76f4\u9694\u677f\u7684\u9ad8\u5ea6\u3002\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u9694\u677f\uff0c\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u7a7a\u95f4\u53ef\u4ee5\u7ec4\u6210\u4e00\u4e2a\u5bb9\u5668\u3002</p> <p>\u5bb9\u5668\u7684\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u548c\u5bbd\u5ea6\u7684\u4e58\u79ef\uff08\u9762\u79ef\uff09\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u8f83\u77ed\u7684\u9694\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u4e2a\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002</p> <p>\u8bf7\u5728\u6570\u7ec4\u4e2d\u9009\u62e9\u4e24\u4e2a\u9694\u677f\uff0c\u4f7f\u5f97\u7ec4\u6210\u7684\u5bb9\u5668\u7684\u5bb9\u91cf\u6700\u5927\uff0c\u8fd4\u56de\u6700\u5927\u5bb9\u91cf\u3002\u793a\u4f8b\u5982\u56fe 15-7 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-7 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u793a\u4f8b\u6570\u636e </p> <p>\u5bb9\u5668\u7531\u4efb\u610f\u4e24\u4e2a\u9694\u677f\u56f4\u6210\uff0c\u56e0\u6b64\u672c\u9898\u7684\u72b6\u6001\u4e3a\u4e24\u4e2a\u9694\u677f\u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\([i, j]\\) \u3002</p> <p>\u6839\u636e\u9898\u610f\uff0c\u5bb9\u91cf\u7b49\u4e8e\u9ad8\u5ea6\u4e58\u4ee5\u5bbd\u5ea6\uff0c\u5176\u4e2d\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u5bbd\u5ea6\u662f\u4e24\u9694\u677f\u7684\u6570\u7ec4\u7d22\u5f15\u4e4b\u5dee\u3002\u8bbe\u5bb9\u91cf\u4e3a \\(cap[i, j]\\) \uff0c\u5219\u53ef\u5f97\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ cap[i, j] = \\min(ht[i], ht[j]) \\times (j - i) \\] <p>\u8bbe\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u4e24\u4e2a\u9694\u677f\u7684\u7ec4\u5408\u6570\u91cf\uff08\u72b6\u6001\u603b\u6570\uff09\u4e3a \\(C_n^2 = \\frac{n(n - 1)}{2}\\) \u4e2a\u3002\u6700\u76f4\u63a5\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u7a77\u4e3e\u6240\u6709\u72b6\u6001\uff0c\u4ece\u800c\u6c42\u5f97\u6700\u5927\u5bb9\u91cf\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_greedy/max_capacity_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u8fd9\u9053\u9898\u8fd8\u6709\u66f4\u9ad8\u6548\u7387\u7684\u89e3\u6cd5\u3002\u5982\u56fe 15-8 \u6240\u793a\uff0c\u73b0\u9009\u53d6\u4e00\u4e2a\u72b6\u6001 \\([i, j]\\) \uff0c\u5176\u6ee1\u8db3\u7d22\u5f15 \\(i &lt; j\\) \u4e14\u9ad8\u5ea6 \\(ht[i] &lt; ht[j]\\) \uff0c\u5373 \\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002</p> <p></p> <p> \u56fe 15-8 \u00a0 \u521d\u59cb\u72b6\u6001 </p> <p>\u5982\u56fe 15-9 \u6240\u793a\uff0c\u82e5\u6b64\u65f6\u5c06\u957f\u677f \\(j\\) \u5411\u77ed\u677f \\(i\\) \u9760\u8fd1\uff0c\u5219\u5bb9\u91cf\u4e00\u5b9a\u53d8\u5c0f\u3002</p> <p>\u8fd9\u662f\u56e0\u4e3a\u5728\u79fb\u52a8\u957f\u677f \\(j\\) \u540e\uff0c\u5bbd\u5ea6 \\(j-i\\) \u80af\u5b9a\u53d8\u5c0f\uff1b\u800c\u9ad8\u5ea6\u7531\u77ed\u677f\u51b3\u5b9a\uff0c\u56e0\u6b64\u9ad8\u5ea6\u53ea\u53ef\u80fd\u4e0d\u53d8\uff08 \\(i\\) \u4ecd\u4e3a\u77ed\u677f\uff09\u6216\u53d8\u5c0f\uff08\u79fb\u52a8\u540e\u7684 \\(j\\) \u6210\u4e3a\u77ed\u677f\uff09\u3002</p> <p></p> <p> \u56fe 15-9 \u00a0 \u5411\u5185\u79fb\u52a8\u957f\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u53cd\u5411\u601d\u8003\uff0c\u6211\u4eec\u53ea\u6709\u5411\u5185\u6536\u7f29\u77ed\u677f \\(i\\) \uff0c\u624d\u6709\u53ef\u80fd\u4f7f\u5bb9\u91cf\u53d8\u5927\u3002\u56e0\u4e3a\u867d\u7136\u5bbd\u5ea6\u4e00\u5b9a\u53d8\u5c0f\uff0c\u4f46\u9ad8\u5ea6\u53ef\u80fd\u4f1a\u53d8\u5927\uff08\u79fb\u52a8\u540e\u7684\u77ed\u677f \\(i\\) \u53ef\u80fd\u4f1a\u53d8\u957f\uff09\u3002\u4f8b\u5982\u5728\u56fe 15-10 \u4e2d\uff0c\u79fb\u52a8\u77ed\u677f\u540e\u9762\u79ef\u53d8\u5927\u3002</p> <p></p> <p> \u56fe 15-10 \u00a0 \u5411\u5185\u79fb\u52a8\u77ed\u677f\u540e\u7684\u72b6\u6001 </p> <p>\u7531\u6b64\u4fbf\u53ef\u63a8\u51fa\u672c\u9898\u7684\u8d2a\u5fc3\u7b56\u7565\uff1a\u521d\u59cb\u5316\u4e24\u6307\u9488\u5206\u5217\u5bb9\u5668\u4e24\u7aef\uff0c\u6bcf\u8f6e\u5411\u5185\u6536\u7f29\u77ed\u677f\u5bf9\u5e94\u7684\u6307\u9488\uff0c\u76f4\u81f3\u4e24\u6307\u9488\u76f8\u9047\u3002</p> <p>\u56fe 15-11 \u5c55\u793a\u4e86\u8d2a\u5fc3\u7b56\u7565\u7684\u6267\u884c\u8fc7\u7a0b\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u5217\u4e0e\u6570\u7ec4\u4e24\u7aef\u3002</li> <li>\u8ba1\u7b97\u5f53\u524d\u72b6\u6001\u7684\u5bb9\u91cf \\(cap[i, j]\\) \uff0c\u5e76\u66f4\u65b0\u6700\u5927\u5bb9\u91cf\u3002</li> <li>\u6bd4\u8f83\u677f \\(i\\) \u548c \u677f \\(j\\) \u7684\u9ad8\u5ea6\uff0c\u5e76\u5c06\u77ed\u677f\u5411\u5185\u79fb\u52a8\u4e00\u683c\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u76f4\u81f3 \\(i\\) \u548c \\(j\\) \u76f8\u9047\u65f6\u7ed3\u675f\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 15-11 \u00a0 \u6700\u5927\u5bb9\u91cf\u95ee\u9898\u7684\u8d2a\u5fc3\u8fc7\u7a0b </p>"},{"location":"chapter_greedy/max_capacity_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u4ee3\u7801\u5faa\u73af\u6700\u591a \\(n\\) \u8f6e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002</p> <p>\u53d8\u91cf \\(i\\)\u3001\\(j\\)\u3001\\(res\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig max_capacity.py<pre><code>def max_capacity(ht: list[int]) -&gt; int:\n    \"\"\"\u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3\"\"\"\n    # \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    i, j = 0, len(ht) - 1\n    # \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    res = 0\n    # \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j:\n        # \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        # \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j]:\n            i += 1\n        else:\n            j -= 1\n    return res\n</code></pre> max_capacity.cpp<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(vector&lt;int&gt; &amp;ht) {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.size() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = min(ht[i], ht[j]) * (j - i);\n        res = max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.java<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.cs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint MaxCapacity(int[] ht) {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0, j = ht.Length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int cap = Math.Min(ht[i], ht[j]) * (j - i);\n        res = Math.Max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.go<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht []int) int {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    i, j := 0, len(ht)-1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    res := 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    for i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        capacity := int(math.Min(float64(ht[i]), float64(ht[j]))) * (j - i)\n        res = int(math.Max(float64(res), float64(capacity)))\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i++\n        } else {\n            j--\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.swift<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunc maxCapacity(ht: [Int]) -&gt; Int {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    var i = 0, j = ht.count - 1\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    var res = 0\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = min(ht[i], ht[j]) * (j - i)\n        res = max(res, cap)\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1\n        } else {\n            j -= 1\n        }\n    }\n    return res\n}\n</code></pre> max_capacity.js<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht) {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.ts<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfunction maxCapacity(ht: number[]): number {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let i = 0,\n        j = ht.length - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        const cap: number = Math.min(ht[i], ht[j]) * (j - i);\n        res = Math.max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.dart<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(List&lt;int&gt; ht) {\n  // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n  int i = 0, j = ht.length - 1;\n  // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n  int res = 0;\n  // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n  while (i &lt; j) {\n    // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n    int cap = min(ht[i], ht[j]) * (j - i);\n    res = max(res, cap);\n    // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n    if (ht[i] &lt; ht[j]) {\n      i++;\n    } else {\n      j--;\n    }\n  }\n  return res;\n}\n</code></pre> max_capacity.rs<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nfn max_capacity(ht: &amp;[i32]) -&gt; i32 {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    let mut i = 0;\n    let mut j = ht.len() - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    let mut res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while i &lt; j {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        let cap = std::cmp::min(ht[i], ht[j]) * (j - i) as i32;\n        res = std::cmp::max(res, cap);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if ht[i] &lt; ht[j] {\n            i += 1;\n        } else {\n            j -= 1;\n        }\n    }\n    res\n}\n</code></pre> max_capacity.c<pre><code>/* \u6700\u5927\u5bb9\u91cf\uff1a\u8d2a\u5fc3 */\nint maxCapacity(int ht[], int htLength) {\n    // \u521d\u59cb\u5316 i, j \u5206\u5217\u6570\u7ec4\u4e24\u7aef\n    int i = 0;\n    int j = htLength - 1;\n    // \u521d\u59cb\u6700\u5927\u5bb9\u91cf\u4e3a 0\n    int res = 0;\n    // \u5faa\u73af\u8d2a\u5fc3\u9009\u62e9\uff0c\u76f4\u81f3\u4e24\u677f\u76f8\u9047\n    while (i &lt; j) {\n        // \u66f4\u65b0\u6700\u5927\u5bb9\u91cf\n        int capacity = myMin(ht[i], ht[j]) * (j - i);\n        res = myMax(res, capacity);\n        // \u5411\u5185\u79fb\u52a8\u77ed\u677f\n        if (ht[i] &lt; ht[j]) {\n            i++;\n        } else {\n            j--;\n        }\n    }\n    return res;\n}\n</code></pre> max_capacity.zig<pre><code>[class]{}-[func]{maxCapacity}\n</code></pre>"},{"location":"chapter_greedy/max_capacity_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4e4b\u6240\u4ee5\u8d2a\u5fc3\u6bd4\u7a77\u4e3e\u66f4\u5feb\uff0c\u662f\u56e0\u4e3a\u6bcf\u8f6e\u7684\u8d2a\u5fc3\u9009\u62e9\u90fd\u4f1a\u201c\u8df3\u8fc7\u201d\u4e00\u4e9b\u72b6\u6001\u3002</p> <p>\u6bd4\u5982\u5728\u72b6\u6001 \\(cap[i, j]\\) \u4e0b\uff0c\\(i\\) \u4e3a\u77ed\u677f\u3001\\(j\\) \u4e3a\u957f\u677f\u3002\u82e5\u8d2a\u5fc3\u5730\u5c06\u77ed\u677f \\(i\\) \u5411\u5185\u79fb\u52a8\u4e00\u683c\uff0c\u4f1a\u5bfc\u81f4\u56fe 15-12 \u6240\u793a\u7684\u72b6\u6001\u88ab\u201c\u8df3\u8fc7\u201d\u3002\u8fd9\u610f\u5473\u7740\u4e4b\u540e\u65e0\u6cd5\u9a8c\u8bc1\u8fd9\u4e9b\u72b6\u6001\u7684\u5bb9\u91cf\u5927\u5c0f\u3002</p> \\[ cap[i, i+1], cap[i, i+2], \\dots, cap[i, j-2], cap[i, j-1] \\] <p></p> <p> \u56fe 15-12 \u00a0 \u79fb\u52a8\u77ed\u677f\u5bfc\u81f4\u88ab\u8df3\u8fc7\u7684\u72b6\u6001 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u8fd9\u4e9b\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u5b9e\u9645\u4e0a\u5c31\u662f\u5c06\u957f\u677f \\(j\\) \u5411\u5185\u79fb\u52a8\u7684\u6240\u6709\u72b6\u6001\u3002\u524d\u9762\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u5185\u79fb\u957f\u677f\u4e00\u5b9a\u4f1a\u5bfc\u81f4\u5bb9\u91cf\u53d8\u5c0f\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u88ab\u8df3\u8fc7\u7684\u72b6\u6001\u90fd\u4e0d\u53ef\u80fd\u662f\u6700\u4f18\u89e3\uff0c\u8df3\u8fc7\u5b83\u4eec\u4e0d\u4f1a\u5bfc\u81f4\u9519\u8fc7\u6700\u4f18\u89e3\u3002</p> <p>\u4ee5\u4e0a\u5206\u6790\u8bf4\u660e\uff0c\u79fb\u52a8\u77ed\u677f\u7684\u64cd\u4f5c\u662f\u201c\u5b89\u5168\u201d\u7684\uff0c\u8d2a\u5fc3\u7b56\u7565\u662f\u6709\u6548\u7684\u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/","title":"15.4 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6b63\u6574\u6570 \\(n\\) \uff0c\u5c06\u5176\u5207\u5206\u4e3a\u81f3\u5c11\u4e24\u4e2a\u6b63\u6574\u6570\u7684\u548c\uff0c\u6c42\u5207\u5206\u540e\u6240\u6709\u6574\u6570\u7684\u4e58\u79ef\u6700\u5927\u662f\u591a\u5c11\uff0c\u5982\u56fe 15-13 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 15-13 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u95ee\u9898\u5b9a\u4e49 </p> <p>\u5047\u8bbe\u6211\u4eec\u5c06 \\(n\\) \u5207\u5206\u4e3a \\(m\\) \u4e2a\u6574\u6570\u56e0\u5b50\uff0c\u5176\u4e2d\u7b2c \\(i\\) \u4e2a\u56e0\u5b50\u8bb0\u4e3a \\(n_i\\) \uff0c\u5373</p> \\[ n = \\sum_{i=1}^{m}n_i \\] <p>\u672c\u9898\u7684\u76ee\u6807\u662f\u6c42\u5f97\u6240\u6709\u6574\u6570\u56e0\u5b50\u7684\u6700\u5927\u4e58\u79ef\uff0c\u5373</p> \\[ \\max(\\prod_{i=1}^{m}n_i) \\] <p>\u6211\u4eec\u9700\u8981\u601d\u8003\u7684\u662f\uff1a\u5207\u5206\u6570\u91cf \\(m\\) \u5e94\u8be5\u591a\u5927\uff0c\u6bcf\u4e2a \\(n_i\\) \u5e94\u8be5\u662f\u591a\u5c11\uff1f</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#1","title":"1. \u00a0 \u8d2a\u5fc3\u7b56\u7565\u786e\u5b9a","text":"<p>\u6839\u636e\u7ecf\u9a8c\uff0c\u4e24\u4e2a\u6574\u6570\u7684\u4e58\u79ef\u5f80\u5f80\u6bd4\u5b83\u4eec\u7684\u52a0\u548c\u66f4\u5927\u3002\u5047\u8bbe\u4ece \\(n\\) \u4e2d\u5206\u51fa\u4e00\u4e2a\u56e0\u5b50 \\(2\\) \uff0c\u5219\u5b83\u4eec\u7684\u4e58\u79ef\u4e3a \\(2(n-2)\\) \u3002\u6211\u4eec\u5c06\u8be5\u4e58\u79ef\u4e0e \\(n\\) \u4f5c\u6bd4\u8f83\uff1a</p> \\[ \\begin{aligned} 2(n-2) &amp; \\geq n \\newline 2n - n - 4 &amp; \\geq 0 \\newline n &amp; \\geq 4 \\end{aligned} \\] <p>\u5982\u56fe 15-14 \u6240\u793a\uff0c\u5f53 \\(n \\geq 4\\) \u65f6\uff0c\u5207\u5206\u51fa\u4e00\u4e2a \\(2\\) \u540e\u4e58\u79ef\u4f1a\u53d8\u5927\uff0c\u8fd9\u8bf4\u660e\u5927\u4e8e\u7b49\u4e8e \\(4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u88ab\u5207\u5206\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e00\uff1a\u5982\u679c\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b \\(\\geq 4\\) \u7684\u56e0\u5b50\uff0c\u90a3\u4e48\u5b83\u5c31\u5e94\u8be5\u88ab\u7ee7\u7eed\u5207\u5206\u3002\u6700\u7ec8\u7684\u5207\u5206\u65b9\u6848\u53ea\u5e94\u51fa\u73b0 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u79cd\u56e0\u5b50\u3002</p> <p></p> <p> \u56fe 15-14 \u00a0 \u5207\u5206\u5bfc\u81f4\u4e58\u79ef\u53d8\u5927 </p> <p>\u63a5\u4e0b\u6765\u601d\u8003\u54ea\u4e2a\u56e0\u5b50\u662f\u6700\u4f18\u7684\u3002\u5728 \\(1\\)\u3001\\(2\\)\u3001\\(3\\) \u8fd9\u4e09\u4e2a\u56e0\u5b50\u4e2d\uff0c\u663e\u7136 \\(1\\) \u662f\u6700\u5dee\u7684\uff0c\u56e0\u4e3a \\(1 \\times (n-1) &lt; n\\) \u6052\u6210\u7acb\uff0c\u5373\u5207\u5206\u51fa \\(1\\) \u53cd\u800c\u4f1a\u5bfc\u81f4\u4e58\u79ef\u51cf\u5c0f\u3002</p> <p>\u5982\u56fe 15-15 \u6240\u793a\uff0c\u5f53 \\(n = 6\\) \u65f6\uff0c\u6709 \\(3 \\times 3 &gt; 2 \\times 2 \\times 2\\) \u3002\u8fd9\u610f\u5473\u7740\u5207\u5206\u51fa \\(3\\) \u6bd4\u5207\u5206\u51fa \\(2\\) \u66f4\u4f18\u3002</p> <p>\u8d2a\u5fc3\u7b56\u7565\u4e8c\uff1a\u5728\u5207\u5206\u65b9\u6848\u4e2d\uff0c\u6700\u591a\u53ea\u5e94\u5b58\u5728\u4e24\u4e2a \\(2\\) \u3002\u56e0\u4e3a\u4e09\u4e2a \\(2\\) \u603b\u662f\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002</p> <p></p> <p> \u56fe 15-15 \u00a0 \u6700\u4f18\u5207\u5206\u56e0\u5b50 </p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u53ef\u63a8\u7406\u51fa\u4ee5\u4e0b\u8d2a\u5fc3\u7b56\u7565\u3002</p> <ol> <li>\u8f93\u5165\u6574\u6570 \\(n\\) \uff0c\u4ece\u5176\u4e0d\u65ad\u5730\u5207\u5206\u51fa\u56e0\u5b50 \\(3\\) \uff0c\u76f4\u81f3\u4f59\u6570\u4e3a \\(0\\)\u3001\\(1\\)\u3001\\(2\\) \u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(0\\) \u65f6\uff0c\u4ee3\u8868 \\(n\\) \u662f \\(3\\) \u7684\u500d\u6570\uff0c\u56e0\u6b64\u4e0d\u505a\u4efb\u4f55\u5904\u7406\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(2\\) \u65f6\uff0c\u4e0d\u7ee7\u7eed\u5212\u5206\uff0c\u4fdd\u7559\u4e4b\u3002</li> <li>\u5f53\u4f59\u6570\u4e3a \\(1\\) \u65f6\uff0c\u7531\u4e8e \\(2 \\times 2 &gt; 1 \\times 3\\) \uff0c\u56e0\u6b64\u5e94\u5c06\u6700\u540e\u4e00\u4e2a \\(3\\) \u66ff\u6362\u4e3a \\(2\\) \u3002</li> </ol>"},{"location":"chapter_greedy/max_product_cutting_problem/#2","title":"2. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5982\u56fe 15-16 \u6240\u793a\uff0c\u6211\u4eec\u65e0\u987b\u901a\u8fc7\u5faa\u73af\u6765\u5207\u5206\u6574\u6570\uff0c\u800c\u53ef\u4ee5\u5229\u7528\u5411\u4e0b\u6574\u9664\u8fd0\u7b97\u5f97\u5230 \\(3\\) \u7684\u4e2a\u6570 \\(a\\) \uff0c\u7528\u53d6\u6a21\u8fd0\u7b97\u5f97\u5230\u4f59\u6570 \\(b\\) \uff0c\u6b64\u65f6\u6709\uff1a</p> \\[ n = 3 a + b \\] <p>\u8bf7\u6ce8\u610f\uff0c\u5bf9\u4e8e \\(n \\leq 3\\) \u7684\u8fb9\u754c\u60c5\u51b5\uff0c\u5fc5\u987b\u62c6\u5206\u51fa\u4e00\u4e2a \\(1\\) \uff0c\u4e58\u79ef\u4e3a \\(1 \\times (n - 1)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig max_product_cutting.py<pre><code>def max_product_cutting(n: int) -&gt; int:\n    \"\"\"\u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3\"\"\"\n    # \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3:\n        return 1 * (n - 1)\n    # \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    a, b = n // 3, n % 3\n    if b == 1:\n        # \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return int(math.pow(3, a - 1)) * 2 * 2\n    if b == 2:\n        # \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return int(math.pow(3, a)) * 2\n    # \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return int(math.pow(3, a))\n</code></pre> max_product_cutting.cpp<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int)pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int)pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int)pow(3, a);\n}\n</code></pre> max_product_cutting.java<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int) Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int) Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int) Math.pow(3, a);\n}\n</code></pre> max_product_cutting.cs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint MaxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return (int)Math.Pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return (int)Math.Pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (int)Math.Pow(3, a);\n}\n</code></pre> max_product_cutting.go<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n int) int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    a := n / 3\n    b := n % 3\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return int(math.Pow(3, float64(a-1))) * 2 * 2\n    }\n    if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return int(math.Pow(3, float64(a))) * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return int(math.Pow(3, float64(a)))\n}\n</code></pre> max_product_cutting.swift<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunc maxProductCutting(n: Int) -&gt; Int {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1)\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = n / 3\n    let b = n % 3\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return pow(3, a - 1) * 2 * 2\n    }\n    if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return pow(3, a) * 2\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return pow(3, a)\n}\n</code></pre> max_product_cutting.js<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = Math.floor(n / 3);\n    let b = n % 3;\n    if (b === 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.ts<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfunction maxProductCutting(n: number): number {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a: number = Math.floor(n / 3);\n    let b: number = n % 3;\n    if (b === 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return Math.pow(3, a - 1) * 2 * 2;\n    }\n    if (b === 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return Math.pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return Math.pow(3, a);\n}\n</code></pre> max_product_cutting.dart<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n  // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n  if (n &lt;= 3) {\n    return 1 * (n - 1);\n  }\n  // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n  int a = n ~/ 3;\n  int b = n % 3;\n  if (b == 1) {\n    // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n    return (pow(3, a - 1) * 2 * 2).toInt();\n  }\n  if (b == 2) {\n    // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return (pow(3, a) * 2).toInt();\n  }\n  // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n  return pow(3, a).toInt();\n}\n</code></pre> max_product_cutting.rs<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nfn max_product_cutting(n: i32) -&gt; i32 {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if n &lt;= 3 {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    let a = n / 3;\n    let b = n % 3;\n    if b == 1 {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        3_i32.pow(a as u32 - 1) * 2 * 2\n    } else if b == 2 {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        3_i32.pow(a as u32) * 2\n    } else {\n        // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        3_i32.pow(a as u32)\n    }\n}\n</code></pre> max_product_cutting.c<pre><code>/* \u6700\u5927\u5207\u5206\u4e58\u79ef\uff1a\u8d2a\u5fc3 */\nint maxProductCutting(int n) {\n    // \u5f53 n &lt;= 3 \u65f6\uff0c\u5fc5\u987b\u5207\u5206\u51fa\u4e00\u4e2a 1\n    if (n &lt;= 3) {\n        return 1 * (n - 1);\n    }\n    // \u8d2a\u5fc3\u5730\u5207\u5206\u51fa 3 \uff0ca \u4e3a 3 \u7684\u4e2a\u6570\uff0cb \u4e3a\u4f59\u6570\n    int a = n / 3;\n    int b = n % 3;\n    if (b == 1) {\n        // \u5f53\u4f59\u6570\u4e3a 1 \u65f6\uff0c\u5c06\u4e00\u5bf9 1 * 3 \u8f6c\u5316\u4e3a 2 * 2\n        return pow(3, a - 1) * 2 * 2;\n    }\n    if (b == 2) {\n        // \u5f53\u4f59\u6570\u4e3a 2 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n        return pow(3, a) * 2;\n    }\n    // \u5f53\u4f59\u6570\u4e3a 0 \u65f6\uff0c\u4e0d\u505a\u5904\u7406\n    return pow(3, a);\n}\n</code></pre> max_product_cutting.zig<pre><code>[class]{}-[func]{maxProductCutting}\n</code></pre> <p></p> <p> \u56fe 15-16 \u00a0 \u6700\u5927\u5207\u5206\u4e58\u79ef\u7684\u8ba1\u7b97\u65b9\u6cd5 </p> <p>\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u5e42\u8fd0\u7b97\u7684\u5b9e\u73b0\u65b9\u6cd5\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u5e38\u7528\u7684\u5e42\u8ba1\u7b97\u51fd\u6570\u6709\u4e09\u79cd\u3002</p> <ul> <li>\u8fd0\u7b97\u7b26 <code>**</code> \u548c\u51fd\u6570 <code>pow()</code> \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log\u2061 a)\\) \u3002</li> <li>\u51fd\u6570 <code>math.pow()</code> \u5185\u90e8\u8c03\u7528 C \u8bed\u8a00\u5e93\u7684 <code>pow()</code> \u51fd\u6570\uff0c\u5176\u6267\u884c\u6d6e\u70b9\u53d6\u5e42\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> </ul> <p>\u53d8\u91cf \\(a\\) \u548c \\(b\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</p>"},{"location":"chapter_greedy/max_product_cutting_problem/#3","title":"3. \u00a0 \u6b63\u786e\u6027\u8bc1\u660e","text":"<p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u53ea\u5206\u6790 \\(n \\geq 3\\) \u7684\u60c5\u51b5\u3002</p> <ol> <li>\u6240\u6709\u56e0\u5b50 \\(\\leq 3\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728 \\(\\geq 4\\) \u7684\u56e0\u5b50 \\(x\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u5c06\u5176\u7ee7\u7eed\u5212\u5206\u4e3a \\(2(x-2)\\) \uff0c\u4ece\u800c\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u4e0d\u5305\u542b \\(1\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5b58\u5728\u4e00\u4e2a\u56e0\u5b50 \\(1\\) \uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u53ef\u4ee5\u5408\u5e76\u5165\u53e6\u5916\u4e00\u4e2a\u56e0\u5b50\u4e2d\uff0c\u4ee5\u83b7\u5f97\u66f4\u5927\u7684\u4e58\u79ef\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> <li>\u5207\u5206\u65b9\u6848\u6700\u591a\u5305\u542b\u4e24\u4e2a \\(2\\) \uff1a\u5047\u8bbe\u6700\u4f18\u5207\u5206\u65b9\u6848\u4e2d\u5305\u542b\u4e09\u4e2a \\(2\\) \uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u66ff\u6362\u4e3a\u4e24\u4e2a \\(3\\) \uff0c\u4e58\u79ef\u66f4\u5927\u3002\u8fd9\u4e0e\u5047\u8bbe\u77db\u76fe\u3002</li> </ol>"},{"location":"chapter_greedy/summary/","title":"15.5 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u8d2a\u5fc3\u7b97\u6cd5\u901a\u5e38\u7528\u4e8e\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\uff0c\u5176\u539f\u7406\u662f\u5728\u6bcf\u4e2a\u51b3\u7b56\u9636\u6bb5\u90fd\u505a\u51fa\u5c40\u90e8\u6700\u4f18\u7684\u51b3\u7b56\uff0c\u4ee5\u671f\u83b7\u5f97\u5168\u5c40\u6700\u4f18\u89e3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4f1a\u8fed\u4ee3\u5730\u505a\u51fa\u4e00\u4e2a\u53c8\u4e00\u4e2a\u7684\u8d2a\u5fc3\u9009\u62e9\uff0c\u6bcf\u8f6e\u90fd\u5c06\u95ee\u9898\u8f6c\u5316\u6210\u4e00\u4e2a\u89c4\u6a21\u66f4\u5c0f\u7684\u5b50\u95ee\u9898\uff0c\u76f4\u5230\u95ee\u9898\u88ab\u89e3\u51b3\u3002</li> <li>\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u4ec5\u5b9e\u73b0\u7b80\u5355\uff0c\u8fd8\u5177\u6709\u5f88\u9ad8\u7684\u89e3\u9898\u6548\u7387\u3002\u76f8\u6bd4\u4e8e\u52a8\u6001\u89c4\u5212\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u66f4\u4f4e\u3002</li> <li>\u5728\u96f6\u94b1\u5151\u6362\u95ee\u9898\u4e2d\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u786c\u5e01\u7ec4\u5408\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u4fdd\u8bc1\u627e\u5230\u6700\u4f18\u89e3\uff1b\u5bf9\u4e8e\u53e6\u5916\u4e00\u4e9b\u786c\u5e01\u7ec4\u5408\u5219\u4e0d\u7136\uff0c\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u80fd\u627e\u5230\u5f88\u5dee\u7684\u89e3\u3002</li> <li>\u9002\u5408\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u7684\u95ee\u9898\u5177\u6709\u4e24\u5927\u6027\u8d28\uff1a\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u548c\u6700\u4f18\u5b50\u7ed3\u6784\u3002\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u4ee3\u8868\u8d2a\u5fc3\u7b56\u7565\u7684\u6709\u6548\u6027\u3002</li> <li>\u5bf9\u4e8e\u67d0\u4e9b\u590d\u6742\u95ee\u9898\uff0c\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u7684\u8bc1\u660e\u5e76\u4e0d\u7b80\u5355\u3002\u76f8\u5bf9\u6765\u8bf4\uff0c\u8bc1\u4f2a\u66f4\u52a0\u5bb9\u6613\uff0c\u4f8b\u5982\u96f6\u94b1\u5151\u6362\u95ee\u9898\u3002</li> <li>\u6c42\u89e3\u8d2a\u5fc3\u95ee\u9898\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\uff1a\u95ee\u9898\u5206\u6790\u3001\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u3001\u6b63\u786e\u6027\u8bc1\u660e\u3002\u5176\u4e2d\uff0c\u786e\u5b9a\u8d2a\u5fc3\u7b56\u7565\u662f\u6838\u5fc3\u6b65\u9aa4\uff0c\u6b63\u786e\u6027\u8bc1\u660e\u5f80\u5f80\u662f\u96be\u70b9\u3002</li> <li>\u5206\u6570\u80cc\u5305\u95ee\u9898\u5728 0-1 \u80cc\u5305\u7684\u57fa\u7840\u4e0a\uff0c\u5141\u8bb8\u9009\u62e9\u7269\u54c1\u7684\u4e00\u90e8\u5206\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6c42\u89e3\u3002\u8d2a\u5fc3\u7b56\u7565\u7684\u6b63\u786e\u6027\u53ef\u4ee5\u4f7f\u7528\u53cd\u8bc1\u6cd5\u6765\u8bc1\u660e\u3002</li> <li>\u6700\u5927\u5bb9\u91cf\u95ee\u9898\u53ef\u4f7f\u7528\u7a77\u4e3e\u6cd5\u6c42\u89e3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u901a\u8fc7\u8bbe\u8ba1\u8d2a\u5fc3\u7b56\u7565\uff0c\u6bcf\u8f6e\u5411\u5185\u79fb\u52a8\u77ed\u677f\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u5728\u6700\u5927\u5207\u5206\u4e58\u79ef\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5148\u540e\u63a8\u7406\u51fa\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff1a\\(\\geq 4\\) \u7684\u6574\u6570\u90fd\u5e94\u8be5\u7ee7\u7eed\u5207\u5206\u3001\u6700\u4f18\u5207\u5206\u56e0\u5b50\u4e3a \\(3\\) \u3002\u4ee3\u7801\u4e2d\u5305\u542b\u5e42\u8fd0\u7b97\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53d6\u51b3\u4e8e\u5e42\u8fd0\u7b97\u5b9e\u73b0\u65b9\u6cd5\uff0c\u901a\u5e38\u4e3a \\(O(1)\\) \u6216 \\(O(\\log n)\\) \u3002</li> </ul>"},{"location":"chapter_hashing/","title":"\u7b2c 6 \u7ae0 \u00a0 \u54c8\u5e0c\u8868","text":"<p>Abstract</p> <p>\u5728\u8ba1\u7b97\u673a\u4e16\u754c\u4e2d\uff0c\u54c8\u5e0c\u8868\u5982\u540c\u4e00\u4f4d\u806a\u6167\u7684\u56fe\u4e66\u7ba1\u7406\u5458\u3002</p> <p>\u4ed6\u77e5\u9053\u5982\u4f55\u8ba1\u7b97\u7d22\u4e66\u53f7\uff0c\u4ece\u800c\u53ef\u4ee5\u5feb\u901f\u627e\u5230\u76ee\u6807\u56fe\u4e66\u3002</p>"},{"location":"chapter_hashing/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>6.1 \u00a0 \u54c8\u5e0c\u8868</li> <li>6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81</li> <li>6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5</li> <li>6.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_hashing/hash_algorithm/","title":"6.3 \u00a0 \u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u524d\u4e24\u8282\u4ecb\u7ecd\u4e86\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u539f\u7406\u548c\u54c8\u5e0c\u51b2\u7a81\u7684\u5904\u7406\u65b9\u6cd5\u3002\u7136\u800c\u65e0\u8bba\u662f\u5f00\u653e\u5bfb\u5740\u8fd8\u662f\u94fe\u5f0f\u5730\u5740\uff0c\u5b83\u4eec\u53ea\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u53d1\u751f\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\uff0c\u800c\u65e0\u6cd5\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u3002</p> <p>\u5982\u679c\u54c8\u5e0c\u51b2\u7a81\u8fc7\u4e8e\u9891\u7e41\uff0c\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u5219\u4f1a\u6025\u5267\u52a3\u5316\u3002\u5982\u56fe 6-8 \u6240\u793a\uff0c\u5bf9\u4e8e\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\uff0c\u7406\u60f3\u60c5\u51b5\u4e0b\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u5404\u4e2a\u6876\u4e2d\uff0c\u8fbe\u5230\u6700\u4f73\u67e5\u8be2\u6548\u7387\uff1b\u6700\u5dee\u60c5\u51b5\u4e0b\u6240\u6709\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 6-8 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u7684\u6700\u4f73\u60c5\u51b5\u4e0e\u6700\u5dee\u60c5\u51b5 </p> <p>\u952e\u503c\u5bf9\u7684\u5206\u5e03\u60c5\u51b5\u7531\u54c8\u5e0c\u51fd\u6570\u51b3\u5b9a\u3002\u56de\u5fc6\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u6b65\u9aa4\uff0c\u5148\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u518d\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\uff1a</p> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u89c2\u5bdf\u4ee5\u4e0a\u516c\u5f0f\uff0c\u5f53\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u56fa\u5b9a\u65f6\uff0c\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u51b3\u5b9a\u4e86\u8f93\u51fa\u503c\uff0c\u8fdb\u800c\u51b3\u5b9a\u4e86\u952e\u503c\u5bf9\u5728\u54c8\u5e0c\u8868\u4e2d\u7684\u5206\u5e03\u60c5\u51b5\u3002</p> <p>\u8fd9\u610f\u5473\u7740\uff0c\u4e3a\u4e86\u964d\u4f4e\u54c8\u5e0c\u51b2\u7a81\u7684\u53d1\u751f\u6982\u7387\uff0c\u6211\u4eec\u5e94\u5f53\u5c06\u6ce8\u610f\u529b\u96c6\u4e2d\u5728\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u7684\u8bbe\u8ba1\u4e0a\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#631","title":"6.3.1 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u76ee\u6807","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u201c\u65e2\u5feb\u53c8\u7a33\u201d\u7684\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u5177\u5907\u4ee5\u4e0b\u7279\u70b9\u3002</p> <ul> <li>\u786e\u5b9a\u6027\uff1a\u5bf9\u4e8e\u76f8\u540c\u7684\u8f93\u5165\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5e94\u59cb\u7ec8\u4ea7\u751f\u76f8\u540c\u7684\u8f93\u51fa\u3002\u8fd9\u6837\u624d\u80fd\u786e\u4fdd\u54c8\u5e0c\u8868\u662f\u53ef\u9760\u7684\u3002</li> <li>\u6548\u7387\u9ad8\uff1a\u8ba1\u7b97\u54c8\u5e0c\u503c\u7684\u8fc7\u7a0b\u5e94\u8be5\u8db3\u591f\u5feb\u3002\u8ba1\u7b97\u5f00\u9500\u8d8a\u5c0f\uff0c\u54c8\u5e0c\u8868\u7684\u5b9e\u7528\u6027\u8d8a\u9ad8\u3002</li> <li>\u5747\u5300\u5206\u5e03\uff1a\u54c8\u5e0c\u7b97\u6cd5\u5e94\u4f7f\u5f97\u952e\u503c\u5bf9\u5747\u5300\u5206\u5e03\u5728\u54c8\u5e0c\u8868\u4e2d\u3002\u5206\u5e03\u8d8a\u5747\u5300\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002</li> </ul> <p>\u5b9e\u9645\u4e0a\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9664\u4e86\u53ef\u4ee5\u7528\u4e8e\u5b9e\u73b0\u54c8\u5e0c\u8868\uff0c\u8fd8\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5176\u4ed6\u9886\u57df\u4e2d\u3002</p> <ul> <li>\u5bc6\u7801\u5b58\u50a8\uff1a\u4e3a\u4e86\u4fdd\u62a4\u7528\u6237\u5bc6\u7801\u7684\u5b89\u5168\uff0c\u7cfb\u7edf\u901a\u5e38\u4e0d\u4f1a\u76f4\u63a5\u5b58\u50a8\u7528\u6237\u7684\u660e\u6587\u5bc6\u7801\uff0c\u800c\u662f\u5b58\u50a8\u5bc6\u7801\u7684\u54c8\u5e0c\u503c\u3002\u5f53\u7528\u6237\u8f93\u5165\u5bc6\u7801\u65f6\uff0c\u7cfb\u7edf\u4f1a\u5bf9\u8f93\u5165\u7684\u5bc6\u7801\u8ba1\u7b97\u54c8\u5e0c\u503c\uff0c\u7136\u540e\u4e0e\u5b58\u50a8\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u5bc6\u7801\u5c31\u88ab\u89c6\u4e3a\u6b63\u786e\u3002</li> <li>\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5\uff1a\u6570\u636e\u53d1\u9001\u65b9\u53ef\u4ee5\u8ba1\u7b97\u6570\u636e\u7684\u54c8\u5e0c\u503c\u5e76\u5c06\u5176\u4e00\u540c\u53d1\u9001\uff1b\u63a5\u6536\u65b9\u53ef\u4ee5\u91cd\u65b0\u8ba1\u7b97\u63a5\u6536\u5230\u7684\u6570\u636e\u7684\u54c8\u5e0c\u503c\uff0c\u5e76\u4e0e\u63a5\u6536\u5230\u7684\u54c8\u5e0c\u503c\u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u4e24\u8005\u5339\u914d\uff0c\u90a3\u4e48\u6570\u636e\u5c31\u88ab\u89c6\u4e3a\u5b8c\u6574\u3002</li> </ul> <p>\u5bf9\u4e8e\u5bc6\u7801\u5b66\u7684\u76f8\u5173\u5e94\u7528\uff0c\u4e3a\u4e86\u9632\u6b62\u4ece\u54c8\u5e0c\u503c\u63a8\u5bfc\u51fa\u539f\u59cb\u5bc6\u7801\u7b49\u9006\u5411\u5de5\u7a0b\uff0c\u54c8\u5e0c\u7b97\u6cd5\u9700\u8981\u5177\u5907\u66f4\u9ad8\u7b49\u7ea7\u7684\u5b89\u5168\u7279\u6027\u3002</p> <ul> <li>\u5355\u5411\u6027\uff1a\u65e0\u6cd5\u901a\u8fc7\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u5173\u4e8e\u8f93\u5165\u6570\u636e\u7684\u4efb\u4f55\u4fe1\u606f\u3002</li> <li>\u6297\u78b0\u649e\u6027\uff1a\u5e94\u5f53\u6781\u96be\u627e\u5230\u4e24\u4e2a\u4e0d\u540c\u7684\u8f93\u5165\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u54c8\u5e0c\u503c\u76f8\u540c\u3002</li> <li>\u96ea\u5d29\u6548\u5e94\uff1a\u8f93\u5165\u7684\u5fae\u5c0f\u53d8\u5316\u5e94\u5f53\u5bfc\u81f4\u8f93\u51fa\u7684\u663e\u8457\u4e14\u4e0d\u53ef\u9884\u6d4b\u7684\u53d8\u5316\u3002</li> </ul> <p>\u8bf7\u6ce8\u610f\uff0c\u201c\u5747\u5300\u5206\u5e03\u201d\u4e0e\u201c\u6297\u78b0\u649e\u6027\u201d\u662f\u4e24\u4e2a\u72ec\u7acb\u7684\u6982\u5ff5\uff0c\u6ee1\u8db3\u5747\u5300\u5206\u5e03\u4e0d\u4e00\u5b9a\u6ee1\u8db3\u6297\u78b0\u649e\u6027\u3002\u4f8b\u5982\uff0c\u5728\u968f\u673a\u8f93\u5165 <code>key</code> \u4e0b\uff0c\u54c8\u5e0c\u51fd\u6570 <code>key % 100</code> \u53ef\u4ee5\u4ea7\u751f\u5747\u5300\u5206\u5e03\u7684\u8f93\u51fa\u3002\u7136\u800c\u8be5\u54c8\u5e0c\u7b97\u6cd5\u8fc7\u4e8e\u7b80\u5355\uff0c\u6240\u6709\u540e\u4e24\u4f4d\u76f8\u7b49\u7684 <code>key</code> \u7684\u8f93\u51fa\u90fd\u76f8\u540c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u4ece\u54c8\u5e0c\u503c\u53cd\u63a8\u51fa\u53ef\u7528\u7684 <code>key</code> \uff0c\u4ece\u800c\u7834\u89e3\u5bc6\u7801\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#632","title":"6.3.2 \u00a0 \u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1","text":"<p>\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e00\u4e2a\u9700\u8981\u8003\u8651\u8bb8\u591a\u56e0\u7d20\u7684\u590d\u6742\u95ee\u9898\u3002\u7136\u800c\u5bf9\u4e8e\u67d0\u4e9b\u8981\u6c42\u4e0d\u9ad8\u7684\u573a\u666f\uff0c\u6211\u4eec\u4e5f\u80fd\u8bbe\u8ba1\u4e00\u4e9b\u7b80\u5355\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>\u52a0\u6cd5\u54c8\u5e0c\uff1a\u5bf9\u8f93\u5165\u7684\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u8fdb\u884c\u76f8\u52a0\uff0c\u5c06\u5f97\u5230\u7684\u603b\u548c\u4f5c\u4e3a\u54c8\u5e0c\u503c\u3002</li> <li>\u4e58\u6cd5\u54c8\u5e0c\uff1a\u5229\u7528\u4e58\u6cd5\u7684\u4e0d\u76f8\u5173\u6027\uff0c\u6bcf\u8f6e\u4e58\u4ee5\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5404\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u5f02\u6216\u54c8\u5e0c\uff1a\u5c06\u8f93\u5165\u6570\u636e\u7684\u6bcf\u4e2a\u5143\u7d20\u901a\u8fc7\u5f02\u6216\u64cd\u4f5c\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\u3002</li> <li>\u65cb\u8f6c\u54c8\u5e0c\uff1a\u5c06\u6bcf\u4e2a\u5b57\u7b26\u7684 ASCII \u7801\u7d2f\u79ef\u5230\u4e00\u4e2a\u54c8\u5e0c\u503c\u4e2d\uff0c\u6bcf\u6b21\u7d2f\u79ef\u4e4b\u524d\u90fd\u4f1a\u5bf9\u54c8\u5e0c\u503c\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig simple_hash.py<pre><code>def add_hash(key: str) -&gt; int:\n    \"\"\"\u52a0\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash += ord(c)\n    return hash % modulus\n\ndef mul_hash(key: str) -&gt; int:\n    \"\"\"\u4e58\u6cd5\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = 31 * hash + ord(c)\n    return hash % modulus\n\ndef xor_hash(key: str) -&gt; int:\n    \"\"\"\u5f02\u6216\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash ^= ord(c)\n    return hash % modulus\n\ndef rot_hash(key: str) -&gt; int:\n    \"\"\"\u65cb\u8f6c\u54c8\u5e0c\"\"\"\n    hash = 0\n    modulus = 1000000007\n    for c in key:\n        hash = (hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ ord(c)\n    return hash % modulus\n</code></pre> simple_hash.cpp<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = (31 * hash + (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash ^= (int)c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(string key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (unsigned char c : key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int)c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.java<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = (31 * hash + (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n    int hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash ^= (int) c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n    long hash = 0;\n    final int MODULUS = 1000000007;\n    for (char c : key.toCharArray()) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (int) c) % MODULUS;\n    }\n    return (int) hash;\n}\n</code></pre> simple_hash.cs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint AddHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint MulHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = (31 * hash + c) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint XorHash(string key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash ^= c;\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint RotHash(string key) {\n    long hash = 0;\n    const int MODULUS = 1000000007;\n    foreach (char c in key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c) % MODULUS;\n    }\n    return (int)hash;\n}\n</code></pre> simple_hash.go<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = (31*hash + int64(b)) % modulus\n    }\n    return int(hash)\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key string) int {\n    hash := 0\n    modulus := 1000000007\n    for _, b := range []byte(key) {\n        fmt.Println(int(b))\n        hash ^= int(b)\n        hash = (31*hash + int(b)) % modulus\n    }\n    return hash &amp; modulus\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key string) int {\n    var hash int64\n    var modulus int64\n\n    modulus = 1000000007\n    for _, b := range []byte(key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ int64(b)) % modulus\n    }\n    return int(hash)\n}\n</code></pre> simple_hash.swift<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunc addHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunc mulHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = (31 * hash + Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunc xorHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash ^= Int(scalar.value)\n        }\n    }\n    return hash &amp; MODULUS\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunc rotHash(key: String) -&gt; Int {\n    var hash = 0\n    let MODULUS = 1_000_000_007\n    for c in key {\n        for scalar in c.unicodeScalars {\n            hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ Int(scalar.value)) % MODULUS\n        }\n    }\n    return hash\n}\n</code></pre> simple_hash.js<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key) {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.ts<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfunction addHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfunction mulHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = (31 * hash + c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfunction xorHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash ^= c.charCodeAt(0);\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfunction rotHash(key: string): number {\n    let hash = 0;\n    const MODULUS = 1000000007;\n    for (const c of key) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c.charCodeAt(0)) % MODULUS;\n    }\n    return hash;\n}\n</code></pre> simple_hash.dart<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = (31 * hash + key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash ^= key.codeUnitAt(i);\n  }\n  return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(String key) {\n  int hash = 0;\n  final int MODULUS = 1000000007;\n  for (int i = 0; i &lt; key.length; i++) {\n    hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ key.codeUnitAt(i)) % MODULUS;\n  }\n  return hash;\n}\n</code></pre> simple_hash.rs<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nfn add_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n} \n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nfn mul_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = (31 * hash + c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nfn xor_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash ^= c as i64;\n    }\n\n    (hash &amp; MODULUS) as i32\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nfn rot_hash(key: &amp;str) -&gt; i32 {\n    let mut hash = 0_i64;\n    const MODULUS: i64 = 1000000007;\n\n    for c in key.chars() {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ c as i64) % MODULUS;\n    }\n\n    hash as i32\n}\n</code></pre> simple_hash.c<pre><code>/* \u52a0\u6cd5\u54c8\u5e0c */\nint addHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u4e58\u6cd5\u54c8\u5e0c */\nint mulHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = (31 * hash + (unsigned char)key[i]) % MODULUS;\n    }\n    return (int)hash;\n}\n\n/* \u5f02\u6216\u54c8\u5e0c */\nint xorHash(char *key) {\n    int hash = 0;\n    const int MODULUS = 1000000007;\n\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash ^= (unsigned char)key[i];\n    }\n    return hash &amp; MODULUS;\n}\n\n/* \u65cb\u8f6c\u54c8\u5e0c */\nint rotHash(char *key) {\n    long long hash = 0;\n    const int MODULUS = 1000000007;\n    for (int i = 0; i &lt; strlen(key); i++) {\n        hash = ((hash &lt;&lt; 4) ^ (hash &gt;&gt; 28) ^ (unsigned char)key[i]) % MODULUS;\n    }\n\n    return (int)hash;\n}\n</code></pre> simple_hash.zig<pre><code>[class]{}-[func]{addHash}\n\n[class]{}-[func]{mulHash}\n\n[class]{}-[func]{xorHash}\n\n[class]{}-[func]{rotHash}\n</code></pre> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u6bcf\u79cd\u54c8\u5e0c\u7b97\u6cd5\u7684\u6700\u540e\u4e00\u6b65\u90fd\u662f\u5bf9\u5927\u8d28\u6570 \\(1000000007\\) \u53d6\u6a21\uff0c\u4ee5\u786e\u4fdd\u54c8\u5e0c\u503c\u5728\u5408\u9002\u7684\u8303\u56f4\u5185\u3002\u503c\u5f97\u601d\u8003\u7684\u662f\uff0c\u4e3a\u4ec0\u4e48\u8981\u5f3a\u8c03\u5bf9\u8d28\u6570\u53d6\u6a21\uff0c\u6216\u8005\u8bf4\u5bf9\u5408\u6570\u53d6\u6a21\u7684\u5f0a\u7aef\u662f\u4ec0\u4e48\uff1f\u8fd9\u662f\u4e00\u4e2a\u6709\u8da3\u7684\u95ee\u9898\u3002</p> <p>\u5148\u629b\u51fa\u7ed3\u8bba\uff1a\u4f7f\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u53ef\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u7684\u5747\u5300\u5206\u5e03\u3002\u56e0\u4e3a\u8d28\u6570\u4e0d\u4e0e\u5176\u4ed6\u6570\u5b57\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u53ef\u4ee5\u51cf\u5c11\u56e0\u53d6\u6a21\u64cd\u4f5c\u800c\u4ea7\u751f\u7684\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u4ece\u800c\u907f\u514d\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u5408\u6570 \\(9\\) \u4f5c\u4e3a\u6a21\u6570\uff0c\u5b83\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\uff0c\u90a3\u4e48\u6240\u6709\u53ef\u4ee5\u88ab \\(3\\) \u6574\u9664\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230 \\(0\\)\u3001\\(3\\)\u3001\\(6\\) \u8fd9\u4e09\u4e2a\u54c8\u5e0c\u503c\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 9 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 0, 3, 6, 0, 3, 6, 0, 3, 6,\\dots \\} \\end{aligned} \\] <p>\u5982\u679c\u8f93\u5165 <code>key</code> \u6070\u597d\u6ee1\u8db3\u8fd9\u79cd\u7b49\u5dee\u6570\u5217\u7684\u6570\u636e\u5206\u5e03\uff0c\u90a3\u4e48\u54c8\u5e0c\u503c\u5c31\u4f1a\u51fa\u73b0\u805a\u5806\uff0c\u4ece\u800c\u52a0\u91cd\u54c8\u5e0c\u51b2\u7a81\u3002\u73b0\u5728\uff0c\u5047\u8bbe\u5c06 <code>modulus</code> \u66ff\u6362\u4e3a\u8d28\u6570 \\(13\\) \uff0c\u7531\u4e8e <code>key</code> \u548c <code>modulus</code> \u4e4b\u95f4\u4e0d\u5b58\u5728\u516c\u7ea6\u6570\uff0c\u56e0\u6b64\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u7684\u5747\u5300\u6027\u4f1a\u660e\u663e\u63d0\u5347\u3002</p> \\[ \\begin{aligned} \\text{modulus} &amp; = 13 \\newline \\text{key} &amp; = \\{ 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, \\dots \\} \\newline \\text{hash} &amp; = \\{ 0, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, \\dots \\} \\end{aligned} \\] <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5982\u679c\u80fd\u591f\u4fdd\u8bc1 <code>key</code> \u662f\u968f\u673a\u5747\u5300\u5206\u5e03\u7684\uff0c\u90a3\u4e48\u9009\u62e9\u8d28\u6570\u6216\u8005\u5408\u6570\u4f5c\u4e3a\u6a21\u6570\u90fd\u53ef\u4ee5\uff0c\u5b83\u4eec\u90fd\u80fd\u8f93\u51fa\u5747\u5300\u5206\u5e03\u7684\u54c8\u5e0c\u503c\u3002\u800c\u5f53 <code>key</code> \u7684\u5206\u5e03\u5b58\u5728\u67d0\u79cd\u5468\u671f\u6027\u65f6\uff0c\u5bf9\u5408\u6570\u53d6\u6a21\u66f4\u5bb9\u6613\u51fa\u73b0\u805a\u96c6\u73b0\u8c61\u3002</p> <p>\u603b\u800c\u8a00\u4e4b\uff0c\u6211\u4eec\u901a\u5e38\u9009\u53d6\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u5e76\u4e14\u8fd9\u4e2a\u8d28\u6570\u6700\u597d\u8db3\u591f\u5927\uff0c\u4ee5\u5c3d\u53ef\u80fd\u6d88\u9664\u5468\u671f\u6027\u6a21\u5f0f\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002</p>"},{"location":"chapter_hashing/hash_algorithm/#633","title":"6.3.3 \u00a0 \u5e38\u89c1\u54c8\u5e0c\u7b97\u6cd5","text":"<p>\u4e0d\u96be\u53d1\u73b0\uff0c\u4ee5\u4e0a\u4ecb\u7ecd\u7684\u7b80\u5355\u54c8\u5e0c\u7b97\u6cd5\u90fd\u6bd4\u8f83\u201c\u8106\u5f31\u201d\uff0c\u8fdc\u8fdc\u6ca1\u6709\u8fbe\u5230\u54c8\u5e0c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u76ee\u6807\u3002\u4f8b\u5982\uff0c\u7531\u4e8e\u52a0\u6cd5\u548c\u5f02\u6216\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u56e0\u6b64\u52a0\u6cd5\u54c8\u5e0c\u548c\u5f02\u6216\u54c8\u5e0c\u65e0\u6cd5\u533a\u5206\u5185\u5bb9\u76f8\u540c\u4f46\u987a\u5e8f\u4e0d\u540c\u7684\u5b57\u7b26\u4e32\uff0c\u8fd9\u53ef\u80fd\u4f1a\u52a0\u5267\u54c8\u5e0c\u51b2\u7a81\uff0c\u5e76\u5f15\u8d77\u4e00\u4e9b\u5b89\u5168\u95ee\u9898\u3002</p> <p>\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u7528\u4e00\u4e9b\u6807\u51c6\u54c8\u5e0c\u7b97\u6cd5\uff0c\u4f8b\u5982 MD5\u3001SHA-1\u3001SHA-2\u3001SHA-3 \u7b49\u3002\u5b83\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u957f\u5ea6\u7684\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u6052\u5b9a\u957f\u5ea6\u7684\u54c8\u5e0c\u503c\u3002</p> <p>\u8fd1\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5904\u5728\u4e0d\u65ad\u5347\u7ea7\u4e0e\u4f18\u5316\u7684\u8fc7\u7a0b\u4e2d\u3002\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u52aa\u529b\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u6027\u80fd\uff0c\u53e6\u4e00\u90e8\u5206\u7814\u7a76\u4eba\u5458\u548c\u9ed1\u5ba2\u5219\u81f4\u529b\u4e8e\u5bfb\u627e\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u95ee\u9898\u3002\u8868 6-2 \u5c55\u793a\u4e86\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u3002</p> <ul> <li>MD5 \u548c SHA-1 \u5df2\u591a\u6b21\u88ab\u6210\u529f\u653b\u51fb\uff0c\u56e0\u6b64\u5b83\u4eec\u88ab\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u5f03\u7528\u3002</li> <li>SHA-2 \u7cfb\u5217\u4e2d\u7684 SHA-256 \u662f\u6700\u5b89\u5168\u7684\u54c8\u5e0c\u7b97\u6cd5\u4e4b\u4e00\uff0c\u4ecd\u672a\u51fa\u73b0\u6210\u529f\u7684\u653b\u51fb\u6848\u4f8b\uff0c\u56e0\u6b64\u5e38\u7528\u5728\u5404\u7c7b\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u4e2d\u3002</li> <li>SHA-3 \u76f8\u8f83 SHA-2 \u7684\u5b9e\u73b0\u5f00\u9500\u66f4\u4f4e\u3001\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\uff0c\u4f46\u76ee\u524d\u4f7f\u7528\u8986\u76d6\u5ea6\u4e0d\u5982 SHA-2 \u7cfb\u5217\u3002</li> </ul> <p> \u8868 6-2 \u00a0 \u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5 </p> MD5 SHA-1 SHA-2 SHA-3 \u63a8\u51fa\u65f6\u95f4 1992 1995 2002 2008 \u8f93\u51fa\u957f\u5ea6 128 bits 160 bits 256/512 bits 224/256/384/512 bits \u54c8\u5e0c\u51b2\u7a81 \u8f83\u591a \u8f83\u591a \u5f88\u5c11 \u5f88\u5c11 \u5b89\u5168\u7b49\u7ea7 \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u4f4e\uff0c\u5df2\u88ab\u6210\u529f\u653b\u51fb \u9ad8 \u9ad8 \u5e94\u7528 \u5df2\u88ab\u5f03\u7528\uff0c\u4ecd\u7528\u4e8e\u6570\u636e\u5b8c\u6574\u6027\u68c0\u67e5 \u5df2\u88ab\u5f03\u7528 \u52a0\u5bc6\u8d27\u5e01\u4ea4\u6613\u9a8c\u8bc1\u3001\u6570\u5b57\u7b7e\u540d\u7b49 \u53ef\u7528\u4e8e\u66ff\u4ee3 SHA-2"},{"location":"chapter_hashing/hash_algorithm/#634","title":"6.3.4 \u00a0 \u6570\u636e\u7ed3\u6784\u7684\u54c8\u5e0c\u503c","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u54c8\u5e0c\u8868\u7684 <code>key</code> \u53ef\u4ee5\u662f\u6574\u6570\u3001\u5c0f\u6570\u6216\u5b57\u7b26\u4e32\u7b49\u6570\u636e\u7c7b\u578b\u3002\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u8fd9\u4e9b\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u7684\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u4ee5 Python \u4e3a\u4f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u8c03\u7528 <code>hash()</code> \u51fd\u6570\u6765\u8ba1\u7b97\u5404\u79cd\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u3002</p> <ul> <li>\u6574\u6570\u548c\u5e03\u5c14\u91cf\u7684\u54c8\u5e0c\u503c\u5c31\u662f\u5176\u672c\u8eab\u3002</li> <li>\u6d6e\u70b9\u6570\u548c\u5b57\u7b26\u4e32\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u8f83\u4e3a\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u8bf7\u81ea\u884c\u5b66\u4e60\u3002</li> <li>\u5143\u7ec4\u7684\u54c8\u5e0c\u503c\u662f\u5bf9\u5176\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u54c8\u5e0c\uff0c\u7136\u540e\u5c06\u8fd9\u4e9b\u54c8\u5e0c\u503c\u7ec4\u5408\u8d77\u6765\uff0c\u5f97\u5230\u5355\u4e00\u7684\u54c8\u5e0c\u503c\u3002</li> <li>\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u57fa\u4e8e\u5176\u5185\u5b58\u5730\u5740\u751f\u6210\u3002\u901a\u8fc7\u91cd\u5199\u5bf9\u8c61\u7684\u54c8\u5e0c\u65b9\u6cd5\uff0c\u53ef\u5b9e\u73b0\u57fa\u4e8e\u5185\u5bb9\u751f\u6210\u54c8\u5e0c\u503c\u3002</li> </ul> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u54c8\u5e0c\u503c\u8ba1\u7b97\u51fd\u6570\u7684\u5b9a\u4e49\u548c\u65b9\u6cd5\u4e0d\u540c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig built_in_hash.py<pre><code>num = 3\nhash_num = hash(num)\n# \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbol = True\nhash_bol = hash(bol)\n# \u5e03\u5c14\u91cf True \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndec = 3.14159\nhash_dec = hash(dec)\n# \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 326484311674566659\n\nstr = \"Hello \u7b97\u6cd5\"\nhash_str = hash(str)\n# \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 4617003410720528961\n\ntup = (12836, \"\u5c0f\u54c8\")\nhash_tup = hash(tup)\n# \u5143\u7ec4 (12836, '\u5c0f\u54c8') \u7684\u54c8\u5e0c\u503c\u4e3a 1029005403108185979\n\nobj = ListNode(0)\nhash_obj = hash(obj)\n# \u8282\u70b9\u5bf9\u8c61 &lt;ListNode object at 0x1058fd810&gt; \u7684\u54c8\u5e0c\u503c\u4e3a 274267521\n</code></pre> built_in_hash.cpp<pre><code>int num = 3;\nsize_t hashNum = hash&lt;int&gt;()(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nbool bol = true;\nsize_t hashBol = hash&lt;bool&gt;()(bol);\n// \u5e03\u5c14\u91cf 1 \u7684\u54c8\u5e0c\u503c\u4e3a 1\n\ndouble dec = 3.14159;\nsize_t hashDec = hash&lt;double&gt;()(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 4614256650576692846\n\nstring str = \"Hello \u7b97\u6cd5\";\nsize_t hashStr = hash&lt;string&gt;()(str);\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 15466937326284535026\n\n// \u5728 C++ \u4e2d\uff0c\u5185\u7f6e std:hash() \u4ec5\u63d0\u4f9b\u57fa\u672c\u6570\u636e\u7c7b\u578b\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\n// \u6570\u7ec4\u3001\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u8ba1\u7b97\u9700\u8981\u81ea\u884c\u5b9e\u73b0\n</code></pre> built_in_hash.java<pre><code>int num = 3;\nint hashNum = Integer.hashCode(num);\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3\n\nboolean bol = true;\nint hashBol = Boolean.hashCode(bol);\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = Double.hashCode(dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -727081396\n\nObject[] arr = { 12836, \"\u5c0f\u54c8\" };\nint hashTup = Arrays.hashCode(arr);\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 1151158\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode();\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode@7dc5e7b4 \u7684\u54c8\u5e0c\u503c\u4e3a 2110121908\n</code></pre> built_in_hash.cs<pre><code>int num = 3;\nint hashNum = num.GetHashCode();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 3;\n\nbool bol = true;\nint hashBol = bol.GetHashCode();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 1;\n\ndouble dec = 3.14159;\nint hashDec = dec.GetHashCode();\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -1340954729;\n\nstring str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.GetHashCode();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -586107568;\n\nobject[] arr = [12836, \"\u5c0f\u54c8\"];\nint hashTup = arr.GetHashCode();\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 42931033;\n\nListNode obj = new(0);\nint hashObj = obj.GetHashCode();\n// \u8282\u70b9\u5bf9\u8c61 0 \u7684\u54c8\u5e0c\u503c\u4e3a 39053774;\n</code></pre> built_in_hash.go<pre><code>// Go \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.swift<pre><code>let num = 3\nlet hashNum = num.hashValue\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 9047044699613009734\n\nlet bol = true\nlet hashBol = bol.hashValue\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a -4431640247352757451\n\nlet dec = 3.14159\nlet hashDec = dec.hashValue\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a -2465384235396674631\n\nlet str = \"Hello \u7b97\u6cd5\"\nlet hashStr = str.hashValue\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a -7850626797806988787\n\nlet arr = [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")]\nlet hashTup = arr.hashValue\n// \u6570\u7ec4 [AnyHashable(12836), AnyHashable(\"\u5c0f\u54c8\")] \u7684\u54c8\u5e0c\u503c\u4e3a -2308633508154532996\n\nlet obj = ListNode(x: 0)\nlet hashObj = obj.hashValue\n// \u8282\u70b9\u5bf9\u8c61 utils.ListNode \u7684\u54c8\u5e0c\u503c\u4e3a -2434780518035996159\n</code></pre> built_in_hash.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.dart<pre><code>int num = 3;\nint hashNum = num.hashCode;\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 34803\n\nbool bol = true;\nint hashBol = bol.hashCode;\n// \u5e03\u5c14\u503c true \u7684\u54c8\u5e0c\u503c\u4e3a 1231\n\ndouble dec = 3.14159;\nint hashDec = dec.hashCode;\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2570631074981783\n\nString str = \"Hello \u7b97\u6cd5\";\nint hashStr = str.hashCode;\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 468167534\n\nList arr = [12836, \"\u5c0f\u54c8\"];\nint hashArr = arr.hashCode;\n// \u6570\u7ec4 [12836, \u5c0f\u54c8] \u7684\u54c8\u5e0c\u503c\u4e3a 976512528\n\nListNode obj = new ListNode(0);\nint hashObj = obj.hashCode;\n// \u8282\u70b9\u5bf9\u8c61 Instance of 'ListNode' \u7684\u54c8\u5e0c\u503c\u4e3a 1033450432\n</code></pre> built_in_hash.rs<pre><code>use std::collections::hash_map::DefaultHasher;\nuse std::hash::{Hash, Hasher};\n\nlet num = 3;\nlet mut num_hasher = DefaultHasher::new();\nnum.hash(&amp;mut num_hasher);\nlet hash_num = num_hasher.finish();\n// \u6574\u6570 3 \u7684\u54c8\u5e0c\u503c\u4e3a 568126464209439262\n\nlet bol = true;\nlet mut bol_hasher = DefaultHasher::new();\nbol.hash(&amp;mut bol_hasher);\nlet hash_bol = bol_hasher.finish();\n// \u5e03\u5c14\u91cf true \u7684\u54c8\u5e0c\u503c\u4e3a 4952851536318644461\n\nlet dec: f32 = 3.14159;\nlet mut dec_hasher = DefaultHasher::new();\ndec.to_bits().hash(&amp;mut dec_hasher);\nlet hash_dec = dec_hasher.finish();\nprintln!(\"\u5c0f\u6570 {} \u7684\u54c8\u5e0c\u503c\u4e3a {}\", dec, hash_dec);\n// \u5c0f\u6570 3.14159 \u7684\u54c8\u5e0c\u503c\u4e3a 2566941990314602357\n\nlet str = \"Hello \u7b97\u6cd5\";\nlet mut str_hasher = DefaultHasher::new();\nstr.hash(&amp;mut str_hasher);\nlet hash_str = str_hasher.finish();\n// \u5b57\u7b26\u4e32\u201cHello \u7b97\u6cd5\u201d\u7684\u54c8\u5e0c\u503c\u4e3a 16092673739211250988\n\nlet arr = (&amp;12836, &amp;\"\u5c0f\u54c8\");\nlet mut tup_hasher = DefaultHasher::new();\narr.hash(&amp;mut tup_hasher);\nlet hash_tup = tup_hasher.finish();\n// \u5143\u7ec4 (12836, \"\u5c0f\u54c8\") \u7684\u54c8\u5e0c\u503c\u4e3a 1885128010422702749\n\nlet node = ListNode::new(42);\nlet mut hasher = DefaultHasher::new();\nnode.borrow().val.hash(&amp;mut hasher);\nlet hash = hasher.finish();\n// \u8282\u70b9\u5bf9\u8c61 RefCell { value: ListNode { val: 42, next: None } } \u7684\u54c8\u5e0c\u503c\u4e3a15387811073369036852\n</code></pre> built_in_hash.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e hash code \u51fd\u6570\n</code></pre> built_in_hash.zig<pre><code>\n</code></pre> <p>\u5728\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u4e2d\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u624d\u53ef\u4f5c\u4e3a\u54c8\u5e0c\u8868\u7684 <code>key</code> \u3002\u5047\u5982\u6211\u4eec\u5c06\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4f5c\u4e3a <code>key</code> \uff0c\u5f53\u5217\u8868\u7684\u5185\u5bb9\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u5b83\u7684\u54c8\u5e0c\u503c\u4e5f\u968f\u4e4b\u6539\u53d8\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u54c8\u5e0c\u8868\u4e2d\u67e5\u8be2\u5230\u539f\u5148\u7684 <code>value</code> \u4e86\u3002</p> <p>\u867d\u7136\u81ea\u5b9a\u4e49\u5bf9\u8c61\uff08\u6bd4\u5982\u94fe\u8868\u8282\u70b9\uff09\u7684\u6210\u5458\u53d8\u91cf\u662f\u53ef\u53d8\u7684\uff0c\u4f46\u5b83\u662f\u53ef\u54c8\u5e0c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a\u5bf9\u8c61\u7684\u54c8\u5e0c\u503c\u901a\u5e38\u662f\u57fa\u4e8e\u5185\u5b58\u5730\u5740\u751f\u6210\u7684\uff0c\u5373\u4f7f\u5bf9\u8c61\u7684\u5185\u5bb9\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u4f46\u5b83\u7684\u5185\u5b58\u5730\u5740\u4e0d\u53d8\uff0c\u54c8\u5e0c\u503c\u4ecd\u7136\u662f\u4e0d\u53d8\u7684\u3002</p> <p>\u7ec6\u5fc3\u7684\u4f60\u53ef\u80fd\u53d1\u73b0\u5728\u4e0d\u540c\u63a7\u5236\u53f0\u4e2d\u8fd0\u884c\u7a0b\u5e8f\u65f6\uff0c\u8f93\u51fa\u7684\u54c8\u5e0c\u503c\u662f\u4e0d\u540c\u7684\u3002\u8fd9\u662f\u56e0\u4e3a Python \u89e3\u91ca\u5668\u5728\u6bcf\u6b21\u542f\u52a8\u65f6\uff0c\u90fd\u4f1a\u4e3a\u5b57\u7b26\u4e32\u54c8\u5e0c\u51fd\u6570\u52a0\u5165\u4e00\u4e2a\u968f\u673a\u7684\u76d0\uff08Salt\uff09\u503c\u3002\u8fd9\u79cd\u505a\u6cd5\u53ef\u4ee5\u6709\u6548\u9632\u6b62 HashDoS \u653b\u51fb\uff0c\u63d0\u5347\u54c8\u5e0c\u7b97\u6cd5\u7684\u5b89\u5168\u6027\u3002</p>"},{"location":"chapter_hashing/hash_collision/","title":"6.2 \u00a0 \u54c8\u5e0c\u51b2\u7a81","text":"<p>\u4e0a\u4e00\u8282\u63d0\u5230\uff0c\u901a\u5e38\u60c5\u51b5\u4e0b\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u5165\u7a7a\u95f4\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff0c\u56e0\u6b64\u7406\u8bba\u4e0a\u54c8\u5e0c\u51b2\u7a81\u662f\u4e0d\u53ef\u907f\u514d\u7684\u3002\u6bd4\u5982\uff0c\u8f93\u5165\u7a7a\u95f4\u4e3a\u5168\u4f53\u6574\u6570\uff0c\u8f93\u51fa\u7a7a\u95f4\u4e3a\u6570\u7ec4\u5bb9\u91cf\u5927\u5c0f\uff0c\u5219\u5fc5\u7136\u6709\u591a\u4e2a\u6574\u6570\u6620\u5c04\u81f3\u540c\u4e00\u6876\u7d22\u5f15\u3002</p> <p>\u54c8\u5e0c\u51b2\u7a81\u4f1a\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u9519\u8bef\uff0c\u4e25\u91cd\u5f71\u54cd\u54c8\u5e0c\u8868\u7684\u53ef\u7528\u6027\u3002\u4e3a\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u6bcf\u5f53\u9047\u5230\u54c8\u5e0c\u51b2\u7a81\u5c31\u8fdb\u884c\u54c8\u5e0c\u8868\u6269\u5bb9\uff0c\u76f4\u81f3\u51b2\u7a81\u6d88\u5931\u4e3a\u6b62\u3002\u6b64\u65b9\u6cd5\u7b80\u5355\u7c97\u66b4\u4e14\u6709\u6548\uff0c\u4f46\u6548\u7387\u592a\u4f4e\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u6570\u636e\u642c\u8fd0\u4e0e\u54c8\u5e0c\u503c\u8ba1\u7b97\u3002\u4e3a\u4e86\u63d0\u5347\u6548\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u7b56\u7565\u3002</p> <ol> <li>\u6539\u826f\u54c8\u5e0c\u8868\u6570\u636e\u7ed3\u6784\uff0c\u4f7f\u5f97\u54c8\u5e0c\u8868\u53ef\u4ee5\u5728\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u65f6\u6b63\u5e38\u5de5\u4f5c\u3002</li> <li>\u4ec5\u5728\u5fc5\u8981\u65f6\uff0c\u5373\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u624d\u6267\u884c\u6269\u5bb9\u64cd\u4f5c\u3002</li> </ol> <p>\u54c8\u5e0c\u8868\u7684\u7ed3\u6784\u6539\u826f\u65b9\u6cd5\u4e3b\u8981\u5305\u62ec\u201c\u94fe\u5f0f\u5730\u5740\u201d\u548c\u201c\u5f00\u653e\u5bfb\u5740\u201d\u3002</p>"},{"location":"chapter_hashing/hash_collision/#621","title":"6.2.1 \u00a0 \u94fe\u5f0f\u5730\u5740","text":"<p>\u5728\u539f\u59cb\u54c8\u5e0c\u8868\u4e2d\uff0c\u6bcf\u4e2a\u6876\u4ec5\u80fd\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u300c\u94fe\u5f0f\u5730\u5740 separate chaining\u300d\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u6362\u4e3a\u94fe\u8868\uff0c\u5c06\u952e\u503c\u5bf9\u4f5c\u4e3a\u94fe\u8868\u8282\u70b9\uff0c\u5c06\u6240\u6709\u53d1\u751f\u51b2\u7a81\u7684\u952e\u503c\u5bf9\u90fd\u5b58\u50a8\u5728\u540c\u4e00\u94fe\u8868\u4e2d\u3002\u56fe 6-5 \u5c55\u793a\u4e86\u4e00\u4e2a\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 6-5 \u00a0 \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 </p> <p>\u57fa\u4e8e\u94fe\u5f0f\u5730\u5740\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\u7684\u64cd\u4f5c\u65b9\u6cd5\u53d1\u751f\u4e86\u4ee5\u4e0b\u53d8\u5316\u3002</p> <ul> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u8f93\u5165 <code>key</code> \uff0c\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u6876\u7d22\u5f15\uff0c\u5373\u53ef\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u904d\u5386\u94fe\u8868\u5e76\u5bf9\u6bd4 <code>key</code> \u4ee5\u67e5\u627e\u76ee\u6807\u952e\u503c\u5bf9\u3002</li> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u9996\u5148\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8bbf\u95ee\u94fe\u8868\u5934\u8282\u70b9\uff0c\u7136\u540e\u5c06\u8282\u70b9\uff08\u952e\u503c\u5bf9\uff09\u6dfb\u52a0\u5230\u94fe\u8868\u4e2d\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u6839\u636e\u54c8\u5e0c\u51fd\u6570\u7684\u7ed3\u679c\u8bbf\u95ee\u94fe\u8868\u5934\u90e8\uff0c\u63a5\u7740\u904d\u5386\u94fe\u8868\u4ee5\u67e5\u627e\u76ee\u6807\u8282\u70b9\u5e76\u5c06\u5176\u5220\u9664\u3002</li> </ul> <p>\u94fe\u5f0f\u5730\u5740\u5b58\u5728\u4ee5\u4e0b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u5360\u7528\u7a7a\u95f4\u589e\u5927\uff0c\u94fe\u8868\u5305\u542b\u8282\u70b9\u6307\u9488\uff0c\u5b83\u76f8\u6bd4\u6570\u7ec4\u66f4\u52a0\u8017\u8d39\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u67e5\u8be2\u6548\u7387\u964d\u4f4e\uff0c\u56e0\u4e3a\u9700\u8981\u7ebf\u6027\u904d\u5386\u94fe\u8868\u6765\u67e5\u627e\u5bf9\u5e94\u5143\u7d20\u3002</li> </ul> <p>\u4ee5\u4e0b\u4ee3\u7801\u7ed9\u51fa\u4e86\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\u7684\u7b80\u5355\u5b9e\u73b0\uff0c\u9700\u8981\u6ce8\u610f\u4e24\u70b9\u3002</p> <ul> <li>\u4f7f\u7528\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\u4ee3\u66ff\u94fe\u8868\uff0c\u4ece\u800c\u7b80\u5316\u4ee3\u7801\u3002\u5728\u8fd9\u79cd\u8bbe\u5b9a\u4e0b\uff0c\u54c8\u5e0c\u8868\uff08\u6570\u7ec4\uff09\u5305\u542b\u591a\u4e2a\u6876\uff0c\u6bcf\u4e2a\u6876\u90fd\u662f\u4e00\u4e2a\u5217\u8868\u3002</li> <li>\u4ee5\u4e0b\u5b9e\u73b0\u5305\u542b\u54c8\u5e0c\u8868\u6269\u5bb9\u65b9\u6cd5\u3002\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(\\frac{2}{3}\\) \u65f6\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map_chaining.py<pre><code>class HashMapChaining:\n    \"\"\"\u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets = [[] for _ in range(self.capacity)]  # \u6876\u6570\u7ec4\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def get(self, key: int) -&gt; str | None:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket:\n            if pair.key == key:\n                return pair.val\n        # \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket:\n            if pair.key == key:\n                pair.val = val\n                return\n        # \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        pair = Pair(key, val)\n        bucket.append(pair)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index = self.hash_func(key)\n        bucket = self.buckets[index]\n        # \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for pair in bucket:\n            if pair.key == key:\n                bucket.remove(pair)\n                self.size -= 1\n                break\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [[] for _ in range(self.capacity)]\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets:\n            for pair in bucket:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for bucket in self.buckets:\n            res = []\n            for pair in bucket:\n                res.append(str(pair.key) + \" -&gt; \" + pair.val)\n            print(res)\n</code></pre> hash_map_chaining.cpp<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  private:\n    int size;                       // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;                   // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres;               // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;                // \u6269\u5bb9\u500d\u6570\n    vector&lt;vector&lt;Pair *&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapChaining() : size(0), capacity(4), loadThres(2.0 / 3.0), extendRatio(2) {\n        buckets.resize(capacity);\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapChaining() {\n        for (auto &amp;bucket : buckets) {\n            for (Pair *pair : bucket) {\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / (double)capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                return pair-&gt;val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair *pair : buckets[index]) {\n            if (pair-&gt;key == key) {\n                pair-&gt;val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].push_back(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        auto &amp;bucket = buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (int i = 0; i &lt; bucket.size(); i++) {\n            if (bucket[i]-&gt;key == key) {\n                Pair *tmp = bucket[i];\n                bucket.erase(bucket.begin() + i); // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n                delete tmp;                       // \u91ca\u653e\u5185\u5b58\n                size--;\n                return;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;vector&lt;Pair *&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets.clear();\n        buckets.resize(capacity);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (auto &amp;bucket : bucketsTmp) {\n            for (Pair *pair : bucket) {\n                put(pair-&gt;key, pair-&gt;val);\n                // \u91ca\u653e\u5185\u5b58\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (auto &amp;bucket : buckets) {\n            cout &lt;&lt; \"[\";\n            for (Pair *pair : bucket) {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; \", \";\n            }\n            cout &lt;&lt; \"]\\n\";\n        }\n    }\n};\n</code></pre> hash_map_chaining.java<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    String get(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        Pair pair = new Pair(key, val);\n        bucket.add(pair);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        List&lt;Pair&gt; bucket = buckets.get(index);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (Pair pair : bucket) {\n            if (pair.key == key) {\n                bucket.remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new ArrayList&lt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.add(new ArrayList&lt;&gt;());\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (List&lt;Pair&gt; bucket : bucketsTmp) {\n            for (Pair pair : bucket) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (List&lt;Pair&gt; bucket : buckets) {\n            List&lt;String&gt; res = new ArrayList&lt;&gt;();\n            for (Pair pair : bucket) {\n                res.add(pair.key + \" -&gt; \" + pair.val);\n            }\n            System.out.println(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.cs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio; // \u6269\u5bb9\u500d\u6570\n    List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapChaining() {\n        size = 0;\n        capacity = 4;\n        loadThres = 2.0 / 3.0;\n        extendRatio = 2;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        foreach (Pair pair in buckets[index]) {\n            if (pair.key == key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        buckets[index].Add(new Pair(key, val));\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        foreach (Pair pair in buckets[index].ToList()) {\n            if (pair.key == key) {\n                buckets[index].Remove(pair);\n                size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new List&lt;List&lt;Pair&gt;&gt;(capacity);\n        for (int i = 0; i &lt; capacity; i++) {\n            buckets.Add([]);\n        }\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (List&lt;Pair&gt; bucket in bucketsTmp) {\n            foreach (Pair pair in bucket) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (List&lt;Pair&gt; bucket in buckets) {\n            List&lt;string&gt; res = [];\n            foreach (Pair pair in bucket) {\n                res.Add(pair.key + \" -&gt; \" + pair.val);\n            }\n            foreach (string kv in res) {\n                Console.WriteLine(kv);\n            }\n        }\n    }\n}\n</code></pre> hash_map_chaining.go<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntype hashMapChaining struct {\n    size        int      // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int      // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64  // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int      // \u6269\u5bb9\u500d\u6570\n    buckets     [][]pair // \u6876\u6570\u7ec4\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapChaining() *hashMapChaining {\n    buckets := make([][]pair, 4)\n    for i := 0; i &lt; 4; i++ {\n        buckets[i] = make([]pair, 0)\n    }\n    return &amp;hashMapChaining{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapChaining) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapChaining) loadFactor() float64 {\n    return float64(m.size / m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapChaining) get(key int) string {\n    idx := m.hashFunc(key)\n    bucket := m.buckets[idx]\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for _, p := range bucket {\n        if p.key == key {\n            return p.val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapChaining) put(key int, val string) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for _, p := range m.buckets[idx] {\n        if p.key == key {\n            p.val = val\n            return\n        }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    p := pair{\n        key: key,\n        val: val,\n    }\n    m.buckets[idx] = append(m.buckets[idx], p)\n    m.size += 1\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapChaining) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for i, p := range m.buckets[idx] {\n        if p.key == key {\n            // \u5207\u7247\u5220\u9664\n            m.buckets[idx] = append(m.buckets[idx][:i], m.buckets[idx][i+1:]...)\n            m.size -= 1\n            break\n        }\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    tmpBuckets := make([][]pair, len(m.buckets))\n    for i := 0; i &lt; len(m.buckets); i++ {\n        tmpBuckets[i] = make([]pair, len(m.buckets[i]))\n        copy(tmpBuckets[i], m.buckets[i])\n    }\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([][]pair, m.capacity)\n    for i := 0; i &lt; m.capacity; i++ {\n        m.buckets[i] = make([]pair, 0)\n    }\n    m.size = 0\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, bucket := range tmpBuckets {\n        for _, p := range bucket {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapChaining) print() {\n    var builder strings.Builder\n\n    for _, bucket := range m.buckets {\n        builder.WriteString(\"[\")\n        for _, p := range bucket {\n            builder.WriteString(strconv.Itoa(p.key) + \" -&gt; \" + p.val + \" \")\n        }\n        builder.WriteString(\"]\")\n        fmt.Println(builder.String())\n        builder.Reset()\n    }\n}\n</code></pre> hash_map_chaining.swift<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [[Pair]] // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: [], count: capacity)\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size / capacity)\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return pair.val\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de nil\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val\n                return\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair(key: key, val: val)\n        buckets[index].append(pair)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        let bucket = buckets[index]\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (pairIndex, pair) in bucket.enumerated() {\n            if pair.key == key {\n                buckets[index].remove(at: pairIndex)\n            }\n        }\n        size -= 1\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: [], count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in bucketsTmp {\n            for pair in bucket {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for bucket in buckets {\n            let res = bucket.map { \"\\($0.key) -&gt; \\($0.val)\" }\n            Swift.print(res)\n        }\n    }\n}\n</code></pre> hash_map_chaining.js<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.ts<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n    #size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    #buckets: Pair[][]; // \u6876\u6570\u7ec4\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0;\n        this.#capacity = 4;\n        this.#loadThres = 2.0 / 3.0;\n        this.#extendRatio = 2;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key: number): number {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor(): number {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                return pair.val;\n            }\n        }\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        const index = this.#hashFunc(key);\n        const bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for (const pair of bucket) {\n            if (pair.key === key) {\n                pair.val = val;\n                return;\n            }\n        }\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        const pair = new Pair(key, val);\n        bucket.push(pair);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        const index = this.#hashFunc(key);\n        let bucket = this.#buckets[index];\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for (let i = 0; i &lt; bucket.length; i++) {\n            if (bucket[i].key === key) {\n                bucket.splice(i, 1);\n                this.#size--;\n                break;\n            }\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = new Array(this.#capacity).fill(null).map((x) =&gt; []);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const bucket of bucketsTmp) {\n            for (const pair of bucket) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const bucket of this.#buckets) {\n            let res = [];\n            for (const pair of bucket) {\n                res.push(pair.key + ' -&gt; ' + pair.val);\n            }\n            console.log(res);\n        }\n    }\n}\n</code></pre> hash_map_chaining.dart<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nclass HashMapChaining {\n  late int size; // \u952e\u503c\u5bf9\u6570\u91cf\n  late int capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  late double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  late int extendRatio; // \u6269\u5bb9\u500d\u6570\n  late List&lt;List&lt;Pair&gt;&gt; buckets; // \u6876\u6570\u7ec4\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapChaining() {\n    size = 0;\n    capacity = 4;\n    loadThres = 2.0 / 3.0;\n    extendRatio = 2;\n    buckets = List.generate(capacity, (_) =&gt; []);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return size / capacity;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        return pair.val;\n      }\n    }\n    // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; loadThres) {\n      extend();\n    }\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        pair.val = val;\n        return;\n      }\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n    Pair pair = Pair(key, val);\n    bucket.add(pair);\n    size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    int index = hashFunc(key);\n    List&lt;Pair&gt; bucket = buckets[index];\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    for (Pair pair in bucket) {\n      if (pair.key == key) {\n        bucket.remove(pair);\n        size--;\n        break;\n      }\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;List&lt;Pair&gt;&gt; bucketsTmp = buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    capacity *= extendRatio;\n    buckets = List.generate(capacity, (_) =&gt; []);\n    size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (List&lt;Pair&gt; bucket in bucketsTmp) {\n      for (Pair pair in bucket) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (List&lt;Pair&gt; bucket in buckets) {\n      List&lt;String&gt; res = [];\n      for (Pair pair in bucket) {\n        res.add(\"${pair.key} -&gt; ${pair.val}\");\n      }\n      print(res);\n    }\n  }\n}\n</code></pre> hash_map_chaining.rs<pre><code>/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapChaining {\n    size: i32,\n    capacity: i32,\n    load_thres: f32,\n    extend_ratio: i32,\n    buckets: Vec&lt;Vec&lt;Pair&gt;&gt;,\n}\n\nimpl HashMapChaining {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![vec![]; 4],\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % self.capacity as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f32 {\n        self.size as f32 / self.capacity as f32\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) -&gt; Option&lt;String&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n        for i in 0..bucket.len() {\n            if bucket[i].key == key {\n                let pair = bucket.remove(i);\n                self.size -= 1;\n                return Some(pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n        None\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = std::mem::replace(&amp;mut self.buckets, vec![]);\n\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![Vec::new(); self.capacity as usize];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for bucket in buckets_tmp {\n            for pair in bucket {\n                self.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for bucket in &amp;self.buckets {\n            let mut res = Vec::new();\n            for pair in bucket {\n                res.push(format!(\"{} -&gt; {}\", pair.key, pair.val));\n            }\n            println!(\"{:?}\", res);\n        }\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n\n        let index = self.hash_func(key);\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n        for pair in bucket {\n            if pair.key == key {\n                pair.val = val.clone();\n                return;\n            }\n        }\n        let bucket = &amp;mut self.buckets[index];\n\n        // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u5c3e\u90e8\n        let pair = Pair {\n            key,\n            val: val.clone(),\n        };\n        bucket.push(pair);\n        self.size += 1;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        let index = self.hash_func(key);\n        let bucket = &amp;self.buckets[index];\n\n        // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n        for pair in bucket {\n            if pair.key == key {\n                return Some(&amp;pair.val);\n            }\n        }\n\n        // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de None\n        None\n    }\n}\n</code></pre> hash_map_chaining.c<pre><code>/* \u94fe\u8868\u8282\u70b9 */\ntypedef struct Node {\n    Pair *pair;\n    struct Node *next;\n} Node;\n\n/* \u94fe\u5f0f\u5730\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Node **buckets;   // \u6876\u6570\u7ec4\n} HashMapChaining;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapChaining *newHashMapChaining() {\n    HashMapChaining *hashMap = (HashMapChaining *)malloc(sizeof(HashMapChaining));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapChaining(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        while (cur) {\n            Node *tmp = cur;\n            cur = cur-&gt;next;\n            free(tmp-&gt;pair);\n            free(tmp);\n        }\n    }\n    free(hashMap-&gt;buckets);\n    free(hashMap);\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapChaining *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapChaining *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u627e\u5230 key \u5219\u8fd4\u56de\u5bf9\u5e94 val\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            return cur-&gt;pair-&gt;val;\n        }\n        cur = cur-&gt;next;\n    }\n    return \"\"; // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapChaining *hashMap, int key, const char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    int index = hashFunc(hashMap, key);\n    // \u904d\u5386\u6876\uff0c\u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n    Node *cur = hashMap-&gt;buckets[index];\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            strcpy(cur-&gt;pair-&gt;val, val); // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val \u5e76\u8fd4\u56de\n            return;\n        }\n        cur = cur-&gt;next;\n    }\n    // \u82e5\u65e0\u8be5 key \uff0c\u5219\u5c06\u952e\u503c\u5bf9\u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n    Pair *newPair = (Pair *)malloc(sizeof(Pair));\n    newPair-&gt;key = key;\n    strcpy(newPair-&gt;val, val);\n    Node *newNode = (Node *)malloc(sizeof(Node));\n    newNode-&gt;pair = newPair;\n    newNode-&gt;next = hashMap-&gt;buckets[index];\n    hashMap-&gt;buckets[index] = newNode;\n    hashMap-&gt;size++;\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapChaining *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    int oldCapacity = hashMap-&gt;capacity;\n    Node **oldBuckets = hashMap-&gt;buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Node **)malloc(hashMap-&gt;capacity * sizeof(Node *));\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        hashMap-&gt;buckets[i] = NULL;\n    }\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Node *cur = oldBuckets[i];\n        while (cur) {\n            put(hashMap, cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            Node *temp = cur;\n            cur = cur-&gt;next;\n            // \u91ca\u653e\u5185\u5b58\n            free(temp-&gt;pair);\n            free(temp);\n        }\n    }\n\n    free(oldBuckets);\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapChaining *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    Node *cur = hashMap-&gt;buckets[index];\n    Node *pre = NULL;\n    while (cur) {\n        if (cur-&gt;pair-&gt;key == key) {\n            // \u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n            if (pre) {\n                pre-&gt;next = cur-&gt;next;\n            } else {\n                hashMap-&gt;buckets[index] = cur-&gt;next;\n            }\n            // \u91ca\u653e\u5185\u5b58\n            free(cur-&gt;pair);\n            free(cur);\n            hashMap-&gt;size--;\n            return;\n        }\n        pre = cur;\n        cur = cur-&gt;next;\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapChaining *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Node *cur = hashMap-&gt;buckets[i];\n        printf(\"[\");\n        while (cur) {\n            printf(\"%d -&gt; %s, \", cur-&gt;pair-&gt;key, cur-&gt;pair-&gt;val);\n            cur = cur-&gt;next;\n        }\n        printf(\"]\\n\");\n    }\n}\n</code></pre> hash_map_chaining.zig<pre><code>[class]{HashMapChaining}-[func]{}\n</code></pre> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u5f53\u94fe\u8868\u5f88\u957f\u65f6\uff0c\u67e5\u8be2\u6548\u7387 \\(O(n)\\) \u5f88\u5dee\u3002\u6b64\u65f6\u53ef\u4ee5\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u201cAVL \u6811\u201d\u6216\u201c\u7ea2\u9ed1\u6811\u201d\uff0c\u4ece\u800c\u5c06\u67e5\u8be2\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002</p>"},{"location":"chapter_hashing/hash_collision/#622","title":"6.2.2 \u00a0 \u5f00\u653e\u5bfb\u5740","text":"<p>\u300c\u5f00\u653e\u5bfb\u5740 open addressing\u300d\u4e0d\u5f15\u5165\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u800c\u662f\u901a\u8fc7\u201c\u591a\u6b21\u63a2\u6d4b\u201d\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\uff0c\u63a2\u6d4b\u65b9\u5f0f\u4e3b\u8981\u5305\u62ec\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u3001\u591a\u6b21\u54c8\u5e0c\u7b49\u3002</p> <p>\u4e0b\u9762\u4ee5\u7ebf\u6027\u63a2\u6d4b\u4e3a\u4f8b\uff0c\u4ecb\u7ecd\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u7684\u5de5\u4f5c\u673a\u5236\u3002</p>"},{"location":"chapter_hashing/hash_collision/#1","title":"1. \u00a0 \u7ebf\u6027\u63a2\u6d4b","text":"<p>\u7ebf\u6027\u63a2\u6d4b\u91c7\u7528\u56fa\u5b9a\u6b65\u957f\u7684\u7ebf\u6027\u641c\u7d22\u6765\u8fdb\u884c\u63a2\u6d4b\uff0c\u5176\u64cd\u4f5c\u65b9\u6cd5\u4e0e\u666e\u901a\u54c8\u5e0c\u8868\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u82e5\u53d1\u73b0\u6876\u5185\u5df2\u6709\u5143\u7d20\uff0c\u5219\u4ece\u51b2\u7a81\u4f4d\u7f6e\u5411\u540e\u7ebf\u6027\u904d\u5386\uff08\u6b65\u957f\u901a\u5e38\u4e3a \\(1\\) \uff09\uff0c\u76f4\u81f3\u627e\u5230\u7a7a\u6876\uff0c\u5c06\u5143\u7d20\u63d2\u5165\u5176\u4e2d\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u82e5\u53d1\u73b0\u54c8\u5e0c\u51b2\u7a81\uff0c\u5219\u4f7f\u7528\u76f8\u540c\u6b65\u957f\u5411\u540e\u7ebf\u6027\u904d\u5386\uff0c\u76f4\u5230\u627e\u5230\u5bf9\u5e94\u5143\u7d20\uff0c\u8fd4\u56de <code>value</code> \u5373\u53ef\uff1b\u5982\u679c\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u76ee\u6807\u5143\u7d20\u4e0d\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8fd4\u56de \\(\\text{None}\\) \u3002</li> </ul> <p>\u56fe 6-6 \u5c55\u793a\u4e86\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u7684\u952e\u503c\u5bf9\u5206\u5e03\u3002\u6839\u636e\u6b64\u54c8\u5e0c\u51fd\u6570\uff0c\u6700\u540e\u4e24\u4f4d\u76f8\u540c\u7684 <code>key</code> \u90fd\u4f1a\u88ab\u6620\u5c04\u5230\u76f8\u540c\u7684\u6876\u3002\u800c\u901a\u8fc7\u7ebf\u6027\u63a2\u6d4b\uff0c\u5b83\u4eec\u88ab\u4f9d\u6b21\u5b58\u50a8\u5728\u8be5\u6876\u4ee5\u53ca\u4e4b\u4e0b\u7684\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 6-6 \u00a0 \u5f00\u653e\u5bfb\u5740\u548c\u7ebf\u6027\u63a2\u6d4b </p> <p>\u7136\u800c\uff0c\u7ebf\u6027\u63a2\u6d4b\u5bb9\u6613\u4ea7\u751f\u201c\u805a\u96c6\u73b0\u8c61\u201d\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u6570\u7ec4\u4e2d\u8fde\u7eed\u88ab\u5360\u7528\u7684\u4f4d\u7f6e\u8d8a\u957f\uff0c\u8fd9\u4e9b\u8fde\u7eed\u4f4d\u7f6e\u53d1\u751f\u54c8\u5e0c\u51b2\u7a81\u7684\u53ef\u80fd\u6027\u8d8a\u5927\uff0c\u4ece\u800c\u8fdb\u4e00\u6b65\u4fc3\u4f7f\u8be5\u4f4d\u7f6e\u7684\u805a\u5806\u751f\u957f\uff0c\u5f62\u6210\u6076\u6027\u5faa\u73af\uff0c\u6700\u7ec8\u5bfc\u81f4\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u6548\u7387\u52a3\u5316\u3002</p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6211\u4eec\u4e0d\u80fd\u5728\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\u4e2d\u76f4\u63a5\u5220\u9664\u5143\u7d20\u3002\u8fd9\u662f\u56e0\u4e3a\u5220\u9664\u5143\u7d20\u4f1a\u5728\u6570\u7ec4\u5185\u4ea7\u751f\u4e00\u4e2a\u7a7a\u6876 \\(\\text{None}\\) \uff0c\u800c\u5f53\u67e5\u8be2\u5143\u7d20\u65f6\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230\u8be5\u7a7a\u6876\u5c31\u4f1a\u8fd4\u56de\uff0c\u56e0\u6b64\u5728\u8be5\u7a7a\u6876\u4e4b\u4e0b\u7684\u5143\u7d20\u90fd\u65e0\u6cd5\u518d\u88ab\u8bbf\u95ee\u5230\uff0c\u7a0b\u5e8f\u53ef\u80fd\u8bef\u5224\u8fd9\u4e9b\u5143\u7d20\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u56fe 6-7 \u00a0 \u5728\u5f00\u653e\u5bfb\u5740\u4e2d\u5220\u9664\u5143\u7d20\u5bfc\u81f4\u7684\u67e5\u8be2\u95ee\u9898 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u300c\u61d2\u5220\u9664 lazy deletion\u300d\u673a\u5236\uff1a\u5b83\u4e0d\u76f4\u63a5\u4ece\u54c8\u5e0c\u8868\u4e2d\u79fb\u9664\u5143\u7d20\uff0c\u800c\u662f\u5229\u7528\u4e00\u4e2a\u5e38\u91cf <code>TOMBSTONE</code> \u6765\u6807\u8bb0\u8fd9\u4e2a\u6876\u3002\u5728\u8be5\u673a\u5236\u4e0b\uff0c\\(\\text{None}\\) \u548c <code>TOMBSTONE</code> \u90fd\u4ee3\u8868\u7a7a\u6876\uff0c\u90fd\u53ef\u4ee5\u653e\u7f6e\u952e\u503c\u5bf9\u3002\u4f46\u4e0d\u540c\u7684\u662f\uff0c\u7ebf\u6027\u63a2\u6d4b\u5230 <code>TOMBSTONE</code> \u65f6\u5e94\u8be5\u7ee7\u7eed\u904d\u5386\uff0c\u56e0\u4e3a\u5176\u4e4b\u4e0b\u53ef\u80fd\u8fd8\u5b58\u5728\u952e\u503c\u5bf9\u3002</p> <p>\u7136\u800c\uff0c\u61d2\u5220\u9664\u53ef\u80fd\u4f1a\u52a0\u901f\u54c8\u5e0c\u8868\u7684\u6027\u80fd\u9000\u5316\u3002\u8fd9\u662f\u56e0\u4e3a\u6bcf\u6b21\u5220\u9664\u64cd\u4f5c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5220\u9664\u6807\u8bb0\uff0c\u968f\u7740 <code>TOMBSTONE</code> \u7684\u589e\u52a0\uff0c\u641c\u7d22\u65f6\u95f4\u4e5f\u4f1a\u589e\u52a0\uff0c\u56e0\u4e3a\u7ebf\u6027\u63a2\u6d4b\u53ef\u80fd\u9700\u8981\u8df3\u8fc7\u591a\u4e2a <code>TOMBSTONE</code> \u624d\u80fd\u627e\u5230\u76ee\u6807\u5143\u7d20\u3002</p> <p>\u4e3a\u6b64\uff0c\u8003\u8651\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a <code>TOMBSTONE</code> \u7684\u7d22\u5f15\uff0c\u5e76\u5c06\u641c\u7d22\u5230\u7684\u76ee\u6807\u5143\u7d20\u4e0e\u8be5 <code>TOMBSTONE</code> \u4ea4\u6362\u4f4d\u7f6e\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u5f53\u6bcf\u6b21\u67e5\u8be2\u6216\u6dfb\u52a0\u5143\u7d20\u65f6\uff0c\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u81f3\u8ddd\u79bb\u7406\u60f3\u4f4d\u7f6e\uff08\u63a2\u6d4b\u8d77\u59cb\u70b9\uff09\u66f4\u8fd1\u7684\u6876\uff0c\u4ece\u800c\u4f18\u5316\u67e5\u8be2\u6548\u7387\u3002</p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u5305\u542b\u61d2\u5220\u9664\u7684\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\uff09\u54c8\u5e0c\u8868\u3002\u4e3a\u4e86\u66f4\u52a0\u5145\u5206\u5730\u4f7f\u7528\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\uff0c\u6211\u4eec\u5c06\u54c8\u5e0c\u8868\u770b\u4f5c\u4e00\u4e2a\u201c\u73af\u5f62\u6570\u7ec4\u201d\uff0c\u5f53\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map_open_addressing.py<pre><code>class HashMapOpenAddressing:\n    \"\"\"\u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.size = 0  # \u952e\u503c\u5bf9\u6570\u91cf\n        self.capacity = 4  # \u54c8\u5e0c\u8868\u5bb9\u91cf\n        self.load_thres = 2.0 / 3.0  # \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        self.extend_ratio = 2  # \u6269\u5bb9\u500d\u6570\n        self.buckets: list[Pair | None] = [None] * self.capacity  # \u6876\u6570\u7ec4\n        self.TOMBSTONE = Pair(-1, \"-1\")  # \u5220\u9664\u6807\u8bb0\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        return key % self.capacity\n\n    def load_factor(self) -&gt; float:\n        \"\"\"\u8d1f\u8f7d\u56e0\u5b50\"\"\"\n        return self.size / self.capacity\n\n    def find_bucket(self, key: int) -&gt; int:\n        \"\"\"\u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\"\"\"\n        index = self.hash_func(key)\n        first_tombstone = -1\n        # \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index] is not None:\n            # \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n            if self.buckets[index].key == key:\n                # \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if first_tombstone != -1:\n                    self.buckets[first_tombstone] = self.buckets[index]\n                    self.buckets[index] = self.TOMBSTONE\n                    return first_tombstone  # \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                return index  # \u8fd4\u56de\u6876\u7d22\u5f15\n            # \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 and self.buckets[index] is self.TOMBSTONE:\n                first_tombstone = index\n            # \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity\n        # \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return index if first_tombstone == -1 else first_tombstone\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            return self.buckets[index].val\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de None\n        return None\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        # \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres:\n            self.extend()\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index].val = val\n            return\n        # \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Pair(key, val)\n        self.size += 1\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        # \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        index = self.find_bucket(key)\n        # \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index] not in [None, self.TOMBSTONE]:\n            self.buckets[index] = self.TOMBSTONE\n            self.size -= 1\n\n    def extend(self):\n        \"\"\"\u6269\u5bb9\u54c8\u5e0c\u8868\"\"\"\n        # \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        buckets_tmp = self.buckets\n        # \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio\n        self.buckets = [None] * self.capacity\n        self.size = 0\n        # \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp:\n            if pair not in [None, self.TOMBSTONE]:\n                self.put(pair.key, pair.val)\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is None:\n                print(\"None\")\n            elif pair is self.TOMBSTONE:\n                print(\"TOMBSTONE\")\n            else:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> hash_map_open_addressing.cpp<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  private:\n    int size;                             // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4;                     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    const double loadThres = 2.0 / 3.0;     // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    const int extendRatio = 2;            // \u6269\u5bb9\u500d\u6570\n    vector&lt;Pair *&gt; buckets;               // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    HashMapOpenAddressing() : size(0), buckets(capacity, nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~HashMapOpenAddressing() {\n        for (Pair *pair : buckets) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                delete pair;\n            }\n        }\n        delete TOMBSTONE;\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double loadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != nullptr) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n            if (buckets[index]-&gt;key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index]-&gt;val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n        return \"\";\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index]-&gt;val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != nullptr &amp;&amp; buckets[index] != TOMBSTONE) {\n            delete buckets[index];\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        vector&lt;Pair *&gt; bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = vector&lt;Pair *&gt;(capacity, nullptr);\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair *pair : bucketsTmp) {\n            if (pair != nullptr &amp;&amp; pair != TOMBSTONE) {\n                put(pair-&gt;key, pair-&gt;val);\n                delete pair;\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *pair : buckets) {\n            if (pair == nullptr) {\n                cout &lt;&lt; \"nullptr\" &lt;&lt; endl;\n            } else if (pair == TOMBSTONE) {\n                cout &lt;&lt; \"TOMBSTONE\" &lt;&lt; endl;\n            } else {\n                cout &lt;&lt; pair-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; pair-&gt;val &lt;&lt; endl;\n            }\n        }\n    }\n};\n</code></pre> hash_map_open_addressing.java<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    private int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private final double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private final int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    private Pair[] buckets; // \u6876\u6570\u7ec4\n    private final Pair TOMBSTONE = new Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private double loadFactor() {\n        return (double) size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private int findBucket(int key) {\n        int index = hashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (loadFactor() &gt; loadThres) {\n            extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private void extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (Pair pair : bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair pair : buckets) {\n            if (pair == null) {\n                System.out.println(\"null\");\n            } else if (pair == TOMBSTONE) {\n                System.out.println(\"TOMBSTONE\");\n            } else {\n                System.out.println(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.cs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    int size; // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n    Pair[] buckets; // \u6876\u6570\u7ec4\n    Pair TOMBSTONE = new(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public HashMapOpenAddressing() {\n        size = 0;\n        buckets = new Pair[capacity];\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        return key % capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    double LoadFactor() {\n        return (double)size / capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    int FindBucket(int key) {\n        int index = HashFunc(key);\n        int firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (buckets[index] != null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n            if (buckets[index].key == key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if (firstTombstone != -1) {\n                    buckets[firstTombstone] = buckets[index];\n                    buckets[index] = TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            return buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (LoadFactor() &gt; loadThres) {\n            Extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = new Pair(key, val);\n        size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        int index = FindBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (buckets[index] != null &amp;&amp; buckets[index] != TOMBSTONE) {\n            buckets[index] = TOMBSTONE;\n            size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    void Extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        Pair[] bucketsTmp = buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio;\n        buckets = new Pair[capacity];\n        size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        foreach (Pair pair in bucketsTmp) {\n            if (pair != null &amp;&amp; pair != TOMBSTONE) {\n                Put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair pair in buckets) {\n            if (pair == null) {\n                Console.WriteLine(\"null\");\n            } else if (pair == TOMBSTONE) {\n                Console.WriteLine(\"TOMBSTONE\");\n            } else {\n                Console.WriteLine(pair.key + \" -&gt; \" + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.go<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntype hashMapOpenAddressing struct {\n    size        int     // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity    int     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    loadThres   float64 // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extendRatio int     // \u6269\u5bb9\u500d\u6570\n    buckets     []pair  // \u6876\u6570\u7ec4\n    removed     pair    // \u5220\u9664\u6807\u8bb0\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newHashMapOpenAddressing() *hashMapOpenAddressing {\n    buckets := make([]pair, 4)\n    return &amp;hashMapOpenAddressing{\n        size:        0,\n        capacity:    4,\n        loadThres:   2.0 / 3.0,\n        extendRatio: 2,\n        buckets:     buckets,\n        removed: pair{\n            key: -1,\n            val: \"-1\",\n        },\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (m *hashMapOpenAddressing) hashFunc(key int) int {\n    return key % m.capacity\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\nfunc (m *hashMapOpenAddressing) loadFactor() float64 {\n    return float64(m.size) / float64(m.capacity)\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) get(key int) string {\n    idx := m.hashFunc(key)\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n    for i := 0; i &lt; m.capacity; i++ {\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n        j := (idx + i) % m.capacity\n        // \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u8fd4\u56de null\n        if m.buckets[j] == (pair{}) {\n            return \"\"\n        }\n        // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if m.buckets[j].key == key &amp;&amp; m.buckets[j] != m.removed {\n            return m.buckets[j].val\n        }\n    }\n    // \u82e5\u672a\u627e\u5230 key \u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\"\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) put(key int, val string) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if m.loadFactor() &gt; m.loadThres {\n        m.extend()\n    }\n    idx := m.hashFunc(key)\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n    for i := 0; i &lt; m.capacity; i++ {\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n        j := (idx + i) % m.capacity\n        // \u82e5\u9047\u5230\u7a7a\u6876\u3001\u6216\u5e26\u6709\u5220\u9664\u6807\u8bb0\u7684\u6876\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u653e\u5165\u8be5\u6876\n        if m.buckets[j] == (pair{}) || m.buckets[j] == m.removed {\n            m.buckets[j] = pair{\n                key: key,\n                val: val,\n            }\n            m.size += 1\n            return\n        }\n        // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u66f4\u65b0\u5bf9\u5e94 val\n        if m.buckets[j].key == key {\n            m.buckets[j].val = val\n        }\n    }\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (m *hashMapOpenAddressing) remove(key int) {\n    idx := m.hashFunc(key)\n    // \u904d\u5386\u6876\uff0c\u4ece\u4e2d\u5220\u9664\u952e\u503c\u5bf9\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u4ece index \u5f00\u59cb\u5411\u540e\u904d\u5386\n    for i := 0; i &lt; m.capacity; i++ {\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n        j := (idx + i) % m.capacity\n        // \u82e5\u9047\u5230\u7a7a\u6876\uff0c\u8bf4\u660e\u65e0\u6b64 key \uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n        if m.buckets[j] == (pair{}) {\n            return\n        }\n        // \u82e5\u9047\u5230\u6307\u5b9a key \uff0c\u5219\u6807\u8bb0\u5220\u9664\u5e76\u8fd4\u56de\n        if m.buckets[j].key == key {\n            m.buckets[j] = m.removed\n            m.size -= 1\n        }\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    tmpBuckets := make([]pair, len(m.buckets))\n    copy(tmpBuckets, m.buckets)\n\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    m.capacity *= m.extendRatio\n    m.buckets = make([]pair, m.capacity)\n    m.size = 0\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for _, p := range tmpBuckets {\n        if p != (pair{}) &amp;&amp; p != m.removed {\n            m.put(p.key, p.val)\n        }\n    }\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (m *hashMapOpenAddressing) print() {\n    for _, p := range m.buckets {\n        if p != (pair{}) {\n            fmt.Println(strconv.Itoa(p.key) + \" -&gt; \" + p.val)\n        } else {\n            fmt.Println(\"nil\")\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.swift<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    var size: Int // \u952e\u503c\u5bf9\u6570\u91cf\n    var capacity: Int // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    var loadThres: Double // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    var extendRatio: Int // \u6269\u5bb9\u500d\u6570\n    var buckets: [Pair?] // \u6876\u6570\u7ec4\n    var TOMBSTONE: Pair // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init() {\n        size = 0\n        capacity = 4\n        loadThres = 2.0 / 3.0\n        extendRatio = 2\n        buckets = Array(repeating: nil, count: capacity)\n        TOMBSTONE = Pair(key: -1, val: \"-1\")\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    func hashFunc(key: Int) -&gt; Int {\n        key % capacity\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    func loadFactor() -&gt; Double {\n        Double(size / capacity)\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    func findBucket(key: Int) -&gt; Int {\n        var index = hashFunc(key: key)\n        var firstTombstone = -1\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while buckets[index] != nil {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n            if buckets[index]!.key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if firstTombstone != -1 {\n                    buckets[firstTombstone] = buckets[index]\n                    buckets[index] = TOMBSTONE\n                    return firstTombstone // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if firstTombstone == -1 &amp;&amp; buckets[index] == TOMBSTONE {\n                firstTombstone = index\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % capacity\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone == -1 ? index : firstTombstone\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            return buckets[index]!.val\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return nil\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if loadFactor() &gt; loadThres {\n            extend()\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index]!.val = val\n            return\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        buckets[index] = Pair(key: key, val: val)\n        size += 1\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = findBucket(key: key)\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if buckets[index] != nil, buckets[index] != TOMBSTONE {\n            buckets[index] = TOMBSTONE\n            size -= 1\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    func extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let bucketsTmp = buckets\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        capacity *= extendRatio\n        buckets = Array(repeating: nil, count: capacity)\n        size = 0\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in bucketsTmp {\n            if let pair, pair != TOMBSTONE {\n                put(key: pair.key, val: pair.val)\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in buckets {\n            if pair == nil {\n                Swift.print(\"null\")\n            } else if pair == TOMBSTONE {\n                Swift.print(\"TOMBSTONE\")\n            } else {\n                Swift.print(\"\\(pair!.key) -&gt; \\(pair!.val)\")\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.js<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    #size; // \u952e\u503c\u5bf9\u6570\u91cf\n    #capacity; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    #loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    #extendRatio; // \u6269\u5bb9\u500d\u6570\n    #buckets; // \u6876\u6570\u7ec4\n    #TOMBSTONE; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.#size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.#capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.#loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.#extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.#buckets = Array(this.#capacity).fill(null); // \u6876\u6570\u7ec4\n        this.#TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % this.#capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    #loadFactor() {\n        return this.#size / this.#capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    #findBucket(key) {\n        let index = this.#hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.#buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n            if (this.#buckets[index].key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if (firstTombstone !== -1) {\n                    this.#buckets[firstTombstone] = this.#buckets[index];\n                    this.#buckets[index] = this.#TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.#buckets[index] === this.#TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.#capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            return this.#buckets[index].val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key, val) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.#loadFactor() &gt; this.#loadThres) {\n            this.#extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index].val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.#buckets[index] = new Pair(key, val);\n        this.#size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.#findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.#buckets[index] !== null &amp;&amp;\n            this.#buckets[index] !== this.#TOMBSTONE\n        ) {\n            this.#buckets[index] = this.#TOMBSTONE;\n            this.#size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    #extend() {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.#buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.#capacity *= this.#extendRatio;\n        this.#buckets = Array(this.#capacity).fill(null);\n        this.#size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.#TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        for (const pair of this.#buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.#TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.ts<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n    private size: number; // \u952e\u503c\u5bf9\u6570\u91cf\n    private capacity: number; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    private loadThres: number; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    private extendRatio: number; // \u6269\u5bb9\u500d\u6570\n    private buckets: Array&lt;Pair | null&gt;; // \u6876\u6570\u7ec4\n    private TOMBSTONE: Pair; // \u5220\u9664\u6807\u8bb0\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor() {\n        this.size = 0; // \u952e\u503c\u5bf9\u6570\u91cf\n        this.capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n        this.loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n        this.extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n        this.buckets = Array(this.capacity).fill(null); // \u6876\u6570\u7ec4\n        this.TOMBSTONE = new Pair(-1, '-1'); // \u5220\u9664\u6807\u8bb0\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % this.capacity;\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    private loadFactor(): number {\n        return this.size / this.capacity;\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    private findBucket(key: number): number {\n        let index = this.hashFunc(key);\n        let firstTombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while (this.buckets[index] !== null) {\n            // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n            if (this.buckets[index]!.key === key) {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if (firstTombstone !== -1) {\n                    this.buckets[firstTombstone] = this.buckets[index];\n                    this.buckets[index] = this.TOMBSTONE;\n                    return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if (\n                firstTombstone === -1 &amp;&amp;\n                this.buckets[index] === this.TOMBSTONE\n            ) {\n                firstTombstone = index;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % this.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        return firstTombstone === -1 ? index : firstTombstone;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key: number): string | null {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            return this.buckets[index]!.val;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        return null;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    put(key: number, val: string): void {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if (this.loadFactor() &gt; this.loadThres) {\n            this.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index]!.val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        this.buckets[index] = new Pair(key, val);\n        this.size++;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    remove(key: number): void {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        const index = this.findBucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if (\n            this.buckets[index] !== null &amp;&amp;\n            this.buckets[index] !== this.TOMBSTONE\n        ) {\n            this.buckets[index] = this.TOMBSTONE;\n            this.size--;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    private extend(): void {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        const bucketsTmp = this.buckets;\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        this.capacity *= this.extendRatio;\n        this.buckets = Array(this.capacity).fill(null);\n        this.size = 0;\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for (const pair of bucketsTmp) {\n            if (pair !== null &amp;&amp; pair !== this.TOMBSTONE) {\n                this.put(pair.key, pair.val);\n            }\n        }\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print(): void {\n        for (const pair of this.buckets) {\n            if (pair === null) {\n                console.log('null');\n            } else if (pair === this.TOMBSTONE) {\n                console.log('TOMBSTONE');\n            } else {\n                console.log(pair.key + ' -&gt; ' + pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.dart<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nclass HashMapOpenAddressing {\n  late int _size; // \u952e\u503c\u5bf9\u6570\u91cf\n  int _capacity = 4; // \u54c8\u5e0c\u8868\u5bb9\u91cf\n  double _loadThres = 2.0 / 3.0; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n  int _extendRatio = 2; // \u6269\u5bb9\u500d\u6570\n  late List&lt;Pair?&gt; _buckets; // \u6876\u6570\u7ec4\n  Pair _TOMBSTONE = Pair(-1, \"-1\"); // \u5220\u9664\u6807\u8bb0\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  HashMapOpenAddressing() {\n    _size = 0;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int hashFunc(int key) {\n    return key % _capacity;\n  }\n\n  /* \u8d1f\u8f7d\u56e0\u5b50 */\n  double loadFactor() {\n    return _size / _capacity;\n  }\n\n  /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n  int findBucket(int key) {\n    int index = hashFunc(key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (_buckets[index] != null) {\n      // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n      if (_buckets[index]!.key == key) {\n        // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n        if (firstTombstone != -1) {\n          _buckets[firstTombstone] = _buckets[index];\n          _buckets[index] = _TOMBSTONE;\n          return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n        }\n        return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n      }\n      // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n      if (firstTombstone == -1 &amp;&amp; _buckets[index] == _TOMBSTONE) {\n        firstTombstone = index;\n      }\n      // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n      index = (index + 1) % _capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      return _buckets[index]!.val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n    return null;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor() &gt; _loadThres) {\n      extend();\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index]!.val = val;\n      return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    _buckets[index] = new Pair(key, val);\n    _size++;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (_buckets[index] != null &amp;&amp; _buckets[index] != _TOMBSTONE) {\n      _buckets[index] = _TOMBSTONE;\n      _size--;\n    }\n  }\n\n  /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n  void extend() {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    List&lt;Pair?&gt; bucketsTmp = _buckets;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    _capacity *= _extendRatio;\n    _buckets = List.generate(_capacity, (index) =&gt; null);\n    _size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (Pair? pair in bucketsTmp) {\n      if (pair != null &amp;&amp; pair != _TOMBSTONE) {\n        put(pair.key, pair.val);\n      }\n    }\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (Pair? pair in _buckets) {\n      if (pair == null) {\n        print(\"null\");\n      } else if (pair == _TOMBSTONE) {\n        print(\"TOMBSTONE\");\n      } else {\n        print(\"${pair.key} -&gt; ${pair.val}\");\n      }\n    }\n  }\n}\n</code></pre> hash_map_open_addressing.rs<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\nstruct HashMapOpenAddressing {\n    size: usize,                    // \u952e\u503c\u5bf9\u6570\u91cf\n    capacity: usize,                // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    load_thres: f64,                // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    extend_ratio: usize,            // \u6269\u5bb9\u500d\u6570\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;,     // \u6876\u6570\u7ec4\n    TOMBSTONE: Option&lt;Pair&gt;,        // \u5220\u9664\u6807\u8bb0\n}\n\n\nimpl HashMapOpenAddressing {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new() -&gt; Self {\n        Self {\n            size: 0,\n            capacity: 4,\n            load_thres: 2.0 / 3.0,\n            extend_ratio: 2,\n            buckets: vec![None; 4],\n            TOMBSTONE: Some(Pair {key: -1, val: \"-1\".to_string()}),\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        (key % self.capacity as i32) as usize\n    }\n\n    /* \u8d1f\u8f7d\u56e0\u5b50 */\n    fn load_factor(&amp;self) -&gt; f64 {\n        self.size as f64 / self.capacity as f64\n    }\n\n    /* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\n    fn find_bucket(&amp;mut self, key: i32) -&gt; usize {\n        let mut index = self.hash_func(key);\n        let mut first_tombstone = -1;\n        // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n        while self.buckets[index].is_some() {\n            // \u82e5\u9047\u5230 key\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n            if self.buckets[index].as_ref().unwrap().key == key {\n                // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u5efa\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n                if first_tombstone != -1 {\n                    self.buckets[first_tombstone as usize] = self.buckets[index].take();\n                    self.buckets[index] = self.TOMBSTONE.clone();\n                    return first_tombstone as usize;    // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n                }\n                return index;   // \u8fd4\u56de\u6876\u7d22\u5f15\n            }\n            // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n            if first_tombstone == -1 &amp;&amp; self.buckets[index] == self.TOMBSTONE {\n                first_tombstone = index as i32;\n            }\n            // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n            index = (index + 1) % self.capacity;\n        }\n        // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n        if first_tombstone == -1 { index } else { first_tombstone as usize }\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    fn get(&amp;mut self, key: i32) -&gt; Option&lt;&amp;str&gt; {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            return self.buckets[index].as_ref().map(|pair| &amp;pair.val as &amp;str);\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de null\n        None\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    fn put(&amp;mut self, key: i32, val: String) {\n        // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n        if self.load_factor() &gt; self.load_thres {\n            self.extend();\n        }\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index].as_mut().unwrap().val = val;\n            return;\n        }\n        // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n        self.buckets[index] = Some(Pair { key, val });\n        self.size += 1;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    fn remove(&amp;mut self, key: i32) {\n        // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n        let index = self.find_bucket(key);\n        // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n        if self.buckets[index].is_some() &amp;&amp; self.buckets[index] != self.TOMBSTONE {\n            self.buckets[index] = self.TOMBSTONE.clone();\n            self.size -= 1;\n        }\n    }\n\n    /* \u6269\u5bb9\u54c8\u5e0c\u8868 */\n    fn extend(&amp;mut self) {\n        // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n        let buckets_tmp = self.buckets.clone();\n        // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n        self.capacity *= self.extend_ratio;\n        self.buckets = vec![None; self.capacity];\n        self.size = 0;\n\n        // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n        for pair in buckets_tmp {\n            if pair.is_none() || pair == self.TOMBSTONE {\n                continue;\n            }\n            let pair = pair.unwrap();\n\n            self.put(pair.key, pair.val);\n        }\n    }\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    fn print(&amp;self) {\n        for pair in &amp;self.buckets {\n            if pair.is_none() {\n                println!(\"null\");\n            } else if pair == &amp;self.TOMBSTONE {\n                println!(\"TOMBSTONE\");\n            } else {\n                let pair = pair.as_ref().unwrap();\n                println!(\"{} -&gt; {}\", pair.key, pair.val);\n            }\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.c<pre><code>/* \u5f00\u653e\u5bfb\u5740\u54c8\u5e0c\u8868 */\ntypedef struct {\n    int size;         // \u952e\u503c\u5bf9\u6570\u91cf\n    int capacity;     // \u54c8\u5e0c\u8868\u5bb9\u91cf\n    double loadThres; // \u89e6\u53d1\u6269\u5bb9\u7684\u8d1f\u8f7d\u56e0\u5b50\u9608\u503c\n    int extendRatio;  // \u6269\u5bb9\u500d\u6570\n    Pair **buckets;   // \u6876\u6570\u7ec4\n    Pair *TOMBSTONE;  // \u5220\u9664\u6807\u8bb0\n} HashMapOpenAddressing;\n\n/* \u6784\u9020\u51fd\u6570 */\nHashMapOpenAddressing *newHashMapOpenAddressing() {\n    HashMapOpenAddressing *hashMap = (HashMapOpenAddressing *)malloc(sizeof(HashMapOpenAddressing));\n    hashMap-&gt;size = 0;\n    hashMap-&gt;capacity = 4;\n    hashMap-&gt;loadThres = 2.0 / 3.0;\n    hashMap-&gt;extendRatio = 2;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;TOMBSTONE = (Pair *)malloc(sizeof(Pair));\n    hashMap-&gt;TOMBSTONE-&gt;key = -1;\n    hashMap-&gt;TOMBSTONE-&gt;val = \"-1\";\n\n    return hashMap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delHashMapOpenAddressing(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nint hashFunc(HashMapOpenAddressing *hashMap, int key) {\n    return key % hashMap-&gt;capacity;\n}\n\n/* \u8d1f\u8f7d\u56e0\u5b50 */\ndouble loadFactor(HashMapOpenAddressing *hashMap) {\n    return (double)hashMap-&gt;size / (double)hashMap-&gt;capacity;\n}\n\n/* \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15 */\nint findBucket(HashMapOpenAddressing *hashMap, int key) {\n    int index = hashFunc(hashMap, key);\n    int firstTombstone = -1;\n    // \u7ebf\u6027\u63a2\u6d4b\uff0c\u5f53\u9047\u5230\u7a7a\u6876\u65f6\u8df3\u51fa\n    while (hashMap-&gt;buckets[index] != NULL) {\n        // \u82e5\u9047\u5230 key \uff0c\u8fd4\u56de\u5bf9\u5e94\u6876\u7d22\u5f15\n        if (hashMap-&gt;buckets[index]-&gt;key == key) {\n            // \u82e5\u4e4b\u524d\u9047\u5230\u4e86\u5220\u9664\u6807\u8bb0\uff0c\u5219\u5c06\u952e\u503c\u5bf9\u79fb\u52a8\u81f3\u8be5\u7d22\u5f15\n            if (firstTombstone != -1) {\n                hashMap-&gt;buckets[firstTombstone] = hashMap-&gt;buckets[index];\n                hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n                return firstTombstone; // \u8fd4\u56de\u79fb\u52a8\u540e\u7684\u6876\u7d22\u5f15\n            }\n            return index; // \u8fd4\u56de\u6876\u7d22\u5f15\n        }\n        // \u8bb0\u5f55\u9047\u5230\u7684\u9996\u4e2a\u5220\u9664\u6807\u8bb0\n        if (firstTombstone == -1 &amp;&amp; hashMap-&gt;buckets[index] == hashMap-&gt;TOMBSTONE) {\n            firstTombstone = index;\n        }\n        // \u8ba1\u7b97\u6876\u7d22\u5f15\uff0c\u8d8a\u8fc7\u5c3e\u90e8\u8fd4\u56de\u5934\u90e8\n        index = (index + 1) % hashMap-&gt;capacity;\n    }\n    // \u82e5 key \u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u6dfb\u52a0\u70b9\u7684\u7d22\u5f15\n    return firstTombstone == -1 ? index : firstTombstone;\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nchar *get(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8fd4\u56de\u5bf9\u5e94 val\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        return hashMap-&gt;buckets[index]-&gt;val;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u8fd4\u56de\u7a7a\u5b57\u7b26\u4e32\n    return \"\";\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(HashMapOpenAddressing *hashMap, int key, char *val) {\n    // \u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7\u9608\u503c\u65f6\uff0c\u6267\u884c\u6269\u5bb9\n    if (loadFactor(hashMap) &gt; hashMap-&gt;loadThres) {\n        extend(hashMap);\n    }\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u8986\u76d6 val \u5e76\u8fd4\u56de\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        free(hashMap-&gt;buckets[index]-&gt;val);\n        hashMap-&gt;buckets[index]-&gt;val = (char *)malloc(sizeof(strlen(val + 1)));\n        strcpy(hashMap-&gt;buckets[index]-&gt;val, val);\n        hashMap-&gt;buckets[index]-&gt;val[strlen(val)] = '\\0';\n        return;\n    }\n    // \u82e5\u952e\u503c\u5bf9\u4e0d\u5b58\u5728\uff0c\u5219\u6dfb\u52a0\u8be5\u952e\u503c\u5bf9\n    Pair *pair = (Pair *)malloc(sizeof(Pair));\n    pair-&gt;key = key;\n    pair-&gt;val = (char *)malloc(sizeof(strlen(val + 1)));\n    strcpy(pair-&gt;val, val);\n    pair-&gt;val[strlen(val)] = '\\0';\n\n    hashMap-&gt;buckets[index] = pair;\n    hashMap-&gt;size++;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(HashMapOpenAddressing *hashMap, int key) {\n    // \u641c\u7d22 key \u5bf9\u5e94\u7684\u6876\u7d22\u5f15\n    int index = findBucket(hashMap, key);\n    // \u82e5\u627e\u5230\u952e\u503c\u5bf9\uff0c\u5219\u7528\u5220\u9664\u6807\u8bb0\u8986\u76d6\u5b83\n    if (hashMap-&gt;buckets[index] != NULL &amp;&amp; hashMap-&gt;buckets[index] != hashMap-&gt;TOMBSTONE) {\n        Pair *pair = hashMap-&gt;buckets[index];\n        free(pair-&gt;val);\n        free(pair);\n        hashMap-&gt;buckets[index] = hashMap-&gt;TOMBSTONE;\n        hashMap-&gt;size--;\n    }\n}\n\n/* \u6269\u5bb9\u54c8\u5e0c\u8868 */\nvoid extend(HashMapOpenAddressing *hashMap) {\n    // \u6682\u5b58\u539f\u54c8\u5e0c\u8868\n    Pair **bucketsTmp = hashMap-&gt;buckets;\n    int oldCapacity = hashMap-&gt;capacity;\n    // \u521d\u59cb\u5316\u6269\u5bb9\u540e\u7684\u65b0\u54c8\u5e0c\u8868\n    hashMap-&gt;capacity *= hashMap-&gt;extendRatio;\n    hashMap-&gt;buckets = (Pair **)malloc(sizeof(Pair *) * hashMap-&gt;capacity);\n    hashMap-&gt;size = 0;\n    // \u5c06\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u642c\u8fd0\u81f3\u65b0\u54c8\u5e0c\u8868\n    for (int i = 0; i &lt; oldCapacity; i++) {\n        Pair *pair = bucketsTmp[i];\n        if (pair != NULL &amp;&amp; pair != hashMap-&gt;TOMBSTONE) {\n            put(hashMap, pair-&gt;key, pair-&gt;val);\n            free(pair-&gt;val);\n            free(pair);\n        }\n    }\n    free(bucketsTmp);\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(HashMapOpenAddressing *hashMap) {\n    for (int i = 0; i &lt; hashMap-&gt;capacity; i++) {\n        Pair *pair = hashMap-&gt;buckets[i];\n        if (pair == NULL) {\n            printf(\"NULL\\n\");\n        } else if (pair == hashMap-&gt;TOMBSTONE) {\n            printf(\"TOMBSTONE\\n\");\n        } else {\n            printf(\"%d -&gt; %s\\n\", pair-&gt;key, pair-&gt;val);\n        }\n    }\n}\n</code></pre> hash_map_open_addressing.zig<pre><code>[class]{HashMapOpenAddressing}-[func]{}\n</code></pre>"},{"location":"chapter_hashing/hash_collision/#2","title":"2. \u00a0 \u5e73\u65b9\u63a2\u6d4b","text":"<p>\u5e73\u65b9\u63a2\u6d4b\u4e0e\u7ebf\u6027\u63a2\u6d4b\u7c7b\u4f3c\uff0c\u90fd\u662f\u5f00\u653e\u5bfb\u5740\u7684\u5e38\u89c1\u7b56\u7565\u4e4b\u4e00\u3002\u5f53\u53d1\u751f\u51b2\u7a81\u65f6\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e0d\u662f\u7b80\u5355\u5730\u8df3\u8fc7\u4e00\u4e2a\u56fa\u5b9a\u7684\u6b65\u6570\uff0c\u800c\u662f\u8df3\u8fc7\u201c\u63a2\u6d4b\u6b21\u6570\u7684\u5e73\u65b9\u201d\u7684\u6b65\u6570\uff0c\u5373 \\(1, 4, 9, \\dots\\) \u6b65\u3002</p> <p>\u5e73\u65b9\u63a2\u6d4b\u4e3b\u8981\u5177\u6709\u4ee5\u4e0b\u4f18\u52bf\u3002</p> <ul> <li>\u5e73\u65b9\u63a2\u6d4b\u901a\u8fc7\u8df3\u8fc7\u5e73\u65b9\u7684\u8ddd\u79bb\uff0c\u8bd5\u56fe\u7f13\u89e3\u7ebf\u6027\u63a2\u6d4b\u7684\u805a\u96c6\u6548\u5e94\u3002</li> <li>\u5e73\u65b9\u63a2\u6d4b\u4f1a\u8df3\u8fc7\u66f4\u5927\u7684\u8ddd\u79bb\u6765\u5bfb\u627e\u7a7a\u4f4d\u7f6e\uff0c\u6709\u52a9\u4e8e\u6570\u636e\u5206\u5e03\u5f97\u66f4\u52a0\u5747\u5300\u3002</li> </ul> <p>\u7136\u800c\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u5e76\u4e0d\u662f\u5b8c\u7f8e\u7684\u3002</p> <ul> <li>\u4ecd\u7136\u5b58\u5728\u805a\u96c6\u73b0\u8c61\uff0c\u5373\u67d0\u4e9b\u4f4d\u7f6e\u6bd4\u5176\u4ed6\u4f4d\u7f6e\u66f4\u5bb9\u6613\u88ab\u5360\u7528\u3002</li> <li>\u7531\u4e8e\u5e73\u65b9\u7684\u589e\u957f\uff0c\u5e73\u65b9\u63a2\u6d4b\u53ef\u80fd\u4e0d\u4f1a\u63a2\u6d4b\u6574\u4e2a\u54c8\u5e0c\u8868\uff0c\u8fd9\u610f\u5473\u7740\u5373\u4f7f\u54c8\u5e0c\u8868\u4e2d\u6709\u7a7a\u6876\uff0c\u5e73\u65b9\u63a2\u6d4b\u4e5f\u53ef\u80fd\u65e0\u6cd5\u8bbf\u95ee\u5230\u5b83\u3002</li> </ul>"},{"location":"chapter_hashing/hash_collision/#3","title":"3. \u00a0 \u591a\u6b21\u54c8\u5e0c","text":"<p>\u987e\u540d\u601d\u4e49\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\)\u3001\\(f_2(x)\\)\u3001\\(f_3(x)\\)\u3001\\(\\dots\\) \u8fdb\u884c\u63a2\u6d4b\u3002</p> <ul> <li>\u63d2\u5165\u5143\u7d20\uff1a\u82e5\u54c8\u5e0c\u51fd\u6570 \\(f_1(x)\\) \u51fa\u73b0\u51b2\u7a81\uff0c\u5219\u5c1d\u8bd5 \\(f_2(x)\\) \uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u627e\u5230\u7a7a\u6876\u540e\u63d2\u5165\u5143\u7d20\u3002</li> <li>\u67e5\u627e\u5143\u7d20\uff1a\u5728\u76f8\u540c\u7684\u54c8\u5e0c\u51fd\u6570\u987a\u5e8f\u4e0b\u8fdb\u884c\u67e5\u627e\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u65f6\u8fd4\u56de\uff1b\u82e5\u9047\u5230\u7a7a\u6876\u6216\u5df2\u5c1d\u8bd5\u6240\u6709\u54c8\u5e0c\u51fd\u6570\uff0c\u8bf4\u660e\u54c8\u5e0c\u8868\u4e2d\u4e0d\u5b58\u5728\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(\\text{None}\\) \u3002</li> </ul> <p>\u4e0e\u7ebf\u6027\u63a2\u6d4b\u76f8\u6bd4\uff0c\u591a\u6b21\u54c8\u5e0c\u65b9\u6cd5\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u4f1a\u5e26\u6765\u989d\u5916\u7684\u8ba1\u7b97\u91cf\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5f00\u653e\u5bfb\u5740\uff08\u7ebf\u6027\u63a2\u6d4b\u3001\u5e73\u65b9\u63a2\u6d4b\u548c\u591a\u6b21\u54c8\u5e0c\uff09\u54c8\u5e0c\u8868\u90fd\u5b58\u5728\u201c\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u201d\u7684\u95ee\u9898\u3002</p>"},{"location":"chapter_hashing/hash_collision/#623","title":"6.2.3 \u00a0 \u7f16\u7a0b\u8bed\u8a00\u7684\u9009\u62e9","text":"<p>\u5404\u4e2a\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u7b56\u7565\uff0c\u4ee5\u4e0b\u4e3e\u51e0\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>Python \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002\u5b57\u5178 dict \u4f7f\u7528\u4f2a\u968f\u673a\u6570\u8fdb\u884c\u63a2\u6d4b\u3002</li> <li>Java \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002\u81ea JDK 1.8 \u4ee5\u6765\uff0c\u5f53 HashMap \u5185\u6570\u7ec4\u957f\u5ea6\u8fbe\u5230 64 \u4e14\u94fe\u8868\u957f\u5ea6\u8fbe\u5230 8 \u65f6\uff0c\u94fe\u8868\u4f1a\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u4ee5\u63d0\u5347\u67e5\u627e\u6027\u80fd\u3002</li> <li>Go \u91c7\u7528\u94fe\u5f0f\u5730\u5740\u3002Go \u89c4\u5b9a\u6bcf\u4e2a\u6876\u6700\u591a\u5b58\u50a8 8 \u4e2a\u952e\u503c\u5bf9\uff0c\u8d85\u51fa\u5bb9\u91cf\u5219\u8fde\u63a5\u4e00\u4e2a\u6ea2\u51fa\u6876\u3002\u5f53\u6ea2\u51fa\u6876\u8fc7\u591a\u65f6\uff0c\u4f1a\u6267\u884c\u4e00\u6b21\u7279\u6b8a\u7684\u7b49\u91cf\u6269\u5bb9\u64cd\u4f5c\uff0c\u4ee5\u786e\u4fdd\u6027\u80fd\u3002</li> </ul>"},{"location":"chapter_hashing/hash_map/","title":"6.1 \u00a0 \u54c8\u5e0c\u8868","text":"<p>\u300c\u54c8\u5e0c\u8868 hash table\u300d\uff0c\u53c8\u79f0\u300c\u6563\u5217\u8868\u300d\uff0c\u5176\u901a\u8fc7\u5efa\u7acb\u952e <code>key</code> \u4e0e\u503c <code>value</code> \u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u8be2\u3002\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u5411\u54c8\u5e0c\u8868\u8f93\u5165\u4e00\u4e2a\u952e <code>key</code> \uff0c\u5219\u53ef\u4ee5\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u83b7\u53d6\u5bf9\u5e94\u7684\u503c <code>value</code> \u3002</p> <p>\u5982\u56fe 6-1 \u6240\u793a\uff0c\u7ed9\u5b9a \\(n\\) \u4e2a\u5b66\u751f\uff0c\u6bcf\u4e2a\u5b66\u751f\u90fd\u6709\u201c\u59d3\u540d\u201d\u548c\u201c\u5b66\u53f7\u201d\u4e24\u9879\u6570\u636e\u3002\u5047\u5982\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u201c\u8f93\u5165\u4e00\u4e2a\u5b66\u53f7\uff0c\u8fd4\u56de\u5bf9\u5e94\u7684\u59d3\u540d\u201d\u7684\u67e5\u8be2\u529f\u80fd\uff0c\u5219\u53ef\u4ee5\u91c7\u7528\u56fe 6-1 \u6240\u793a\u7684\u54c8\u5e0c\u8868\u6765\u5b9e\u73b0\u3002</p> <p></p> <p> \u56fe 6-1 \u00a0 \u54c8\u5e0c\u8868\u7684\u62bd\u8c61\u8868\u793a </p> <p>\u9664\u54c8\u5e0c\u8868\u5916\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u4e5f\u53ef\u4ee5\u5b9e\u73b0\u67e5\u8be2\u529f\u80fd\uff0c\u5b83\u4eec\u7684\u6548\u7387\u5bf9\u6bd4\u5982\u8868 6-1 \u6240\u793a\u3002</p> <ul> <li>\u6dfb\u52a0\u5143\u7d20\uff1a\u4ec5\u9700\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u6570\u7ec4\uff08\u94fe\u8868\uff09\u7684\u5c3e\u90e8\u5373\u53ef\uff0c\u4f7f\u7528 \\(O(1)\\) \u65f6\u95f4\u3002</li> <li>\u67e5\u8be2\u5143\u7d20\uff1a\u7531\u4e8e\u6570\u7ec4\uff08\u94fe\u8868\uff09\u662f\u4e71\u5e8f\u7684\uff0c\u56e0\u6b64\u9700\u8981\u904d\u5386\u5176\u4e2d\u7684\u6240\u6709\u5143\u7d20\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u5220\u9664\u5143\u7d20\uff1a\u9700\u8981\u5148\u67e5\u8be2\u5230\u5143\u7d20\uff0c\u518d\u4ece\u6570\u7ec4\uff08\u94fe\u8868\uff09\u4e2d\u5220\u9664\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> </ul> <p> \u8868 6-1 \u00a0 \u5143\u7d20\u67e5\u8be2\u6548\u7387\u5bf9\u6bd4 </p> \u6570\u7ec4 \u94fe\u8868 \u54c8\u5e0c\u8868 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) \u6dfb\u52a0\u5143\u7d20 \\(O(1)\\) \\(O(1)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(1)\\) <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5728\u54c8\u5e0c\u8868\u4e2d\u8fdb\u884c\u589e\u5220\u67e5\u6539\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(1)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_hashing/hash_map/#611","title":"6.1.1 \u00a0 \u54c8\u5e0c\u8868\u5e38\u7528\u64cd\u4f5c","text":"<p>\u54c8\u5e0c\u8868\u7684\u5e38\u89c1\u64cd\u4f5c\u5305\u62ec\uff1a\u521d\u59cb\u5316\u3001\u67e5\u8be2\u64cd\u4f5c\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u548c\u5220\u9664\u952e\u503c\u5bf9\u7b49\uff0c\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py<pre><code># \u521d\u59cb\u5316\u54c8\u5e0c\u8868\nhmap: dict = {}\n\n# \u6dfb\u52a0\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n# \u67e5\u8be2\u64cd\u4f5c\n# \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname: str = hmap[15937]\n\n# \u5220\u9664\u64cd\u4f5c\n# \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nhmap.pop(10583)\n</code></pre> hash_map.cpp<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nunordered_map&lt;int, string&gt; map;\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.erase(10583);\n</code></pre> hash_map.java<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;Integer, String&gt; map = new HashMap&lt;&gt;();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.put(12836, \"\u5c0f\u54c8\");   \nmap.put(15937, \"\u5c0f\u5570\");   \nmap.put(16750, \"\u5c0f\u7b97\");   \nmap.put(13276, \"\u5c0f\u6cd5\");\nmap.put(10583, \"\u5c0f\u9e2d\");\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.cs<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nDictionary&lt;int, string&gt; map = new() {\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    // \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\n    { 12836, \"\u5c0f\u54c8\" },\n    { 15937, \"\u5c0f\u5570\" },\n    { 16750, \"\u5c0f\u7b97\" },\n    { 13276, \"\u5c0f\u6cd5\" },\n    { 10583, \"\u5c0f\u9e2d\" }\n};\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nstring name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.Remove(10583);\n</code></pre> hash_map_test.go<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nhmap := make(map[int]string)\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nhmap[12836] = \"\u5c0f\u54c8\"\nhmap[15937] = \"\u5c0f\u5570\"\nhmap[16750] = \"\u5c0f\u7b97\"\nhmap[13276] = \"\u5c0f\u6cd5\"\nhmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nname := hmap[15937]\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\ndelete(hmap, 10583)\n</code></pre> hash_map.swift<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nvar map: [Int: String] = [:]\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\"\nmap[15937] = \"\u5c0f\u5570\"\nmap[16750] = \"\u5c0f\u7b97\"\nmap[13276] = \"\u5c0f\u6cd5\"\nmap[10583] = \"\u5c0f\u9e2d\"\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map[15937]!\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.removeValue(forKey: 10583)\n</code></pre> hash_map.js<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\n</code></pre> hash_map.ts<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nconst map = new Map&lt;number, string&gt;();\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.set(12836, '\u5c0f\u54c8');\nmap.set(15937, '\u5c0f\u5570');\nmap.set(16750, '\u5c0f\u7b97');\nmap.set(13276, '\u5c0f\u6cd5');\nmap.set(10583, '\u5c0f\u9e2d');\nconsole.info('\\n\u6dfb\u52a0\u5b8c\u6210\u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet name = map.get(15937);\nconsole.info('\\n\u8f93\u5165\u5b66\u53f7 15937 \uff0c\u67e5\u8be2\u5230\u59d3\u540d ' + name);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.delete(10583);\nconsole.info('\\n\u5220\u9664 10583 \u540e\uff0c\u54c8\u5e0c\u8868\u4e3a\\nKey -&gt; Value');\nconsole.info(map);\n</code></pre> hash_map.dart<pre><code>/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nMap&lt;int, String&gt; map = {};\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap[12836] = \"\u5c0f\u54c8\";\nmap[15937] = \"\u5c0f\u5570\";\nmap[16750] = \"\u5c0f\u7b97\";\nmap[13276] = \"\u5c0f\u6cd5\";\nmap[10583] = \"\u5c0f\u9e2d\";\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nString name = map[15937];\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nmap.remove(10583);\n</code></pre> hash_map.rs<pre><code>use std::collections::HashMap;\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nlet mut map: HashMap&lt;i32, String&gt; = HashMap::new();\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u6dfb\u52a0\u952e\u503c\u5bf9 (key, value)\nmap.insert(12836, \"\u5c0f\u54c8\".to_string());\nmap.insert(15937, \"\u5c0f\u5570\".to_string());\nmap.insert(16750, \"\u5c0f\u7b97\".to_string());\nmap.insert(13279, \"\u5c0f\u6cd5\".to_string());\nmap.insert(10583, \"\u5c0f\u9e2d\".to_string());\n\n/* \u67e5\u8be2\u64cd\u4f5c */\n// \u5411\u54c8\u5e0c\u8868\u4e2d\u8f93\u5165\u952e key \uff0c\u5f97\u5230\u503c value\nlet _name: Option&lt;&amp;String&gt; = map.get(&amp;15937);\n\n/* \u5220\u9664\u64cd\u4f5c */\n// \u5728\u54c8\u5e0c\u8868\u4e2d\u5220\u9664\u952e\u503c\u5bf9 (key, value)\nlet _removed_value: Option&lt;String&gt; = map.remove(&amp;10583);\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.zig<pre><code>\n</code></pre> <p>\u54c8\u5e0c\u8868\u6709\u4e09\u79cd\u5e38\u7528\u7684\u904d\u5386\u65b9\u5f0f\uff1a\u904d\u5386\u952e\u503c\u5bf9\u3001\u904d\u5386\u952e\u548c\u904d\u5386\u503c\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig hash_map.py<pre><code># \u904d\u5386\u54c8\u5e0c\u8868\n# \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value in hmap.items():\n    print(key, \"-&gt;\", value)\n# \u5355\u72ec\u904d\u5386\u952e key\nfor key in hmap.keys():\n    print(key)\n# \u5355\u72ec\u904d\u5386\u503c value\nfor value in hmap.values():\n    print(value)\n</code></pre> hash_map.cpp<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (auto kv: map) {\n    cout &lt;&lt; kv.first &lt;&lt; \" -&gt; \" &lt;&lt; kv.second &lt;&lt; endl;\n}\n// \u4f7f\u7528\u8fed\u4ee3\u5668\u904d\u5386 key-&gt;value\nfor (auto iter = map.begin(); iter != map.end(); iter++) {\n    cout &lt;&lt; iter-&gt;first &lt;&lt; \"-&gt;\" &lt;&lt; iter-&gt;second &lt;&lt; endl;\n}\n</code></pre> hash_map.java<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor (Map.Entry &lt;Integer, String&gt; kv: map.entrySet()) {\n    System.out.println(kv.getKey() + \" -&gt; \" + kv.getValue());\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor (int key: map.keySet()) {\n    System.out.println(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor (String val: map.values()) {\n    System.out.println(val);\n}\n</code></pre> hash_map.cs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nforeach (var kv in map) {\n    Console.WriteLine(kv.Key + \" -&gt; \" + kv.Value);\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nforeach (int key in map.Keys) {\n    Console.WriteLine(key);\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nforeach (string val in map.Values) {\n    Console.WriteLine(val);\n}\n</code></pre> hash_map_test.go<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 key-&gt;value\nfor key, value := range hmap {\n    fmt.Println(key, \"-&gt;\", value)\n}\n// \u5355\u72ec\u904d\u5386\u952e key\nfor key := range hmap {\n    fmt.Println(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c value\nfor _, value := range hmap {\n    fmt.Println(value)\n}\n</code></pre> hash_map.swift<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in map {\n    print(\"\\(key) -&gt; \\(value)\")\n}\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys {\n    print(key)\n}\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values {\n    print(value)\n}\n</code></pre> hash_map.js<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.ts<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\nconsole.info('\\n\u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value');\nfor (const [k, v] of map.entries()) {\n    console.info(k + ' -&gt; ' + v);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u952e Key');\nfor (const k of map.keys()) {\n    console.info(k);\n}\nconsole.info('\\n\u5355\u72ec\u904d\u5386\u503c Value');\nfor (const v of map.values()) {\n    console.info(v);\n}\n</code></pre> hash_map.dart<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nmap.forEach((key, value) {\n  print('$key -&gt; $value');\n});\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nmap.keys.forEach((key) {\n  print(key);\n});\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nmap.values.forEach((value) {\n  print(value);\n});\n</code></pre> hash_map.rs<pre><code>/* \u904d\u5386\u54c8\u5e0c\u8868 */\n// \u904d\u5386\u952e\u503c\u5bf9 Key-&gt;Value\nfor (key, value) in &amp;map {\n    println!(\"{key} -&gt; {value}\");\n}\n\n// \u5355\u72ec\u904d\u5386\u952e Key\nfor key in map.keys() {\n    println!(\"{key}\"); \n}\n\n// \u5355\u72ec\u904d\u5386\u503c Value\nfor value in map.values() {\n    println!(\"{value}\");\n}\n</code></pre> hash_map.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u8868\n</code></pre> hash_map.zig<pre><code>\n</code></pre>"},{"location":"chapter_hashing/hash_map/#612","title":"6.1.2 \u00a0 \u54c8\u5e0c\u8868\u7b80\u5355\u5b9e\u73b0","text":"<p>\u6211\u4eec\u5148\u8003\u8651\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4ec5\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u5b9e\u73b0\u54c8\u5e0c\u8868\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5c06\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u7a7a\u4f4d\u79f0\u4e3a\u300c\u6876 bucket\u300d\uff0c\u6bcf\u4e2a\u6876\u53ef\u5b58\u50a8\u4e00\u4e2a\u952e\u503c\u5bf9\u3002\u56e0\u6b64\uff0c\u67e5\u8be2\u64cd\u4f5c\u5c31\u662f\u627e\u5230 <code>key</code> \u5bf9\u5e94\u7684\u6876\uff0c\u5e76\u5728\u6876\u4e2d\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u90a3\u4e48\uff0c\u5982\u4f55\u57fa\u4e8e <code>key</code> \u5b9a\u4f4d\u5bf9\u5e94\u7684\u6876\u5462\uff1f\u8fd9\u662f\u901a\u8fc7\u300c\u54c8\u5e0c\u51fd\u6570 hash function\u300d\u5b9e\u73b0\u7684\u3002\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u8f93\u51fa\u7a7a\u95f4\u3002\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u8f93\u5165\u7a7a\u95f4\u662f\u6240\u6709 <code>key</code> \uff0c\u8f93\u51fa\u7a7a\u95f4\u662f\u6240\u6709\u6876\uff08\u6570\u7ec4\u7d22\u5f15\uff09\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u5f97\u5230\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u952e\u503c\u5bf9\u5728\u6570\u7ec4\u4e2d\u7684\u5b58\u50a8\u4f4d\u7f6e\u3002</p> <p>\u8f93\u5165\u4e00\u4e2a <code>key</code> \uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u901a\u8fc7\u67d0\u79cd\u54c8\u5e0c\u7b97\u6cd5 <code>hash()</code> \u8ba1\u7b97\u5f97\u5230\u54c8\u5e0c\u503c\u3002</li> <li>\u5c06\u54c8\u5e0c\u503c\u5bf9\u6876\u6570\u91cf\uff08\u6570\u7ec4\u957f\u5ea6\uff09<code>capacity</code> \u53d6\u6a21\uff0c\u4ece\u800c\u83b7\u53d6\u8be5 <code>key</code> \u5bf9\u5e94\u7684\u6570\u7ec4\u7d22\u5f15 <code>index</code> \u3002</li> </ol> <pre><code>index = hash(key) % capacity\n</code></pre> <p>\u968f\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5229\u7528 <code>index</code> \u5728\u54c8\u5e0c\u8868\u4e2d\u8bbf\u95ee\u5bf9\u5e94\u7684\u6876\uff0c\u4ece\u800c\u83b7\u53d6 <code>value</code> \u3002</p> <p>\u8bbe\u6570\u7ec4\u957f\u5ea6 <code>capacity = 100</code>\u3001\u54c8\u5e0c\u7b97\u6cd5 <code>hash(key) = key</code> \uff0c\u6613\u5f97\u54c8\u5e0c\u51fd\u6570\u4e3a <code>key % 100</code> \u3002\u56fe 6-2 \u4ee5 <code>key</code> \u5b66\u53f7\u548c <code>value</code> \u59d3\u540d\u4e3a\u4f8b\uff0c\u5c55\u793a\u4e86\u54c8\u5e0c\u51fd\u6570\u7684\u5de5\u4f5c\u539f\u7406\u3002</p> <p></p> <p> \u56fe 6-2 \u00a0 \u54c8\u5e0c\u51fd\u6570\u5de5\u4f5c\u539f\u7406 </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7b80\u5355\u54c8\u5e0c\u8868\u3002\u5176\u4e2d\uff0c\u6211\u4eec\u5c06 <code>key</code> \u548c <code>value</code> \u5c01\u88c5\u6210\u4e00\u4e2a\u7c7b <code>Pair</code> \uff0c\u4ee5\u8868\u793a\u952e\u503c\u5bf9\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_hash_map.py<pre><code>class Pair:\n    \"\"\"\u952e\u503c\u5bf9\"\"\"\n\n    def __init__(self, key: int, val: str):\n        self.key = key\n        self.val = val\n\nclass ArrayHashMap:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        # \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        self.buckets: list[Pair | None] = [None] * 100\n\n    def hash_func(self, key: int) -&gt; int:\n        \"\"\"\u54c8\u5e0c\u51fd\u6570\"\"\"\n        index = key % 100\n        return index\n\n    def get(self, key: int) -&gt; str:\n        \"\"\"\u67e5\u8be2\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        pair: Pair = self.buckets[index]\n        if pair is None:\n            return None\n        return pair.val\n\n    def put(self, key: int, val: str):\n        \"\"\"\u6dfb\u52a0\u64cd\u4f5c\"\"\"\n        pair = Pair(key, val)\n        index: int = self.hash_func(key)\n        self.buckets[index] = pair\n\n    def remove(self, key: int):\n        \"\"\"\u5220\u9664\u64cd\u4f5c\"\"\"\n        index: int = self.hash_func(key)\n        # \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None\n\n    def entry_set(self) -&gt; list[Pair]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\"\"\"\n        result: list[Pair] = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair)\n        return result\n\n    def key_set(self) -&gt; list[int]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u952e\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.key)\n        return result\n\n    def value_set(self) -&gt; list[str]:\n        \"\"\"\u83b7\u53d6\u6240\u6709\u503c\"\"\"\n        result = []\n        for pair in self.buckets:\n            if pair is not None:\n                result.append(pair.val)\n        return result\n\n    def print(self):\n        \"\"\"\u6253\u5370\u54c8\u5e0c\u8868\"\"\"\n        for pair in self.buckets:\n            if pair is not None:\n                print(pair.key, \"-&gt;\", pair.val)\n</code></pre> array_hash_map.cpp<pre><code>/* \u952e\u503c\u5bf9 */\nstruct Pair {\n  public:\n    int key;\n    string val;\n    Pair(int key, string val) {\n        this-&gt;key = key;\n        this-&gt;val = val;\n    }\n};\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  private:\n    vector&lt;Pair *&gt; buckets;\n\n  public:\n    ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = vector&lt;Pair *&gt;(100);\n    }\n\n    ~ArrayHashMap() {\n        // \u91ca\u653e\u5185\u5b58\n        for (const auto &amp;bucket : buckets) {\n            delete bucket;\n        }\n        buckets.clear();\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    string get(int key) {\n        int index = hashFunc(key);\n        Pair *pair = buckets[index];\n        if (pair == nullptr)\n            return \"\";\n        return pair-&gt;val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    void put(int key, string val) {\n        Pair *pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    void remove(int key) {\n        int index = hashFunc(key);\n        // \u91ca\u653e\u5185\u5b58\u5e76\u7f6e\u4e3a nullptr\n        delete buckets[index];\n        buckets[index] = nullptr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    vector&lt;Pair *&gt; pairSet() {\n        vector&lt;Pair *&gt; pairSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                pairSet.push_back(pair);\n            }\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    vector&lt;int&gt; keySet() {\n        vector&lt;int&gt; keySet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                keySet.push_back(pair-&gt;key);\n            }\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    vector&lt;string&gt; valueSet() {\n        vector&lt;string&gt; valueSet;\n        for (Pair *pair : buckets) {\n            if (pair != nullptr) {\n                valueSet.push_back(pair-&gt;val);\n            }\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    void print() {\n        for (Pair *kv : pairSet()) {\n            cout &lt;&lt; kv-&gt;key &lt;&lt; \" -&gt; \" &lt;&lt; kv-&gt;val &lt;&lt; endl;\n        }\n    }\n};\n</code></pre> array_hash_map.java<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n    public int key;\n    public String val;\n\n    public Pair(int key, String val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private List&lt;Pair&gt; buckets;\n\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = new ArrayList&lt;&gt;();\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private int hashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public String get(int key) {\n        int index = hashFunc(key);\n        Pair pair = buckets.get(index);\n        if (pair == null)\n            return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void put(int key, String val) {\n        Pair pair = new Pair(key, val);\n        int index = hashFunc(key);\n        buckets.set(index, pair);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void remove(int key) {\n        int index = hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets.set(index, null);\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; pairSet() {\n        List&lt;Pair&gt; pairSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                pairSet.add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;Integer&gt; keySet() {\n        List&lt;Integer&gt; keySet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                keySet.add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;String&gt; valueSet() {\n        List&lt;String&gt; valueSet = new ArrayList&lt;&gt;();\n        for (Pair pair : buckets) {\n            if (pair != null)\n                valueSet.add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void print() {\n        for (Pair kv : pairSet()) {\n            System.out.println(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.cs<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\nclass Pair(int key, string val) {\n    public int key = key;\n    public string val = val;\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    List&lt;Pair?&gt; buckets;\n    public ArrayHashMap() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        buckets = [];\n        for (int i = 0; i &lt; 100; i++) {\n            buckets.Add(null);\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    int HashFunc(int key) {\n        int index = key % 100;\n        return index;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public string? Get(int key) {\n        int index = HashFunc(key);\n        Pair? pair = buckets[index];\n        if (pair == null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public void Put(int key, string val) {\n        Pair pair = new(key, val);\n        int index = HashFunc(key);\n        buckets[index] = pair;\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public void Remove(int key) {\n        int index = HashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public List&lt;Pair&gt; PairSet() {\n        List&lt;Pair&gt; pairSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                pairSet.Add(pair);\n        }\n        return pairSet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public List&lt;int&gt; KeySet() {\n        List&lt;int&gt; keySet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                keySet.Add(pair.key);\n        }\n        return keySet;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public List&lt;string&gt; ValueSet() {\n        List&lt;string&gt; valueSet = [];\n        foreach (Pair? pair in buckets) {\n            if (pair != null)\n                valueSet.Add(pair.val);\n        }\n        return valueSet;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public void Print() {\n        foreach (Pair kv in PairSet()) {\n            Console.WriteLine(kv.key + \" -&gt; \" + kv.val);\n        }\n    }\n}\n</code></pre> array_hash_map.go<pre><code>/* \u952e\u503c\u5bf9 */\ntype pair struct {\n    key int\n    val string\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntype arrayHashMap struct {\n    buckets []*pair\n}\n\n/* \u521d\u59cb\u5316\u54c8\u5e0c\u8868 */\nfunc newArrayHashMap() *arrayHashMap {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    buckets := make([]*pair, 100)\n    return &amp;arrayHashMap{buckets: buckets}\n}\n\n/* \u54c8\u5e0c\u51fd\u6570 */\nfunc (a *arrayHashMap) hashFunc(key int) int {\n    index := key % 100\n    return index\n}\n\n/* \u67e5\u8be2\u64cd\u4f5c */\nfunc (a *arrayHashMap) get(key int) string {\n    index := a.hashFunc(key)\n    pair := a.buckets[index]\n    if pair == nil {\n        return \"Not Found\"\n    }\n    return pair.val\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nfunc (a *arrayHashMap) put(key int, val string) {\n    pair := &amp;pair{key: key, val: val}\n    index := a.hashFunc(key)\n    a.buckets[index] = pair\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nfunc (a *arrayHashMap) remove(key int) {\n    index := a.hashFunc(key)\n    // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n    a.buckets[index] = nil\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u5bf9 */\nfunc (a *arrayHashMap) pairSet() []*pair {\n    var pairs []*pair\n    for _, pair := range a.buckets {\n        if pair != nil {\n            pairs = append(pairs, pair)\n        }\n    }\n    return pairs\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nfunc (a *arrayHashMap) keySet() []int {\n    var keys []int\n    for _, pair := range a.buckets {\n        if pair != nil {\n            keys = append(keys, pair.key)\n        }\n    }\n    return keys\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nfunc (a *arrayHashMap) valueSet() []string {\n    var values []string\n    for _, pair := range a.buckets {\n        if pair != nil {\n            values = append(values, pair.val)\n        }\n    }\n    return values\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nfunc (a *arrayHashMap) print() {\n    for _, pair := range a.buckets {\n        if pair != nil {\n            fmt.Println(pair.key, \"-&gt;\", pair.val)\n        }\n    }\n}\n</code></pre> array_hash_map.swift<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair: Equatable {\n    public var key: Int\n    public var val: String\n\n    public init(key: Int, val: String) {\n        self.key = key\n        self.val = val\n    }\n\n    public static func == (lhs: Pair, rhs: Pair) -&gt; Bool {\n        lhs.key == rhs.key &amp;&amp; lhs.val == rhs.val\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private var buckets: [Pair?] = []\n\n    init() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        for _ in 0 ..&lt; 100 {\n            buckets.append(nil)\n        }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private func hashFunc(key: Int) -&gt; Int {\n        let index = key % 100\n        return index\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    func get(key: Int) -&gt; String? {\n        let index = hashFunc(key: key)\n        let pair = buckets[index]\n        return pair?.val\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    func put(key: Int, val: String) {\n        let pair = Pair(key: key, val: val)\n        let index = hashFunc(key: key)\n        buckets[index] = pair\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    func remove(key: Int) {\n        let index = hashFunc(key: key)\n        // \u7f6e\u4e3a nil \uff0c\u4ee3\u8868\u5220\u9664\n        buckets[index] = nil\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    func pairSet() -&gt; [Pair] {\n        var pairSet: [Pair] = []\n        for pair in buckets {\n            if let pair = pair {\n                pairSet.append(pair)\n            }\n        }\n        return pairSet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    func keySet() -&gt; [Int] {\n        var keySet: [Int] = []\n        for pair in buckets {\n            if let pair = pair {\n                keySet.append(pair.key)\n            }\n        }\n        return keySet\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    func valueSet() -&gt; [String] {\n        var valueSet: [String] = []\n        for pair in buckets {\n            if let pair = pair {\n                valueSet.append(pair.val)\n            }\n        }\n        return valueSet\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    func print() {\n        for pair in pairSet() {\n            Swift.print(\"\\(pair.key) -&gt; \\(pair.val)\")\n        }\n    }\n}\n</code></pre> array_hash_map.js<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    constructor(key, val) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    #buckets;\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.#buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    #hashFunc(key) {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    get(key) {\n        let index = this.#hashFunc(key);\n        let pair = this.#buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    set(key, val) {\n        let index = this.#hashFunc(key);\n        this.#buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    delete(key) {\n        let index = this.#hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.#buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    entries() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    keys() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    values() {\n        let arr = [];\n        for (let i = 0; i &lt; this.#buckets.length; i++) {\n            if (this.#buckets[i]) {\n                arr.push(this.#buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.ts<pre><code>/* \u952e\u503c\u5bf9 Number -&gt; String */\nclass Pair {\n    public key: number;\n    public val: string;\n\n    constructor(key: number, val: string) {\n        this.key = key;\n        this.val = val;\n    }\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n    private readonly buckets: (Pair | null)[];\n\n    constructor() {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        this.buckets = new Array(100).fill(null);\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    private hashFunc(key: number): number {\n        return key % 100;\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    public get(key: number): string | null {\n        let index = this.hashFunc(key);\n        let pair = this.buckets[index];\n        if (pair === null) return null;\n        return pair.val;\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    public set(key: number, val: string) {\n        let index = this.hashFunc(key);\n        this.buckets[index] = new Pair(key, val);\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    public delete(key: number) {\n        let index = this.hashFunc(key);\n        // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n        this.buckets[index] = null;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    public entries(): (Pair | null)[] {\n        let arr: (Pair | null)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i]);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    public keys(): (number | undefined)[] {\n        let arr: (number | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].key);\n            }\n        }\n        return arr;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    public values(): (string | undefined)[] {\n        let arr: (string | undefined)[] = [];\n        for (let i = 0; i &lt; this.buckets.length; i++) {\n            if (this.buckets[i]) {\n                arr.push(this.buckets[i].val);\n            }\n        }\n        return arr;\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    public print() {\n        let pairSet = this.entries();\n        for (const pair of pairSet) {\n            console.info(`${pair.key} -&gt; ${pair.val}`);\n        }\n    }\n}\n</code></pre> array_hash_map.dart<pre><code>/* \u952e\u503c\u5bf9 */\nclass Pair {\n  int key;\n  String val;\n  Pair(this.key, this.val);\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\nclass ArrayHashMap {\n  late List&lt;Pair?&gt; _buckets;\n\n  ArrayHashMap() {\n    // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n    _buckets = List.filled(100, null);\n  }\n\n  /* \u54c8\u5e0c\u51fd\u6570 */\n  int _hashFunc(int key) {\n    final int index = key % 100;\n    return index;\n  }\n\n  /* \u67e5\u8be2\u64cd\u4f5c */\n  String? get(int key) {\n    final int index = _hashFunc(key);\n    final Pair? pair = _buckets[index];\n    if (pair == null) {\n      return null;\n    }\n    return pair.val;\n  }\n\n  /* \u6dfb\u52a0\u64cd\u4f5c */\n  void put(int key, String val) {\n    final Pair pair = Pair(key, val);\n    final int index = _hashFunc(key);\n    _buckets[index] = pair;\n  }\n\n  /* \u5220\u9664\u64cd\u4f5c */\n  void remove(int key) {\n    final int index = _hashFunc(key);\n    _buckets[index] = null;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n  List&lt;Pair&gt; pairSet() {\n    List&lt;Pair&gt; pairSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        pairSet.add(pair);\n      }\n    }\n    return pairSet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u952e */\n  List&lt;int&gt; keySet() {\n    List&lt;int&gt; keySet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        keySet.add(pair.key);\n      }\n    }\n    return keySet;\n  }\n\n  /* \u83b7\u53d6\u6240\u6709\u503c */\n  List&lt;String&gt; values() {\n    List&lt;String&gt; valueSet = [];\n    for (final Pair? pair in _buckets) {\n      if (pair != null) {\n        valueSet.add(pair.val);\n      }\n    }\n    return valueSet;\n  }\n\n  /* \u6253\u5370\u54c8\u5e0c\u8868 */\n  void printHashMap() {\n    for (final Pair kv in pairSet()) {\n      print(\"${kv.key} -&gt; ${kv.val}\");\n    }\n  }\n}\n</code></pre> array_hash_map.rs<pre><code>/* \u952e\u503c\u5bf9 */\n#[derive(Debug, Clone, PartialEq)]\npub struct Pair {\n    pub key: i32,\n    pub val: String,\n}\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\npub struct ArrayHashMap {\n    buckets: Vec&lt;Option&lt;Pair&gt;&gt;\n}\n\nimpl ArrayHashMap {\n    pub fn new() -&gt; ArrayHashMap {\n        // \u521d\u59cb\u5316\u6570\u7ec4\uff0c\u5305\u542b 100 \u4e2a\u6876\n        Self { buckets: vec![None; 100] }\n    }\n\n    /* \u54c8\u5e0c\u51fd\u6570 */\n    fn hash_func(&amp;self, key: i32) -&gt; usize {\n        key as usize % 100\n    }\n\n    /* \u67e5\u8be2\u64cd\u4f5c */\n    pub fn get(&amp;self, key: i32) -&gt; Option&lt;&amp;String&gt; {\n        let index = self.hash_func(key);\n        self.buckets[index].as_ref().map(|pair| &amp;pair.val)\n    }\n\n    /* \u6dfb\u52a0\u64cd\u4f5c */\n    pub fn put(&amp;mut self, key: i32, val: &amp;str) {\n        let index = self.hash_func(key);\n        self.buckets[index] = Some(Pair {\n            key,\n            val: val.to_string(),\n        });\n    }\n\n    /* \u5220\u9664\u64cd\u4f5c */\n    pub fn remove(&amp;mut self, key: i32) {\n        let index = self.hash_func(key);\n        // \u7f6e\u4e3a None \uff0c\u4ee3\u8868\u5220\u9664\n        self.buckets[index] = None;\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\n    pub fn entry_set(&amp;self) -&gt; Vec&lt;&amp;Pair&gt; {\n        self.buckets.iter().filter_map(|pair| pair.as_ref()).collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u952e */\n    pub fn key_set(&amp;self) -&gt; Vec&lt;&amp;i32&gt; {\n        self.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.key)).collect()\n    }\n\n    /* \u83b7\u53d6\u6240\u6709\u503c */\n    pub fn value_set(&amp;self) -&gt; Vec&lt;&amp;String&gt; {\n        self.buckets.iter().filter_map(|pair| pair.as_ref().map(|pair| &amp;pair.val)).collect()\n    }\n\n    /* \u6253\u5370\u54c8\u5e0c\u8868 */\n    pub fn print(&amp;self) {\n        for pair in self.entry_set() {\n            println!(\"{} -&gt; {}\", pair.key, pair.val);\n        }\n    }\n}\n</code></pre> array_hash_map.c<pre><code>/* \u952e\u503c\u5bf9 int-&gt;string */\ntypedef struct {\n    int key;\n    char *val;\n} Pair;\n\n/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868 */\ntypedef struct {\n    Pair *buckets[HASHTABLE_CAPACITY];\n} ArrayHashMap;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayHashMap *newArrayHashMap() {\n    ArrayHashMap *hmap = malloc(sizeof(ArrayHashMap));\n    return hmap;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayHashMap(ArrayHashMap *hmap) {\n    for (int i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            free(hmap-&gt;buckets[i]-&gt;val);\n            free(hmap-&gt;buckets[i]);\n        }\n    }\n    free(hmap);\n}\n\n/* \u6dfb\u52a0\u64cd\u4f5c */\nvoid put(ArrayHashMap *hmap, const int key, const char *val) {\n    Pair *Pair = malloc(sizeof(Pair));\n    Pair-&gt;key = key;\n    Pair-&gt;val = malloc(strlen(val) + 1);\n    strcpy(Pair-&gt;val, val);\n\n    int index = hashFunc(key);\n    hmap-&gt;buckets[index] = Pair;\n}\n\n/* \u5220\u9664\u64cd\u4f5c */\nvoid removeItem(ArrayHashMap *hmap, const int key) {\n    int index = hashFunc(key);\n    free(hmap-&gt;buckets[index]-&gt;val);\n    free(hmap-&gt;buckets[index]);\n    hmap-&gt;buckets[index] = NULL;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9 */\nvoid pairSet(ArrayHashMap *hmap, MapSet *set) {\n    Pair *entries;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    entries = malloc(sizeof(Pair) * total);\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            entries[index].key = hmap-&gt;buckets[i]-&gt;key;\n            entries[index].val = malloc(strlen(hmap-&gt;buckets[i]-&gt;val) + 1);\n            strcpy(entries[index].val, hmap-&gt;buckets[i]-&gt;val);\n            index++;\n        }\n    }\n    set-&gt;set = entries;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u952e */\nvoid keySet(ArrayHashMap *hmap, MapSet *set) {\n    int *keys;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    keys = malloc(total * sizeof(int));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            keys[index] = hmap-&gt;buckets[i]-&gt;key;\n            index++;\n        }\n    }\n    set-&gt;set = keys;\n    set-&gt;len = total;\n}\n\n/* \u83b7\u53d6\u6240\u6709\u503c */\nvoid valueSet(ArrayHashMap *hmap, MapSet *set) {\n    char **vals;\n    int i = 0, index = 0;\n    int total = 0;\n    /* \u7edf\u8ba1\u6709\u6548\u952e\u503c\u5bf9\u6570\u91cf */\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            total++;\n        }\n    }\n    vals = malloc(total * sizeof(char *));\n    for (i = 0; i &lt; HASHTABLE_CAPACITY; i++) {\n        if (hmap-&gt;buckets[i] != NULL) {\n            vals[index] = hmap-&gt;buckets[i]-&gt;val;\n            index++;\n        }\n    }\n    set-&gt;set = vals;\n    set-&gt;len = total;\n}\n\n/* \u6253\u5370\u54c8\u5e0c\u8868 */\nvoid print(ArrayHashMap *hmap) {\n    int i;\n    MapSet set;\n    pairSet(hmap, &amp;set);\n    Pair *entries = (Pair *)set.set;\n    for (i = 0; i &lt; set.len; i++) {\n        printf(\"%d -&gt; %s\\n\", entries[i].key, entries[i].val);\n    }\n    free(set.set);\n}\n</code></pre> array_hash_map.zig<pre><code>// \u952e\u503c\u5bf9\nconst Pair = struct {\n    key: usize = undefined,\n    val: []const u8 = undefined,\n\n   pub fn init(key: usize, val: []const u8) Pair {\n        return Pair {\n            .key = key,\n            .val = val,\n        };\n    }\n};\n\n// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u54c8\u5e0c\u8868\nfn ArrayHashMap(comptime T: type) type {\n    return struct {\n        bucket: ?std.ArrayList(?T) = null,\n        mem_allocator: std.mem.Allocator = undefined,\n\n        const Self = @This();\n\n        // \u6784\u9020\u51fd\u6570\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            self.mem_allocator = allocator;\n            // \u521d\u59cb\u5316\u4e00\u4e2a\u957f\u5ea6\u4e3a 100 \u7684\u6876\uff08\u6570\u7ec4\uff09\n            self.bucket = std.ArrayList(?T).init(self.mem_allocator);\n            var i: i32 = 0;\n            while (i &lt; 100) : (i += 1) {\n                try self.bucket.?.append(null);\n            }\n        }\n\n        // \u6790\u6784\u51fd\u6570\n        pub fn deinit(self: *Self) void {\n            if (self.bucket != null) self.bucket.?.deinit();\n        }\n\n        // \u54c8\u5e0c\u51fd\u6570\n        fn hashFunc(key: usize) usize {\n            var index = key % 100;\n            return index;\n        }\n\n        // \u67e5\u8be2\u64cd\u4f5c\n        pub fn get(self: *Self, key: usize) []const u8 {\n            var index = hashFunc(key);\n            var pair = self.bucket.?.items[index];\n            return pair.?.val;\n        }\n\n        // \u6dfb\u52a0\u64cd\u4f5c\n        pub fn put(self: *Self, key: usize, val: []const u8) !void {\n            var pair = Pair.init(key, val);\n            var index = hashFunc(key);\n            self.bucket.?.items[index] = pair;\n        }\n\n        // \u5220\u9664\u64cd\u4f5c\n        pub fn remove(self: *Self, key: usize) !void {\n            var index = hashFunc(key);\n            // \u7f6e\u4e3a null \uff0c\u4ee3\u8868\u5220\u9664\n            self.bucket.?.items[index] = null;\n        }       \n\n        // \u83b7\u53d6\u6240\u6709\u952e\u503c\u5bf9\n        pub fn pairSet(self: *Self) !std.ArrayList(T) {\n            var entry_set = std.ArrayList(T).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try entry_set.append(item.?);\n            }\n            return entry_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u952e\n        pub fn keySet(self: *Self) !std.ArrayList(usize) {\n            var key_set = std.ArrayList(usize).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try key_set.append(item.?.key);\n            }\n            return key_set;\n        }  \n\n        // \u83b7\u53d6\u6240\u6709\u503c\n        pub fn valueSet(self: *Self) !std.ArrayList([]const u8) {\n            var value_set = std.ArrayList([]const u8).init(self.mem_allocator);\n            for (self.bucket.?.items) |item| {\n                if (item == null) continue;\n                try value_set.append(item.?.val);\n            }\n            return value_set;\n        }\n\n        // \u6253\u5370\u54c8\u5e0c\u8868\n        pub fn print(self: *Self) !void {\n            var entry_set = try self.pairSet();\n            defer entry_set.deinit();\n            for (entry_set.items) |item| {\n                std.debug.print(\"{} -&gt; {s}\\n\", .{item.key, item.val});\n            }\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_hashing/hash_map/#613","title":"6.1.3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u4e0e\u6269\u5bb9","text":"<p>\u4ece\u672c\u8d28\u4e0a\u770b\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u4f5c\u7528\u662f\u5c06\u6240\u6709 <code>key</code> \u6784\u6210\u7684\u8f93\u5165\u7a7a\u95f4\u6620\u5c04\u5230\u6570\u7ec4\u6240\u6709\u7d22\u5f15\u6784\u6210\u7684\u8f93\u51fa\u7a7a\u95f4\uff0c\u800c\u8f93\u5165\u7a7a\u95f4\u5f80\u5f80\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\u3002\u56e0\u6b64\uff0c\u7406\u8bba\u4e0a\u4e00\u5b9a\u5b58\u5728\u201c\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u76f8\u540c\u8f93\u51fa\u201d\u7684\u60c5\u51b5\u3002</p> <p>\u5bf9\u4e8e\u4e0a\u8ff0\u793a\u4f8b\u4e2d\u7684\u54c8\u5e0c\u51fd\u6570\uff0c\u5f53\u8f93\u5165\u7684 <code>key</code> \u540e\u4e24\u4f4d\u76f8\u540c\u65f6\uff0c\u54c8\u5e0c\u51fd\u6570\u7684\u8f93\u51fa\u7ed3\u679c\u4e5f\u76f8\u540c\u3002\u4f8b\u5982\uff0c\u67e5\u8be2\u5b66\u53f7\u4e3a 12836 \u548c 20336 \u7684\u4e24\u4e2a\u5b66\u751f\u65f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a</p> <pre><code>12836 % 100 = 36\n20336 % 100 = 36\n</code></pre> <p>\u5982\u56fe 6-3 \u6240\u793a\uff0c\u4e24\u4e2a\u5b66\u53f7\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u59d3\u540d\uff0c\u8fd9\u663e\u7136\u662f\u4e0d\u5bf9\u7684\u3002\u6211\u4eec\u5c06\u8fd9\u79cd\u591a\u4e2a\u8f93\u5165\u5bf9\u5e94\u540c\u4e00\u8f93\u51fa\u7684\u60c5\u51b5\u79f0\u4e3a\u300c\u54c8\u5e0c\u51b2\u7a81 hash collision\u300d\u3002</p> <p></p> <p> \u56fe 6-3 \u00a0 \u54c8\u5e0c\u51b2\u7a81\u793a\u4f8b </p> <p>\u5bb9\u6613\u60f3\u5230\uff0c\u54c8\u5e0c\u8868\u5bb9\u91cf \\(n\\) \u8d8a\u5927\uff0c\u591a\u4e2a <code>key</code> \u88ab\u5206\u914d\u5230\u540c\u4e00\u4e2a\u6876\u4e2d\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\uff0c\u51b2\u7a81\u5c31\u8d8a\u5c11\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</p> <p>\u5982\u56fe 6-4 \u6240\u793a\uff0c\u6269\u5bb9\u524d\u952e\u503c\u5bf9 <code>(136, A)</code> \u548c <code>(236, D)</code> \u53d1\u751f\u51b2\u7a81\uff0c\u6269\u5bb9\u540e\u51b2\u7a81\u6d88\u5931\u3002</p> <p></p> <p> \u56fe 6-4 \u00a0 \u54c8\u5e0c\u8868\u6269\u5bb9 </p> <p>\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u6269\u5bb9\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u9700\u5c06\u6240\u6709\u952e\u503c\u5bf9\u4ece\u539f\u54c8\u5e0c\u8868\u8fc1\u79fb\u81f3\u65b0\u54c8\u5e0c\u8868\uff0c\u975e\u5e38\u8017\u65f6\uff1b\u5e76\u4e14\u7531\u4e8e\u54c8\u5e0c\u8868\u5bb9\u91cf <code>capacity</code> \u6539\u53d8\uff0c\u6211\u4eec\u9700\u8981\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u6765\u91cd\u65b0\u8ba1\u7b97\u6240\u6709\u952e\u503c\u5bf9\u7684\u5b58\u50a8\u4f4d\u7f6e\uff0c\u8fd9\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u6269\u5bb9\u8fc7\u7a0b\u7684\u8ba1\u7b97\u5f00\u9500\u3002\u4e3a\u6b64\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u9884\u7559\u8db3\u591f\u5927\u7684\u54c8\u5e0c\u8868\u5bb9\u91cf\uff0c\u9632\u6b62\u9891\u7e41\u6269\u5bb9\u3002</p> <p>\u300c\u8d1f\u8f7d\u56e0\u5b50 load factor\u300d\u662f\u54c8\u5e0c\u8868\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u5176\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u7684\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u7528\u4e8e\u8861\u91cf\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u4e5f\u5e38\u4f5c\u4e3a\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u89e6\u53d1\u6761\u4ef6\u3002\u4f8b\u5982\u5728 Java \u4e2d\uff0c\u5f53\u8d1f\u8f7d\u56e0\u5b50\u8d85\u8fc7 \\(0.75\\) \u65f6\uff0c\u7cfb\u7edf\u4f1a\u5c06\u54c8\u5e0c\u8868\u6269\u5bb9\u81f3\u539f\u5148\u7684 \\(2\\) \u500d\u3002</p>"},{"location":"chapter_hashing/summary/","title":"6.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_hashing/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u8f93\u5165 <code>key</code> \uff0c\u54c8\u5e0c\u8868\u80fd\u591f\u5728 \\(O(1)\\) \u65f6\u95f4\u5185\u67e5\u8be2\u5230 <code>value</code> \uff0c\u6548\u7387\u975e\u5e38\u9ad8\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u8868\u64cd\u4f5c\u5305\u62ec\u67e5\u8be2\u3001\u6dfb\u52a0\u952e\u503c\u5bf9\u3001\u5220\u9664\u952e\u503c\u5bf9\u548c\u904d\u5386\u54c8\u5e0c\u8868\u7b49\u3002</li> <li>\u54c8\u5e0c\u51fd\u6570\u5c06 <code>key</code> \u6620\u5c04\u4e3a\u6570\u7ec4\u7d22\u5f15\uff0c\u4ece\u800c\u8bbf\u95ee\u5bf9\u5e94\u6876\u5e76\u83b7\u53d6 <code>value</code> \u3002</li> <li>\u4e24\u4e2a\u4e0d\u540c\u7684 <code>key</code> \u53ef\u80fd\u5728\u7ecf\u8fc7\u54c8\u5e0c\u51fd\u6570\u540e\u5f97\u5230\u76f8\u540c\u7684\u6570\u7ec4\u7d22\u5f15\uff0c\u5bfc\u81f4\u67e5\u8be2\u7ed3\u679c\u51fa\u9519\uff0c\u8fd9\u79cd\u73b0\u8c61\u88ab\u79f0\u4e3a\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u54c8\u5e0c\u8868\u5bb9\u91cf\u8d8a\u5927\uff0c\u54c8\u5e0c\u51b2\u7a81\u7684\u6982\u7387\u5c31\u8d8a\u4f4e\u3002\u56e0\u6b64\u53ef\u4ee5\u901a\u8fc7\u6269\u5bb9\u54c8\u5e0c\u8868\u6765\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\u3002\u4e0e\u6570\u7ec4\u6269\u5bb9\u7c7b\u4f3c\uff0c\u54c8\u5e0c\u8868\u6269\u5bb9\u64cd\u4f5c\u7684\u5f00\u9500\u5f88\u5927\u3002</li> <li>\u8d1f\u8f7d\u56e0\u5b50\u5b9a\u4e49\u4e3a\u54c8\u5e0c\u8868\u4e2d\u5143\u7d20\u6570\u91cf\u9664\u4ee5\u6876\u6570\u91cf\uff0c\u53cd\u6620\u4e86\u54c8\u5e0c\u51b2\u7a81\u7684\u4e25\u91cd\u7a0b\u5ea6\uff0c\u5e38\u7528\u4f5c\u89e6\u53d1\u54c8\u5e0c\u8868\u6269\u5bb9\u7684\u6761\u4ef6\u3002</li> <li>\u94fe\u5f0f\u5730\u5740\u901a\u8fc7\u5c06\u5355\u4e2a\u5143\u7d20\u8f6c\u5316\u4e3a\u94fe\u8868\uff0c\u5c06\u6240\u6709\u51b2\u7a81\u5143\u7d20\u5b58\u50a8\u5728\u540c\u4e00\u4e2a\u94fe\u8868\u4e2d\u3002\u7136\u800c\uff0c\u94fe\u8868\u8fc7\u957f\u4f1a\u964d\u4f4e\u67e5\u8be2\u6548\u7387\uff0c\u53ef\u4ee5\u901a\u8fc7\u8fdb\u4e00\u6b65\u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u7ea2\u9ed1\u6811\u6765\u63d0\u9ad8\u6548\u7387\u3002</li> <li>\u5f00\u653e\u5bfb\u5740\u901a\u8fc7\u591a\u6b21\u63a2\u6d4b\u6765\u5904\u7406\u54c8\u5e0c\u51b2\u7a81\u3002\u7ebf\u6027\u63a2\u6d4b\u4f7f\u7528\u56fa\u5b9a\u6b65\u957f\uff0c\u7f3a\u70b9\u662f\u4e0d\u80fd\u5220\u9664\u5143\u7d20\uff0c\u4e14\u5bb9\u6613\u4ea7\u751f\u805a\u96c6\u3002\u591a\u6b21\u54c8\u5e0c\u4f7f\u7528\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u8fdb\u884c\u63a2\u6d4b\uff0c\u76f8\u8f83\u7ebf\u6027\u63a2\u6d4b\u66f4\u4e0d\u6613\u4ea7\u751f\u805a\u96c6\uff0c\u4f46\u591a\u4e2a\u54c8\u5e0c\u51fd\u6570\u589e\u52a0\u4e86\u8ba1\u7b97\u91cf\u3002</li> <li>\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u91c7\u53d6\u4e86\u4e0d\u540c\u7684\u54c8\u5e0c\u8868\u5b9e\u73b0\u3002\u4f8b\u5982\uff0cJava \u7684 <code>HashMap</code> \u4f7f\u7528\u94fe\u5f0f\u5730\u5740\uff0c\u800c Python \u7684 <code>Dict</code> \u91c7\u7528\u5f00\u653e\u5bfb\u5740\u3002</li> <li>\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u54c8\u5e0c\u7b97\u6cd5\u5177\u6709\u786e\u5b9a\u6027\u3001\u9ad8\u6548\u7387\u548c\u5747\u5300\u5206\u5e03\u7684\u7279\u70b9\u3002\u5728\u5bc6\u7801\u5b66\u4e2d\uff0c\u54c8\u5e0c\u7b97\u6cd5\u8fd8\u5e94\u8be5\u5177\u5907\u6297\u78b0\u649e\u6027\u548c\u96ea\u5d29\u6548\u5e94\u3002</li> <li>\u54c8\u5e0c\u7b97\u6cd5\u901a\u5e38\u91c7\u7528\u5927\u8d28\u6570\u4f5c\u4e3a\u6a21\u6570\uff0c\u4ee5\u6700\u5927\u5316\u5730\u4fdd\u8bc1\u54c8\u5e0c\u503c\u5747\u5300\u5206\u5e03\uff0c\u51cf\u5c11\u54c8\u5e0c\u51b2\u7a81\u3002</li> <li>\u5e38\u89c1\u7684\u54c8\u5e0c\u7b97\u6cd5\u5305\u62ec MD5\u3001SHA-1\u3001SHA-2 \u548c SHA-3 \u7b49\u3002MD5 \u5e38\u7528\u4e8e\u6821\u9a8c\u6587\u4ef6\u5b8c\u6574\u6027\uff0cSHA-2 \u5e38\u7528\u4e8e\u5b89\u5168\u5e94\u7528\u4e0e\u534f\u8bae\u3002</li> <li>\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u4f1a\u4e3a\u6570\u636e\u7c7b\u578b\u63d0\u4f9b\u5185\u7f6e\u54c8\u5e0c\u7b97\u6cd5\uff0c\u7528\u4e8e\u8ba1\u7b97\u54c8\u5e0c\u8868\u4e2d\u7684\u6876\u7d22\u5f15\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u53ea\u6709\u4e0d\u53ef\u53d8\u5bf9\u8c61\u662f\u53ef\u54c8\u5e0c\u7684\u3002</li> </ul>"},{"location":"chapter_hashing/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u4e0d\u662f \\(O(n)\\) \uff1f</p> <p>\u5f53\u54c8\u5e0c\u51b2\u7a81\u6bd4\u8f83\u4e25\u91cd\u65f6\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u9000\u5316\u81f3 \\(O(n)\\) \u3002\u5f53\u54c8\u5e0c\u51fd\u6570\u8bbe\u8ba1\u5f97\u6bd4\u8f83\u597d\u3001\u5bb9\u91cf\u8bbe\u7f6e\u6bd4\u8f83\u5408\u7406\u3001\u51b2\u7a81\u6bd4\u8f83\u5e73\u5747\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002\u6211\u4eec\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u54c8\u5e0c\u8868\u65f6\uff0c\u901a\u5e38\u8ba4\u4e3a\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(1)\\) \u3002</p> <p>\u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u7528\u54c8\u5e0c\u51fd\u6570 \\(f(x) = x\\) \u5462\uff1f\u8fd9\u6837\u5c31\u4e0d\u4f1a\u6709\u51b2\u7a81\u4e86</p> <p>\u5728 \\(f(x) = x\\) \u54c8\u5e0c\u51fd\u6570\u4e0b\uff0c\u6bcf\u4e2a\u5143\u7d20\u5bf9\u5e94\u552f\u4e00\u7684\u6876\u7d22\u5f15\uff0c\u8fd9\u4e0e\u6570\u7ec4\u7b49\u4ef7\u3002\u7136\u800c\uff0c\u8f93\u5165\u7a7a\u95f4\u901a\u5e38\u8fdc\u5927\u4e8e\u8f93\u51fa\u7a7a\u95f4\uff08\u6570\u7ec4\u957f\u5ea6\uff09\uff0c\u56e0\u6b64\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6\u53d6\u6a21\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u54c8\u5e0c\u8868\u7684\u76ee\u6807\u662f\u5c06\u4e00\u4e2a\u8f83\u5927\u7684\u72b6\u6001\u7a7a\u95f4\u6620\u5c04\u5230\u4e00\u4e2a\u8f83\u5c0f\u7684\u7a7a\u95f4\uff0c\u5e76\u63d0\u4f9b \\(O(1)\\) \u7684\u67e5\u8be2\u6548\u7387\u3002</p> <p>\u54c8\u5e0c\u8868\u5e95\u5c42\u5b9e\u73b0\u662f\u6570\u7ec4\u3001\u94fe\u8868\u3001\u4e8c\u53c9\u6811\uff0c\u4f46\u4e3a\u4ec0\u4e48\u6548\u7387\u53ef\u4ee5\u6bd4\u5b83\u4eec\u66f4\u9ad8\u5462\uff1f</p> <p>\u9996\u5148\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\uff0c\u4f46\u7a7a\u95f4\u6548\u7387\u53d8\u4f4e\u4e86\u3002\u54c8\u5e0c\u8868\u6709\u76f8\u5f53\u4e00\u90e8\u5206\u5185\u5b58\u672a\u4f7f\u7528\u3002</p> <p>\u5176\u6b21\uff0c\u53ea\u662f\u5728\u7279\u5b9a\u4f7f\u7528\u573a\u666f\u4e0b\u65f6\u95f4\u6548\u7387\u53d8\u9ad8\u4e86\u3002\u5982\u679c\u4e00\u4e2a\u529f\u80fd\u80fd\u591f\u5728\u76f8\u540c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u4f7f\u7528\u6570\u7ec4\u6216\u94fe\u8868\u5b9e\u73b0\uff0c\u90a3\u4e48\u901a\u5e38\u6bd4\u54c8\u5e0c\u8868\u66f4\u5feb\u3002\u8fd9\u662f\u56e0\u4e3a\u54c8\u5e0c\u51fd\u6570\u8ba1\u7b97\u9700\u8981\u5f00\u9500\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5e38\u6570\u9879\u66f4\u5927\u3002</p> <p>\u6700\u540e\uff0c\u54c8\u5e0c\u8868\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u80fd\u53d1\u751f\u52a3\u5316\u3002\u4f8b\u5982\u5728\u94fe\u5f0f\u5730\u5740\u4e2d\uff0c\u6211\u4eec\u91c7\u53d6\u5728\u94fe\u8868\u6216\u7ea2\u9ed1\u6811\u4e2d\u6267\u884c\u67e5\u627e\u64cd\u4f5c\uff0c\u4ecd\u7136\u6709\u9000\u5316\u81f3 \\(O(n)\\) \u65f6\u95f4\u7684\u98ce\u9669\u3002</p> <p>\u591a\u6b21\u54c8\u5e0c\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\u5417\uff1f\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u8fd8\u80fd\u518d\u6b21\u4f7f\u7528\u5417\uff1f</p> <p>\u591a\u6b21\u54c8\u5e0c\u662f\u5f00\u653e\u5bfb\u5740\u7684\u4e00\u79cd\uff0c\u5f00\u653e\u5bfb\u5740\u6cd5\u90fd\u6709\u4e0d\u80fd\u76f4\u63a5\u5220\u9664\u5143\u7d20\u7684\u7f3a\u9677\uff0c\u9700\u8981\u901a\u8fc7\u6807\u8bb0\u5220\u9664\u3002\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u7a7a\u95f4\u53ef\u4ee5\u518d\u6b21\u4f7f\u7528\u3002\u5f53\u5c06\u65b0\u5143\u7d20\u63d2\u5165\u54c8\u5e0c\u8868\uff0c\u5e76\u4e14\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u6807\u8bb0\u4e3a\u5df2\u5220\u9664\u7684\u4f4d\u7f6e\u65f6\uff0c\u8be5\u4f4d\u7f6e\u53ef\u4ee5\u88ab\u65b0\u5143\u7d20\u4f7f\u7528\u3002\u8fd9\u6837\u505a\u65e2\u80fd\u4fdd\u6301\u54c8\u5e0c\u8868\u7684\u63a2\u6d4b\u5e8f\u5217\u4e0d\u53d8\uff0c\u53c8\u80fd\u4fdd\u8bc1\u54c8\u5e0c\u8868\u7684\u7a7a\u95f4\u4f7f\u7528\u7387\u3002</p> <p>\u4e3a\u4ec0\u4e48\u5728\u7ebf\u6027\u63a2\u6d4b\u4e2d\uff0c\u67e5\u627e\u5143\u7d20\u7684\u65f6\u5019\u4f1a\u51fa\u73b0\u54c8\u5e0c\u51b2\u7a81\u5462\uff1f</p> <p>\u67e5\u627e\u7684\u65f6\u5019\u901a\u8fc7\u54c8\u5e0c\u51fd\u6570\u627e\u5230\u5bf9\u5e94\u7684\u6876\u548c\u952e\u503c\u5bf9\uff0c\u53d1\u73b0 <code>key</code> \u4e0d\u5339\u914d\uff0c\u8fd9\u5c31\u4ee3\u8868\u6709\u54c8\u5e0c\u51b2\u7a81\u3002\u56e0\u6b64\uff0c\u7ebf\u6027\u63a2\u6d4b\u6cd5\u4f1a\u6839\u636e\u9884\u5148\u8bbe\u5b9a\u7684\u6b65\u957f\u4f9d\u6b21\u5411\u4e0b\u67e5\u627e\uff0c\u76f4\u81f3\u627e\u5230\u6b63\u786e\u7684\u952e\u503c\u5bf9\u6216\u65e0\u6cd5\u627e\u5230\u8df3\u51fa\u4e3a\u6b62\u3002</p> <p>\u4e3a\u4ec0\u4e48\u54c8\u5e0c\u8868\u6269\u5bb9\u80fd\u591f\u7f13\u89e3\u54c8\u5e0c\u51b2\u7a81\uff1f</p> <p>\u54c8\u5e0c\u51fd\u6570\u7684\u6700\u540e\u4e00\u6b65\u5f80\u5f80\u662f\u5bf9\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d6\u4f59\uff0c\u8ba9\u8f93\u51fa\u503c\u843d\u5728\u6570\u7ec4\u7d22\u5f15\u8303\u56f4\u5185\uff1b\u5728\u6269\u5bb9\u540e\uff0c\u6570\u7ec4\u957f\u5ea6 \\(n\\) \u53d1\u751f\u53d8\u5316\uff0c\u800c <code>key</code> \u5bf9\u5e94\u7684\u7d22\u5f15\u4e5f\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002\u539f\u5148\u843d\u5728\u540c\u4e00\u4e2a\u6876\u7684\u591a\u4e2a <code>key</code> \uff0c\u5728\u6269\u5bb9\u540e\u53ef\u80fd\u4f1a\u88ab\u5206\u914d\u5230\u591a\u4e2a\u6876\u4e2d\uff0c\u4ece\u800c\u5b9e\u73b0\u54c8\u5e0c\u51b2\u7a81\u7684\u7f13\u89e3\u3002</p>"},{"location":"chapter_heap/","title":"\u7b2c 8 \u7ae0 \u00a0 \u5806","text":"<p>Abstract</p> <p>\u5806\u5c31\u50cf\u662f\u5c71\u5cb3\u5cf0\u5ce6\uff0c\u5c42\u53e0\u8d77\u4f0f\u3001\u5f62\u6001\u5404\u5f02\u3002</p> <p>\u5ea7\u5ea7\u5c71\u5cf0\u9ad8\u4f4e\u9519\u843d\uff0c\u800c\u6700\u9ad8\u7684\u5c71\u5cf0\u603b\u662f\u6700\u5148\u6620\u5165\u773c\u5e18\u3002</p>"},{"location":"chapter_heap/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>8.1 \u00a0 \u5806</li> <li>8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c</li> <li>8.3 \u00a0 Top-K \u95ee\u9898</li> <li>8.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_heap/build_heap/","title":"8.2 \u00a0 \u5efa\u5806\u64cd\u4f5c","text":"<p>\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u5e0c\u671b\u4f7f\u7528\u4e00\u4e2a\u5217\u8868\u7684\u6240\u6709\u5143\u7d20\u6765\u6784\u5efa\u4e00\u4e2a\u5806\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u88ab\u79f0\u4e3a\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u3002</p>"},{"location":"chapter_heap/build_heap/#821","title":"8.2.1 \u00a0 \u501f\u52a9\u5165\u5806\u64cd\u4f5c\u5b9e\u73b0","text":"<p>\u6211\u4eec\u9996\u5148\u521b\u5efa\u4e00\u4e2a\u7a7a\u5806\uff0c\u7136\u540e\u904d\u5386\u5217\u8868\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u201c\u5165\u5806\u64cd\u4f5c\u201d\uff0c\u5373\u5148\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u5806\u7684\u5c3e\u90e8\uff0c\u518d\u5bf9\u8be5\u5143\u7d20\u6267\u884c\u201c\u4ece\u5e95\u81f3\u9876\u201d\u5806\u5316\u3002</p> <p>\u6bcf\u5f53\u4e00\u4e2a\u5143\u7d20\u5165\u5806\uff0c\u5806\u7684\u957f\u5ea6\u5c31\u52a0\u4e00\u3002\u7531\u4e8e\u8282\u70b9\u662f\u4ece\u9876\u5230\u5e95\u4f9d\u6b21\u88ab\u6dfb\u52a0\u8fdb\u4e8c\u53c9\u6811\u7684\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0a\u800c\u4e0b\u201d\u6784\u5efa\u7684\u3002</p> <p>\u8bbe\u5143\u7d20\u6570\u91cf\u4e3a \\(n\\) \uff0c\u6bcf\u4e2a\u5143\u7d20\u7684\u5165\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(\\log{n})\\) \u65f6\u95f4\uff0c\u56e0\u6b64\u8be5\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p>"},{"location":"chapter_heap/build_heap/#822","title":"8.2.2 \u00a0 \u901a\u8fc7\u904d\u5386\u5806\u5316\u5b9e\u73b0","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u4e00\u79cd\u66f4\u4e3a\u9ad8\u6548\u7684\u5efa\u5806\u65b9\u6cd5\uff0c\u5171\u5206\u4e3a\u4e24\u6b65\u3002</p> <ol> <li>\u5c06\u5217\u8868\u6240\u6709\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u5230\u5806\u4e2d\uff0c\u6b64\u65f6\u5806\u7684\u6027\u8d28\u5c1a\u672a\u5f97\u5230\u6ee1\u8db3\u3002</li> <li>\u5012\u5e8f\u904d\u5386\u5806\uff08\u5373\u5c42\u5e8f\u904d\u5386\u7684\u5012\u5e8f\uff09\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e2a\u975e\u53f6\u8282\u70b9\u6267\u884c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u3002</li> </ol> <p>\u6bcf\u5f53\u5806\u5316\u4e00\u4e2a\u8282\u70b9\u540e\uff0c\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\u5c31\u5f62\u6210\u4e00\u4e2a\u5408\u6cd5\u7684\u5b50\u5806\u3002\u800c\u7531\u4e8e\u662f\u5012\u5e8f\u904d\u5386\uff0c\u56e0\u6b64\u5806\u662f\u201c\u81ea\u4e0b\u800c\u4e0a\u201d\u6784\u5efa\u7684\u3002</p> <p>\u4e4b\u6240\u4ee5\u9009\u62e9\u5012\u5e8f\u904d\u5386\uff0c\u662f\u56e0\u4e3a\u8fd9\u6837\u80fd\u591f\u4fdd\u8bc1\u5f53\u524d\u8282\u70b9\u4e4b\u4e0b\u7684\u5b50\u6811\u5df2\u7ecf\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u8fd9\u6837\u5806\u5316\u5f53\u524d\u8282\u70b9\u624d\u662f\u6709\u6548\u7684\u3002</p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u53f6\u8282\u70b9\u6ca1\u6709\u5b50\u8282\u70b9\uff0c\u5929\u7136\u5c31\u662f\u5408\u6cd5\u7684\u5b50\u5806\uff0c\u56e0\u6b64\u65e0\u987b\u5806\u5316\u3002\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\uff0c\u6700\u540e\u4e00\u4e2a\u975e\u53f6\u8282\u70b9\u662f\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u7684\u7236\u8282\u70b9\uff0c\u6211\u4eec\u4ece\u5b83\u5f00\u59cb\u5012\u5e8f\u904d\u5386\u5e76\u6267\u884c\u5806\u5316\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def __init__(self, nums: list[int]):\n    \"\"\"\u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\"\"\"\n    # \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    self.max_heap = nums\n    # \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(self.parent(self.size() - 1), -1, -1):\n        self.sift_down(i)\n</code></pre> my_heap.cpp<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(vector&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums;\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;Integer&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new ArrayList&lt;&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = parent(size() - 1); i &gt;= 0; i--) {\n        siftDown(i);\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(IEnumerable&lt;int&gt; nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = new List&lt;int&gt;(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var size = Parent(this.Size() - 1);\n    for (int i = size; i &gt;= 0; i--) {\n        SiftDown(i);\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nfunc newMaxHeap(nums []any) *maxHeap {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    h := &amp;maxHeap{data: nums}\n    for i := h.parent(len(h.data) - 1); i &gt;= 0; i-- {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        h.siftDown(i)\n    }\n    return h\n}\n</code></pre> my_heap.swift<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\ninit(nums: [Int]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    maxHeap = nums\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in stride(from: parent(i: size() - 1), through: 0, by: -1) {\n        siftDown(i: i)\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.#maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.#parent(this.size() - 1); i &gt;= 0; i--) {\n        this.#siftDown(i);\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u5efa\u7acb\u7a7a\u5806\u6216\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nconstructor(nums?: number[]) {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    this.maxHeap = nums === undefined ? [] : [...nums];\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = this.parent(this.size() - 1); i &gt;= 0; i--) {\n        this.siftDown(i);\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nMaxHeap(List&lt;int&gt; nums) {\n  // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n  _maxHeap = nums;\n  // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = _parent(size() - 1); i &gt;= 0; i--) {\n    siftDown(i);\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806 */\nfn new(nums: Vec&lt;i32&gt;) -&gt; Self {\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    let mut heap = MaxHeap { max_heap: nums };\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=Self::parent(heap.size() - 1)).rev() {\n        heap.sift_down(i);\n    }\n    heap\n}\n</code></pre> my_heap.c<pre><code>/* \u6784\u9020\u51fd\u6570\uff0c\u6839\u636e\u5207\u7247\u5efa\u5806 */\nMaxHeap *newMaxHeap(int nums[], int size) {\n    // \u6240\u6709\u5143\u7d20\u5165\u5806\n    MaxHeap *maxHeap = (MaxHeap *)malloc(sizeof(MaxHeap));\n    maxHeap-&gt;size = size;\n    memcpy(maxHeap-&gt;data, nums, size * sizeof(int));\n    for (int i = parent(maxHeap, size - 1); i &gt;= 0; i--) {\n        // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n        siftDown(maxHeap, i);\n    }\n    return maxHeap;\n}\n</code></pre> my_heap.zig<pre><code>// \u6784\u9020\u65b9\u6cd5\uff0c\u6839\u636e\u8f93\u5165\u5217\u8868\u5efa\u5806\nfn init(self: *Self, allocator: std.mem.Allocator, nums: []const T) !void {\n    if (self.max_heap != null) return;\n    self.max_heap = std.ArrayList(T).init(allocator);\n    // \u5c06\u5217\u8868\u5143\u7d20\u539f\u5c01\u4e0d\u52a8\u6dfb\u52a0\u8fdb\u5806\n    try self.max_heap.?.appendSlice(nums);\n    // \u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    var i: usize = parent(self.size() - 1) + 1;\n    while (i &gt; 0) : (i -= 1) {\n        try self.siftDown(i - 1);\n    }\n}\n</code></pre>"},{"location":"chapter_heap/build_heap/#823","title":"8.2.3 \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<p>\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u63a8\u7b97\u7b2c\u4e8c\u79cd\u5efa\u5806\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> <ul> <li>\u5047\u8bbe\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \uff0c\u5219\u53f6\u8282\u70b9\u6570\u91cf\u4e3a \\((n + 1) / 2\\) \uff0c\u5176\u4e2d \\(/\\) \u4e3a\u5411\u4e0b\u6574\u9664\u3002\u56e0\u6b64\u9700\u8981\u5806\u5316\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\((n - 1) / 2\\) \u3002</li> <li>\u5728\u4ece\u9876\u81f3\u5e95\u5806\u5316\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u6700\u591a\u5806\u5316\u5230\u53f6\u8282\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u4e3a\u4e8c\u53c9\u6811\u9ad8\u5ea6 \\(\\log n\\) \u3002</li> </ul> <p>\u5c06\u4e0a\u8ff0\u4e24\u8005\u76f8\u4e58\uff0c\u53ef\u5f97\u5230\u5efa\u5806\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u4f46\u8fd9\u4e2a\u4f30\u7b97\u7ed3\u679c\u5e76\u4e0d\u51c6\u786e\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u6709\u8003\u8651\u5230\u4e8c\u53c9\u6811\u5e95\u5c42\u8282\u70b9\u6570\u91cf\u8fdc\u591a\u4e8e\u9876\u5c42\u8282\u70b9\u7684\u6027\u8d28\u3002</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u6765\u8fdb\u884c\u66f4\u4e3a\u51c6\u786e\u7684\u8ba1\u7b97\u3002\u4e3a\u4e86\u964d\u4f4e\u8ba1\u7b97\u96be\u5ea6\uff0c\u5047\u8bbe\u7ed9\u5b9a\u4e00\u4e2a\u8282\u70b9\u6570\u91cf\u4e3a \\(n\\) \u3001\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff0c\u8be5\u5047\u8bbe\u4e0d\u4f1a\u5f71\u54cd\u8ba1\u7b97\u7ed3\u679c\u7684\u6b63\u786e\u6027\u3002</p> <p></p> <p> \u56fe 8-5 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u5404\u5c42\u8282\u70b9\u6570\u91cf </p> <p>\u5982\u56fe 8-5 \u6240\u793a\uff0c\u8282\u70b9\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u7b49\u4e8e\u8be5\u8282\u70b9\u5230\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u800c\u8be5\u8ddd\u79bb\u6b63\u662f\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5bf9\u5404\u5c42\u7684\u201c\u8282\u70b9\u6570\u91cf \\(\\times\\) \u8282\u70b9\u9ad8\u5ea6\u201d\u6c42\u548c\uff0c\u5f97\u5230\u6240\u6709\u8282\u70b9\u7684\u5806\u5316\u8fed\u4ee3\u6b21\u6570\u7684\u603b\u548c\u3002</p> \\[ T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{(h-1)}\\times1 \\] <p>\u5316\u7b80\u4e0a\u5f0f\u9700\u8981\u501f\u52a9\u4e2d\u5b66\u7684\u6570\u5217\u77e5\u8bc6\uff0c\u5148\u5bf9 \\(T(h)\\) \u4e58\u4ee5 \\(2\\) \uff0c\u5f97\u5230\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2^0h + 2^1(h-1) + 2^2(h-2) + \\dots + 2^{h-1}\\times1 \\newline 2 T(h) &amp; = 2^1h + 2^2(h-1) + 2^3(h-2) + \\dots + 2^{h}\\times1 \\newline \\end{aligned} \\] <p>\u4f7f\u7528\u9519\u4f4d\u76f8\u51cf\u6cd5\uff0c\u7528\u4e0b\u5f0f \\(2 T(h)\\) \u51cf\u53bb\u4e0a\u5f0f \\(T(h)\\) \uff0c\u53ef\u5f97\uff1a</p> \\[ 2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \\dots + 2^{h-1} + 2^h \\] <p>\u89c2\u5bdf\u4e0a\u5f0f\uff0c\u53d1\u73b0 \\(T(h)\\) \u662f\u4e00\u4e2a\u7b49\u6bd4\u6570\u5217\uff0c\u53ef\u76f4\u63a5\u4f7f\u7528\u6c42\u548c\u516c\u5f0f\uff0c\u5f97\u5230\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\uff1a</p> \\[ \\begin{aligned} T(h) &amp; = 2 \\frac{1 - 2^h}{1 - 2} - h \\newline &amp; = 2^{h+1} - h - 2 \\newline &amp; = O(2^h) \\end{aligned} \\] <p>\u8fdb\u4e00\u6b65\u5730\uff0c\u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u6570\u91cf\u4e3a \\(n = 2^{h+1} - 1\\) \uff0c\u6613\u5f97\u590d\u6742\u5ea6\u4e3a \\(O(2^h) = O(n)\\) \u3002\u4ee5\u4e0a\u63a8\u7b97\u8868\u660e\uff0c\u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</p>"},{"location":"chapter_heap/heap/","title":"8.1 \u00a0 \u5806","text":"<p>\u300c\u5806 heap\u300d\u662f\u4e00\u79cd\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u4e3b\u8981\u53ef\u5206\u4e3a\u4e24\u79cd\u7c7b\u578b\uff0c\u5982\u56fe 8-1 \u6240\u793a\u3002</p> <ul> <li>\u300c\u5927\u9876\u5806 max heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\geq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> <li>\u300c\u5c0f\u9876\u5806 min heap\u300d\uff1a\u4efb\u610f\u8282\u70b9\u7684\u503c \\(\\leq\\) \u5176\u5b50\u8282\u70b9\u7684\u503c\u3002</li> </ul> <p></p> <p> \u56fe 8-1 \u00a0 \u5c0f\u9876\u5806\u4e0e\u5927\u9876\u5806 </p> <p>\u5806\u4f5c\u4e3a\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u6700\u5e95\u5c42\u8282\u70b9\u9760\u5de6\u586b\u5145\uff0c\u5176\u4ed6\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u3002</li> <li>\u6211\u4eec\u5c06\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u9876\u201d\uff0c\u5c06\u5e95\u5c42\u6700\u9760\u53f3\u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5806\u5e95\u201d\u3002</li> <li>\u5bf9\u4e8e\u5927\u9876\u5806\uff08\u5c0f\u9876\u5806\uff09\uff0c\u5806\u9876\u5143\u7d20\uff08\u6839\u8282\u70b9\uff09\u7684\u503c\u5206\u522b\u662f\u6700\u5927\uff08\u6700\u5c0f\uff09\u7684\u3002</li> </ul>"},{"location":"chapter_heap/heap/#811","title":"8.1.1 \u00a0 \u5806\u5e38\u7528\u64cd\u4f5c","text":"<p>\u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u662f\u300c\u4f18\u5148\u961f\u5217 priority queue\u300d\uff0c\u8fd9\u662f\u4e00\u79cd\u62bd\u8c61\u6570\u636e\u7ed3\u6784\uff0c\u5b9a\u4e49\u4e3a\u5177\u6709\u4f18\u5148\u7ea7\u6392\u5e8f\u7684\u961f\u5217\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5806\u901a\u5e38\u7528\u4e8e\u5b9e\u73b0\u4f18\u5148\u961f\u5217\uff0c\u5927\u9876\u5806\u76f8\u5f53\u4e8e\u5143\u7d20\u6309\u4ece\u5927\u5230\u5c0f\u7684\u987a\u5e8f\u51fa\u961f\u7684\u4f18\u5148\u961f\u5217\u3002\u4ece\u4f7f\u7528\u89d2\u5ea6\u6765\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u201c\u4f18\u5148\u961f\u5217\u201d\u548c\u201c\u5806\u201d\u770b\u4f5c\u7b49\u4ef7\u7684\u6570\u636e\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u672c\u4e66\u5bf9\u4e24\u8005\u4e0d\u505a\u7279\u522b\u533a\u5206\uff0c\u7edf\u4e00\u79f0\u4f5c\u201c\u5806\u201d\u3002</p> <p>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u89c1\u8868 8-1 \uff0c\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 8-1 \u00a0 \u5806\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u5806 \\(O(\\log n)\\) pop() \u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) peek() \u8bbf\u95ee\u5806\u9876\u5143\u7d20\uff08\u5927 / \u5c0f\u9876\u5806\u5206\u522b\u4e3a\u6700\u5927 / \u5c0f\u503c\uff09 \\(O(1)\\) size() \u83b7\u53d6\u5806\u7684\u5143\u7d20\u6570\u91cf \\(O(1)\\) isEmpty() \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a \\(O(1)\\) <p>\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u63d0\u4f9b\u7684\u5806\u7c7b\uff08\u6216\u4f18\u5148\u961f\u5217\u7c7b\uff09\u3002</p> <p>\u7c7b\u4f3c\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u201c\u4ece\u5c0f\u5230\u5927\u6392\u5217\u201d\u548c\u201c\u4ece\u5927\u5230\u5c0f\u6392\u5217\u201d\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e2a <code>flag</code> \u6216\u4fee\u6539 <code>Comparator</code> \u5b9e\u73b0\u201c\u5c0f\u9876\u5806\u201d\u4e0e\u201c\u5927\u9876\u5806\u201d\u4e4b\u95f4\u7684\u8f6c\u6362\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig heap.py<pre><code># \u521d\u59cb\u5316\u5c0f\u9876\u5806\nmin_heap, flag = [], 1\n# \u521d\u59cb\u5316\u5927\u9876\u5806\nmax_heap, flag = [], -1\n\n# Python \u7684 heapq \u6a21\u5757\u9ed8\u8ba4\u5b9e\u73b0\u5c0f\u9876\u5806\n# \u8003\u8651\u5c06\u201c\u5143\u7d20\u53d6\u8d1f\u201d\u540e\u518d\u5165\u5806\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5c06\u5927\u5c0f\u5173\u7cfb\u98a0\u5012\uff0c\u4ece\u800c\u5b9e\u73b0\u5927\u9876\u5806\n# \u5728\u672c\u793a\u4f8b\u4e2d\uff0cflag = 1 \u65f6\u5bf9\u5e94\u5c0f\u9876\u5806\uff0cflag = -1 \u65f6\u5bf9\u5e94\u5927\u9876\u5806\n\n# \u5143\u7d20\u5165\u5806\nheapq.heappush(max_heap, flag * 1)\nheapq.heappush(max_heap, flag * 3)\nheapq.heappush(max_heap, flag * 2)\nheapq.heappush(max_heap, flag * 5)\nheapq.heappush(max_heap, flag * 4)\n\n# \u83b7\u53d6\u5806\u9876\u5143\u7d20\npeek: int = flag * max_heap[0] # 5\n\n# \u5806\u9876\u5143\u7d20\u51fa\u5806\n# \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nval = flag * heapq.heappop(max_heap) # 5\nval = flag * heapq.heappop(max_heap) # 4\nval = flag * heapq.heappop(max_heap) # 3\nval = flag * heapq.heappop(max_heap) # 2\nval = flag * heapq.heappop(max_heap) # 1\n\n# \u83b7\u53d6\u5806\u5927\u5c0f\nsize: int = len(max_heap)\n\n# \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = not max_heap\n\n# \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806\nmin_heap: list[int] = [1, 3, 2, 5, 4]\nheapq.heapify(min_heap)\n</code></pre> heap.cpp<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap;\n// \u521d\u59cb\u5316\u5927\u9876\u5806\npriority_queue&lt;int, vector&lt;int&gt;, less&lt;int&gt;&gt; maxHeap;\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.push(1);\nmaxHeap.push(3);\nmaxHeap.push(2);\nmaxHeap.push(5);\nmaxHeap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.top(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nmaxHeap.pop(); // 5\nmaxHeap.pop(); // 4\nmaxHeap.pop(); // 3\nmaxHeap.pop(); // 2\nmaxHeap.pop(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nvector&lt;int&gt; input{1, 3, 2, 5, 4};\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; minHeap(input.begin(), input.end());\n</code></pre> heap.java<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nQueue&lt;Integer&gt; minHeap = new PriorityQueue&lt;&gt;();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nQueue&lt;Integer&gt; maxHeap = new PriorityQueue&lt;&gt;((a, b) -&gt; b - a);\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.offer(1);\nmaxHeap.offer(3);\nmaxHeap.offer(2);\nmaxHeap.offer(5);\nmaxHeap.offer(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.peek(); // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.poll(); // 5\npeek = maxHeap.poll(); // 4\npeek = maxHeap.poll(); // 3\npeek = maxHeap.poll(); // 2\npeek = maxHeap.poll(); // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.size();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = maxHeap.isEmpty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;&gt;(Arrays.asList(1, 3, 2, 5, 4));\n</code></pre> heap.cs<pre><code>/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nPriorityQueue&lt;int, int&gt; minHeap = new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\uff08\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f\u4fee\u6539 Comparator \u5373\u53ef\uff09\nPriorityQueue&lt;int, int&gt; maxHeap = new(Comparer&lt;int&gt;.Create((x, y) =&gt; y - x));\n\n/* \u5143\u7d20\u5165\u5806 */\nmaxHeap.Enqueue(1, 1);\nmaxHeap.Enqueue(3, 3);\nmaxHeap.Enqueue(2, 2);\nmaxHeap.Enqueue(5, 5);\nmaxHeap.Enqueue(4, 4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nint peek = maxHeap.Peek();//5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\npeek = maxHeap.Dequeue();  // 5\npeek = maxHeap.Dequeue();  // 4\npeek = maxHeap.Dequeue();  // 3\npeek = maxHeap.Dequeue();  // 2\npeek = maxHeap.Dequeue();  // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nint size = maxHeap.Count;\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = maxHeap.Count == 0;\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nminHeap = new PriorityQueue&lt;int, int&gt;([(1, 1), (3, 3), (2, 2), (5, 5), (4, 4)]);\n</code></pre> heap.go<pre><code>// Go \u8bed\u8a00\u4e2d\u53ef\u4ee5\u901a\u8fc7\u5b9e\u73b0 heap.Interface \u6765\u6784\u5efa\u6574\u6570\u5927\u9876\u5806\n// \u5b9e\u73b0 heap.Interface \u9700\u8981\u540c\u65f6\u5b9e\u73b0 sort.Interface\ntype intHeap []any\n\n// Push heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u63a8\u5165\u5143\u7d20\u5230\u5806\nfunc (h *intHeap) Push(x any) {\n    // Push \u548c Pop \u4f7f\u7528 pointer receiver \u4f5c\u4e3a\u53c2\u6570\n    // \u56e0\u4e3a\u5b83\u4eec\u4e0d\u4ec5\u4f1a\u5bf9\u5207\u7247\u7684\u5185\u5bb9\u8fdb\u884c\u8c03\u6574\uff0c\u8fd8\u4f1a\u4fee\u6539\u5207\u7247\u7684\u957f\u5ea6\u3002\n    *h = append(*h, x.(int))\n}\n\n// Pop heap.Interface \u7684\u65b9\u6cd5\uff0c\u5b9e\u73b0\u5f39\u51fa\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Pop() any {\n    // \u5f85\u51fa\u5806\u5143\u7d20\u5b58\u653e\u5728\u6700\u540e\n    last := (*h)[len(*h)-1]\n    *h = (*h)[:len(*h)-1]\n    return last\n}\n\n// Len sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Len() int {\n    return len(*h)\n}\n\n// Less sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Less(i, j int) bool {\n    // \u5982\u679c\u5b9e\u73b0\u5c0f\u9876\u5806\uff0c\u5219\u9700\u8981\u8c03\u6574\u4e3a\u5c0f\u4e8e\u53f7\n    return (*h)[i].(int) &gt; (*h)[j].(int)\n}\n\n// Swap sort.Interface \u7684\u65b9\u6cd5\nfunc (h *intHeap) Swap(i, j int) {\n    (*h)[i], (*h)[j] = (*h)[j], (*h)[i]\n}\n\n// Top \u83b7\u53d6\u5806\u9876\u5143\u7d20\nfunc (h *intHeap) Top() any {\n    return (*h)[0]\n}\n\n/* Driver Code */\nfunc TestHeap(t *testing.T) {\n    /* \u521d\u59cb\u5316\u5806 */\n    // \u521d\u59cb\u5316\u5927\u9876\u5806\n    maxHeap := &amp;intHeap{}\n    heap.Init(maxHeap)\n    /* \u5143\u7d20\u5165\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u6dfb\u52a0\u5143\u7d20\n    heap.Push(maxHeap, 1)\n    heap.Push(maxHeap, 3)\n    heap.Push(maxHeap, 2)\n    heap.Push(maxHeap, 4)\n    heap.Push(maxHeap, 5)\n\n    /* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\n    top := maxHeap.Top()\n    fmt.Printf(\"\u5806\u9876\u5143\u7d20\u4e3a %d\\n\", top)\n\n    /* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n    // \u8c03\u7528 heap.Interface \u7684\u65b9\u6cd5\uff0c\u6765\u79fb\u9664\u5143\u7d20\n    heap.Pop(maxHeap) // 5\n    heap.Pop(maxHeap) // 4\n    heap.Pop(maxHeap) // 3\n    heap.Pop(maxHeap) // 2\n    heap.Pop(maxHeap) // 1\n\n    /* \u83b7\u53d6\u5806\u5927\u5c0f */\n    size := len(*maxHeap)\n    fmt.Printf(\"\u5806\u5143\u7d20\u6570\u91cf\u4e3a %d\\n\", size)\n\n    /* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\n    isEmpty := len(*maxHeap) == 0\n    fmt.Printf(\"\u5806\u662f\u5426\u4e3a\u7a7a %t\\n\", isEmpty)\n}\n</code></pre> heap.swift<pre><code>// Swift \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.js<pre><code>// JavaScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.ts<pre><code>// TypeScript \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.dart<pre><code>// Dart \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.rs<pre><code>use std::collections::BinaryHeap;\nuse std::cmp::Reverse;\n\n/* \u521d\u59cb\u5316\u5806 */\n// \u521d\u59cb\u5316\u5c0f\u9876\u5806\nlet mut min_heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n// \u521d\u59cb\u5316\u5927\u9876\u5806\nlet mut max_heap = BinaryHeap::new();\n\n/* \u5143\u7d20\u5165\u5806 */\nmax_heap.push(1);\nmax_heap.push(3);\nmax_heap.push(2);\nmax_heap.push(5);\nmax_heap.push(4);\n\n/* \u83b7\u53d6\u5806\u9876\u5143\u7d20 */\nlet peek = max_heap.peek().unwrap();  // 5\n\n/* \u5806\u9876\u5143\u7d20\u51fa\u5806 */\n// \u51fa\u5806\u5143\u7d20\u4f1a\u5f62\u6210\u4e00\u4e2a\u4ece\u5927\u5230\u5c0f\u7684\u5e8f\u5217\nlet peek = max_heap.pop().unwrap();   // 5\nlet peek = max_heap.pop().unwrap();   // 4\nlet peek = max_heap.pop().unwrap();   // 3\nlet peek = max_heap.pop().unwrap();   // 2\nlet peek = max_heap.pop().unwrap();   // 1\n\n/* \u83b7\u53d6\u5806\u5927\u5c0f */\nlet size = max_heap.len();\n\n/* \u5224\u65ad\u5806\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = max_heap.is_empty();\n\n/* \u8f93\u5165\u5217\u8868\u5e76\u5efa\u5806 */\nlet min_heap = BinaryHeap::from(vec![Reverse(1), Reverse(3), Reverse(2), Reverse(5), Reverse(4)]);\n</code></pre> heap.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e Heap \u7c7b\n</code></pre> heap.zig<pre><code>\n</code></pre>"},{"location":"chapter_heap/heap/#812","title":"8.1.2 \u00a0 \u5806\u7684\u5b9e\u73b0","text":"<p>\u4e0b\u6587\u5b9e\u73b0\u7684\u662f\u5927\u9876\u5806\u3002\u82e5\u8981\u5c06\u5176\u8f6c\u6362\u4e3a\u5c0f\u9876\u5806\uff0c\u53ea\u9700\u5c06\u6240\u6709\u5927\u5c0f\u903b\u8f91\u5224\u65ad\u53d6\u9006\uff08\u4f8b\u5982\uff0c\u5c06 \\(\\geq\\) \u66ff\u6362\u4e3a \\(\\leq\\) \uff09\u3002\u611f\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p>"},{"location":"chapter_heap/heap/#1","title":"1. \u00a0 \u5806\u7684\u5b58\u50a8\u4e0e\u8868\u793a","text":"<p>\u201c\u4e8c\u53c9\u6811\u201d\u7ae0\u8282\u8bb2\u8fc7\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u7531\u4e8e\u5806\u6b63\u662f\u4e00\u79cd\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u56e0\u6b64\u6211\u4eec\u5c06\u91c7\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</p> <p>\u5f53\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u4e8c\u53c9\u6811\u65f6\uff0c\u5143\u7d20\u4ee3\u8868\u8282\u70b9\u503c\uff0c\u7d22\u5f15\u4ee3\u8868\u8282\u70b9\u5728\u4e8c\u53c9\u6811\u4e2d\u7684\u4f4d\u7f6e\u3002\u8282\u70b9\u6307\u9488\u901a\u8fc7\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u6765\u5b9e\u73b0\u3002</p> <p>\u5982\u56fe 8-2 \u6240\u793a\uff0c\u7ed9\u5b9a\u7d22\u5f15 \\(i\\) \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \uff0c\u7236\u8282\u70b9\u7d22\u5f15\u4e3a \\((i - 1) / 2\\)\uff08\u5411\u4e0b\u6574\u9664\uff09\u3002\u5f53\u7d22\u5f15\u8d8a\u754c\u65f6\uff0c\u8868\u793a\u7a7a\u8282\u70b9\u6216\u8282\u70b9\u4e0d\u5b58\u5728\u3002</p> <p></p> <p> \u56fe 8-2 \u00a0 \u5806\u7684\u8868\u793a\u4e0e\u5b58\u50a8 </p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u7d22\u5f15\u6620\u5c04\u516c\u5f0f\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def left(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\n    return 2 * i + 1\n\ndef right(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\"\"\"\n    return 2 * i + 2\n\ndef parent(self, i: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\"\"\"\n    return (i - 1) // 2  # \u5411\u4e0b\u6574\u9664\n</code></pre> my_heap.cpp<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.java<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.cs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Left(int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint Right(int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint Parent(int i) {\n    return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.go<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc (h *maxHeap) parent(i int) int {\n    // \u5411\u4e0b\u6574\u9664\n    return (i - 1) / 2\n}\n</code></pre> my_heap.swift<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc left(i: Int) -&gt; Int {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfunc right(i: Int) -&gt; Int {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfunc parent(i: Int) -&gt; Int {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.js<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\n#left(i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\n#right(i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\n#parent(i) {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.ts<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nleft(i: number): number {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nright(i: number): number {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nparent(i: number): number {\n    return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.dart<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _left(int i) {\n  return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint _right(int i) {\n  return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint _parent(int i) {\n  return (i - 1) ~/ 2; // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.rs<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn left(i: usize) -&gt; usize {\n    2 * i + 1\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nfn right(i: usize) -&gt; usize {\n    2 * i + 2\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nfn parent(i: usize) -&gt; usize {\n    (i - 1) / 2 // \u5411\u4e0b\u6574\u9664\n}\n</code></pre> my_heap.c<pre><code>/* \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15 */\nint left(MaxHeap *maxHeap, int i) {\n    return 2 * i + 1;\n}\n\n/* \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15 */\nint right(MaxHeap *maxHeap, int i) {\n    return 2 * i + 2;\n}\n\n/* \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15 */\nint parent(MaxHeap *maxHeap, int i) {\n    return (i - 1) / 2;\n}\n</code></pre> my_heap.zig<pre><code>// \u83b7\u53d6\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\nfn left(i: usize) usize {\n    return 2 * i + 1;\n}\n\n// \u83b7\u53d6\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\nfn right(i: usize) usize {\n    return 2 * i + 2;\n}\n\n// \u83b7\u53d6\u7236\u8282\u70b9\u7d22\u5f15\nfn parent(i: usize) usize {\n    // return (i - 1) / 2; // \u5411\u4e0b\u6574\u9664\n    return @divFloor(i - 1, 2);\n}\n</code></pre>"},{"location":"chapter_heap/heap/#2","title":"2. \u00a0 \u8bbf\u95ee\u5806\u9876\u5143\u7d20","text":"<p>\u5806\u9876\u5143\u7d20\u5373\u4e3a\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u4e5f\u5c31\u662f\u5217\u8868\u7684\u9996\u4e2a\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def peek(self) -&gt; int:\n    \"\"\"\u8bbf\u95ee\u5806\u9876\u5143\u7d20\"\"\"\n    return self.max_heap[0]\n</code></pre> my_heap.cpp<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.java<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n    return maxHeap.get(0);\n}\n</code></pre> my_heap.cs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint Peek() {\n    return maxHeap[0];\n}\n</code></pre> my_heap.go<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc (h *maxHeap) peek() any {\n    return h.data[0]\n}\n</code></pre> my_heap.swift<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunc peek() -&gt; Int {\n    maxHeap[0]\n}\n</code></pre> my_heap.js<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek() {\n    return this.#maxHeap[0];\n}\n</code></pre> my_heap.ts<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\npeek(): number {\n    return this.maxHeap[0];\n}\n</code></pre> my_heap.dart<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek() {\n  return _maxHeap[0];\n}\n</code></pre> my_heap.rs<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfn peek(&amp;self) -&gt; Option&lt;i32&gt; {\n    self.max_heap.first().copied()\n}\n</code></pre> my_heap.c<pre><code>/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peek(MaxHeap *maxHeap) {\n    return maxHeap-&gt;data[0];\n}\n</code></pre> my_heap.zig<pre><code>// \u8bbf\u95ee\u5806\u9876\u5143\u7d20\nfn peek(self: *Self) T {\n    return self.max_heap.?.items[0];\n}  \n</code></pre>"},{"location":"chapter_heap/heap/#3","title":"3. \u00a0 \u5143\u7d20\u5165\u5806","text":"<p>\u7ed9\u5b9a\u5143\u7d20 <code>val</code> \uff0c\u6211\u4eec\u9996\u5148\u5c06\u5176\u6dfb\u52a0\u5230\u5806\u5e95\u3002\u6dfb\u52a0\u4e4b\u540e\uff0c\u7531\u4e8e val \u53ef\u80fd\u5927\u4e8e\u5806\u4e2d\u5176\u4ed6\u5143\u7d20\uff0c\u5806\u7684\u6210\u7acb\u6761\u4ef6\u53ef\u80fd\u5df2\u88ab\u7834\u574f\uff0c\u56e0\u6b64\u9700\u8981\u4fee\u590d\u4ece\u63d2\u5165\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u88ab\u79f0\u4e3a\u300c\u5806\u5316 heapify\u300d\u3002</p> <p>\u8003\u8651\u4ece\u5165\u5806\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u6267\u884c\u5806\u5316\u3002\u5982\u56fe 8-3 \u6240\u793a\uff0c\u6211\u4eec\u6bd4\u8f83\u63d2\u5165\u8282\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u7684\u503c\uff0c\u5982\u679c\u63d2\u5165\u8282\u70b9\u66f4\u5927\uff0c\u5219\u5c06\u5b83\u4eec\u4ea4\u6362\u3002\u7136\u540e\u7ee7\u7eed\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u4ece\u5e95\u81f3\u9876\u4fee\u590d\u5806\u4e2d\u7684\u5404\u4e2a\u8282\u70b9\uff0c\u76f4\u81f3\u8d8a\u8fc7\u6839\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-3 \u00a0 \u5143\u7d20\u5165\u5806\u6b65\u9aa4 </p> <p>\u8bbe\u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \uff0c\u5219\u6811\u7684\u9ad8\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u7531\u6b64\u53ef\u77e5\uff0c\u5806\u5316\u64cd\u4f5c\u7684\u5faa\u73af\u8f6e\u6570\u6700\u591a\u4e3a \\(O(\\log n)\\) \uff0c\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def push(self, val: int):\n    \"\"\"\u5143\u7d20\u5165\u5806\"\"\"\n    # \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.append(val)\n    # \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1)\n\ndef sift_up(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\"\"\"\n    while True:\n        # \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p = self.parent(i)\n        # \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 or self.max_heap[i] &lt;= self.max_heap[p]:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p)\n        # \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.push_back(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap[i], maxHeap[p]);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap.get(i) &lt;= maxHeap.get(p))\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid Push(int val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.Add(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    SiftUp(Size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid SiftUp(int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = Parent(i);\n        // \u82e5\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap[i] &lt;= maxHeap[p])\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc (h *maxHeap) push(val any) {\n    // \u6dfb\u52a0\u8282\u70b9\n    h.data = append(h.data, val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    h.siftUp(len(h.data) - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc (h *maxHeap) siftUp(i int) {\n    for true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        p := h.parent(i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || h.data[i].(int) &lt;= h.data[p].(int) {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunc push(val: Int) {\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap.append(val)\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(i: size() - 1)\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfunc siftUp(i: Int) {\n    var i = i\n    while true {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = parent(i: i)\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if p &lt; 0 || maxHeap[i] &lt;= maxHeap[p] {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: p)\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val) {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.#maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.#siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\n#siftUp(i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.#parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.#maxHeap[i] &lt;= this.#maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\npush(val: number): void {\n    // \u6dfb\u52a0\u8282\u70b9\n    this.maxHeap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    this.siftUp(this.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nsiftUp(i: number): void {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        const p = this.parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || this.maxHeap[i] &lt;= this.maxHeap[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(int val) {\n  // \u6dfb\u52a0\u8282\u70b9\n  _maxHeap.add(val);\n  // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n  siftUp(size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(int i) {\n  while (true) {\n    // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n    int p = _parent(i);\n    // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n    if (p &lt; 0 || _maxHeap[i] &lt;= _maxHeap[p]) {\n      break;\n    }\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, p);\n    // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n    i = p;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfn push(&amp;mut self, val: i32) {\n    // \u6dfb\u52a0\u8282\u70b9\n    self.max_heap.push(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    self.sift_up(self.size() - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nfn sift_up(&amp;mut self, mut i: usize) {\n    loop {\n        // \u8282\u70b9 i \u5df2\u7ecf\u662f\u5806\u9876\u8282\u70b9\u4e86\uff0c\u7ed3\u675f\u5806\u5316\n        if i == 0 {\n            break;\n        }\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        let p = Self::parent(i);\n        // \u5f53\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if self.max_heap[i] &lt;= self.max_heap[p] {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid push(MaxHeap *maxHeap, int val) {\n    // \u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u4e0d\u5e94\u8be5\u6dfb\u52a0\u8fd9\u4e48\u591a\u8282\u70b9\n    if (maxHeap-&gt;size == MAX_SIZE) {\n        printf(\"heap is full!\");\n        return;\n    }\n    // \u6dfb\u52a0\u8282\u70b9\n    maxHeap-&gt;data[maxHeap-&gt;size] = val;\n    maxHeap-&gt;size++;\n\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    siftUp(maxHeap, maxHeap-&gt;size - 1);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316 */\nvoid siftUp(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        int p = parent(maxHeap, i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 || maxHeap-&gt;data[i] &lt;= maxHeap-&gt;data[p]) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u5165\u5806\nfn push(self: *Self, val: T) !void {\n    // \u6dfb\u52a0\u8282\u70b9\n    try self.max_heap.?.append(val);\n    // \u4ece\u5e95\u81f3\u9876\u5806\u5316\n    try self.siftUp(self.size() - 1);\n}  \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u5e95\u81f3\u9876\u5806\u5316\nfn siftUp(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u83b7\u53d6\u8282\u70b9 i \u7684\u7236\u8282\u70b9\n        var p = parent(i);\n        // \u5f53\u201c\u8d8a\u8fc7\u6839\u8282\u70b9\u201d\u6216\u201c\u8282\u70b9\u65e0\u987b\u4fee\u590d\u201d\u65f6\uff0c\u7ed3\u675f\u5806\u5316\n        if (p &lt; 0 or self.max_heap.?.items[i] &lt;= self.max_heap.?.items[p]) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, p);\n        // \u5faa\u73af\u5411\u4e0a\u5806\u5316\n        i = p;\n    }\n}\n</code></pre>"},{"location":"chapter_heap/heap/#4","title":"4. \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806","text":"<p>\u5806\u9876\u5143\u7d20\u662f\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9\uff0c\u5373\u5217\u8868\u9996\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u76f4\u63a5\u4ece\u5217\u8868\u4e2d\u5220\u9664\u9996\u5143\u7d20\uff0c\u90a3\u4e48\u4e8c\u53c9\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u7d22\u5f15\u90fd\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u8fd9\u5c06\u4f7f\u5f97\u540e\u7eed\u4f7f\u7528\u5806\u5316\u8fdb\u884c\u4fee\u590d\u53d8\u5f97\u56f0\u96be\u3002\u4e3a\u4e86\u5c3d\u91cf\u51cf\u5c11\u5143\u7d20\u7d22\u5f15\u7684\u53d8\u52a8\uff0c\u6211\u4eec\u91c7\u7528\u4ee5\u4e0b\u64cd\u4f5c\u6b65\u9aa4\u3002</p> <ol> <li>\u4ea4\u6362\u5806\u9876\u5143\u7d20\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff09\u3002</li> <li>\u4ea4\u6362\u5b8c\u6210\u540e\uff0c\u5c06\u5806\u5e95\u4ece\u5217\u8868\u4e2d\u5220\u9664\uff08\u6ce8\u610f\uff0c\u7531\u4e8e\u5df2\u7ecf\u4ea4\u6362\uff0c\u56e0\u6b64\u5b9e\u9645\u4e0a\u5220\u9664\u7684\u662f\u539f\u6765\u7684\u5806\u9876\u5143\u7d20\uff09\u3002</li> <li>\u4ece\u6839\u8282\u70b9\u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u6267\u884c\u5806\u5316\u3002</li> </ol> <p>\u5982\u56fe 8-4 \u6240\u793a\uff0c\u201c\u4ece\u9876\u81f3\u5e95\u5806\u5316\u201d\u7684\u64cd\u4f5c\u65b9\u5411\u4e0e\u201c\u4ece\u5e95\u81f3\u9876\u5806\u5316\u201d\u76f8\u53cd\uff0c\u6211\u4eec\u5c06\u6839\u8282\u70b9\u7684\u503c\u4e0e\u5176\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u503c\u8fdb\u884c\u6bd4\u8f83\uff0c\u5c06\u6700\u5927\u7684\u5b50\u8282\u70b9\u4e0e\u6839\u8282\u70b9\u4ea4\u6362\u3002\u7136\u540e\u5faa\u73af\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\u6216\u9047\u5230\u65e0\u987b\u4ea4\u6362\u7684\u8282\u70b9\u65f6\u7ed3\u675f\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-4 \u00a0 \u5806\u9876\u5143\u7d20\u51fa\u5806\u6b65\u9aa4 </p> <p>\u4e0e\u5143\u7d20\u5165\u5806\u64cd\u4f5c\u76f8\u4f3c\uff0c\u5806\u9876\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4e3a \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig my_heap.py<pre><code>def pop(self) -&gt; int:\n    \"\"\"\u5143\u7d20\u51fa\u5806\"\"\"\n    # \u5224\u7a7a\u5904\u7406\n    if self.is_empty():\n        raise IndexError(\"\u5806\u4e3a\u7a7a\")\n    # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1)\n    # \u5220\u9664\u8282\u70b9\n    val = self.max_heap.pop()\n    # \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0)\n    # \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n\ndef sift_down(self, i: int):\n    \"\"\"\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l, r, ma = self.left(i), self.right(i), i\n        if l &lt; self.size() and self.max_heap[l] &gt; self.max_heap[ma]:\n            ma = l\n        if r &lt; self.size() and self.max_heap[r] &gt; self.max_heap[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma)\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n</code></pre> my_heap.cpp<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nvoid pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty()) {\n        throw out_of_range(\"\u5806\u4e3a\u7a7a\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap[0], maxHeap[size() - 1]);\n    // \u5220\u9664\u8282\u70b9\n    maxHeap.pop_back();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        swap(maxHeap[i], maxHeap[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.java<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty())\n        throw new IndexOutOfBoundsException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(0, size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.remove(size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = left(i), r = right(i), ma = i;\n        if (l &lt; size() &amp;&amp; maxHeap.get(l) &gt; maxHeap.get(ma))\n            ma = l;\n        if (r &lt; size() &amp;&amp; maxHeap.get(r) &gt; maxHeap.get(ma))\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.cs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint Pop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (IsEmpty())\n        throw new IndexOutOfRangeException();\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    Swap(0, Size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap.Last();\n    maxHeap.RemoveAt(Size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    SiftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = Left(i), r = Right(i), ma = i;\n        if (l &lt; Size() &amp;&amp; maxHeap[l] &gt; maxHeap[ma])\n            ma = l;\n        if (r &lt; Size() &amp;&amp; maxHeap[r] &gt; maxHeap[ma])\n            ma = r;\n        // \u82e5\u201c\u8282\u70b9 i \u6700\u5927\u201d\u6216\u201c\u8d8a\u8fc7\u53f6\u8282\u70b9\u201d\uff0c\u5219\u7ed3\u675f\u5806\u5316\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        Swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.go<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc (h *maxHeap) pop() any {\n    // \u5224\u7a7a\u5904\u7406\n    if h.isEmpty() {\n        fmt.Println(\"error\")\n        return nil\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    h.swap(0, h.size()-1)\n    // \u5220\u9664\u8282\u70b9\n    val := h.data[len(h.data)-1]\n    h.data = h.data[:len(h.data)-1]\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    h.siftDown(0)\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc (h *maxHeap) siftDown(i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        l, r, max := h.left(i), h.right(i), i\n        if l &lt; h.size() &amp;&amp; h.data[l].(int) &gt; h.data[max].(int) {\n            max = l\n        }\n        if r &lt; h.size() &amp;&amp; h.data[r].(int) &gt; h.data[max].(int) {\n            max = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if max == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        h.swap(i, max)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max\n    }\n}\n</code></pre> my_heap.swift<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfunc pop() -&gt; Int {\n    // \u5224\u7a7a\u5904\u7406\n    if isEmpty() {\n        fatalError(\"\u5806\u4e3a\u7a7a\")\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(i: 0, j: size() - 1)\n    // \u5220\u9664\u8282\u70b9\n    let val = maxHeap.remove(at: size() - 1)\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(i: 0)\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = left(i: i)\n        let r = right(i: i)\n        var ma = i\n        if l &lt; size(), maxHeap[l] &gt; maxHeap[ma] {\n            ma = l\n        }\n        if r &lt; size(), maxHeap[r] &gt; maxHeap[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(i: i, j: ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n</code></pre> my_heap.js<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop() {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new Error('\u5806\u4e3a\u7a7a');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.#swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.#maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.#siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\n#siftDown(i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.#left(i),\n            r = this.#right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.#maxHeap[l] &gt; this.#maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.#maxHeap[r] &gt; this.#maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.#swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.ts<pre><code>/* \u5143\u7d20\u51fa\u5806 */\npop(): number {\n    // \u5224\u7a7a\u5904\u7406\n    if (this.isEmpty()) throw new RangeError('Heap is empty.');\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    this.swap(0, this.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    const val = this.maxHeap.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    this.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nsiftDown(i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        const l = this.left(i),\n            r = this.right(i);\n        let ma = i;\n        if (l &lt; this.size() &amp;&amp; this.maxHeap[l] &gt; this.maxHeap[ma]) ma = l;\n        if (r &lt; this.size() &amp;&amp; this.maxHeap[r] &gt; this.maxHeap[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        this.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.dart<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop() {\n  // \u5224\u7a7a\u5904\u7406\n  if (isEmpty()) throw Exception('\u5806\u4e3a\u7a7a');\n  // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n  _swap(0, size() - 1);\n  // \u5220\u9664\u8282\u70b9\n  int val = _maxHeap.removeLast();\n  // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n  siftDown(0);\n  // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n  return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = _left(i);\n    int r = _right(i);\n    int ma = i;\n    if (l &lt; size() &amp;&amp; _maxHeap[l] &gt; _maxHeap[ma]) ma = l;\n    if (r &lt; size() &amp;&amp; _maxHeap[r] &gt; _maxHeap[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    _swap(i, ma);\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n</code></pre> my_heap.rs<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nfn pop(&amp;mut self) -&gt; i32 {\n    // \u5224\u7a7a\u5904\u7406\n    if self.is_empty() {\n        panic!(\"index out of bounds\");\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    let val = self.max_heap.remove(self.size() - 1);\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    self.sift_down(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    val\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(&amp;mut self, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let (l, r, mut ma) = (Self::left(i), Self::right(i), i);\n        if l &lt; self.size() &amp;&amp; self.max_heap[l] &gt; self.max_heap[ma] {\n            ma = l;\n        }\n        if r &lt; self.size() &amp;&amp; self.max_heap[r] &gt; self.max_heap[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre> my_heap.c<pre><code>/* \u5143\u7d20\u51fa\u5806 */\nint pop(MaxHeap *maxHeap) {\n    // \u5224\u7a7a\u5904\u7406\n    if (isEmpty(maxHeap)) {\n        printf(\"heap is empty!\");\n        return INT_MAX;\n    }\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    swap(maxHeap, 0, size(maxHeap) - 1);\n    // \u5220\u9664\u8282\u70b9\n    int val = maxHeap-&gt;data[maxHeap-&gt;size - 1];\n    maxHeap-&gt;size--;\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    siftDown(maxHeap, 0);\n\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n}\n\n/* \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(MaxHeap *maxHeap, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a max\n        int l = left(maxHeap, i);\n        int r = right(maxHeap, i);\n        int max = i;\n        if (l &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[l] &gt; maxHeap-&gt;data[max]) {\n            max = l;\n        }\n        if (r &lt; size(maxHeap) &amp;&amp; maxHeap-&gt;data[r] &gt; maxHeap-&gt;data[max]) {\n            max = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (max == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(maxHeap, i, max);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = max;\n    }\n}\n</code></pre> my_heap.zig<pre><code>// \u5143\u7d20\u51fa\u5806\nfn pop(self: *Self) !T {\n    // \u5224\u65ad\u5904\u7406\n    if (self.isEmpty()) unreachable;\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    try self.swap(0, self.size() - 1);\n    // \u5220\u9664\u8282\u70b9\n    var val = self.max_heap.?.pop();\n    // \u4ece\u9876\u81f3\u5e95\u5806\u5316\n    try self.siftDown(0);\n    // \u8fd4\u56de\u5806\u9876\u5143\u7d20\n    return val;\n} \n\n// \u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\nfn siftDown(self: *Self, i_: usize) !void {\n    var i = i_;\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        var l = left(i);\n        var r = right(i);\n        var ma = i;\n        if (l &lt; self.size() and self.max_heap.?.items[l] &gt; self.max_heap.?.items[ma]) ma = l;\n        if (r &lt; self.size() and self.max_heap.?.items[r] &gt; self.max_heap.?.items[ma]) ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        try self.swap(i, ma);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n</code></pre>"},{"location":"chapter_heap/heap/#813","title":"8.1.3 \u00a0 \u5806\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u4f18\u5148\u961f\u5217\uff1a\u5806\u901a\u5e38\u4f5c\u4e3a\u5b9e\u73b0\u4f18\u5148\u961f\u5217\u7684\u9996\u9009\u6570\u636e\u7ed3\u6784\uff0c\u5176\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \uff0c\u800c\u5efa\u961f\u64cd\u4f5c\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4e9b\u64cd\u4f5c\u90fd\u975e\u5e38\u9ad8\u6548\u3002</li> <li>\u5806\u6392\u5e8f\uff1a\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5b83\u4eec\u5efa\u7acb\u4e00\u4e2a\u5806\uff0c\u7136\u540e\u4e0d\u65ad\u5730\u6267\u884c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\uff0c\u4ece\u800c\u5f97\u5230\u6709\u5e8f\u6570\u636e\u3002\u7136\u800c\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u4f7f\u7528\u4e00\u79cd\u66f4\u4f18\u96c5\u7684\u65b9\u5f0f\u5b9e\u73b0\u5806\u6392\u5e8f\uff0c\u8be6\u89c1\u201c\u5806\u6392\u5e8f\u201d\u7ae0\u8282\u3002</li> <li>\u83b7\u53d6\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff1a\u8fd9\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7b97\u6cd5\u95ee\u9898\uff0c\u540c\u65f6\u4e5f\u662f\u4e00\u79cd\u5178\u578b\u5e94\u7528\uff0c\u4f8b\u5982\u9009\u62e9\u70ed\u5ea6\u524d 10 \u7684\u65b0\u95fb\u4f5c\u4e3a\u5fae\u535a\u70ed\u641c\uff0c\u9009\u53d6\u9500\u91cf\u524d 10 \u7684\u5546\u54c1\u7b49\u3002</li> </ul>"},{"location":"chapter_heap/summary/","title":"8.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_heap/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5806\u662f\u4e00\u68f5\u5b8c\u5168\u4e8c\u53c9\u6811\uff0c\u6839\u636e\u6210\u7acb\u6761\u4ef6\u53ef\u5206\u4e3a\u5927\u9876\u5806\u548c\u5c0f\u9876\u5806\u3002\u5927\uff08\u5c0f\uff09\u9876\u5806\u7684\u5806\u9876\u5143\u7d20\u662f\u6700\u5927\uff08\u5c0f\uff09\u7684\u3002</li> <li>\u4f18\u5148\u961f\u5217\u7684\u5b9a\u4e49\u662f\u5177\u6709\u51fa\u961f\u4f18\u5148\u7ea7\u7684\u961f\u5217\uff0c\u901a\u5e38\u4f7f\u7528\u5806\u6765\u5b9e\u73b0\u3002</li> <li>\u5806\u7684\u5e38\u7528\u64cd\u4f5c\u53ca\u5176\u5bf9\u5e94\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5305\u62ec\uff1a\u5143\u7d20\u5165\u5806 \\(O(\\log n)\\)\u3001\u5806\u9876\u5143\u7d20\u51fa\u5806 \\(O(\\log n)\\) \u548c\u8bbf\u95ee\u5806\u9876\u5143\u7d20 \\(O(1)\\) \u7b49\u3002</li> <li>\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u7528\u6570\u7ec4\u8868\u793a\uff0c\u56e0\u6b64\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u6570\u7ec4\u6765\u5b58\u50a8\u5806\u3002</li> <li>\u5806\u5316\u64cd\u4f5c\u7528\u4e8e\u7ef4\u62a4\u5806\u7684\u6027\u8d28\uff0c\u5728\u5165\u5806\u548c\u51fa\u5806\u64cd\u4f5c\u4e2d\u90fd\u4f1a\u7528\u5230\u3002</li> <li>\u8f93\u5165 \\(n\\) \u4e2a\u5143\u7d20\u5e76\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(n)\\) \uff0c\u975e\u5e38\u9ad8\u6548\u3002</li> <li>Top-K \u662f\u4e00\u4e2a\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528\u5806\u6570\u636e\u7ed3\u6784\u9ad8\u6548\u89e3\u51b3\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002</li> </ul>"},{"location":"chapter_heap/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6570\u636e\u7ed3\u6784\u7684\u201c\u5806\u201d\u4e0e\u5185\u5b58\u7ba1\u7406\u7684\u201c\u5806\u201d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\u5417\uff1f</p> <p>\u4e24\u8005\u4e0d\u662f\u540c\u4e00\u4e2a\u6982\u5ff5\uff0c\u53ea\u662f\u78b0\u5de7\u90fd\u53eb\u5806\u3002\u8ba1\u7b97\u673a\u7cfb\u7edf\u5185\u5b58\u4e2d\u7684\u5806\u662f\u52a8\u6001\u5185\u5b58\u5206\u914d\u7684\u4e00\u90e8\u5206\uff0c\u7a0b\u5e8f\u5728\u8fd0\u884c\u65f6\u53ef\u4ee5\u4f7f\u7528\u5b83\u6765\u5b58\u50a8\u6570\u636e\u3002\u7a0b\u5e8f\u53ef\u4ee5\u8bf7\u6c42\u4e00\u5b9a\u91cf\u7684\u5806\u5185\u5b58\uff0c\u7528\u4e8e\u5b58\u50a8\u5982\u5bf9\u8c61\u548c\u6570\u7ec4\u7b49\u590d\u6742\u7ed3\u6784\u3002\u5f53\u8fd9\u4e9b\u6570\u636e\u4e0d\u518d\u9700\u8981\u65f6\uff0c\u7a0b\u5e8f\u9700\u8981\u91ca\u653e\u8fd9\u4e9b\u5185\u5b58\uff0c\u4ee5\u9632\u6b62\u5185\u5b58\u6cc4\u6f0f\u3002\u76f8\u8f83\u4e8e\u6808\u5185\u5b58\uff0c\u5806\u5185\u5b58\u7684\u7ba1\u7406\u548c\u4f7f\u7528\u9700\u8981\u66f4\u8c28\u614e\uff0c\u4f7f\u7528\u4e0d\u5f53\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5185\u5b58\u6cc4\u6f0f\u548c\u91ce\u6307\u9488\u7b49\u95ee\u9898\u3002</p>"},{"location":"chapter_heap/top_k/","title":"8.3 \u00a0 Top-K \u95ee\u9898","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u65e0\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u524d \\(k\\) \u5927\u7684\u5143\u7d20\u3002</p> <p>\u5bf9\u4e8e\u8be5\u95ee\u9898\uff0c\u6211\u4eec\u5148\u4ecb\u7ecd\u4e24\u79cd\u601d\u8def\u6bd4\u8f83\u76f4\u63a5\u7684\u89e3\u6cd5\uff0c\u518d\u4ecb\u7ecd\u6548\u7387\u66f4\u9ad8\u7684\u5806\u89e3\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#831","title":"8.3.1 \u00a0 \u65b9\u6cd5\u4e00\uff1a\u904d\u5386\u9009\u62e9","text":"<p>\u6211\u4eec\u53ef\u4ee5\u8fdb\u884c\u56fe 8-6 \u6240\u793a\u7684 \\(k\\) \u8f6e\u904d\u5386\uff0c\u5206\u522b\u5728\u6bcf\u8f6e\u4e2d\u63d0\u53d6\u7b2c \\(1\\)\u3001\\(2\\)\u3001\\(\\dots\\)\u3001\\(k\\) \u5927\u7684\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(nk)\\) \u3002</p> <p>\u6b64\u65b9\u6cd5\u53ea\u9002\u7528\u4e8e \\(k \\ll n\\) \u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u5f53 \\(k\\) \u4e0e \\(n\\) \u6bd4\u8f83\u63a5\u8fd1\u65f6\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411\u4e8e \\(O(n^2)\\) \uff0c\u975e\u5e38\u8017\u65f6\u3002</p> <p></p> <p> \u56fe 8-6 \u00a0 \u904d\u5386\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>Tip</p> <p>\u5f53 \\(k = n\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5b8c\u6574\u7684\u6709\u5e8f\u5e8f\u5217\uff0c\u6b64\u65f6\u7b49\u4ef7\u4e8e\u201c\u9009\u62e9\u6392\u5e8f\u201d\u7b97\u6cd5\u3002</p>"},{"location":"chapter_heap/top_k/#832","title":"8.3.2 \u00a0 \u65b9\u6cd5\u4e8c\uff1a\u6392\u5e8f","text":"<p>\u5982\u56fe 8-7 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u5bf9\u6570\u7ec4 <code>nums</code> \u8fdb\u884c\u6392\u5e8f\uff0c\u518d\u8fd4\u56de\u6700\u53f3\u8fb9\u7684 \\(k\\) \u4e2a\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</p> <p>\u663e\u7136\uff0c\u8be5\u65b9\u6cd5\u201c\u8d85\u989d\u201d\u5b8c\u6210\u4efb\u52a1\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u53ea\u9700\u627e\u51fa\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u5373\u53ef\uff0c\u800c\u4e0d\u9700\u8981\u6392\u5e8f\u5176\u4ed6\u5143\u7d20\u3002</p> <p></p> <p> \u56fe 8-7 \u00a0 \u6392\u5e8f\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p>"},{"location":"chapter_heap/top_k/#833","title":"8.3.3 \u00a0 \u65b9\u6cd5\u4e09\uff1a\u5806","text":"<p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u5806\u66f4\u52a0\u9ad8\u6548\u5730\u89e3\u51b3 Top-K \u95ee\u9898\uff0c\u6d41\u7a0b\u5982\u56fe 8-8 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5176\u5806\u9876\u5143\u7d20\u6700\u5c0f\u3002</li> <li>\u5148\u5c06\u6570\u7ec4\u7684\u524d \\(k\\) \u4e2a\u5143\u7d20\u4f9d\u6b21\u5165\u5806\u3002</li> <li>\u4ece\u7b2c \\(k + 1\\) \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\uff0c\u5e76\u5c06\u5f53\u524d\u5143\u7d20\u5165\u5806\u3002</li> <li>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u5806\u4e2d\u4fdd\u5b58\u7684\u5c31\u662f\u6700\u5927\u7684 \\(k\\) \u4e2a\u5143\u7d20\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 8-8 \u00a0 \u57fa\u4e8e\u5806\u5bfb\u627e\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig top_k.py<pre><code>def top_k_heap(nums: list[int], k: int) -&gt; list[int]:\n    \"\"\"\u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\"\"\"\n    # \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    heap = []\n    # \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i in range(k):\n        heapq.heappush(heap, nums[i])\n    # \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in range(k, len(nums)):\n        # \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap[0]:\n            heapq.heappop(heap)\n            heapq.heappush(heap, nums[i])\n    return heap\n</code></pre> top_k.cpp<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\npriority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; topKHeap(vector&lt;int&gt; &amp;nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    priority_queue&lt;int, vector&lt;int&gt;, greater&lt;int&gt;&gt; heap;\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.push(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.size(); i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.top()) {\n            heap.pop();\n            heap.push(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.java<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nQueue&lt;Integer&gt; topKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    Queue&lt;Integer&gt; heap = new PriorityQueue&lt;Integer&gt;();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.offer(nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.peek()) {\n            heap.poll();\n            heap.offer(nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.cs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nPriorityQueue&lt;int, int&gt; TopKHeap(int[] nums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    PriorityQueue&lt;int, int&gt; heap = new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        heap.Enqueue(nums[i], nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; nums.Length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; heap.Peek()) {\n            heap.Dequeue();\n            heap.Enqueue(nums[i], nums[i]);\n        }\n    }\n    return heap;\n}\n</code></pre> top_k.go<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums []int, k int) *minHeap {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    h := &amp;minHeap{}\n    heap.Init(h)\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for i := 0; i &lt; k; i++ {\n        heap.Push(h, nums[i])\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i := k; i &lt; len(nums); i++ {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; h.Top().(int) {\n            heap.Pop(h)\n            heap.Push(h, nums[i])\n        }\n    }\n    return h\n}\n</code></pre> top_k.swift<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunc topKHeap(nums: [Int], k: Int) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5c0f\u9876\u5806\uff0c\u5e76\u5c06\u524d k \u4e2a\u5143\u7d20\u5efa\u5806\n    var heap = Heap(nums.prefix(k))\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for i in stride(from: k, to: nums.count, by: 1) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if nums[i] &gt; heap.min()! {\n            _ = heap.removeMin()\n            heap.insert(nums[i])\n        }\n    }\n    return heap.unordered\n}\n</code></pre> top_k.js<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap, val) {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums, k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.ts<pre><code>/* \u5143\u7d20\u5165\u5806 */\nfunction pushMinHeap(maxHeap: MaxHeap, val: number): void {\n    // \u5143\u7d20\u53d6\u53cd\n    maxHeap.push(-val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nfunction popMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.pop();\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nfunction peekMinHeap(maxHeap: MaxHeap): number {\n    // \u5143\u7d20\u53d6\u53cd\n    return -maxHeap.peek();\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nfunction getMinHeap(maxHeap: MaxHeap): number[] {\n    // \u5143\u7d20\u53d6\u53cd\n    return maxHeap.getMaxHeap().map((num: number) =&gt; -num);\n}\n\n/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfunction topKHeap(nums: number[], k: number): number[] {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    const maxHeap = new MaxHeap([]);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (let i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (let i = k; i &lt; nums.length; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    // \u8fd4\u56de\u5806\u4e2d\u5143\u7d20\n    return getMinHeap(maxHeap);\n}\n</code></pre> top_k.dart<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nMinHeap topKHeap(List&lt;int&gt; nums, int k) {\n  // \u521d\u59cb\u5316\u5c0f\u9876\u5806\uff0c\u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n  MinHeap heap = MinHeap(nums.sublist(0, k));\n  // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n  for (int i = k; i &lt; nums.length; i++) {\n    // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n    if (nums[i] &gt; heap.peek()) {\n      heap.pop();\n      heap.push(nums[i]);\n    }\n  }\n  return heap;\n}\n</code></pre> top_k.rs<pre><code>/* \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20 */\nfn top_k_heap(nums: Vec&lt;i32&gt;, k: usize) -&gt; BinaryHeap&lt;Reverse&lt;i32&gt;&gt; {\n    // BinaryHeap \u662f\u5927\u9876\u5806\uff0c\u4f7f\u7528 Reverse \u5c06\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u5b9e\u73b0\u5c0f\u9876\u5806\n    let mut heap = BinaryHeap::&lt;Reverse&lt;i32&gt;&gt;::new();\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for &amp;num in nums.iter().take(k) {\n        heap.push(Reverse(num));\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for &amp;num in nums.iter().skip(k) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if num &gt; heap.peek().unwrap().0 {\n            heap.pop();\n            heap.push(Reverse(num));\n        }\n    }\n    heap\n}\n</code></pre> top_k.c<pre><code>/* \u5143\u7d20\u5165\u5806 */\nvoid pushMinHeap(MaxHeap *maxHeap, int val) {\n    // \u5143\u7d20\u53d6\u53cd\n    push(maxHeap, -val);\n}\n\n/* \u5143\u7d20\u51fa\u5806 */\nint popMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -pop(maxHeap);\n}\n\n/* \u8bbf\u95ee\u5806\u9876\u5143\u7d20 */\nint peekMinHeap(MaxHeap *maxHeap) {\n    // \u5143\u7d20\u53d6\u53cd\n    return -peek(maxHeap);\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n/* \u53d6\u51fa\u5806\u4e2d\u5143\u7d20 */\nint *getMinHeap(MaxHeap *maxHeap) {\n    // \u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\u5e76\u5b58\u5165 res \u6570\u7ec4\n    int *res = (int *)malloc(maxHeap-&gt;size * sizeof(int));\n    for (int i = 0; i &lt; maxHeap-&gt;size; i++) {\n        res[i] = -maxHeap-&gt;data[i];\n    }\n    return res;\n}\n\n// \u57fa\u4e8e\u5806\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5927\u7684 k \u4e2a\u5143\u7d20\u7684\u51fd\u6570\nint *topKHeap(int *nums, int sizeNums, int k) {\n    // \u521d\u59cb\u5316\u5c0f\u9876\u5806\n    // \u8bf7\u6ce8\u610f\uff1a\u6211\u4eec\u5c06\u5806\u4e2d\u6240\u6709\u5143\u7d20\u53d6\u53cd\uff0c\u4ece\u800c\u7528\u5927\u9876\u5806\u6765\u6a21\u62df\u5c0f\u9876\u5806\n    int *empty = (int *)malloc(0);\n    MaxHeap *maxHeap = newMaxHeap(empty, 0);\n    // \u5c06\u6570\u7ec4\u7684\u524d k \u4e2a\u5143\u7d20\u5165\u5806\n    for (int i = 0; i &lt; k; i++) {\n        pushMinHeap(maxHeap, nums[i]);\n    }\n    // \u4ece\u7b2c k+1 \u4e2a\u5143\u7d20\u5f00\u59cb\uff0c\u4fdd\u6301\u5806\u7684\u957f\u5ea6\u4e3a k\n    for (int i = k; i &lt; sizeNums; i++) {\n        // \u82e5\u5f53\u524d\u5143\u7d20\u5927\u4e8e\u5806\u9876\u5143\u7d20\uff0c\u5219\u5c06\u5806\u9876\u5143\u7d20\u51fa\u5806\u3001\u5f53\u524d\u5143\u7d20\u5165\u5806\n        if (nums[i] &gt; peekMinHeap(maxHeap)) {\n            popMinHeap(maxHeap);\n            pushMinHeap(maxHeap, nums[i]);\n        }\n    }\n    int *res = getMinHeap(maxHeap);\n    // \u91ca\u653e\u5185\u5b58\n    delMaxHeap(maxHeap);\n    return res;\n}\n</code></pre> top_k.zig<pre><code>[class]{}-[func]{topKHeap}\n</code></pre> <p>\u603b\u5171\u6267\u884c\u4e86 \\(n\\) \u8f6e\u5165\u5806\u548c\u51fa\u5806\uff0c\u5806\u7684\u6700\u5927\u957f\u5ea6\u4e3a \\(k\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log k)\\) \u3002\u8be5\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u9ad8\uff0c\u5f53 \\(k\\) \u8f83\u5c0f\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \uff1b\u5f53 \\(k\\) \u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(O(n \\log n)\\) \u3002</p> <p>\u53e6\u5916\uff0c\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u52a8\u6001\u6570\u636e\u6d41\u7684\u4f7f\u7528\u573a\u666f\u3002\u5728\u4e0d\u65ad\u52a0\u5165\u6570\u636e\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6301\u7eed\u7ef4\u62a4\u5806\u5185\u7684\u5143\u7d20\uff0c\u4ece\u800c\u5b9e\u73b0\u6700\u5927 \\(k\\) \u4e2a\u5143\u7d20\u7684\u52a8\u6001\u66f4\u65b0\u3002</p>"},{"location":"chapter_introduction/","title":"\u7b2c 1 \u7ae0 \u00a0 \u521d\u8bc6\u7b97\u6cd5","text":"<p>Abstract</p> <p>\u4e00\u4f4d\u5c11\u5973\u7fe9\u7fe9\u8d77\u821e\uff0c\u4e0e\u6570\u636e\u4ea4\u7ec7\u5728\u4e00\u8d77\uff0c\u88d9\u6446\u4e0a\u98d8\u626c\u7740\u7b97\u6cd5\u7684\u65cb\u5f8b\u3002</p> <p>\u5979\u9080\u8bf7\u4f60\u5171\u821e\uff0c\u8bf7\u7d27\u8ddf\u5979\u7684\u6b65\u4f10\uff0c\u8e0f\u5165\u5145\u6ee1\u903b\u8f91\u4e0e\u7f8e\u611f\u7684\u7b97\u6cd5\u4e16\u754c\u3002</p>"},{"location":"chapter_introduction/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728</li> <li>1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48</li> <li>1.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_introduction/algorithms_are_everywhere/","title":"1.1 \u00a0 \u7b97\u6cd5\u65e0\u5904\u4e0d\u5728","text":"<p>\u5f53\u6211\u4eec\u542c\u5230\u201c\u7b97\u6cd5\u201d\u8fd9\u4e2a\u8bcd\u65f6\uff0c\u5f88\u81ea\u7136\u5730\u4f1a\u60f3\u5230\u6570\u5b66\u3002\u7136\u800c\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7b97\u6cd5\u5e76\u4e0d\u6d89\u53ca\u590d\u6742\u6570\u5b66\uff0c\u800c\u662f\u66f4\u591a\u5730\u4f9d\u8d56\u57fa\u672c\u903b\u8f91\uff0c\u8fd9\u4e9b\u903b\u8f91\u5728\u6211\u4eec\u7684\u65e5\u5e38\u751f\u6d3b\u4e2d\u5904\u5904\u53ef\u89c1\u3002</p> <p>\u5728\u6b63\u5f0f\u63a2\u8ba8\u7b97\u6cd5\u4e4b\u524d\uff0c\u6709\u4e00\u4e2a\u6709\u8da3\u7684\u4e8b\u5b9e\u503c\u5f97\u5206\u4eab\uff1a\u4f60\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u5e76\u4e60\u60ef\u5c06\u5b83\u4eec\u5e94\u7528\u5230\u65e5\u5e38\u751f\u6d3b\u4e2d\u4e86\u3002\u4e0b\u9762\u6211\u5c06\u4e3e\u51e0\u4e2a\u5177\u4f53\u7684\u4f8b\u5b50\u6765\u8bc1\u5b9e\u8fd9\u4e00\u70b9\u3002</p> <p>\u4f8b\u4e00\uff1a\u67e5\u9605\u5b57\u5178\u3002\u5728\u5b57\u5178\u91cc\uff0c\u6bcf\u4e2a\u6c49\u5b57\u90fd\u5bf9\u5e94\u4e00\u4e2a\u62fc\u97f3\uff0c\u800c\u5b57\u5178\u662f\u6309\u7167\u62fc\u97f3\u5b57\u6bcd\u987a\u5e8f\u6392\u5217\u7684\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u67e5\u627e\u4e00\u4e2a\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u5b57\uff0c\u901a\u5e38\u4f1a\u6309\u7167\u56fe 1-1 \u6240\u793a\u7684\u65b9\u5f0f\u5b9e\u73b0\u3002</p> <ol> <li>\u7ffb\u5f00\u5b57\u5178\u7ea6\u4e00\u534a\u7684\u9875\u6570\uff0c\u67e5\u770b\u8be5\u9875\u7684\u9996\u5b57\u6bcd\u662f\u4ec0\u4e48\uff0c\u5047\u8bbe\u9996\u5b57\u6bcd\u4e3a \\(m\\) \u3002</li> <li>\u7531\u4e8e\u5728\u62fc\u97f3\u5b57\u6bcd\u8868\u4e2d \\(r\\) \u4f4d\u4e8e \\(m\\) \u4e4b\u540e\uff0c\u6240\u4ee5\u6392\u9664\u5b57\u5178\u524d\u534a\u90e8\u5206\uff0c\u67e5\u627e\u8303\u56f4\u7f29\u5c0f\u5230\u540e\u534a\u90e8\u5206\u3002</li> <li>\u4e0d\u65ad\u91cd\u590d\u6b65\u9aa4 <code>1.</code> \u548c \u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u81f3\u627e\u5230\u62fc\u97f3\u9996\u5b57\u6bcd\u4e3a \\(r\\) \u7684\u9875\u7801\u4e3a\u6b62\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 1-1 \u00a0 \u67e5\u5b57\u5178\u6b65\u9aa4 </p> <p>\u67e5\u5b57\u5178\u8fd9\u4e2a\u5c0f\u5b66\u751f\u5fc5\u5907\u6280\u80fd\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u8457\u540d\u7684\u201c\u4e8c\u5206\u67e5\u627e\u201d\u7b97\u6cd5\u3002\u4ece\u6570\u636e\u7ed3\u6784\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b57\u5178\u89c6\u4e3a\u4e00\u4e2a\u5df2\u6392\u5e8f\u7684\u201c\u6570\u7ec4\u201d\uff1b\u4ece\u7b97\u6cd5\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e0a\u8ff0\u67e5\u5b57\u5178\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u770b\u4f5c\u201c\u4e8c\u5206\u67e5\u627e\u201d\u3002</p> <p>\u4f8b\u4e8c\uff1a\u6574\u7406\u6251\u514b\u3002\u6211\u4eec\u5728\u6253\u724c\u65f6\uff0c\u6bcf\u5c40\u90fd\u9700\u8981\u6574\u7406\u624b\u4e2d\u7684\u6251\u514b\u724c\uff0c\u4f7f\u5176\u4ece\u5c0f\u5230\u5927\u6392\u5217\uff0c\u5b9e\u73b0\u6d41\u7a0b\u5982\u56fe 1-2 \u6240\u793a\u3002</p> <ol> <li>\u5c06\u6251\u514b\u724c\u5212\u5206\u4e3a\u201c\u6709\u5e8f\u201d\u548c\u201c\u65e0\u5e8f\u201d\u4e24\u90e8\u5206\uff0c\u5e76\u5047\u8bbe\u521d\u59cb\u72b6\u6001\u4e0b\u6700\u5de6 1 \u5f20\u6251\u514b\u724c\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u5728\u65e0\u5e8f\u90e8\u5206\u62bd\u51fa\u4e00\u5f20\u6251\u514b\u724c\uff0c\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\uff1b\u5b8c\u6210\u540e\u6700\u5de6 2 \u5f20\u6251\u514b\u5df2\u7ecf\u6709\u5e8f\u3002</li> <li>\u4e0d\u65ad\u5faa\u73af\u6b65\u9aa4 <code>2.</code> \uff0c\u6bcf\u4e00\u8f6e\u5c06\u4e00\u5f20\u6251\u514b\u724c\u4ece\u65e0\u5e8f\u90e8\u5206\u63d2\u5165\u81f3\u6709\u5e8f\u90e8\u5206\uff0c\u76f4\u81f3\u6240\u6709\u6251\u514b\u724c\u90fd\u6709\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 1-2 \u00a0 \u6251\u514b\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u4e0a\u8ff0\u6574\u7406\u6251\u514b\u724c\u7684\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u63d2\u5165\u6392\u5e8f\u201d\u7b97\u6cd5\uff0c\u5b83\u5728\u5904\u7406\u5c0f\u578b\u6570\u636e\u96c6\u65f6\u975e\u5e38\u9ad8\u6548\u3002\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\u7684\u6392\u5e8f\u5e93\u51fd\u6570\u4e2d\u90fd\u6709\u63d2\u5165\u6392\u5e8f\u7684\u8eab\u5f71\u3002</p> <p>\u4f8b\u4e09\uff1a\u8d27\u5e01\u627e\u96f6\u3002\u5047\u8bbe\u6211\u4eec\u5728\u8d85\u5e02\u8d2d\u4e70\u4e86 \\(69\\) \u5143\u7684\u5546\u54c1\uff0c\u7ed9\u4e86\u6536\u94f6\u5458 \\(100\\) \u5143\uff0c\u5219\u6536\u94f6\u5458\u9700\u8981\u627e\u6211\u4eec \\(31\\) \u5143\u3002\u4ed6\u4f1a\u5f88\u81ea\u7136\u5730\u5b8c\u6210\u5982\u56fe 1-3 \u6240\u793a\u7684\u601d\u8003\u3002</p> <ol> <li>\u53ef\u9009\u9879\u662f\u6bd4 \\(31\\) \u5143\u9762\u503c\u66f4\u5c0f\u7684\u8d27\u5e01\uff0c\u5305\u62ec \\(1\\) \u5143\u3001\\(5\\) \u5143\u3001\\(10\\) \u5143\u3001\\(20\\) \u5143\u3002</li> <li>\u4ece\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(20\\) \u5143\uff0c\u5269\u4f59 \\(31 - 20 = 11\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(10\\) \u5143\uff0c\u5269\u4f59 \\(11 - 10 = 1\\) \u5143\u3002</li> <li>\u4ece\u5269\u4f59\u53ef\u9009\u9879\u4e2d\u62ff\u51fa\u6700\u5927\u7684 \\(1\\) \u5143\uff0c\u5269\u4f59 \\(1 - 1 = 0\\) \u5143\u3002</li> <li>\u5b8c\u6210\u627e\u96f6\uff0c\u65b9\u6848\u4e3a \\(20 + 10 + 1 = 31\\) \u5143\u3002</li> </ol> <p></p> <p> \u56fe 1-3 \u00a0 \u8d27\u5e01\u627e\u96f6\u8fc7\u7a0b </p> <p>\u5728\u4ee5\u4e0a\u6b65\u9aa4\u4e2d\uff0c\u6211\u4eec\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\uff08\u5c3d\u53ef\u80fd\u7528\u5927\u9762\u989d\u7684\u8d27\u5e01\uff09\uff0c\u6700\u7ec8\u5f97\u5230\u4e86\u53ef\u884c\u7684\u627e\u96f6\u65b9\u6848\u3002\u4ece\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u89d2\u5ea6\u770b\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u201c\u8d2a\u5fc3\u201d\u7b97\u6cd5\u3002</p> <p>\u5c0f\u5230\u70f9\u996a\u4e00\u9053\u83dc\uff0c\u5927\u5230\u661f\u9645\u822a\u884c\uff0c\u51e0\u4e4e\u6240\u6709\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u79bb\u4e0d\u5f00\u7b97\u6cd5\u3002\u8ba1\u7b97\u673a\u7684\u51fa\u73b0\u4f7f\u5f97\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7f16\u7a0b\u5c06\u6570\u636e\u7ed3\u6784\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u540c\u65f6\u7f16\u5199\u4ee3\u7801\u8c03\u7528 CPU \u548c GPU \u6267\u884c\u7b97\u6cd5\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u628a\u751f\u6d3b\u4e2d\u7684\u95ee\u9898\u8f6c\u79fb\u5230\u8ba1\u7b97\u673a\u4e0a\uff0c\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u89e3\u51b3\u5404\u79cd\u590d\u6742\u95ee\u9898\u3002</p> <p>Tip</p> <p>\u5982\u679c\u4f60\u5bf9\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u3001\u6570\u7ec4\u548c\u4e8c\u5206\u67e5\u627e\u7b49\u6982\u5ff5\u4ecd\u611f\u5230\u4e00\u77e5\u534a\u89e3\uff0c\u8bf7\u7ee7\u7eed\u5f80\u4e0b\u9605\u8bfb\uff0c\u672c\u4e66\u5c06\u5f15\u5bfc\u4f60\u8fc8\u5165\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u77e5\u8bc6\u6bbf\u5802\u3002</p>"},{"location":"chapter_introduction/summary/","title":"1.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u7b97\u6cd5\u5728\u65e5\u5e38\u751f\u6d3b\u4e2d\u65e0\u5904\u4e0d\u5728\uff0c\u5e76\u4e0d\u662f\u9065\u4e0d\u53ef\u53ca\u7684\u9ad8\u6df1\u77e5\u8bc6\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u5df2\u7ecf\u5728\u4e0d\u77e5\u4e0d\u89c9\u4e2d\u5b66\u4f1a\u4e86\u8bb8\u591a\u7b97\u6cd5\uff0c\u7528\u4ee5\u89e3\u51b3\u751f\u6d3b\u4e2d\u7684\u5927\u5c0f\u95ee\u9898\u3002</li> <li>\u67e5\u9605\u5b57\u5178\u7684\u539f\u7406\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u76f8\u4e00\u81f4\u3002\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u4f53\u73b0\u4e86\u5206\u800c\u6cbb\u4e4b\u7684\u91cd\u8981\u7b97\u6cd5\u601d\u60f3\u3002</li> <li>\u6574\u7406\u6251\u514b\u7684\u8fc7\u7a0b\u4e0e\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u975e\u5e38\u7c7b\u4f3c\u3002\u63d2\u5165\u6392\u5e8f\u7b97\u6cd5\u9002\u5408\u6392\u5e8f\u5c0f\u578b\u6570\u636e\u96c6\u3002</li> <li>\u8d27\u5e01\u627e\u96f6\u7684\u6b65\u9aa4\u672c\u8d28\u4e0a\u662f\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u6bcf\u4e00\u6b65\u90fd\u91c7\u53d6\u5f53\u524d\u770b\u6765\u6700\u597d\u7684\u9009\u62e9\u3002</li> <li>\u7b97\u6cd5\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u800c\u6570\u636e\u7ed3\u6784\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\u3002</li> <li>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7d27\u5bc6\u76f8\u8fde\u3002\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\uff0c\u800c\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002</li> <li>\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728\uff0c\u79ef\u6728\u4ee3\u8868\u6570\u636e\uff0c\u79ef\u6728\u7684\u5f62\u72b6\u548c\u8fde\u63a5\u65b9\u5f0f\u7b49\u4ee3\u8868\u6570\u636e\u7ed3\u6784\uff0c\u62fc\u88c5\u79ef\u6728\u7684\u6b65\u9aa4\u5219\u5bf9\u5e94\u7b97\u6cd5\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/","title":"1.2 \u00a0 \u7b97\u6cd5\u662f\u4ec0\u4e48","text":""},{"location":"chapter_introduction/what_is_dsa/#121","title":"1.2.1 \u00a0 \u7b97\u6cd5\u5b9a\u4e49","text":"<p>\u300c\u7b97\u6cd5 algorithm\u300d\u662f\u5728\u6709\u9650\u65f6\u95f4\u5185\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u7684\u4e00\u7ec4\u6307\u4ee4\u6216\u64cd\u4f5c\u6b65\u9aa4\uff0c\u5b83\u5177\u6709\u4ee5\u4e0b\u7279\u6027\u3002</p> <ul> <li>\u95ee\u9898\u662f\u660e\u786e\u7684\uff0c\u5305\u542b\u6e05\u6670\u7684\u8f93\u5165\u548c\u8f93\u51fa\u5b9a\u4e49\u3002</li> <li>\u5177\u6709\u53ef\u884c\u6027\uff0c\u80fd\u591f\u5728\u6709\u9650\u6b65\u9aa4\u3001\u65f6\u95f4\u548c\u5185\u5b58\u7a7a\u95f4\u4e0b\u5b8c\u6210\u3002</li> <li>\u5404\u6b65\u9aa4\u90fd\u6709\u786e\u5b9a\u7684\u542b\u4e49\uff0c\u5728\u76f8\u540c\u7684\u8f93\u5165\u548c\u8fd0\u884c\u6761\u4ef6\u4e0b\uff0c\u8f93\u51fa\u59cb\u7ec8\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#122","title":"1.2.2 \u00a0 \u6570\u636e\u7ed3\u6784\u5b9a\u4e49","text":"<p>\u300c\u6570\u636e\u7ed3\u6784 data structure\u300d\u662f\u8ba1\u7b97\u673a\u4e2d\u7ec4\u7ec7\u548c\u5b58\u50a8\u6570\u636e\u7684\u65b9\u5f0f\uff0c\u5177\u6709\u4ee5\u4e0b\u8bbe\u8ba1\u76ee\u6807\u3002</p> <ul> <li>\u7a7a\u95f4\u5360\u7528\u5c3d\u91cf\u5c11\uff0c\u4ee5\u8282\u7701\u8ba1\u7b97\u673a\u5185\u5b58\u3002</li> <li>\u6570\u636e\u64cd\u4f5c\u5c3d\u53ef\u80fd\u5feb\u901f\uff0c\u6db5\u76d6\u6570\u636e\u8bbf\u95ee\u3001\u6dfb\u52a0\u3001\u5220\u9664\u3001\u66f4\u65b0\u7b49\u3002</li> <li>\u63d0\u4f9b\u7b80\u6d01\u7684\u6570\u636e\u8868\u793a\u548c\u903b\u8f91\u4fe1\u606f\uff0c\u4ee5\u4fbf\u7b97\u6cd5\u9ad8\u6548\u8fd0\u884c\u3002</li> </ul> <p>\u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u662f\u4e00\u4e2a\u5145\u6ee1\u6743\u8861\u7684\u8fc7\u7a0b\u3002\u5982\u679c\u60f3\u5728\u67d0\u65b9\u9762\u53d6\u5f97\u63d0\u5347\uff0c\u5f80\u5f80\u9700\u8981\u5728\u53e6\u4e00\u65b9\u9762\u4f5c\u51fa\u59a5\u534f\u3002\u4e0b\u9762\u4e3e\u4e24\u4e2a\u4f8b\u5b50\u3002</p> <ul> <li>\u94fe\u8868\u76f8\u8f83\u4e8e\u6570\u7ec4\uff0c\u5728\u6570\u636e\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u4e0a\u66f4\u52a0\u4fbf\u6377\uff0c\u4f46\u727a\u7272\u4e86\u6570\u636e\u8bbf\u95ee\u901f\u5ea6\u3002</li> <li>\u56fe\u76f8\u8f83\u4e8e\u94fe\u8868\uff0c\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u903b\u8f91\u4fe1\u606f\uff0c\u4f46\u9700\u8981\u5360\u7528\u66f4\u5927\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_introduction/what_is_dsa/#123","title":"1.2.3 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb","text":"<p>\u5982\u56fe 1-4 \u6240\u793a\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u9ad8\u5ea6\u76f8\u5173\u3001\u7d27\u5bc6\u7ed3\u5408\uff0c\u5177\u4f53\u8868\u73b0\u5728\u4ee5\u4e0b\u4e09\u4e2a\u65b9\u9762\u3002</p> <ul> <li>\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u7684\u57fa\u77f3\u3002\u6570\u636e\u7ed3\u6784\u4e3a\u7b97\u6cd5\u63d0\u4f9b\u4e86\u7ed3\u6784\u5316\u5b58\u50a8\u7684\u6570\u636e\uff0c\u4ee5\u53ca\u64cd\u4f5c\u6570\u636e\u7684\u65b9\u6cd5\u3002</li> <li>\u7b97\u6cd5\u662f\u6570\u636e\u7ed3\u6784\u53d1\u6325\u4f5c\u7528\u7684\u821e\u53f0\u3002\u6570\u636e\u7ed3\u6784\u672c\u8eab\u4ec5\u5b58\u50a8\u6570\u636e\u4fe1\u606f\uff0c\u7ed3\u5408\u7b97\u6cd5\u624d\u80fd\u89e3\u51b3\u7279\u5b9a\u95ee\u9898\u3002</li> <li>\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u57fa\u4e8e\u4e0d\u540c\u7684\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\uff0c\u4f46\u6267\u884c\u6548\u7387\u53ef\u80fd\u76f8\u5dee\u5f88\u5927\uff0c\u9009\u62e9\u5408\u9002\u7684\u6570\u636e\u7ed3\u6784\u662f\u5173\u952e\u3002</li> </ul> <p></p> <p> \u56fe 1-4 \u00a0 \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u5173\u7cfb </p> <p>\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u72b9\u5982\u56fe 1-5 \u6240\u793a\u7684\u62fc\u88c5\u79ef\u6728\u3002\u4e00\u5957\u79ef\u6728\uff0c\u9664\u4e86\u5305\u542b\u8bb8\u591a\u96f6\u4ef6\u4e4b\u5916\uff0c\u8fd8\u9644\u6709\u8be6\u7ec6\u7684\u7ec4\u88c5\u8bf4\u660e\u4e66\u3002\u6211\u4eec\u6309\u7167\u8bf4\u660e\u4e66\u4e00\u6b65\u6b65\u64cd\u4f5c\uff0c\u5c31\u80fd\u7ec4\u88c5\u51fa\u7cbe\u7f8e\u7684\u79ef\u6728\u6a21\u578b\u3002</p> <p></p> <p> \u56fe 1-5 \u00a0 \u62fc\u88c5\u79ef\u6728 </p> <p>\u4e24\u8005\u7684\u8be6\u7ec6\u5bf9\u5e94\u5173\u7cfb\u5982\u8868 1-1 \u6240\u793a\u3002</p> <p> \u8868 1-1 \u00a0 \u5c06\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7c7b\u6bd4\u4e3a\u62fc\u88c5\u79ef\u6728 </p> \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5 \u62fc\u88c5\u79ef\u6728 \u8f93\u5165\u6570\u636e \u672a\u62fc\u88c5\u7684\u79ef\u6728 \u6570\u636e\u7ed3\u6784 \u79ef\u6728\u7ec4\u7ec7\u5f62\u5f0f\uff0c\u5305\u62ec\u5f62\u72b6\u3001\u5927\u5c0f\u3001\u8fde\u63a5\u65b9\u5f0f\u7b49 \u7b97\u6cd5 \u628a\u79ef\u6728\u62fc\u6210\u76ee\u6807\u5f62\u6001\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6b65\u9aa4 \u8f93\u51fa\u6570\u636e \u79ef\u6728\u6a21\u578b <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u662f\u72ec\u7acb\u4e8e\u7f16\u7a0b\u8bed\u8a00\u7684\u3002\u6b63\u56e0\u5982\u6b64\uff0c\u672c\u4e66\u5f97\u4ee5\u63d0\u4f9b\u57fa\u4e8e\u591a\u79cd\u7f16\u7a0b\u8bed\u8a00\u7684\u5b9e\u73b0\u3002</p> <p>\u7ea6\u5b9a\u4fd7\u6210\u7684\u7b80\u79f0</p> <p>\u5728\u5b9e\u9645\u8ba8\u8bba\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u201c\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u201d\u7b80\u79f0\u4e3a\u201c\u7b97\u6cd5\u201d\u3002\u6bd4\u5982\u4f17\u6240\u5468\u77e5\u7684 LeetCode \u7b97\u6cd5\u9898\u76ee\uff0c\u5b9e\u9645\u4e0a\u540c\u65f6\u8003\u67e5\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e24\u65b9\u9762\u7684\u77e5\u8bc6\u3002</p>"},{"location":"chapter_preface/","title":"\u7b2c 0 \u7ae0 \u00a0 \u524d\u8a00","text":"<p>Abstract</p> <p>\u7b97\u6cd5\u72b9\u5982\u7f8e\u5999\u7684\u4ea4\u54cd\u4e50\uff0c\u6bcf\u4e00\u884c\u4ee3\u7801\u90fd\u50cf\u97f5\u5f8b\u822c\u6d41\u6dcc\u3002</p> <p>\u613f\u8fd9\u672c\u4e66\u5728\u4f60\u7684\u8111\u6d77\u4e2d\u8f7b\u8f7b\u54cd\u8d77\uff0c\u7559\u4e0b\u72ec\u7279\u800c\u6df1\u523b\u7684\u65cb\u5f8b\u3002</p>"},{"location":"chapter_preface/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>0.1 \u00a0 \u5173\u4e8e\u672c\u4e66</li> <li>0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66</li> <li>0.3 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_preface/about_the_book/","title":"0.1 \u00a0 \u5173\u4e8e\u672c\u4e66","text":"<p>\u672c\u9879\u76ee\u65e8\u5728\u521b\u5efa\u4e00\u672c\u5f00\u6e90\u3001\u514d\u8d39\u3001\u65b0\u624b\u53cb\u597d\u7684\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5165\u95e8\u6559\u7a0b\u3002</p> <ul> <li>\u5168\u4e66\u91c7\u7528\u52a8\u753b\u56fe\u89e3\uff0c\u7ed3\u6784\u5316\u5730\u8bb2\u89e3\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u5185\u5bb9\u6e05\u6670\u6613\u61c2\u3001\u5b66\u4e60\u66f2\u7ebf\u5e73\u6ed1\u3002</li> <li>\u7b97\u6cd5\u6e90\u4ee3\u7801\u7686\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u652f\u6301 Python\u3001C++\u3001Java\u3001C#\u3001Go\u3001Swift\u3001JavaScript\u3001TypeScript\u3001Dart\u3001Rust\u3001C\u3001Zig \u7b49\u8bed\u8a00\u3002</li> <li>\u9f13\u52b1\u8bfb\u8005\u5728\u7ae0\u8282\u8ba8\u8bba\u533a\u4e92\u5e2e\u4e92\u52a9\u3001\u5171\u540c\u8fdb\u6b65\uff0c\u63d0\u95ee\u4e0e\u8bc4\u8bba\u901a\u5e38\u53ef\u5728\u4e24\u65e5\u5185\u5f97\u5230\u56de\u590d\u3002</li> </ul>"},{"location":"chapter_preface/about_the_book/#011","title":"0.1.1 \u00a0 \u8bfb\u8005\u5bf9\u8c61","text":"<p>\u82e5\u4f60\u662f\u7b97\u6cd5\u521d\u5b66\u8005\uff0c\u4ece\u672a\u63a5\u89e6\u8fc7\u7b97\u6cd5\uff0c\u6216\u8005\u5df2\u7ecf\u6709\u4e00\u4e9b\u5237\u9898\u7ecf\u9a8c\uff0c\u5bf9\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u6709\u6a21\u7cca\u7684\u8ba4\u8bc6\uff0c\u5728\u4f1a\u4e0e\u4e0d\u4f1a\u4e4b\u95f4\u53cd\u590d\u6a2a\u8df3\uff0c\u90a3\u4e48\u672c\u4e66\u6b63\u662f\u4e3a\u4f60\u91cf\u8eab\u5b9a\u5236\u7684\uff01</p> <p>\u5982\u679c\u4f60\u5df2\u7ecf\u79ef\u7d2f\u4e00\u5b9a\u5237\u9898\u91cf\uff0c\u719f\u6089\u5927\u90e8\u5206\u9898\u578b\uff0c\u90a3\u4e48\u672c\u4e66\u53ef\u52a9\u4f60\u56de\u987e\u4e0e\u68b3\u7406\u7b97\u6cd5\u77e5\u8bc6\u4f53\u7cfb\uff0c\u4ed3\u5e93\u6e90\u4ee3\u7801\u53ef\u4ee5\u5f53\u4f5c\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u6216\u201c\u7b97\u6cd5\u5b57\u5178\u201d\u6765\u4f7f\u7528\u3002</p> <p>\u82e5\u4f60\u662f\u7b97\u6cd5\u5927\u795e\uff0c\u6211\u4eec\u671f\u5f85\u6536\u5230\u4f60\u7684\u5b9d\u8d35\u5efa\u8bae\uff0c\u6216\u8005\u4e00\u8d77\u53c2\u4e0e\u521b\u4f5c\u3002</p> <p>\u524d\u7f6e\u6761\u4ef6</p> <p>\u4f60\u9700\u8981\u81f3\u5c11\u5177\u5907\u4efb\u4e00\u8bed\u8a00\u7684\u7f16\u7a0b\u57fa\u7840\uff0c\u80fd\u591f\u9605\u8bfb\u548c\u7f16\u5199\u7b80\u5355\u4ee3\u7801\u3002</p>"},{"location":"chapter_preface/about_the_book/#012","title":"0.1.2 \u00a0 \u5185\u5bb9\u7ed3\u6784","text":"<p>\u672c\u4e66\u7684\u4e3b\u8981\u5185\u5bb9\u5982\u56fe 0-1 \u6240\u793a\u3002</p> <ul> <li>\u590d\u6742\u5ea6\u5206\u6790\uff1a\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u7684\u8bc4\u4ef7\u7ef4\u5ea6\u4e0e\u65b9\u6cd5\u3002\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\u7684\u63a8\u7b97\u65b9\u6cd5\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u793a\u4f8b\u7b49\u3002</li> <li>\u6570\u636e\u7ed3\u6784\uff1a\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c\u6570\u636e\u7ed3\u6784\u7684\u5206\u7c7b\u65b9\u6cd5\u3002\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6808\u3001\u961f\u5217\u3001\u54c8\u5e0c\u8868\u3001\u6811\u3001\u5806\u3001\u56fe\u7b49\u6570\u636e\u7ed3\u6784\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u5e38\u7528\u64cd\u4f5c\u3001\u5e38\u89c1\u7c7b\u578b\u3001\u5178\u578b\u5e94\u7528\u3001\u5b9e\u73b0\u65b9\u6cd5\u7b49\u3002</li> <li>\u7b97\u6cd5\uff1a\u641c\u7d22\u3001\u6392\u5e8f\u3001\u5206\u6cbb\u3001\u56de\u6eaf\u3001\u52a8\u6001\u89c4\u5212\u3001\u8d2a\u5fc3\u7b49\u7b97\u6cd5\u7684\u5b9a\u4e49\u3001\u4f18\u7f3a\u70b9\u3001\u6548\u7387\u3001\u5e94\u7528\u573a\u666f\u3001\u89e3\u9898\u6b65\u9aa4\u548c\u793a\u4f8b\u95ee\u9898\u7b49\u3002</li> </ul> <p></p> <p> \u56fe 0-1 \u00a0 \u672c\u4e66\u4e3b\u8981\u5185\u5bb9 </p>"},{"location":"chapter_preface/about_the_book/#013","title":"0.1.3 \u00a0 \u81f4\u8c22","text":"<p>\u5728\u672c\u4e66\u7684\u521b\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u5f97\u5230\u4e86\u8bb8\u591a\u4eba\u7684\u5e2e\u52a9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a</p> <ul> <li>\u611f\u8c22\u6211\u5728\u516c\u53f8\u7684\u5bfc\u5e08\u674e\u6c50\u535a\u58eb\uff0c\u5728\u4e00\u6b21\u7545\u8c08\u4e2d\u4f60\u9f13\u52b1\u6211\u201c\u5feb\u884c\u52a8\u8d77\u6765\u201d\uff0c\u575a\u5b9a\u4e86\u6211\u5199\u8fd9\u672c\u4e66\u7684\u51b3\u5fc3\uff1b</li> <li>\u611f\u8c22\u6211\u7684\u5973\u670b\u53cb\u6ce1\u6ce1\u4f5c\u4e3a\u672c\u4e66\u7684\u9996\u4f4d\u8bfb\u8005\uff0c\u4ece\u7b97\u6cd5\u5c0f\u767d\u7684\u89d2\u5ea6\u63d0\u51fa\u8bb8\u591a\u5b9d\u8d35\u5efa\u8bae\uff0c\u4f7f\u5f97\u672c\u4e66\u66f4\u9002\u5408\u65b0\u624b\u9605\u8bfb\uff1b</li> <li>\u611f\u8c22\u817e\u5b9d\u3001\u7426\u5b9d\u3001\u98de\u5b9d\u4e3a\u672c\u4e66\u8d77\u4e86\u4e00\u4e2a\u5bcc\u6709\u521b\u610f\u7684\u540d\u5b57\uff0c\u5524\u8d77\u5927\u5bb6\u5199\u4e0b\u7b2c\u4e00\u884c\u4ee3\u7801 \"Hello World!\" \u7684\u7f8e\u597d\u56de\u5fc6\uff1b</li> <li>\u611f\u8c22\u6821\u94e8\u5728\u77e5\u8bc6\u4ea7\u6743\u65b9\u9762\u63d0\u4f9b\u7684\u4e13\u4e1a\u5e2e\u52a9\uff0c\u8fd9\u5bf9\u672c\u5f00\u6e90\u4e66\u7684\u53d1\u5c55\u8d77\u5230\u4e86\u91cd\u8981\u7684\u4f5c\u7528\uff1b</li> <li>\u611f\u8c22\u82cf\u6f7c\u4e3a\u672c\u4e66\u8bbe\u8ba1\u4e86\u7cbe\u7f8e\u7684\u5c01\u9762\u548c logo \uff0c\u5e76\u5728\u6211\u7684\u5f3a\u8feb\u75c7\u4e0b\u591a\u6b21\u8010\u5fc3\u4fee\u6539\uff1b</li> <li>\u611f\u8c22 @squidfunk \u63d0\u4f9b\u7684\u5199\u4f5c\u6392\u7248\u5efa\u8bae\uff0c\u4ee5\u53ca\u4ed6\u5f00\u53d1\u7684\u5f00\u6e90\u6587\u6863\u4e3b\u9898 Material-for-MkDocs \u3002</li> </ul> <p>\u5728\u5199\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u9605\u8bfb\u4e86\u8bb8\u591a\u5173\u4e8e\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u6559\u6750\u548c\u6587\u7ae0\u3002\u8fd9\u4e9b\u4f5c\u54c1\u4e3a\u672c\u4e66\u63d0\u4f9b\u4e86\u4f18\u79c0\u7684\u8303\u672c\uff0c\u786e\u4fdd\u4e86\u672c\u4e66\u5185\u5bb9\u7684\u51c6\u786e\u6027\u4e0e\u54c1\u8d28\u3002\u5728\u6b64\u611f\u8c22\u6240\u6709\u8001\u5e08\u548c\u524d\u8f88\u4eec\u7684\u6770\u51fa\u8d21\u732e\uff01</p> <p>\u672c\u4e66\u5021\u5bfc\u624b\u8111\u5e76\u7528\u7684\u5b66\u4e60\u65b9\u5f0f\uff0c\u5728\u8fd9\u4e00\u70b9\u4e0a\u6df1\u53d7\u300a\u52a8\u624b\u5b66\u6df1\u5ea6\u5b66\u4e60\u300b\u7684\u542f\u53d1\u3002\u5728\u6b64\u5411\u5404\u4f4d\u8bfb\u8005\u5f3a\u70c8\u63a8\u8350\u8fd9\u672c\u4f18\u79c0\u7684\u8457\u4f5c\u3002</p> <p>\u8877\u5fc3\u611f\u8c22\u6211\u7684\u7236\u6bcd\uff0c\u6b63\u662f\u4f60\u4eec\u4e00\u76f4\u4ee5\u6765\u7684\u652f\u6301\u4e0e\u9f13\u52b1\uff0c\u8ba9\u6211\u6709\u673a\u4f1a\u505a\u8fd9\u4ef6\u5bcc\u6709\u8da3\u5473\u7684\u4e8b\u3002</p>"},{"location":"chapter_preface/suggestions/","title":"0.2 \u00a0 \u5982\u4f55\u4f7f\u7528\u672c\u4e66","text":"<p>Tip</p> <p>\u4e3a\u4e86\u83b7\u5f97\u6700\u4f73\u7684\u9605\u8bfb\u4f53\u9a8c\uff0c\u5efa\u8bae\u4f60\u901a\u8bfb\u672c\u8282\u5185\u5bb9\u3002</p>"},{"location":"chapter_preface/suggestions/#021","title":"0.2.1 \u00a0 \u884c\u6587\u98ce\u683c\u7ea6\u5b9a","text":"<ul> <li>\u6807\u9898\u540e\u6807\u6ce8 <code>*</code> \u7684\u662f\u9009\u8bfb\u7ae0\u8282\uff0c\u5185\u5bb9\u76f8\u5bf9\u56f0\u96be\u3002\u5982\u679c\u4f60\u7684\u65f6\u95f4\u6709\u9650\uff0c\u53ef\u4ee5\u5148\u8df3\u8fc7\u3002</li> <li>\u91cd\u8981\u4e13\u6709\u540d\u8bcd\u53ca\u5176\u82f1\u6587\u7ffb\u8bd1\u4f1a\u7528 <code>\u300c \u300d</code> \u62ec\u53f7\u6807\u6ce8\uff0c\u4f8b\u5982 <code>\u300c\u6570\u7ec4 array\u300d</code> \u3002\u5efa\u8bae\u8bb0\u4f4f\u5b83\u4eec\uff0c\u4ee5\u4fbf\u9605\u8bfb\u6587\u732e\u3002</li> <li>\u4e13\u6709\u540d\u8bcd\u548c\u6709\u7279\u6307\u542b\u4e49\u7684\u8bcd\u53e5\u4f1a\u4f7f\u7528 <code>\u201c\u5f15\u53f7\u201d</code> \u6807\u6ce8\uff0c\u4ee5\u907f\u514d\u6b67\u4e49\u3002</li> <li>\u91cd\u8981\u540d\u8bcd\u3001\u91cd\u70b9\u5185\u5bb9\u548c\u603b\u7ed3\u6027\u8bed\u53e5\u4f1a\u88ab \u52a0\u7c97 \uff0c\u8fd9\u7c7b\u6587\u5b57\u503c\u5f97\u7279\u522b\u5173\u6ce8\u3002</li> <li>\u5f53\u6d89\u53ca\u7f16\u7a0b\u8bed\u8a00\u4e4b\u95f4\u4e0d\u4e00\u81f4\u7684\u540d\u8bcd\u65f6\uff0c\u672c\u4e66\u5747\u4ee5 Python \u4e3a\u51c6\uff0c\u4f8b\u5982\u4f7f\u7528 \\(\\text{None}\\) \u6765\u8868\u793a\u201c\u7a7a\u201d\u3002</li> <li>\u672c\u4e66\u90e8\u5206\u653e\u5f03\u4e86\u7f16\u7a0b\u8bed\u8a00\u7684\u6ce8\u91ca\u89c4\u8303\uff0c\u4ee5\u6362\u53d6\u66f4\u52a0\u7d27\u51d1\u7684\u5185\u5bb9\u6392\u7248\u3002\u6ce8\u91ca\u4e3b\u8981\u5206\u4e3a\u4e09\u79cd\u7c7b\u578b\uff1a\u6807\u9898\u6ce8\u91ca\u3001\u5185\u5bb9\u6ce8\u91ca\u3001\u591a\u884c\u6ce8\u91ca\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>\"\"\"\u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\"\"\"\n\n# \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n\"\"\"\n\u591a\u884c\n\u6ce8\u91ca\n\"\"\"\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>/* \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49 */\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n/**\n * \u591a\u884c\n * \u6ce8\u91ca\n */\n</code></pre> <pre><code>// \u6807\u9898\u6ce8\u91ca\uff0c\u7528\u4e8e\u6807\u6ce8\u51fd\u6570\u3001\u7c7b\u3001\u6d4b\u8bd5\u6837\u4f8b\u7b49\n\n// \u5185\u5bb9\u6ce8\u91ca\uff0c\u7528\u4e8e\u8be6\u89e3\u4ee3\u7801\n\n// \u591a\u884c\n// \u6ce8\u91ca\n</code></pre>"},{"location":"chapter_preface/suggestions/#022","title":"0.2.2 \u00a0 \u5728\u52a8\u753b\u56fe\u89e3\u4e2d\u9ad8\u6548\u5b66\u4e60","text":"<p>\u76f8\u8f83\u4e8e\u6587\u5b57\uff0c\u89c6\u9891\u548c\u56fe\u7247\u5177\u6709\u66f4\u9ad8\u7684\u4fe1\u606f\u5bc6\u5ea6\u548c\u7ed3\u6784\u5316\u7a0b\u5ea6\uff0c\u66f4\u6613\u4e8e\u7406\u89e3\u3002\u5728\u672c\u4e66\u4e2d\uff0c\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u5c06\u4e3b\u8981\u901a\u8fc7\u52a8\u753b\u548c\u56fe\u89e3\u5f62\u5f0f\u5c55\u793a\uff0c\u800c\u6587\u5b57\u5219\u4f5c\u4e3a\u52a8\u753b\u548c\u56fe\u7247\u7684\u89e3\u91ca\u4e0e\u8865\u5145\u3002</p> <p>\u5982\u679c\u4f60\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u53d1\u73b0\u67d0\u6bb5\u5185\u5bb9\u63d0\u4f9b\u4e86\u5982\u56fe 0-2 \u6240\u793a\u7684\u52a8\u753b\u6216\u56fe\u89e3\uff0c\u8bf7\u4ee5\u56fe\u4e3a\u4e3b\u3001\u4ee5\u6587\u5b57\u4e3a\u8f85\uff0c\u7efc\u5408\u4e24\u8005\u6765\u7406\u89e3\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-2 \u00a0 \u52a8\u753b\u56fe\u89e3\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#023","title":"0.2.3 \u00a0 \u5728\u4ee3\u7801\u5b9e\u8df5\u4e2d\u52a0\u6df1\u7406\u89e3","text":"<p>\u672c\u4e66\u7684\u914d\u5957\u4ee3\u7801\u6258\u7ba1\u5728 GitHub \u4ed3\u5e93\u3002\u5982\u56fe 0-3 \u6240\u793a\uff0c\u6e90\u4ee3\u7801\u9644\u6709\u6d4b\u8bd5\u6837\u4f8b\uff0c\u53ef\u4e00\u952e\u8fd0\u884c\u3002</p> <p>\u5982\u679c\u65f6\u95f4\u5141\u8bb8\uff0c\u5efa\u8bae\u4f60\u53c2\u7167\u4ee3\u7801\u81ea\u884c\u6572\u4e00\u904d\u3002\u5982\u679c\u5b66\u4e60\u65f6\u95f4\u6709\u9650\uff0c\u8bf7\u81f3\u5c11\u901a\u8bfb\u5e76\u8fd0\u884c\u6240\u6709\u4ee3\u7801\u3002</p> <p>\u4e0e\u9605\u8bfb\u4ee3\u7801\u76f8\u6bd4\uff0c\u7f16\u5199\u4ee3\u7801\u7684\u8fc7\u7a0b\u5f80\u5f80\u80fd\u5e26\u6765\u66f4\u591a\u6536\u83b7\u3002\u52a8\u624b\u5b66\uff0c\u624d\u662f\u771f\u7684\u5b66\u3002</p> <p></p> <p> \u56fe 0-3 \u00a0 \u8fd0\u884c\u4ee3\u7801\u793a\u4f8b </p> <p>\u8fd0\u884c\u4ee3\u7801\u7684\u524d\u7f6e\u5de5\u4f5c\u4e3b\u8981\u5206\u4e3a\u4e09\u6b65\u3002</p> <p>\u7b2c\u4e00\u6b65\uff1a\u5b89\u88c5\u672c\u5730\u7f16\u7a0b\u73af\u5883\u3002\u8bf7\u53c2\u7167\u9644\u5f55\u6240\u793a\u7684\u6559\u7a0b\u8fdb\u884c\u5b89\u88c5\uff0c\u5982\u679c\u5df2\u5b89\u88c5\uff0c\u5219\u53ef\u8df3\u8fc7\u6b64\u6b65\u9aa4\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u514b\u9686\u6216\u4e0b\u8f7d\u4ee3\u7801\u4ed3\u5e93\u3002\u524d\u5f80 GitHub \u4ed3\u5e93\u3002</p> <p>\u5982\u679c\u5df2\u7ecf\u5b89\u88c5 Git \uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u514b\u9686\u672c\u4ed3\u5e93\uff1a</p> <pre><code>git clone https://github.com/krahets/hello-algo.git\n</code></pre> <p>\u5f53\u7136\uff0c\u4f60\u4e5f\u53ef\u4ee5\u5728\u56fe 0-4 \u6240\u793a\u7684\u4f4d\u7f6e\uff0c\u70b9\u51fb\u201cDownload ZIP\u201d\u6309\u94ae\u76f4\u63a5\u4e0b\u8f7d\u4ee3\u7801\u538b\u7f29\u5305\uff0c\u7136\u540e\u5728\u672c\u5730\u89e3\u538b\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 0-4 \u00a0 \u514b\u9686\u4ed3\u5e93\u4e0e\u4e0b\u8f7d\u4ee3\u7801 </p> <p>\u7b2c\u4e09\u6b65\uff1a\u8fd0\u884c\u6e90\u4ee3\u7801\u3002\u5982\u56fe 0-5 \u6240\u793a\uff0c\u5bf9\u4e8e\u9876\u90e8\u6807\u6709\u6587\u4ef6\u540d\u79f0\u7684\u4ee3\u7801\u5757\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4ed3\u5e93\u7684 <code>codes</code> \u6587\u4ef6\u5939\u5185\u627e\u5230\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6\u3002\u6e90\u4ee3\u7801\u6587\u4ef6\u53ef\u4e00\u952e\u8fd0\u884c\uff0c\u5c06\u5e2e\u52a9\u4f60\u8282\u7701\u4e0d\u5fc5\u8981\u7684\u8c03\u8bd5\u65f6\u95f4\uff0c\u8ba9\u4f60\u80fd\u591f\u4e13\u6ce8\u4e8e\u5b66\u4e60\u5185\u5bb9\u3002</p> <p></p> <p> \u56fe 0-5 \u00a0 \u4ee3\u7801\u5757\u4e0e\u5bf9\u5e94\u7684\u6e90\u4ee3\u7801\u6587\u4ef6 </p>"},{"location":"chapter_preface/suggestions/#024","title":"0.2.4 \u00a0 \u5728\u63d0\u95ee\u8ba8\u8bba\u4e2d\u5171\u540c\u6210\u957f","text":"<p>\u5728\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u8bf7\u4e0d\u8981\u8f7b\u6613\u8df3\u8fc7\u90a3\u4e9b\u6ca1\u5b66\u660e\u767d\u7684\u77e5\u8bc6\u70b9\u3002\u6b22\u8fce\u5728\u8bc4\u8bba\u533a\u63d0\u51fa\u4f60\u7684\u95ee\u9898\uff0c\u6211\u548c\u5c0f\u4f19\u4f34\u4eec\u5c06\u7aed\u8bda\u4e3a\u4f60\u89e3\u7b54\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\u53ef\u5728\u4e24\u5929\u5185\u56de\u590d\u3002</p> <p>\u5982\u56fe 0-6 \u6240\u793a\uff0c\u6bcf\u4e2a\u7ae0\u8282\u7684\u5e95\u90e8\u90fd\u914d\u6709\u8bc4\u8bba\u533a\u3002\u5e0c\u671b\u4f60\u80fd\u591a\u5173\u6ce8\u8bc4\u8bba\u533a\u7684\u5185\u5bb9\u3002\u4e00\u65b9\u9762\uff0c\u4f60\u53ef\u4ee5\u4e86\u89e3\u5927\u5bb6\u9047\u5230\u7684\u95ee\u9898\uff0c\u4ece\u800c\u67e5\u6f0f\u8865\u7f3a\uff0c\u6fc0\u53d1\u66f4\u6df1\u5165\u7684\u601d\u8003\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u671f\u5f85\u4f60\u80fd\u6177\u6168\u5730\u56de\u7b54\u5176\u4ed6\u5c0f\u4f19\u4f34\u7684\u95ee\u9898\uff0c\u5206\u4eab\u4f60\u7684\u89c1\u89e3\uff0c\u5e2e\u52a9\u4ed6\u4eba\u8fdb\u6b65\u3002</p> <p></p> <p> \u56fe 0-6 \u00a0 \u8bc4\u8bba\u533a\u793a\u4f8b </p>"},{"location":"chapter_preface/suggestions/#025","title":"0.2.5 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf","text":"<p>\u4ece\u603b\u4f53\u4e0a\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5212\u5206\u4e3a\u4e09\u4e2a\u9636\u6bb5\u3002</p> <ol> <li>\u9636\u6bb5\u4e00\uff1a\u7b97\u6cd5\u5165\u95e8\u3002\u6211\u4eec\u9700\u8981\u719f\u6089\u5404\u79cd\u6570\u636e\u7ed3\u6784\u7684\u7279\u70b9\u548c\u7528\u6cd5\uff0c\u5b66\u4e60\u4e0d\u540c\u7b97\u6cd5\u7684\u539f\u7406\u3001\u6d41\u7a0b\u3001\u7528\u9014\u548c\u6548\u7387\u7b49\u65b9\u9762\u7684\u5185\u5bb9\u3002</li> <li>\u9636\u6bb5\u4e8c\uff1a\u5237\u7b97\u6cd5\u9898\u3002\u5efa\u8bae\u4ece\u70ed\u95e8\u9898\u76ee\u5f00\u5237\uff0c\u5982\u201c\u5251\u6307 Offer\u201d\u548c\u201cLeetCode Hot 100\u201d\uff0c\u5148\u79ef\u7d2f\u81f3\u5c11 100 \u9053\u9898\u76ee\uff0c\u719f\u6089\u4e3b\u6d41\u7684\u7b97\u6cd5\u95ee\u9898\u3002\u521d\u6b21\u5237\u9898\u65f6\uff0c\u201c\u77e5\u8bc6\u9057\u5fd8\u201d\u53ef\u80fd\u662f\u4e00\u4e2a\u6311\u6218\uff0c\u4f46\u8bf7\u653e\u5fc3\uff0c\u8fd9\u662f\u5f88\u6b63\u5e38\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u201c\u827e\u5bbe\u6d69\u65af\u9057\u5fd8\u66f2\u7ebf\u201d\u6765\u590d\u4e60\u9898\u76ee\uff0c\u901a\u5e38\u5728\u8fdb\u884c 3\uff5e5 \u8f6e\u7684\u91cd\u590d\u540e\uff0c\u5c31\u80fd\u5c06\u5176\u7262\u8bb0\u5728\u5fc3\u3002</li> <li>\u9636\u6bb5\u4e09\uff1a\u642d\u5efa\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5b66\u4e60\u65b9\u9762\uff0c\u6211\u4eec\u53ef\u4ee5\u9605\u8bfb\u7b97\u6cd5\u4e13\u680f\u6587\u7ae0\u3001\u89e3\u9898\u6846\u67b6\u548c\u7b97\u6cd5\u6559\u6750\uff0c\u4ee5\u4e0d\u65ad\u4e30\u5bcc\u77e5\u8bc6\u4f53\u7cfb\u3002\u5728\u5237\u9898\u65b9\u9762\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u91c7\u7528\u8fdb\u9636\u5237\u9898\u7b56\u7565\uff0c\u5982\u6309\u4e13\u9898\u5206\u7c7b\u3001\u4e00\u9898\u591a\u89e3\u3001\u4e00\u89e3\u591a\u9898\u7b49\uff0c\u76f8\u5173\u7684\u5237\u9898\u5fc3\u5f97\u53ef\u4ee5\u5728\u5404\u4e2a\u793e\u533a\u627e\u5230\u3002</li> </ol> <p>\u5982\u56fe 0-7 \u6240\u793a\uff0c\u672c\u4e66\u5185\u5bb9\u4e3b\u8981\u6db5\u76d6\u201c\u9636\u6bb5\u4e00\u201d\uff0c\u65e8\u5728\u5e2e\u52a9\u4f60\u66f4\u9ad8\u6548\u5730\u5c55\u5f00\u9636\u6bb5\u4e8c\u548c\u9636\u6bb5\u4e09\u7684\u5b66\u4e60\u3002</p> <p></p> <p> \u56fe 0-7 \u00a0 \u7b97\u6cd5\u5b66\u4e60\u8def\u7ebf </p>"},{"location":"chapter_preface/summary/","title":"0.3 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u672c\u4e66\u7684\u4e3b\u8981\u53d7\u4f17\u662f\u7b97\u6cd5\u521d\u5b66\u8005\u3002\u5982\u679c\u5df2\u6709\u4e00\u5b9a\u57fa\u7840\uff0c\u672c\u4e66\u80fd\u5e2e\u52a9\u4f60\u7cfb\u7edf\u56de\u987e\u7b97\u6cd5\u77e5\u8bc6\uff0c\u4e66\u4e2d\u6e90\u4ee3\u7801\u4e5f\u53ef\u4f5c\u4e3a\u201c\u5237\u9898\u5de5\u5177\u5e93\u201d\u4f7f\u7528\u3002</li> <li>\u4e66\u4e2d\u5185\u5bb9\u4e3b\u8981\u5305\u62ec\u590d\u6742\u5ea6\u5206\u6790\u3001\u6570\u636e\u7ed3\u6784\u3001\u7b97\u6cd5\u4e09\u90e8\u5206\uff0c\u6db5\u76d6\u4e86\u8be5\u9886\u57df\u7684\u5927\u90e8\u5206\u4e3b\u9898\u3002</li> <li>\u5bf9\u4e8e\u7b97\u6cd5\u65b0\u624b\uff0c\u5728\u521d\u5b66\u9636\u6bb5\u9605\u8bfb\u4e00\u672c\u5165\u95e8\u4e66\u81f3\u5173\u91cd\u8981\uff0c\u53ef\u4ee5\u5c11\u8d70\u8bb8\u591a\u5f2f\u8def\u3002</li> <li>\u4e66\u4e2d\u7684\u52a8\u753b\u548c\u56fe\u89e3\u901a\u5e38\u7528\u4e8e\u4ecb\u7ecd\u91cd\u70b9\u548c\u96be\u70b9\u77e5\u8bc6\u3002\u9605\u8bfb\u672c\u4e66\u65f6\uff0c\u5e94\u7ed9\u4e88\u8fd9\u4e9b\u5185\u5bb9\u66f4\u591a\u5173\u6ce8\u3002</li> <li>\u5b9e\u8df5\u4e43\u5b66\u4e60\u7f16\u7a0b\u4e4b\u6700\u4f73\u9014\u5f84\u3002\u5f3a\u70c8\u5efa\u8bae\u8fd0\u884c\u6e90\u4ee3\u7801\u5e76\u4eb2\u81ea\u6572\u4ee3\u7801\u3002</li> <li>\u672c\u4e66\u7f51\u9875\u7248\u7684\u6bcf\u4e2a\u7ae0\u8282\u90fd\u8bbe\u6709\u8ba8\u8bba\u533a\uff0c\u6b22\u8fce\u968f\u65f6\u5206\u4eab\u4f60\u7684\u7591\u60d1\u4e0e\u89c1\u89e3\u3002</li> </ul>"},{"location":"chapter_reference/","title":"\u53c2\u8003\u6587\u732e","text":"<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p> <p>[2] Aditya Bhargava. Grokking Algorithms: An Illustrated Guide for Programmers and Other Curious People (1<sup>st</sup> Edition).</p> <p>[3] \u4e25\u851a\u654f. \u6570\u636e\u7ed3\u6784\uff08C \u8bed\u8a00\u7248\uff09.</p> <p>[4] \u9093\u4fca\u8f89. \u6570\u636e\u7ed3\u6784\uff08C++ \u8bed\u8a00\u7248\uff0c\u7b2c\u4e09\u7248\uff09.</p> <p>[5] \u9a6c\u514b \u827e\u4f26 \u7ef4\u65af\u8457\uff0c\u9648\u8d8a\u8bd1. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\uff1aJava\u8bed\u8a00\u63cf\u8ff0\uff08\u7b2c\u4e09\u7248\uff09.</p> <p>[6] \u7a0b\u6770. \u5927\u8bdd\u6570\u636e\u7ed3\u6784.</p> <p>[7] \u738b\u4e89. \u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u4e4b\u7f8e.</p> <p>[8] Gayle Laakmann McDowell. Cracking the Coding Interview: 189 Programming Questions and Solutions (6<sup>th</sup> Edition). </p> <p>[9] Aston Zhang, et al. Dive into Deep Learning.</p>"},{"location":"chapter_searching/","title":"\u7b2c 10 \u7ae0 \u00a0 \u641c\u7d22","text":"<p>Abstract</p> <p>\u641c\u7d22\u662f\u4e00\u573a\u672a\u77e5\u7684\u5192\u9669\uff0c\u6211\u4eec\u6216\u8bb8\u9700\u8981\u8d70\u904d\u795e\u79d8\u7a7a\u95f4\u7684\u6bcf\u4e2a\u89d2\u843d\uff0c\u53c8\u6216\u8bb8\u53ef\u4ee5\u5feb\u901f\u9501\u5b9a\u76ee\u6807\u3002</p> <p>\u5728\u8fd9\u573a\u5bfb\u89c5\u4e4b\u65c5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u63a2\u7d22\u90fd\u53ef\u80fd\u5f97\u5230\u4e00\u4e2a\u672a\u66fe\u6599\u60f3\u7684\u7b54\u6848\u3002</p>"},{"location":"chapter_searching/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>10.1 \u00a0 \u4e8c\u5206\u67e5\u627e</li> <li>10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9</li> <li>10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c</li> <li>10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565</li> <li>10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5</li> <li>10.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_searching/binary_search/","title":"10.1 \u00a0 \u4e8c\u5206\u67e5\u627e","text":"<p>\u300c\u4e8c\u5206\u67e5\u627e binary search\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u9ad8\u6548\u641c\u7d22\u7b97\u6cd5\u3002\u5b83\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\u641c\u7d22\u8303\u56f4\uff0c\u76f4\u81f3\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u4e3a\u6b62\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5143\u7d20\u6309\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u6392\u5217\u4e14\u4e0d\u91cd\u590d\u3002\u8bf7\u67e5\u627e\u5e76\u8fd4\u56de\u5143\u7d20 <code>target</code> \u5728\u8be5\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002\u793a\u4f8b\u5982\u56fe 10-1 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-1 \u00a0 \u4e8c\u5206\u67e5\u627e\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u56fe 10-2 \u6240\u793a\uff0c\u6211\u4eec\u5148\u521d\u59cb\u5316\u6307\u9488 \\(i = 0\\) \u548c \\(j = n - 1\\) \uff0c\u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u548c\u5c3e\u5143\u7d20\uff0c\u4ee3\u8868\u641c\u7d22\u533a\u95f4 \\([0, n - 1]\\) \u3002\u8bf7\u6ce8\u610f\uff0c\u4e2d\u62ec\u53f7\u8868\u793a\u95ed\u533a\u95f4\uff0c\u5176\u5305\u542b\u8fb9\u754c\u503c\u672c\u8eab\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u5faa\u73af\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m = \\lfloor {(i + j) / 2} \\rfloor\\) \uff0c\u5176\u4e2d \\(\\lfloor \\: \\rfloor\\) \u8868\u793a\u5411\u4e0b\u53d6\u6574\u64cd\u4f5c\u3002</li> <li>\u5224\u65ad <code>nums[m]</code> \u548c <code>target</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5\u3002<ol> <li>\u5f53 <code>nums[m] &lt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([m + 1, j]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(i = m + 1\\) \u3002</li> <li>\u5f53 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e <code>target</code> \u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u6267\u884c \\(j = m - 1\\) \u3002</li> <li>\u5f53 <code>nums[m] = target</code> \u65f6\uff0c\u8bf4\u660e\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de\u7d22\u5f15 \\(m\\) \u3002</li> </ol> </li> </ol> <p>\u82e5\u6570\u7ec4\u4e0d\u5305\u542b\u76ee\u6807\u5143\u7d20\uff0c\u641c\u7d22\u533a\u95f4\u6700\u7ec8\u4f1a\u7f29\u5c0f\u4e3a\u7a7a\u3002\u6b64\u65f6\u8fd4\u56de \\(-1\\) \u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-2 \u00a0 \u4e8c\u5206\u67e5\u627e\u6d41\u7a0b </p> <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e \\(i\\) \u548c \\(j\\) \u90fd\u662f <code>int</code> \u7c7b\u578b\uff0c\u56e0\u6b64 \\(i + j\\) \u53ef\u80fd\u4f1a\u8d85\u51fa <code>int</code> \u7c7b\u578b\u7684\u53d6\u503c\u8303\u56f4\u3002\u4e3a\u4e86\u907f\u514d\u5927\u6570\u8d8a\u754c\uff0c\u6211\u4eec\u901a\u5e38\u91c7\u7528\u516c\u5f0f \\(m = \\lfloor {i + (j - i) / 2} \\rfloor\\) \u6765\u8ba1\u7b97\u4e2d\u70b9\u3002</p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py<pre><code>def binary_search(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\"\"\"\n    # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j = 0, len(nums) - 1\n    # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j:\n        # \u7406\u8bba\u4e0a Python \u7684\u6570\u5b57\u53ef\u4ee5\u65e0\u9650\u5927\uff08\u53d6\u51b3\u4e8e\u5185\u5b58\u5927\u5c0f\uff09\uff0c\u65e0\u987b\u8003\u8651\u5927\u6570\u8d8a\u754c\u95ee\u9898\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.size() - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint BinarySearch(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = nums.Length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    i, j := 0, len(nums)-1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    for i &lt;= j {\n        m := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunc binarySearch(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i = 0\n    var j = nums.count - 1\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums, target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else return m; // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfunction binarySearch(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let i = 0,\n        j = nums.length - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n  int i = 0, j = nums.length - 1;\n  // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n      j = m - 1;\n    } else {\n      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nfn binary_search(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    let mut i = 0;\n    let mut j = nums.len() as i32 - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt;= j {\n        let m = i + (j - i) / 2;      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {         // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {  // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {                      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }                       \n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09 */\nint binarySearch(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    int i = 0, j = len - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u53cc\u95ed\u533a\u95f4\uff09\nfn binarySearch(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1] \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20\n    var i: usize = 0;\n    var j: usize = nums.items.len - 1;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i &gt; j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1;\n        } else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> <p>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(\\log n)\\) \uff1a\u5728\u4e8c\u5206\u5faa\u73af\u4e2d\uff0c\u533a\u95f4\u6bcf\u8f6e\u7f29\u5c0f\u4e00\u534a\uff0c\u5faa\u73af\u6b21\u6570\u4e3a \\(\\log_2 n\\) \u3002</p> <p>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\) \uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7a7a\u95f4\u3002</p>"},{"location":"chapter_searching/binary_search/#1011","title":"10.1.1 \u00a0 \u533a\u95f4\u8868\u793a\u65b9\u6cd5","text":"<p>\u9664\u4e86\u4e0a\u8ff0\u53cc\u95ed\u533a\u95f4\u5916\uff0c\u5e38\u89c1\u7684\u533a\u95f4\u8868\u793a\u8fd8\u6709\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u533a\u95f4\uff0c\u5b9a\u4e49\u4e3a \\([0, n)\\) \uff0c\u5373\u5de6\u8fb9\u754c\u5305\u542b\u81ea\u8eab\uff0c\u53f3\u8fb9\u754c\u4e0d\u5305\u542b\u81ea\u8eab\u3002\u5728\u8be5\u8868\u793a\u4e0b\uff0c\u533a\u95f4 \\([i, j]\\) \u5728 \\(i = j\\) \u65f6\u4e3a\u7a7a\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u57fa\u4e8e\u8be5\u8868\u793a\u5b9e\u73b0\u5177\u6709\u76f8\u540c\u529f\u80fd\u7684\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search.py<pre><code>def binary_search_lcro(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\"\"\"\n    # \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j = 0, len(nums)\n    # \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n        elif nums[m] &gt; target:\n            j = m  # \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n        else:\n            return m  # \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n    return -1  # \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n</code></pre> binary_search.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(vector&lt;int&gt; &amp;nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.size();\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint BinarySearchLCRO(int[] nums, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = nums.Length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2;   // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else                       // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    i, j := 0, len(nums)\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    for i &lt; j {\n        m := i + (j-i)/2      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunc binarySearchLCRO(nums: [Int], target: Int) -&gt; Int {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i = 0\n    var j = nums.count\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target { // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m\n        } else { // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1\n}\n</code></pre> binary_search.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums, target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m \uff0c\u4f7f\u7528 parseInt() \u5411\u4e0b\u53d6\u6574\n        const m = parseInt(i + (j - i) / 2);\n        if (nums[m] &lt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target)\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n        else return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfunction binarySearchLCRO(nums: number[], target: number): number {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let i = 0,\n        j = nums.length;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        const m = Math.floor(i + (j - i) / 2);\n        if (nums[m] &lt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums[m] &gt; target) {\n            // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }\n    }\n    return -1; // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n}\n</code></pre> binary_search.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(List&lt;int&gt; nums, int target) {\n  // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n  int i = 0, j = nums.length;\n  // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n  while (i &lt; j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n      i = m + 1;\n    } else if (nums[m] &gt; target) {\n      // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n      j = m;\n    } else {\n      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n      return m;\n    }\n  }\n  // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n  return -1;\n}\n</code></pre> binary_search.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nfn binary_search_lcro(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    let mut i = 0;\n    let mut j = nums.len() as i32;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while i &lt; j {\n        let m = i + (j - i) / 2;      // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {         // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if nums[m as usize] &gt; target {  // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m - 1;\n        } else {                      // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n        }                       \n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09 */\nint binarySearchLCRO(int *nums, int len, int target) {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    int i = 0, j = len;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt; j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target)    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        else if (nums[m] &gt; target) // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        else // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return m;\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> binary_search.zig<pre><code>// \u4e8c\u5206\u67e5\u627e\uff08\u5de6\u95ed\u53f3\u5f00\u533a\u95f4\uff09\nfn binarySearchLCRO(comptime T: type, nums: std.ArrayList(T), target: T) T {\n    // \u521d\u59cb\u5316\u5de6\u95ed\u53f3\u5f00\u533a\u95f4 [0, n) \uff0c\u5373 i, j \u5206\u522b\u6307\u5411\u6570\u7ec4\u9996\u5143\u7d20\u3001\u5c3e\u5143\u7d20+1\n    var i: usize = 0;\n    var j: usize = nums.items.len;\n    // \u5faa\u73af\uff0c\u5f53\u641c\u7d22\u533a\u95f4\u4e3a\u7a7a\u65f6\u8df3\u51fa\uff08\u5f53 i = j \u65f6\u4e3a\u7a7a\uff09\n    while (i &lt;= j) {\n        var m = i + (j - i) / 2;                // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums.items[m] &lt; target) {           // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [m+1, j) \u4e2d\n            i = m + 1;\n        } else if (nums.items[m] &gt; target) {    // \u6b64\u60c5\u51b5\u8bf4\u660e target \u5728\u533a\u95f4 [i, m) \u4e2d\n            j = m;\n        } else {                                // \u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de\u5176\u7d22\u5f15\n            return @intCast(m);\n        }\n    }\n    // \u672a\u627e\u5230\u76ee\u6807\u5143\u7d20\uff0c\u8fd4\u56de -1\n    return -1;\n}\n</code></pre> <p>\u5982\u56fe 10-3 \u6240\u793a\uff0c\u5728\u4e24\u79cd\u533a\u95f4\u8868\u793a\u4e0b\uff0c\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u521d\u59cb\u5316\u3001\u5faa\u73af\u6761\u4ef6\u548c\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\u7686\u6709\u6240\u4e0d\u540c\u3002</p> <p>\u7531\u4e8e\u201c\u53cc\u95ed\u533a\u95f4\u201d\u8868\u793a\u4e2d\u7684\u5de6\u53f3\u8fb9\u754c\u90fd\u88ab\u5b9a\u4e49\u4e3a\u95ed\u533a\u95f4\uff0c\u56e0\u6b64\u901a\u8fc7\u6307\u9488 \\(i\\) \u548c\u6307\u9488 \\(j\\) \u7f29\u5c0f\u533a\u95f4\u7684\u64cd\u4f5c\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u8fd9\u6837\u66f4\u4e0d\u5bb9\u6613\u51fa\u9519\uff0c\u56e0\u6b64\u4e00\u822c\u5efa\u8bae\u91c7\u7528\u201c\u53cc\u95ed\u533a\u95f4\u201d\u7684\u5199\u6cd5\u3002</p> <p></p> <p> \u56fe 10-3 \u00a0 \u4e24\u79cd\u533a\u95f4\u5b9a\u4e49 </p>"},{"location":"chapter_searching/binary_search/#1012","title":"10.1.2 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u5206\u67e5\u627e\u5728\u65f6\u95f4\u548c\u7a7a\u95f4\u65b9\u9762\u90fd\u6709\u8f83\u597d\u7684\u6027\u80fd\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u7684\u65f6\u95f4\u6548\u7387\u9ad8\u3002\u5728\u5927\u6570\u636e\u91cf\u4e0b\uff0c\u5bf9\u6570\u9636\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002\u4f8b\u5982\uff0c\u5f53\u6570\u636e\u5927\u5c0f \\(n = 2^{20}\\) \u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u9700\u8981 \\(2^{20} = 1048576\\) \u8f6e\u5faa\u73af\uff0c\u800c\u4e8c\u5206\u67e5\u627e\u4ec5\u9700 \\(\\log_2 2^{20} = 20\\) \u8f6e\u5faa\u73af\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u65e0\u987b\u989d\u5916\u7a7a\u95f4\u3002\u76f8\u8f83\u4e8e\u9700\u8981\u501f\u52a9\u989d\u5916\u7a7a\u95f4\u7684\u641c\u7d22\u7b97\u6cd5\uff08\u4f8b\u5982\u54c8\u5e0c\u67e5\u627e\uff09\uff0c\u4e8c\u5206\u67e5\u627e\u66f4\u52a0\u8282\u7701\u7a7a\u95f4\u3002</li> </ul> <p>\u7136\u800c\uff0c\u4e8c\u5206\u67e5\u627e\u5e76\u975e\u9002\u7528\u4e8e\u6240\u6709\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6709\u5e8f\u6570\u636e\u3002\u82e5\u8f93\u5165\u6570\u636e\u65e0\u5e8f\uff0c\u4e3a\u4e86\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u800c\u4e13\u95e8\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u4e0d\u507f\u5931\u3002\u56e0\u4e3a\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u901a\u5e38\u4e3a \\(O(n \\log n)\\) \uff0c\u6bd4\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u90fd\u66f4\u9ad8\u3002\u5bf9\u4e8e\u9891\u7e41\u63d2\u5165\u5143\u7d20\u7684\u573a\u666f\uff0c\u4e3a\u4fdd\u6301\u6570\u7ec4\u6709\u5e8f\u6027\uff0c\u9700\u8981\u5c06\u5143\u7d20\u63d2\u5165\u5230\u7279\u5b9a\u4f4d\u7f6e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u4e5f\u662f\u975e\u5e38\u6602\u8d35\u7684\u3002</li> <li>\u4e8c\u5206\u67e5\u627e\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002\u4e8c\u5206\u67e5\u627e\u9700\u8981\u8df3\u8dc3\u5f0f\uff08\u975e\u8fde\u7eed\u5730\uff09\u8bbf\u95ee\u5143\u7d20\uff0c\u800c\u5728\u94fe\u8868\u4e2d\u6267\u884c\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u7684\u6548\u7387\u8f83\u4f4e\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5e94\u7528\u5728\u94fe\u8868\u6216\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5c0f\u6570\u636e\u91cf\u4e0b\uff0c\u7ebf\u6027\u67e5\u627e\u6027\u80fd\u66f4\u4f73\u3002\u5728\u7ebf\u6027\u67e5\u627e\u4e2d\uff0c\u6bcf\u8f6e\u53ea\u9700 1 \u6b21\u5224\u65ad\u64cd\u4f5c\uff1b\u800c\u5728\u4e8c\u5206\u67e5\u627e\u4e2d\uff0c\u9700\u8981 1 \u6b21\u52a0\u6cd5\u30011 \u6b21\u9664\u6cd5\u30011 ~ 3 \u6b21\u5224\u65ad\u64cd\u4f5c\u30011 \u6b21\u52a0\u6cd5\uff08\u51cf\u6cd5\uff09\uff0c\u5171 4 ~ 6 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u56e0\u6b64\uff0c\u5f53\u6570\u636e\u91cf \\(n\\) \u8f83\u5c0f\u65f6\uff0c\u7ebf\u6027\u67e5\u627e\u53cd\u800c\u6bd4\u4e8c\u5206\u67e5\u627e\u66f4\u5feb\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_edge/","title":"10.3 \u00a0 \u4e8c\u5206\u67e5\u627e\u8fb9\u754c","text":""},{"location":"chapter_searching/binary_search_edge/#1031","title":"10.3.1 \u00a0 \u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\u3002\u8bf7\u8fd4\u56de\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a\u5143\u7d20 <code>target</code> \u7684\u7d22\u5f15\u3002\u82e5\u6570\u7ec4\u4e2d\u4e0d\u5305\u542b\u8be5\u5143\u7d20\uff0c\u5219\u8fd4\u56de \\(-1\\) \u3002</p> <p>\u56de\u5fc6\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u7684\u65b9\u6cd5\uff0c\u641c\u7d22\u5b8c\u6210\u540e \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u67e5\u627e\u63d2\u5165\u70b9\u672c\u8d28\u4e0a\u662f\u5728\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u8003\u8651\u901a\u8fc7\u67e5\u627e\u63d2\u5165\u70b9\u7684\u51fd\u6570\u5b9e\u73b0\u67e5\u627e\u5de6\u8fb9\u754c\u3002\u8bf7\u6ce8\u610f\uff0c\u6570\u7ec4\u4e2d\u53ef\u80fd\u4e0d\u5305\u542b <code>target</code> \uff0c\u8fd9\u79cd\u60c5\u51b5\u53ef\u80fd\u5bfc\u81f4\u4ee5\u4e0b\u4e24\u79cd\u7ed3\u679c\u3002</p> <ul> <li>\u63d2\u5165\u70b9\u7684\u7d22\u5f15 \\(i\\) \u8d8a\u754c\u3002</li> <li>\u5143\u7d20 <code>nums[i]</code> \u4e0e <code>target</code> \u4e0d\u76f8\u7b49\u3002</li> </ul> <p>\u5f53\u9047\u5230\u4ee5\u4e0a\u4e24\u79cd\u60c5\u51b5\u65f6\uff0c\u76f4\u63a5\u8fd4\u56de \\(-1\\) \u5373\u53ef\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py<pre><code>def binary_search_left_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target\"\"\"\n    # \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    i = binary_search_insertion(nums, target)\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) or nums[i] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.size() || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binary_search_insertion.binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint BinarySearchLeftEdge(int[] nums, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == nums.Length || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums []int, target int) int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    i := binarySearchInsertion(nums, target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == len(nums) || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunc binarySearchLeftEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    let i = binarySearchInsertion(nums: nums, target: target)\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.count || nums[i] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums, target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfunction binarySearchLeftEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    const i = binarySearchInsertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i === nums.length || nums[i] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(List&lt;int&gt; nums, int target) {\n  // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n  int i = binarySearchInsertion(nums, target);\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (i == nums.length || nums[i] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n  return i;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nfn binary_search_left_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    let i = binary_search_insertion(nums, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if i == nums.len() as i32 || nums[i as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    i\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target */\nint binarySearchLeftEdge(int *nums, int numSize, int target) {\n    // \u7b49\u4ef7\u4e8e\u67e5\u627e target \u7684\u63d2\u5165\u70b9\n    int i = binarySearchInsertion(nums, numSize, target);\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (i == numSize || nums[i] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 i\n    return i;\n}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchLeftEdge}\n</code></pre>"},{"location":"chapter_searching/binary_search_edge/#1032","title":"10.3.2 \u00a0 \u67e5\u627e\u53f3\u8fb9\u754c","text":"<p>\u90a3\u4e48\u5982\u4f55\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u5462\uff1f\u6700\u76f4\u63a5\u7684\u65b9\u5f0f\u662f\u4fee\u6539\u4ee3\u7801\uff0c\u66ff\u6362\u5728 <code>nums[m] == target</code> \u60c5\u51b5\u4e0b\u7684\u6307\u9488\u6536\u7f29\u64cd\u4f5c\u3002\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e0b\u9762\u6211\u4eec\u4ecb\u7ecd\u4e24\u79cd\u66f4\u52a0\u53d6\u5de7\u7684\u65b9\u6cd5\u3002</p>"},{"location":"chapter_searching/binary_search_edge/#1","title":"1. \u00a0 \u590d\u7528\u67e5\u627e\u5de6\u8fb9\u754c","text":"<p>\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u67e5\u627e\u6700\u5de6\u5143\u7d20\u7684\u51fd\u6570\u6765\u67e5\u627e\u6700\u53f3\u5143\u7d20\uff0c\u5177\u4f53\u65b9\u6cd5\u4e3a\uff1a\u5c06\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\u3002</p> <p>\u5982\u56fe 10-7 \u6240\u793a\uff0c\u67e5\u627e\u5b8c\u6210\u540e\uff0c\u6307\u9488 \\(i\\) \u6307\u5411\u6700\u5de6\u4e00\u4e2a <code>target + 1</code>\uff08\u5982\u679c\u5b58\u5728\uff09\uff0c\u800c \\(j\\) \u6307\u5411\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff0c\u56e0\u6b64\u8fd4\u56de \\(j\\) \u5373\u53ef\u3002</p> <p></p> <p> \u56fe 10-7 \u00a0 \u5c06\u67e5\u627e\u53f3\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5de6\u8fb9\u754c </p> <p>\u8bf7\u6ce8\u610f\uff0c\u8fd4\u56de\u7684\u63d2\u5165\u70b9\u662f \\(i\\) \uff0c\u56e0\u6b64\u9700\u8981\u5c06\u5176\u51cf \\(1\\) \uff0c\u4ece\u800c\u83b7\u5f97 \\(j\\) \uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_edge.py<pre><code>def binary_search_right_edge(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target\"\"\"\n    # \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    i = binary_search_insertion(nums, target + 1)\n    # j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    j = i - 1\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 or nums[j] != target:\n        return -1\n    # \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n</code></pre> binary_search_edge.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(vector&lt;int&gt; &amp;nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int[] nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binary_search_insertion.binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint BinarySearchRightEdge(int[] nums, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binary_search_insertion.BinarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums []int, target int) int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    i := binarySearchInsertion(nums, target+1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    j := i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunc binarySearchRightEdge(nums: [Int], target: Int) -&gt; Int {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    let i = binarySearchInsertion(nums: nums, target: target + 1)\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    let j = i - 1\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j] != target {\n        return -1\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j\n}\n</code></pre> binary_search_edge.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums, target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfunction binarySearchRightEdge(nums: Array&lt;number&gt;, target: number): number {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    const i = binarySearchInsertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    const j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j === -1 || nums[j] !== target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(List&lt;int&gt; nums, int target) {\n  // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n  int i = binarySearchInsertion(nums, target + 1);\n  // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n  int j = i - 1;\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n  if (j == -1 || nums[j] != target) {\n    return -1;\n  }\n  // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n  return j;\n}\n</code></pre> binary_search_edge.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nfn binary_search_right_edge(nums: &amp;[i32], target: i32) -&gt; i32 {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    let i = binary_search_insertion(nums, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    let j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if j == -1 || nums[j as usize] != target {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    j\n}\n</code></pre> binary_search_edge.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u6700\u53f3\u4e00\u4e2a target */\nint binarySearchRightEdge(int *nums, int numSize, int target) {\n    // \u8f6c\u5316\u4e3a\u67e5\u627e\u6700\u5de6\u4e00\u4e2a target + 1\n    int i = binarySearchInsertion(nums, numSize, target + 1);\n    // j \u6307\u5411\u6700\u53f3\u4e00\u4e2a target \uff0ci \u6307\u5411\u9996\u4e2a\u5927\u4e8e target \u7684\u5143\u7d20\n    int j = i - 1;\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de -1\n    if (j == -1 || nums[j] != target) {\n        return -1;\n    }\n    // \u627e\u5230 target \uff0c\u8fd4\u56de\u7d22\u5f15 j\n    return j;\n}\n</code></pre> binary_search_edge.zig<pre><code>[class]{}-[func]{binarySearchRightEdge}\n</code></pre>"},{"location":"chapter_searching/binary_search_edge/#2","title":"2. \u00a0 \u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20","text":"<p>\u6211\u4eec\u77e5\u9053\uff0c\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u6700\u7ec8 \\(i\\) \u548c \\(j\\) \u4f1a\u5206\u522b\u6307\u5411\u9996\u4e2a\u5927\u4e8e\u3001\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002</p> <p>\u56e0\u6b64\uff0c\u5982\u56fe 10-8 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728\u7684\u5143\u7d20\uff0c\u7528\u4e8e\u67e5\u627e\u5de6\u53f3\u8fb9\u754c\u3002</p> <ul> <li>\u67e5\u627e\u6700\u5de6\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target - 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(i\\) \u3002</li> <li>\u67e5\u627e\u6700\u53f3\u4e00\u4e2a <code>target</code> \uff1a\u53ef\u4ee5\u8f6c\u5316\u4e3a\u67e5\u627e <code>target + 0.5</code> \uff0c\u5e76\u8fd4\u56de\u6307\u9488 \\(j\\) \u3002</li> </ul> <p></p> <p> \u56fe 10-8 \u00a0 \u5c06\u67e5\u627e\u8fb9\u754c\u8f6c\u5316\u4e3a\u67e5\u627e\u5143\u7d20 </p> <p>\u4ee3\u7801\u5728\u6b64\u7701\u7565\uff0c\u4ee5\u4e0b\u4e24\u70b9\u503c\u5f97\u6ce8\u610f\u3002</p> <ul> <li>\u7ed9\u5b9a\u6570\u7ec4\u4e0d\u5305\u542b\u5c0f\u6570\uff0c\u8fd9\u610f\u5473\u7740\u6211\u4eec\u65e0\u987b\u5173\u5fc3\u5982\u4f55\u5904\u7406\u76f8\u7b49\u7684\u60c5\u51b5\u3002</li> <li>\u56e0\u4e3a\u8be5\u65b9\u6cd5\u5f15\u5165\u4e86\u5c0f\u6570\uff0c\u6240\u4ee5\u9700\u8981\u5c06\u51fd\u6570\u4e2d\u7684\u53d8\u91cf <code>target</code> \u6539\u4e3a\u6d6e\u70b9\u6570\u7c7b\u578b\u3002</li> </ul>"},{"location":"chapter_searching/binary_search_insertion/","title":"10.2 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9","text":"<p>\u4e8c\u5206\u67e5\u627e\u4e0d\u4ec5\u53ef\u7528\u4e8e\u641c\u7d22\u76ee\u6807\u5143\u7d20\uff0c\u8fd8\u53ef\u7528\u4e8e\u89e3\u51b3\u8bb8\u591a\u53d8\u79cd\u95ee\u9898\uff0c\u6bd4\u5982\u641c\u7d22\u76ee\u6807\u5143\u7d20\u7684\u63d2\u5165\u4f4d\u7f6e\u3002</p>"},{"location":"chapter_searching/binary_search_insertion/#1021","title":"10.2.1 \u00a0 \u65e0\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6709\u5e8f\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u5143\u7d20 <code>target</code> \uff0c\u6570\u7ec4\u4e0d\u5b58\u5728\u91cd\u590d\u5143\u7d20\u3002\u73b0\u5c06 <code>target</code> \u63d2\u5165\u6570\u7ec4 <code>nums</code> \u4e2d\uff0c\u5e76\u4fdd\u6301\u5176\u6709\u5e8f\u6027\u3002\u82e5\u6570\u7ec4\u4e2d\u5df2\u5b58\u5728\u5143\u7d20 <code>target</code> \uff0c\u5219\u63d2\u5165\u5230\u5176\u5de6\u65b9\u3002\u8bf7\u8fd4\u56de\u63d2\u5165\u540e <code>target</code> \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 10-4 \u00a0 \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\u793a\u4f8b\u6570\u636e </p> <p>\u5982\u679c\u60f3\u590d\u7528\u4e0a\u4e00\u8282\u7684\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\uff0c\u5219\u9700\u8981\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\u3002</p> <p>\u95ee\u9898\u4e00\uff1a\u5f53\u6570\u7ec4\u4e2d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u662f\u5426\u662f\u8be5\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u9898\u76ee\u8981\u6c42\u5c06 <code>target</code> \u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u5de6\u8fb9\uff0c\u8fd9\u610f\u5473\u7740\u65b0\u63d2\u5165\u7684 <code>target</code> \u66ff\u6362\u4e86\u539f\u6765 <code>target</code> \u7684\u4f4d\u7f6e\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6570\u7ec4\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u7684\u7d22\u5f15\u5c31\u662f\u8be5 <code>target</code> \u7684\u7d22\u5f15\u3002</p> <p>\u95ee\u9898\u4e8c\uff1a\u5f53\u6570\u7ec4\u4e2d\u4e0d\u5b58\u5728 <code>target</code> \u65f6\uff0c\u63d2\u5165\u70b9\u662f\u54ea\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\uff1f</p> <p>\u8fdb\u4e00\u6b65\u601d\u8003\u4e8c\u5206\u67e5\u627e\u8fc7\u7a0b\uff1a\u5f53 <code>nums[m] &lt; target</code> \u65f6 \\(i\\) \u79fb\u52a8\uff0c\u8fd9\u610f\u5473\u7740\u6307\u9488 \\(i\\) \u5728\u5411\u5927\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002\u540c\u7406\uff0c\u6307\u9488 \\(j\\) \u59cb\u7ec8\u5728\u5411\u5c0f\u4e8e\u7b49\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</p> <p>\u56e0\u6b64\u4e8c\u5206\u7ed3\u675f\u65f6\u4e00\u5b9a\u6709\uff1a\\(i\\) \u6307\u5411\u9996\u4e2a\u5927\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u3002\u6613\u5f97\u5f53\u6570\u7ec4\u4e0d\u5305\u542b <code>target</code> \u65f6\uff0c\u63d2\u5165\u7d22\u5f15\u4e3a \\(i\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.py<pre><code>def binary_search_insertion_simple(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            return m  # \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n    # \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertionSimple(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n            return m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertionSimple(nums: [Int], target: Int) -&gt; Int {\n    var i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertionSimple(\n    nums: Array&lt;number&gt;,\n    target: number\n): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n    } else {\n      return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n    }\n  }\n  // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1);    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1;  // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u65e0\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertionSimple(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            return m; // \u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 m\n        }\n    }\n    // \u672a\u627e\u5230 target \uff0c\u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertionSimple}\n</code></pre>"},{"location":"chapter_searching/binary_search_insertion/#1022","title":"10.2.2 \u00a0 \u5b58\u5728\u91cd\u590d\u5143\u7d20\u7684\u60c5\u51b5","text":"<p>Question</p> <p>\u5728\u4e0a\u4e00\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u89c4\u5b9a\u6570\u7ec4\u53ef\u80fd\u5305\u542b\u91cd\u590d\u5143\u7d20\uff0c\u5176\u4f59\u4e0d\u53d8\u3002</p> <p>\u5047\u8bbe\u6570\u7ec4\u4e2d\u5b58\u5728\u591a\u4e2a <code>target</code> \uff0c\u5219\u666e\u901a\u4e8c\u5206\u67e5\u627e\u53ea\u80fd\u8fd4\u56de\u5176\u4e2d\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u800c\u65e0\u6cd5\u786e\u5b9a\u8be5\u5143\u7d20\u7684\u5de6\u8fb9\u548c\u53f3\u8fb9\u8fd8\u6709\u591a\u5c11 <code>target</code>\u3002</p> <p>\u9898\u76ee\u8981\u6c42\u5c06\u76ee\u6807\u5143\u7d20\u63d2\u5165\u5230\u6700\u5de6\u8fb9\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u67e5\u627e\u6570\u7ec4\u4e2d\u6700\u5de6\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\u3002\u521d\u6b65\u8003\u8651\u901a\u8fc7\u56fe 10-5 \u6240\u793a\u7684\u6b65\u9aa4\u5b9e\u73b0\u3002</p> <ol> <li>\u6267\u884c\u4e8c\u5206\u67e5\u627e\uff0c\u5f97\u5230\u4efb\u610f\u4e00\u4e2a <code>target</code> \u7684\u7d22\u5f15\uff0c\u8bb0\u4e3a \\(k\\) \u3002</li> <li>\u4ece\u7d22\u5f15 \\(k\\) \u5f00\u59cb\uff0c\u5411\u5de6\u8fdb\u884c\u7ebf\u6027\u904d\u5386\uff0c\u5f53\u627e\u5230\u6700\u5de6\u8fb9\u7684 <code>target</code> \u65f6\u8fd4\u56de\u3002</li> </ol> <p></p> <p> \u56fe 10-5 \u00a0 \u7ebf\u6027\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9 </p> <p>\u6b64\u65b9\u6cd5\u867d\u7136\u53ef\u7528\uff0c\u4f46\u5176\u5305\u542b\u7ebf\u6027\u67e5\u627e\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5f53\u6570\u7ec4\u4e2d\u5b58\u5728\u5f88\u591a\u91cd\u590d\u7684 <code>target</code> \u65f6\uff0c\u8be5\u65b9\u6cd5\u6548\u7387\u5f88\u4f4e\u3002</p> <p>\u73b0\u8003\u8651\u62d3\u5c55\u4e8c\u5206\u67e5\u627e\u4ee3\u7801\u3002\u5982\u56fe 10-6 \u6240\u793a\uff0c\u6574\u4f53\u6d41\u7a0b\u4fdd\u6301\u4e0d\u53d8\uff0c\u6bcf\u8f6e\u5148\u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 \\(m\\) \uff0c\u518d\u5224\u65ad <code>target</code> \u548c <code>nums[m]</code> \u7684\u5927\u5c0f\u5173\u7cfb\uff0c\u5206\u4e3a\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5\u3002</p> <ul> <li>\u5f53 <code>nums[m] &lt; target</code> \u6216 <code>nums[m] &gt; target</code> \u65f6\uff0c\u8bf4\u660e\u8fd8\u6ca1\u6709\u627e\u5230 <code>target</code> \uff0c\u56e0\u6b64\u91c7\u7528\u666e\u901a\u4e8c\u5206\u67e5\u627e\u7684\u7f29\u5c0f\u533a\u95f4\u64cd\u4f5c\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5411 <code>target</code> \u9760\u8fd1\u3002</li> <li>\u5f53 <code>nums[m] == target</code> \u65f6\uff0c\u8bf4\u660e\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u5728\u533a\u95f4 \\([i, m - 1]\\) \u4e2d\uff0c\u56e0\u6b64\u91c7\u7528 \\(j = m - 1\\) \u6765\u7f29\u5c0f\u533a\u95f4\uff0c\u4ece\u800c\u4f7f\u6307\u9488 \\(j\\) \u5411\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\u9760\u8fd1\u3002</li> </ul> <p>\u5faa\u73af\u5b8c\u6210\u540e\uff0c\\(i\\) \u6307\u5411\u6700\u5de6\u8fb9\u7684 <code>target</code> \uff0c\\(j\\) \u6307\u5411\u9996\u4e2a\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff0c\u56e0\u6b64\u7d22\u5f15 \\(i\\) \u5c31\u662f\u63d2\u5165\u70b9\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 10-6 \u00a0 \u4e8c\u5206\u67e5\u627e\u91cd\u590d\u5143\u7d20\u7684\u63d2\u5165\u70b9\u7684\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u4ee5\u4e0b\u4ee3\u7801\uff0c\u5224\u65ad\u5206\u652f <code>nums[m] &gt; target</code> \u548c <code>nums[m] == target</code> \u7684\u64cd\u4f5c\u76f8\u540c\uff0c\u56e0\u6b64\u4e24\u8005\u53ef\u4ee5\u5408\u5e76\u3002</p> <p>\u5373\u4fbf\u5982\u6b64\uff0c\u6211\u4eec\u4ecd\u7136\u53ef\u4ee5\u5c06\u5224\u65ad\u6761\u4ef6\u4fdd\u6301\u5c55\u5f00\uff0c\u56e0\u4e3a\u5176\u903b\u8f91\u66f4\u52a0\u6e05\u6670\u3001\u53ef\u8bfb\u6027\u66f4\u597d\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_insertion.py<pre><code>def binary_search_insertion(nums: list[int], target: int) -&gt; int:\n    \"\"\"\u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09\"\"\"\n    i, j = 0, len(nums) - 1  # \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j:\n        m = (i + j) // 2  # \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target:\n            i = m + 1  # target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        elif nums[m] &gt; target:\n            j = m - 1  # target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        else:\n            j = m - 1  # \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n    # \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n</code></pre> binary_search_insertion.cpp<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(vector&lt;int&gt; &amp;nums, int target) {\n    int i = 0, j = nums.size() - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.java<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.cs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint BinarySearchInsertion(int[] nums, int target) {\n    int i = 0, j = nums.Length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.go<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums []int, target int) int {\n    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    i, j := 0, len(nums)-1\n    for i &lt;= j {\n        // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        m := i + (j-i)/2\n        if nums[m] &lt; target {\n            // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n            i = m + 1\n        } else if nums[m] &gt; target {\n            // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        } else {\n            // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n            j = m - 1\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.swift<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunc binarySearchInsertion(nums: [Int], target: Int) -&gt; Int {\n    var i = 0, j = nums.count - 1 // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2 // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m] &lt; target {\n            i = m + 1 // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m] &gt; target {\n            j = m - 1 // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1 // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i\n}\n</code></pre> binary_search_insertion.js<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums, target) {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.ts<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nfunction binarySearchInsertion(nums: Array&lt;number&gt;, target: number): number {\n    let i = 0,\n        j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        const m = Math.floor(i + (j - i) / 2); // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m, \u4f7f\u7528 Math.floor() \u5411\u4e0b\u53d6\u6574\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.dart<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(List&lt;int&gt; nums, int target) {\n  int i = 0, j = nums.length - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n  while (i &lt;= j) {\n    int m = i + (j - i) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n    if (nums[m] &lt; target) {\n      i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n    } else if (nums[m] &gt; target) {\n      j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n    } else {\n      j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n    }\n  }\n  // \u8fd4\u56de\u63d2\u5165\u70b9 i\n  return i;\n}\n</code></pre> binary_search_insertion.rs<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\npub fn binary_search_insertion(nums: &amp;[i32], target: i32) -&gt; i32 {\n    let (mut i, mut j) = (0, nums.len() as i32 - 1);    // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while i &lt;= j {\n        let m = i + (j - i) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if nums[m as usize] &lt; target {\n            i = m + 1;  // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if nums[m as usize] &gt; target {\n            j = m - 1;  // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1;  // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    i\n}\n</code></pre> binary_search_insertion.c<pre><code>/* \u4e8c\u5206\u67e5\u627e\u63d2\u5165\u70b9\uff08\u5b58\u5728\u91cd\u590d\u5143\u7d20\uff09 */\nint binarySearchInsertion(int *nums, int numSize, int target) {\n    int i = 0, j = numSize - 1; // \u521d\u59cb\u5316\u53cc\u95ed\u533a\u95f4 [0, n-1]\n    while (i &lt;= j) {\n        int m = i + (j - i) / 2; // \u8ba1\u7b97\u4e2d\u70b9\u7d22\u5f15 m\n        if (nums[m] &lt; target) {\n            i = m + 1; // target \u5728\u533a\u95f4 [m+1, j] \u4e2d\n        } else if (nums[m] &gt; target) {\n            j = m - 1; // target \u5728\u533a\u95f4 [i, m-1] \u4e2d\n        } else {\n            j = m - 1; // \u9996\u4e2a\u5c0f\u4e8e target \u7684\u5143\u7d20\u5728\u533a\u95f4 [i, m-1] \u4e2d\n        }\n    }\n    // \u8fd4\u56de\u63d2\u5165\u70b9 i\n    return i;\n}\n</code></pre> binary_search_insertion.zig<pre><code>[class]{}-[func]{binarySearchInsertion}\n</code></pre> <p>Tip</p> <p>\u672c\u8282\u7684\u4ee3\u7801\u90fd\u662f\u201c\u53cc\u95ed\u533a\u95f4\u201d\u5199\u6cd5\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u5b9e\u73b0\u201c\u5de6\u95ed\u53f3\u5f00\u201d\u5199\u6cd5\u3002</p> <p>\u603b\u7684\u6765\u770b\uff0c\u4e8c\u5206\u67e5\u627e\u65e0\u975e\u5c31\u662f\u7ed9\u6307\u9488 \\(i\\) \u548c \\(j\\) \u5206\u522b\u8bbe\u5b9a\u641c\u7d22\u76ee\u6807\uff0c\u76ee\u6807\u53ef\u80fd\u662f\u4e00\u4e2a\u5177\u4f53\u7684\u5143\u7d20\uff08\u4f8b\u5982 <code>target</code> \uff09\uff0c\u4e5f\u53ef\u80fd\u662f\u4e00\u4e2a\u5143\u7d20\u8303\u56f4\uff08\u4f8b\u5982\u5c0f\u4e8e <code>target</code> \u7684\u5143\u7d20\uff09\u3002</p> <p>\u5728\u4e0d\u65ad\u7684\u5faa\u73af\u4e8c\u5206\u4e2d\uff0c\u6307\u9488 \\(i\\) \u548c \\(j\\) \u90fd\u9010\u6e10\u903c\u8fd1\u9884\u5148\u8bbe\u5b9a\u7684\u76ee\u6807\u3002\u6700\u7ec8\uff0c\u5b83\u4eec\u6216\u662f\u6210\u529f\u627e\u5230\u7b54\u6848\uff0c\u6216\u662f\u8d8a\u8fc7\u8fb9\u754c\u540e\u505c\u6b62\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/","title":"10.4 \u00a0 \u54c8\u5e0c\u4f18\u5316\u7b56\u7565","text":"<p>\u5728\u7b97\u6cd5\u9898\u4e2d\uff0c\u6211\u4eec\u5e38\u901a\u8fc7\u5c06\u7ebf\u6027\u67e5\u627e\u66ff\u6362\u4e3a\u54c8\u5e0c\u67e5\u627e\u6765\u964d\u4f4e\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u6211\u4eec\u501f\u52a9\u4e00\u4e2a\u7b97\u6cd5\u9898\u6765\u52a0\u6df1\u7406\u89e3\u3002</p> <p>Question</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6574\u6570\u6570\u7ec4 <code>nums</code> \u548c\u4e00\u4e2a\u76ee\u6807\u5143\u7d20 <code>target</code> \uff0c\u8bf7\u5728\u6570\u7ec4\u4e2d\u641c\u7d22\u201c\u548c\u201d\u4e3a <code>target</code> \u7684\u4e24\u4e2a\u5143\u7d20\uff0c\u5e76\u8fd4\u56de\u5b83\u4eec\u7684\u6570\u7ec4\u7d22\u5f15\u3002\u8fd4\u56de\u4efb\u610f\u4e00\u4e2a\u89e3\u5373\u53ef\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1041","title":"10.4.1 \u00a0 \u7ebf\u6027\u67e5\u627e\uff1a\u4ee5\u65f6\u95f4\u6362\u7a7a\u95f4","text":"<p>\u8003\u8651\u76f4\u63a5\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u7ec4\u5408\u3002\u5982\u56fe 10-9 \u6240\u793a\uff0c\u6211\u4eec\u5f00\u542f\u4e00\u4e2a\u4e24\u5c42\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u5224\u65ad\u4e24\u4e2a\u6574\u6570\u7684\u548c\u662f\u5426\u4e3a <code>target</code> \uff0c\u82e5\u662f\uff0c\u5219\u8fd4\u56de\u5b83\u4eec\u7684\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 10-9 \u00a0 \u7ebf\u6027\u67e5\u627e\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py<pre><code>def two_sum_brute_force(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\"\"\"\n    # \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in range(len(nums) - 1):\n        for j in range(i + 1, len(nums)):\n            if nums[i] + nums[j] == target:\n                return [i, j]\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nvector&lt;int&gt; twoSumBruteForce(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return {i, j};\n        }\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] twoSumBruteForce(int[] nums, int target) {\n    int size = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return new int[] { i, j };\n        }\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint[] TwoSumBruteForce(int[] nums, int target) {\n    int size = nums.Length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (int i = 0; i &lt; size - 1; i++) {\n        for (int j = i + 1; j &lt; size; j++) {\n            if (nums[i] + nums[j] == target)\n                return [i, j];\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums []int, target int) []int {\n    size := len(nums)\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i := 0; i &lt; size-1; i++ {\n        for j := i + 1; i &lt; size; j++ {\n            if nums[i]+nums[j] == target {\n                return []int{i, j}\n            }\n        }\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunc twoSumBruteForce(nums: [Int], target: Int) -&gt; [Int] {\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in nums.indices.dropLast() {\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[i] + nums[j] == target {\n                return [i, j]\n            }\n        }\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums, target) {\n    const n = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nfunction twoSumBruteForce(nums: number[], target: number): number[] {\n    const n = nums.length;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for (let i = 0; i &lt; n; i++) {\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[i] + nums[j] === target) {\n                return [i, j];\n            }\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e00\uff1a \u66b4\u529b\u679a\u4e3e */\nList&lt;int&gt; twoSumBruteForce(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n  for (var i = 0; i &lt; size - 1; i++) {\n    for (var j = i + 1; j &lt; size; j++) {\n      if (nums[i] + nums[j] == target) return [i, j];\n    }\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\npub fn two_sum_brute_force(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    let size = nums.len();\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    for i in 0..size - 1 {\n        for j in i + 1..size {\n            if nums[i] + nums[j] == target {\n                return Some(vec![i as i32, j as i32]);\n            }\n        }\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e */\nint *twoSumBruteForce(int *nums, int numsSize, int target, int *returnSize) {\n    for (int i = 0; i &lt; numsSize; ++i) {\n        for (int j = i + 1; j &lt; numsSize; ++j) {\n            if (nums[i] + nums[j] == target) {\n                int *res = malloc(sizeof(int) * 2);\n                res[0] = i, res[1] = j;\n                *returnSize = 2;\n                return res;\n            }\n        }\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e00\uff1a\u66b4\u529b\u679a\u4e3e\nfn twoSumBruteForce(nums: []i32, target: i32) ?[2]i32 {\n    var size: usize = nums.len;\n    var i: usize = 0;\n    // \u4e24\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n^2)\n    while (i &lt; size - 1) : (i += 1) {\n        var j = i + 1;\n        while (j &lt; size) : (j += 1) {\n            if (nums[i] + nums[j] == target) {\n                return [_]i32{@intCast(i), @intCast(j)};\n            }\n        }\n    }\n    return null;\n}\n</code></pre> <p>\u6b64\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \uff0c\u5728\u5927\u6570\u636e\u91cf\u4e0b\u975e\u5e38\u8017\u65f6\u3002</p>"},{"location":"chapter_searching/replace_linear_by_hashing/#1042","title":"10.4.2 \u00a0 \u54c8\u5e0c\u67e5\u627e\uff1a\u4ee5\u7a7a\u95f4\u6362\u65f6\u95f4","text":"<p>\u8003\u8651\u501f\u52a9\u4e00\u4e2a\u54c8\u5e0c\u8868\uff0c\u952e\u503c\u5bf9\u5206\u522b\u4e3a\u6570\u7ec4\u5143\u7d20\u548c\u5143\u7d20\u7d22\u5f15\u3002\u5faa\u73af\u904d\u5386\u6570\u7ec4\uff0c\u6bcf\u8f6e\u6267\u884c\u56fe 10-10 \u6240\u793a\u7684\u6b65\u9aa4\u3002</p> <ol> <li>\u5224\u65ad\u6570\u5b57 <code>target - nums[i]</code> \u662f\u5426\u5728\u54c8\u5e0c\u8868\u4e2d\uff0c\u82e5\u662f\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\u8fd9\u4e24\u4e2a\u5143\u7d20\u7684\u7d22\u5f15\u3002</li> <li>\u5c06\u952e\u503c\u5bf9 <code>nums[i]</code> \u548c\u7d22\u5f15 <code>i</code> \u6dfb\u52a0\u8fdb\u54c8\u5e0c\u8868\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt; <p></p> <p></p> <p></p> <p> \u56fe 10-10 \u00a0 \u8f85\u52a9\u54c8\u5e0c\u8868\u6c42\u89e3\u4e24\u6570\u4e4b\u548c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff0c\u4ec5\u9700\u5355\u5c42\u5faa\u73af\u5373\u53ef\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig two_sum.py<pre><code>def two_sum_hash_table(nums: list[int], target: int) -&gt; list[int]:\n    \"\"\"\u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\"\"\"\n    # \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    dic = {}\n    # \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for i in range(len(nums)):\n        if target - nums[i] in dic:\n            return [dic[target - nums[i]], i]\n        dic[nums[i]] = i\n    return []\n</code></pre> two_sum.cpp<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nvector&lt;int&gt; twoSumHashTable(vector&lt;int&gt; &amp;nums, int target) {\n    int size = nums.size();\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    unordered_map&lt;int, int&gt; dic;\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.find(target - nums[i]) != dic.end()) {\n            return {dic[target - nums[i]], i};\n        }\n        dic.emplace(nums[i], i);\n    }\n    return {};\n}\n</code></pre> two_sum.java<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] twoSumHashTable(int[] nums, int target) {\n    int size = nums.length;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    Map&lt;Integer, Integer&gt; dic = new HashMap&lt;&gt;();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.containsKey(target - nums[i])) {\n            return new int[] { dic.get(target - nums[i]), i };\n        }\n        dic.put(nums[i], i);\n    }\n    return new int[0];\n}\n</code></pre> two_sum.cs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint[] TwoSumHashTable(int[] nums, int target) {\n    int size = nums.Length;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    Dictionary&lt;int, int&gt; dic = [];\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (int i = 0; i &lt; size; i++) {\n        if (dic.ContainsKey(target - nums[i])) {\n            return [dic[target - nums[i]], i];\n        }\n        dic.Add(nums[i], i);\n    }\n    return [];\n}\n</code></pre> two_sum.go<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums []int, target int) []int {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    hashTable := map[int]int{}\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for idx, val := range nums {\n        if preIdx, ok := hashTable[target-val]; ok {\n            return []int{preIdx, idx}\n        }\n        hashTable[val] = idx\n    }\n    return nil\n}\n</code></pre> two_sum.swift<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunc twoSumHashTable(nums: [Int], target: Int) -&gt; [Int] {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    var dic: [Int: Int] = [:]\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for i in nums.indices {\n        if let j = dic[target - nums[i]] {\n            return [j, i]\n        }\n        dic[nums[i]] = i\n    }\n    return [0]\n}\n</code></pre> two_sum.js<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums, target) {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let m = {};\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        if (m[target - nums[i]] !== undefined) {\n            return [m[target - nums[i]], i];\n        } else {\n            m[nums[i]] = i;\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.ts<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nfunction twoSumHashTable(nums: number[], target: number): number[] {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let m: Map&lt;number, number&gt; = new Map();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (let i = 0; i &lt; nums.length; i++) {\n        let index = m.get(target - nums[i]);\n        if (index !== undefined) {\n            return [index, i];\n        } else {\n            m.set(nums[i], i);\n        }\n    }\n    return [];\n}\n</code></pre> two_sum.dart<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a \u8f85\u52a9\u54c8\u5e0c\u8868 */\nList&lt;int&gt; twoSumHashTable(List&lt;int&gt; nums, int target) {\n  int size = nums.length;\n  // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n  Map&lt;int, int&gt; dic = HashMap();\n  // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n  for (var i = 0; i &lt; size; i++) {\n    if (dic.containsKey(target - nums[i])) {\n      return [dic[target - nums[i]]!, i];\n    }\n    dic.putIfAbsent(nums[i], () =&gt; i);\n  }\n  return [0];\n}\n</code></pre> two_sum.rs<pre><code>/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\npub fn two_sum_hash_table(nums: &amp;Vec&lt;i32&gt;, target: i32) -&gt; Option&lt;Vec&lt;i32&gt;&gt; {\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    let mut dic = HashMap::new();\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    for (i, num) in nums.iter().enumerate() {\n        match dic.get(&amp;(target - num)) {\n            Some(v) =&gt; return Some(vec![*v as i32, i as i32]),\n            None =&gt; dic.insert(num, i as i32)\n        };\n    }\n    None\n}\n</code></pre> two_sum.c<pre><code>/* \u54c8\u5e0c\u8868 */\ntypedef struct {\n    int key;\n    int val;\n    UT_hash_handle hh; // \u57fa\u4e8e uthash.h \u5b9e\u73b0\n} HashTable;\n\n/* \u54c8\u5e0c\u8868\u67e5\u8be2 */\nHashTable *find(HashTable *h, int key) {\n    HashTable *tmp;\n    HASH_FIND_INT(h, &amp;key, tmp);\n    return tmp;\n}\n\n/* \u54c8\u5e0c\u8868\u5143\u7d20\u63d2\u5165 */\nvoid insert(HashTable *h, int key, int val) {\n    HashTable *t = find(h, key);\n    if (t == NULL) {\n        HashTable *tmp = malloc(sizeof(HashTable));\n        tmp-&gt;key = key, tmp-&gt;val = val;\n        HASH_ADD_INT(h, key, tmp);\n    } else {\n        t-&gt;val = val;\n    }\n}\n\n/* \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868 */\nint *twoSumHashTable(int *nums, int numsSize, int target, int *returnSize) {\n    HashTable *hashtable = NULL;\n    for (int i = 0; i &lt; numsSize; i++) {\n        HashTable *t = find(hashtable, target - nums[i]);\n        if (t != NULL) {\n            int *res = malloc(sizeof(int) * 2);\n            res[0] = t-&gt;val, res[1] = i;\n            *returnSize = 2;\n            return res;\n        }\n        insert(hashtable, nums[i], i);\n    }\n    *returnSize = 0;\n    return NULL;\n}\n</code></pre> two_sum.zig<pre><code>// \u65b9\u6cd5\u4e8c\uff1a\u8f85\u52a9\u54c8\u5e0c\u8868\nfn twoSumHashTable(nums: []i32, target: i32) !?[2]i32 {\n    var size: usize = nums.len;\n    // \u8f85\u52a9\u54c8\u5e0c\u8868\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    var dic = std.AutoHashMap(i32, i32).init(std.heap.page_allocator);\n    defer dic.deinit();\n    var i: usize = 0;\n    // \u5355\u5c42\u5faa\u73af\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\n    while (i &lt; size) : (i += 1) {\n        if (dic.contains(target - nums[i])) {\n            return [_]i32{dic.get(target - nums[i]).?, @intCast(i)};\n        }\n        try dic.put(nums[i], @intCast(i));\n    }\n    return null;\n}\n</code></pre> <p>\u6b64\u65b9\u6cd5\u901a\u8fc7\u54c8\u5e0c\u67e5\u627e\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n^2)\\) \u964d\u81f3 \\(O(n)\\) \uff0c\u5927\u5e45\u63d0\u5347\u8fd0\u884c\u6548\u7387\u3002</p> <p>\u7531\u4e8e\u9700\u8981\u7ef4\u62a4\u4e00\u4e2a\u989d\u5916\u7684\u54c8\u5e0c\u8868\uff0c\u56e0\u6b64\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8be5\u65b9\u6cd5\u7684\u6574\u4f53\u65f6\u7a7a\u6548\u7387\u66f4\u4e3a\u5747\u8861\uff0c\u56e0\u6b64\u5b83\u662f\u672c\u9898\u7684\u6700\u4f18\u89e3\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/","title":"10.5 \u00a0 \u91cd\u8bc6\u641c\u7d22\u7b97\u6cd5","text":"<p>\u300c\u641c\u7d22\u7b97\u6cd5 searching algorithm\u300d\u7528\u4e8e\u5728\u6570\u636e\u7ed3\u6784\uff08\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u6216\u56fe\uff09\u4e2d\u641c\u7d22\u4e00\u4e2a\u6216\u4e00\u7ec4\u6ee1\u8db3\u7279\u5b9a\u6761\u4ef6\u7684\u5143\u7d20\u3002</p> <p>\u641c\u7d22\u7b97\u6cd5\u53ef\u6839\u636e\u5b9e\u73b0\u601d\u8def\u5206\u4e3a\u4ee5\u4e0b\u4e24\u7c7b\u3002</p> <ul> <li>\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\uff0c\u4f8b\u5982\u6570\u7ec4\u3001\u94fe\u8868\u3001\u6811\u548c\u56fe\u7684\u904d\u5386\u7b49\u3002</li> <li>\u5229\u7528\u6570\u636e\u7ec4\u7ec7\u7ed3\u6784\u6216\u6570\u636e\u5305\u542b\u7684\u5148\u9a8c\u4fe1\u606f\uff0c\u5b9e\u73b0\u9ad8\u6548\u5143\u7d20\u67e5\u627e\uff0c\u4f8b\u5982\u4e8c\u5206\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u548c\u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u7b49\u3002</li> </ul> <p>\u4e0d\u96be\u53d1\u73b0\uff0c\u8fd9\u4e9b\u77e5\u8bc6\u70b9\u90fd\u5df2\u5728\u524d\u9762\u7684\u7ae0\u8282\u4e2d\u4ecb\u7ecd\u8fc7\uff0c\u56e0\u6b64\u641c\u7d22\u7b97\u6cd5\u5bf9\u4e8e\u6211\u4eec\u6765\u8bf4\u5e76\u4e0d\u964c\u751f\u3002\u5728\u672c\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4ece\u66f4\u52a0\u7cfb\u7edf\u7684\u89c6\u89d2\u5207\u5165\uff0c\u91cd\u65b0\u5ba1\u89c6\u641c\u7d22\u7b97\u6cd5\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1051","title":"10.5.1 \u00a0 \u66b4\u529b\u641c\u7d22","text":"<p>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u7684\u6bcf\u4e2a\u5143\u7d20\u6765\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u7ebf\u6027\u641c\u7d22\u201d\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\u7b49\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u5b83\u4ece\u6570\u636e\u7ed3\u6784\u7684\u4e00\u7aef\u5f00\u59cb\uff0c\u9010\u4e2a\u8bbf\u95ee\u5143\u7d20\uff0c\u76f4\u5230\u627e\u5230\u76ee\u6807\u5143\u7d20\u6216\u5230\u8fbe\u53e6\u4e00\u7aef\u4ecd\u6ca1\u6709\u627e\u5230\u76ee\u6807\u5143\u7d20\u4e3a\u6b62\u3002</li> <li>\u201c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u548c\u201c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u201d\u662f\u56fe\u548c\u6811\u7684\u4e24\u79cd\u904d\u5386\u7b56\u7565\u3002\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\u9010\u5c42\u641c\u7d22\uff0c\u7531\u8fd1\u53ca\u8fdc\u5730\u8bbf\u95ee\u5404\u4e2a\u8282\u70b9\u3002\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4ece\u521d\u59cb\u8282\u70b9\u5f00\u59cb\uff0c\u6cbf\u7740\u4e00\u6761\u8def\u5f84\u8d70\u5230\u5934\uff0c\u518d\u56de\u6eaf\u5e76\u5c1d\u8bd5\u5176\u4ed6\u8def\u5f84\uff0c\u76f4\u5230\u904d\u5386\u5b8c\u6574\u4e2a\u6570\u636e\u7ed3\u6784\u3002</li> </ul> <p>\u66b4\u529b\u641c\u7d22\u7684\u4f18\u70b9\u662f\u7b80\u5355\u4e14\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u505a\u9884\u5904\u7406\u548c\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u7136\u800c\uff0c\u6b64\u7c7b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u5176\u4e2d \\(n\\) \u4e3a\u5143\u7d20\u6570\u91cf\uff0c\u56e0\u6b64\u5728\u6570\u636e\u91cf\u8f83\u5927\u7684\u60c5\u51b5\u4e0b\u6027\u80fd\u8f83\u5dee\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1052","title":"10.5.2 \u00a0 \u81ea\u9002\u5e94\u641c\u7d22","text":"<p>\u81ea\u9002\u5e94\u641c\u7d22\u5229\u7528\u6570\u636e\u7684\u7279\u6709\u5c5e\u6027\uff08\u4f8b\u5982\u6709\u5e8f\u6027\uff09\u6765\u4f18\u5316\u641c\u7d22\u8fc7\u7a0b\uff0c\u4ece\u800c\u66f4\u9ad8\u6548\u5730\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</p> <ul> <li>\u201c\u4e8c\u5206\u67e5\u627e\u201d\u5229\u7528\u6570\u636e\u7684\u6709\u5e8f\u6027\u5b9e\u73b0\u9ad8\u6548\u67e5\u627e\uff0c\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u3002</li> <li>\u201c\u54c8\u5e0c\u67e5\u627e\u201d\u5229\u7528\u54c8\u5e0c\u8868\u5c06\u641c\u7d22\u6570\u636e\u548c\u76ee\u6807\u6570\u636e\u5efa\u7acb\u4e3a\u952e\u503c\u5bf9\u6620\u5c04\uff0c\u4ece\u800c\u5b9e\u73b0\u67e5\u8be2\u64cd\u4f5c\u3002</li> <li>\u201c\u6811\u67e5\u627e\u201d\u5728\u7279\u5b9a\u7684\u6811\u7ed3\u6784\uff08\u4f8b\u5982\u4e8c\u53c9\u641c\u7d22\u6811\uff09\u4e2d\uff0c\u57fa\u4e8e\u6bd4\u8f83\u8282\u70b9\u503c\u6765\u5feb\u901f\u6392\u9664\u8282\u70b9\uff0c\u4ece\u800c\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002</li> </ul> <p>\u6b64\u7c7b\u7b97\u6cd5\u7684\u4f18\u70b9\u662f\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \u3002</p> <p>\u7136\u800c\uff0c\u4f7f\u7528\u8fd9\u4e9b\u7b97\u6cd5\u5f80\u5f80\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\u3002\u4f8b\u5982\uff0c\u4e8c\u5206\u67e5\u627e\u9700\u8981\u9884\u5148\u5bf9\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\uff0c\u54c8\u5e0c\u67e5\u627e\u548c\u6811\u67e5\u627e\u90fd\u9700\u8981\u501f\u52a9\u989d\u5916\u7684\u6570\u636e\u7ed3\u6784\uff0c\u7ef4\u62a4\u8fd9\u4e9b\u6570\u636e\u7ed3\u6784\u4e5f\u9700\u8981\u989d\u5916\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u5f00\u9500\u3002</p> <p>Tip</p> <p>\u81ea\u9002\u5e94\u641c\u7d22\u7b97\u6cd5\u5e38\u88ab\u79f0\u4e3a\u67e5\u627e\u7b97\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u68c0\u7d22\u76ee\u6807\u5143\u7d20\u3002</p>"},{"location":"chapter_searching/searching_algorithm_revisited/#1053","title":"10.5.3 \u00a0 \u641c\u7d22\u65b9\u6cd5\u9009\u53d6","text":"<p>\u7ed9\u5b9a\u5927\u5c0f\u4e3a \\(n\\) \u7684\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u7ebf\u6027\u641c\u7d22\u3001\u4e8c\u5206\u67e5\u627e\u3001\u6811\u67e5\u627e\u3001\u54c8\u5e0c\u67e5\u627e\u7b49\u591a\u79cd\u65b9\u6cd5\u4ece\u4e2d\u641c\u7d22\u76ee\u6807\u5143\u7d20\u3002\u5404\u4e2a\u65b9\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u5982\u56fe 10-11 \u6240\u793a\u3002</p> <p></p> <p> \u56fe 10-11 \u00a0 \u591a\u79cd\u641c\u7d22\u7b56\u7565 </p> <p>\u4e0a\u8ff0\u51e0\u79cd\u65b9\u6cd5\u7684\u64cd\u4f5c\u6548\u7387\u4e0e\u7279\u6027\u5982\u8868 10-1 \u6240\u793a\u3002</p> <p> \u8868 10-1 \u00a0 \u67e5\u627e\u7b97\u6cd5\u6548\u7387\u5bf9\u6bd4 </p> \u7ebf\u6027\u641c\u7d22 \u4e8c\u5206\u67e5\u627e \u6811\u67e5\u627e \u54c8\u5e0c\u67e5\u627e \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(n)\\) \\(O(\\log n)\\) \\(O(1)\\) \u989d\u5916\u7a7a\u95f4 \\(O(1)\\) \\(O(1)\\) \\(O(n)\\) \\(O(n)\\) \u6570\u636e\u9884\u5904\u7406 / \u6392\u5e8f \\(O(n \\log n)\\) \u5efa\u6811 \\(O(n \\log n)\\) \u5efa\u54c8\u5e0c\u8868 \\(O(n)\\) \u6570\u636e\u662f\u5426\u6709\u5e8f \u65e0\u5e8f \u6709\u5e8f \u6709\u5e8f \u65e0\u5e8f <p>\u641c\u7d22\u7b97\u6cd5\u7684\u9009\u62e9\u8fd8\u53d6\u51b3\u4e8e\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u4e0e\u66f4\u65b0\u9891\u7387\u7b49\u3002</p> <p>\u7ebf\u6027\u641c\u7d22</p> <ul> <li>\u901a\u7528\u6027\u8f83\u597d\uff0c\u65e0\u987b\u4efb\u4f55\u6570\u636e\u9884\u5904\u7406\u64cd\u4f5c\u3002\u5047\u5982\u6211\u4eec\u4ec5\u9700\u67e5\u8be2\u4e00\u6b21\u6570\u636e\uff0c\u90a3\u4e48\u5176\u4ed6\u4e09\u79cd\u65b9\u6cd5\u7684\u6570\u636e\u9884\u5904\u7406\u7684\u65f6\u95f4\u6bd4\u7ebf\u6027\u641c\u7d22\u7684\u65f6\u95f4\u8fd8\u8981\u66f4\u957f\u3002</li> <li>\u9002\u7528\u4e8e\u4f53\u91cf\u8f83\u5c0f\u7684\u6570\u636e\uff0c\u6b64\u60c5\u51b5\u4e0b\u65f6\u95f4\u590d\u6742\u5ea6\u5bf9\u6548\u7387\u5f71\u54cd\u8f83\u5c0f\u3002</li> <li>\u9002\u7528\u4e8e\u6570\u636e\u66f4\u65b0\u9891\u7387\u8f83\u9ad8\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u8be5\u65b9\u6cd5\u4e0d\u9700\u8981\u5bf9\u6570\u636e\u8fdb\u884c\u4efb\u4f55\u989d\u5916\u7ef4\u62a4\u3002</li> </ul> <p>\u4e8c\u5206\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u6548\u7387\u8868\u73b0\u7a33\u5b9a\uff0c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \u3002</li> <li>\u6570\u636e\u91cf\u4e0d\u80fd\u8fc7\u5927\uff0c\u56e0\u4e3a\u5b58\u50a8\u6570\u7ec4\u9700\u8981\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u3002</li> <li>\u4e0d\u9002\u7528\u4e8e\u9ad8\u9891\u589e\u5220\u6570\u636e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u7ef4\u62a4\u6709\u5e8f\u6570\u7ec4\u7684\u5f00\u9500\u8f83\u5927\u3002</li> </ul> <p>\u54c8\u5e0c\u67e5\u627e</p> <ul> <li>\u9002\u5408\u5bf9\u67e5\u8be2\u6027\u80fd\u8981\u6c42\u5f88\u9ad8\u7684\u573a\u666f\uff0c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\) \u3002</li> <li>\u4e0d\u9002\u5408\u9700\u8981\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u65e0\u6cd5\u7ef4\u62a4\u6570\u636e\u7684\u6709\u5e8f\u6027\u3002</li> <li>\u5bf9\u54c8\u5e0c\u51fd\u6570\u548c\u54c8\u5e0c\u51b2\u7a81\u5904\u7406\u7b56\u7565\u7684\u4f9d\u8d56\u6027\u8f83\u9ad8\uff0c\u5177\u6709\u8f83\u5927\u7684\u6027\u80fd\u52a3\u5316\u98ce\u9669\u3002</li> <li>\u4e0d\u9002\u5408\u6570\u636e\u91cf\u8fc7\u5927\u7684\u60c5\u51b5\uff0c\u56e0\u4e3a\u54c8\u5e0c\u8868\u9700\u8981\u989d\u5916\u7a7a\u95f4\u6765\u6700\u5927\u7a0b\u5ea6\u5730\u51cf\u5c11\u51b2\u7a81\uff0c\u4ece\u800c\u63d0\u4f9b\u826f\u597d\u7684\u67e5\u8be2\u6027\u80fd\u3002</li> </ul> <p>\u6811\u67e5\u627e</p> <ul> <li>\u9002\u7528\u4e8e\u6d77\u91cf\u6570\u636e\uff0c\u56e0\u4e3a\u6811\u8282\u70b9\u5728\u5185\u5b58\u4e2d\u662f\u5206\u6563\u5b58\u50a8\u7684\u3002</li> <li>\u9002\u5408\u9700\u8981\u7ef4\u62a4\u6709\u5e8f\u6570\u636e\u6216\u8303\u56f4\u67e5\u627e\u7684\u573a\u666f\u3002</li> <li>\u5728\u6301\u7eed\u589e\u5220\u8282\u70b9\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u4ea7\u751f\u503e\u659c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>\u82e5\u4f7f\u7528 AVL \u6811\u6216\u7ea2\u9ed1\u6811\uff0c\u5219\u5404\u9879\u64cd\u4f5c\u53ef\u5728 \\(O(\\log n)\\) \u6548\u7387\u4e0b\u7a33\u5b9a\u8fd0\u884c\uff0c\u4f46\u7ef4\u62a4\u6811\u5e73\u8861\u7684\u64cd\u4f5c\u4f1a\u589e\u52a0\u989d\u5916\u5f00\u9500\u3002</li> </ul>"},{"location":"chapter_searching/summary/","title":"10.6 \u00a0 \u5c0f\u7ed3","text":"<ul> <li>\u4e8c\u5206\u67e5\u627e\u4f9d\u8d56\u6570\u636e\u7684\u6709\u5e8f\u6027\uff0c\u901a\u8fc7\u5faa\u73af\u9010\u6b65\u7f29\u51cf\u4e00\u534a\u641c\u7d22\u533a\u95f4\u6765\u8fdb\u884c\u67e5\u627e\u3002\u5b83\u8981\u6c42\u8f93\u5165\u6570\u636e\u6709\u5e8f\uff0c\u4e14\u4ec5\u9002\u7528\u4e8e\u6570\u7ec4\u6216\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u66b4\u529b\u641c\u7d22\u901a\u8fc7\u904d\u5386\u6570\u636e\u7ed3\u6784\u6765\u5b9a\u4f4d\u6570\u636e\u3002\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u6570\u7ec4\u548c\u94fe\u8868\uff0c\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u548c\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u9002\u7528\u4e8e\u56fe\u548c\u6811\u3002\u6b64\u7c7b\u7b97\u6cd5\u901a\u7528\u6027\u597d\uff0c\u65e0\u987b\u5bf9\u6570\u636e\u8fdb\u884c\u9884\u5904\u7406\uff0c\u4f46\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u8f83\u9ad8\u3002</li> <li>\u54c8\u5e0c\u67e5\u627e\u3001\u6811\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u5c5e\u4e8e\u9ad8\u6548\u641c\u7d22\u65b9\u6cd5\uff0c\u53ef\u5728\u7279\u5b9a\u6570\u636e\u7ed3\u6784\u4e2d\u5feb\u901f\u5b9a\u4f4d\u76ee\u6807\u5143\u7d20\u3002\u6b64\u7c7b\u7b97\u6cd5\u6548\u7387\u9ad8\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(\\log n)\\) \u751a\u81f3 \\(O(1)\\) \uff0c\u4f46\u901a\u5e38\u9700\u8981\u501f\u52a9\u989d\u5916\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u6570\u636e\u4f53\u91cf\u3001\u641c\u7d22\u6027\u80fd\u8981\u6c42\u3001\u6570\u636e\u67e5\u8be2\u548c\u66f4\u65b0\u9891\u7387\u7b49\u56e0\u7d20\u8fdb\u884c\u5177\u4f53\u5206\u6790\uff0c\u4ece\u800c\u9009\u62e9\u5408\u9002\u7684\u641c\u7d22\u65b9\u6cd5\u3002</li> <li>\u7ebf\u6027\u641c\u7d22\u9002\u7528\u4e8e\u5c0f\u578b\u6216\u9891\u7e41\u66f4\u65b0\u7684\u6570\u636e\uff1b\u4e8c\u5206\u67e5\u627e\u9002\u7528\u4e8e\u5927\u578b\u3001\u6392\u5e8f\u7684\u6570\u636e\uff1b\u54c8\u5e0c\u67e5\u627e\u9002\u7528\u4e8e\u5bf9\u67e5\u8be2\u6548\u7387\u8981\u6c42\u8f83\u9ad8\u4e14\u65e0\u987b\u8303\u56f4\u67e5\u8be2\u7684\u6570\u636e\uff1b\u6811\u67e5\u627e\u9002\u7528\u4e8e\u9700\u8981\u7ef4\u62a4\u987a\u5e8f\u548c\u652f\u6301\u8303\u56f4\u67e5\u8be2\u7684\u5927\u578b\u52a8\u6001\u6570\u636e\u3002</li> <li>\u7528\u54c8\u5e0c\u67e5\u627e\u66ff\u6362\u7ebf\u6027\u67e5\u627e\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u8fd0\u884c\u65f6\u95f4\u7684\u7b56\u7565\uff0c\u53ef\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4ece \\(O(n)\\) \u964d\u81f3 \\(O(1)\\) \u3002</li> </ul>"},{"location":"chapter_sorting/","title":"\u7b2c 11 \u7ae0 \u00a0 \u6392\u5e8f","text":"<p>Abstract</p> <p>\u6392\u5e8f\u72b9\u5982\u4e00\u628a\u5c06\u6df7\u4e71\u53d8\u4e3a\u79e9\u5e8f\u7684\u9b54\u6cd5\u94a5\u5319\uff0c\u4f7f\u6211\u4eec\u80fd\u4ee5\u66f4\u9ad8\u6548\u7684\u65b9\u5f0f\u7406\u89e3\u4e0e\u5904\u7406\u6570\u636e\u3002</p> <p>\u65e0\u8bba\u662f\u7b80\u5355\u7684\u5347\u5e8f\uff0c\u8fd8\u662f\u590d\u6742\u7684\u5206\u7c7b\u6392\u5217\uff0c\u6392\u5e8f\u90fd\u5411\u6211\u4eec\u5c55\u793a\u4e86\u6570\u636e\u7684\u548c\u8c10\u7f8e\u611f\u3002</p>"},{"location":"chapter_sorting/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5</li> <li>11.2 \u00a0 \u9009\u62e9\u6392\u5e8f</li> <li>11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f</li> <li>11.4 \u00a0 \u63d2\u5165\u6392\u5e8f</li> <li>11.5 \u00a0 \u5feb\u901f\u6392\u5e8f</li> <li>11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f</li> <li>11.7 \u00a0 \u5806\u6392\u5e8f</li> <li>11.8 \u00a0 \u6876\u6392\u5e8f</li> <li>11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f</li> <li>11.10 \u00a0 \u57fa\u6570\u6392\u5e8f</li> <li>11.11 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_sorting/bubble_sort/","title":"11.3 \u00a0 \u5192\u6ce1\u6392\u5e8f","text":"<p>\u300c\u5192\u6ce1\u6392\u5e8f bubble sort\u300d\u901a\u8fc7\u8fde\u7eed\u5730\u6bd4\u8f83\u4e0e\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u5b9e\u73b0\u6392\u5e8f\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u5c31\u50cf\u6c14\u6ce1\u4ece\u5e95\u90e8\u5347\u5230\u9876\u90e8\u4e00\u6837\uff0c\u56e0\u6b64\u5f97\u540d\u5192\u6ce1\u6392\u5e8f\u3002</p> <p>\u5982\u56fe 11-4 \u6240\u793a\uff0c\u5192\u6ce1\u8fc7\u7a0b\u53ef\u4ee5\u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6765\u6a21\u62df\uff1a\u4ece\u6570\u7ec4\u6700\u5de6\u7aef\u5f00\u59cb\u5411\u53f3\u904d\u5386\uff0c\u4f9d\u6b21\u6bd4\u8f83\u76f8\u90bb\u5143\u7d20\u5927\u5c0f\uff0c\u5982\u679c\u201c\u5de6\u5143\u7d20 &gt; \u53f3\u5143\u7d20\u201d\u5c31\u4ea4\u6362\u4e8c\u8005\u3002\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6700\u5927\u7684\u5143\u7d20\u4f1a\u88ab\u79fb\u52a8\u5230\u6570\u7ec4\u7684\u6700\u53f3\u7aef\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-4 \u00a0 \u5229\u7528\u5143\u7d20\u4ea4\u6362\u64cd\u4f5c\u6a21\u62df\u5192\u6ce1 </p>"},{"location":"chapter_sorting/bubble_sort/#1131","title":"11.3.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6b65\u9aa4\u5982\u56fe 11-5 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9 \\(n\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\uff0c</li> <li>\u63a5\u4e0b\u6765\uff0c\u5bf9\u5269\u4f59 \\(n - 1\\) \u4e2a\u5143\u7d20\u6267\u884c\u201c\u5192\u6ce1\u201d\uff0c\u5c06\u7b2c\u4e8c\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u201c\u5192\u6ce1\u201d\u540e\uff0c\u524d \\(n - 1\\) \u5927\u7684\u5143\u7d20\u90fd\u88ab\u4ea4\u6362\u81f3\u6b63\u786e\u4f4d\u7f6e\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5c0f\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> <p></p> <p> \u56fe 11-5 \u00a0 \u5192\u6ce1\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py<pre><code>def bubble_sort(nums: list[int]):\n    \"\"\"\u5192\u6ce1\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(n - 1, 0, -1):\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n                swap(nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid BubbleSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunc bubbleSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in stride(from: nums.count - 1, to: 0, by: -1) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for j in stride(from: 0, to: i, by: 1) {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfunction bubbleSort(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n      }\n    }\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nfn bubble_sort(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f */\nvoid bubbleSort(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = 0; i &lt; size - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; size - 1 - i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n            }\n        }\n    }\n}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\nfn bubbleSort(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n            }\n        }\n    }\n}\n</code></pre>"},{"location":"chapter_sorting/bubble_sort/#1132","title":"11.3.2 \u00a0 \u6548\u7387\u4f18\u5316","text":"<p>\u6211\u4eec\u53d1\u73b0\uff0c\u5982\u679c\u67d0\u8f6e\u201c\u5192\u6ce1\u201d\u4e2d\u6ca1\u6709\u6267\u884c\u4efb\u4f55\u4ea4\u6362\u64cd\u4f5c\uff0c\u8bf4\u660e\u6570\u7ec4\u5df2\u7ecf\u5b8c\u6210\u6392\u5e8f\uff0c\u53ef\u76f4\u63a5\u8fd4\u56de\u7ed3\u679c\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u589e\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d <code>flag</code> \u6765\u76d1\u6d4b\u8fd9\u79cd\u60c5\u51b5\uff0c\u4e00\u65e6\u51fa\u73b0\u5c31\u7acb\u5373\u8fd4\u56de\u3002</p> <p>\u7ecf\u8fc7\u4f18\u5316\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4ecd\u4e3a \\(O(n^2)\\) \uff1b\u4f46\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u53ef\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig bubble_sort.py<pre><code>def bubble_sort_with_flag(nums: list[int]):\n    \"\"\"\u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in range(n - 1, 0, -1):\n        flag = False  # \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        # \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j in range(i):\n            if nums[j] &gt; nums[j + 1]:\n                # \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j + 1] = nums[j + 1], nums[j]\n                flag = True  # \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n        if not flag:\n            break  # \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n</code></pre> bubble_sort.cpp<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.size() - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                // \u8fd9\u91cc\u4f7f\u7528\u4e86 std::swap() \u51fd\u6570\n                swap(nums[j], nums[j + 1]);\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.java<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nvoid bubbleSortWithFlag(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        boolean flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                int tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag)\n            break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.cs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid BubbleSortWithFlag(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for (int j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                (nums[j + 1], nums[j]) = (nums[j], nums[j + 1]);\n                flag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break;     // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.go<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := len(nums) - 1; i &gt; 0; i-- {\n        flag := false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for j := 0; j &lt; i; j++ {\n            if nums[j] &gt; nums[j+1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                nums[j], nums[j+1] = nums[j+1], nums[j]\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if flag == false { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.swift<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunc bubbleSortWithFlag(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in stride(from: nums.count - 1, to: 0, by: -1) {\n        var flag = false // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        for j in stride(from: 0, to: i, by: 1) {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j]\n                nums[j] = nums[j + 1]\n                nums[j + 1] = tmp\n                flag = true // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if !flag { // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n            break\n        }\n    }\n}\n</code></pre> bubble_sort.js<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.ts<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nfunction bubbleSortWithFlag(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        let flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (let j = 0; j &lt; i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.dart<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    bool flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n    // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n    for (int j = 0; j &lt; i; j++) {\n      if (nums[j] &gt; nums[j + 1]) {\n        // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n        int tmp = nums[j];\n        nums[j] = nums[j + 1];\n        nums[j + 1] = tmp;\n        flag = true; // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n      }\n    }\n    if (!flag) break; // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n  }\n}\n</code></pre> bubble_sort.rs<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09 */\nfn bubble_sort_with_flag(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i in (1..nums.len()).rev() {\n        let mut flag = false; // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        for j in 0..i {\n            if nums[j] &gt; nums[j + 1] {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                let tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true;  // \u8bb0\u5f55\u4ea4\u6362\u5143\u7d20\n            }\n        }\n        if !flag {break};  // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre> bubble_sort.c<pre><code>/* \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09*/\nvoid bubbleSortWithFlag(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for (int i = 0; i &lt; size - 1; i++) {\n        bool flag = false;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef\n        for (int j = 0; j &lt; size - 1 - i; j++) {\n            if (nums[j] &gt; nums[j + 1]) {\n                int temp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = temp;\n                flag = true;\n            }\n        }\n        if (!flag)\n            break;\n    }\n}\n</code></pre> bubble_sort.zig<pre><code>// \u5192\u6ce1\u6392\u5e8f\uff08\u6807\u5fd7\u4f18\u5316\uff09\nfn bubbleSortWithFlag(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    var i: usize = nums.len - 1;\n    while (i &gt; 0) : (i -= 1) {\n        var flag = false;   // \u521d\u59cb\u5316\u6807\u5fd7\u4f4d\n        var j: usize = 0;\n        // \u5185\u5faa\u73af\uff1a\u5c06\u672a\u6392\u5e8f\u533a\u95f4 [0, i] \u4e2d\u7684\u6700\u5927\u5143\u7d20\u4ea4\u6362\u81f3\u8be5\u533a\u95f4\u7684\u6700\u53f3\u7aef \n        while (j &lt; i) : (j += 1) {\n            if (nums[j] &gt; nums[j + 1]) {\n                // \u4ea4\u6362 nums[j] \u4e0e nums[j + 1]\n                var tmp = nums[j];\n                nums[j] = nums[j + 1];\n                nums[j + 1] = tmp;\n                flag = true;\n            }\n        }\n        if (!flag) break;   // \u6b64\u8f6e\u5192\u6ce1\u672a\u4ea4\u6362\u4efb\u4f55\u5143\u7d20\uff0c\u76f4\u63a5\u8df3\u51fa\n    }\n}\n</code></pre>"},{"location":"chapter_sorting/bubble_sort/#1133","title":"11.3.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5404\u8f6e\u201c\u5192\u6ce1\u201d\u904d\u5386\u7684\u6570\u7ec4\u957f\u5ea6\u4f9d\u6b21\u4e3a \\(n - 1\\)\u3001\\(n - 2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u603b\u548c\u4e3a \\((n - 1) n / 2\\) \u3002\u5728\u5f15\u5165 <code>flag</code> \u4f18\u5316\u540e\uff0c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe\u5230 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5728\u201c\u5192\u6ce1\u201d\u4e2d\u9047\u5230\u76f8\u7b49\u5143\u7d20\u4e0d\u4ea4\u6362\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/","title":"11.8 \u00a0 \u6876\u6392\u5e8f","text":"<p>\u524d\u8ff0\u51e0\u79cd\u6392\u5e8f\u7b97\u6cd5\u90fd\u5c5e\u4e8e\u201c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u901a\u8fc7\u6bd4\u8f83\u5143\u7d20\u95f4\u7684\u5927\u5c0f\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u6b64\u7c7b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u65e0\u6cd5\u8d85\u8d8a \\(O(n \\log n)\\) \u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u51e0\u79cd\u201c\u975e\u6bd4\u8f83\u6392\u5e8f\u7b97\u6cd5\u201d\uff0c\u5b83\u4eec\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u8fbe\u5230\u7ebf\u6027\u9636\u3002</p> <p>\u300c\u6876\u6392\u5e8f bucket sort\u300d\u662f\u5206\u6cbb\u7b56\u7565\u7684\u4e00\u4e2a\u5178\u578b\u5e94\u7528\u3002\u5b83\u901a\u8fc7\u8bbe\u7f6e\u4e00\u4e9b\u5177\u6709\u5927\u5c0f\u987a\u5e8f\u7684\u6876\uff0c\u6bcf\u4e2a\u6876\u5bf9\u5e94\u4e00\u4e2a\u6570\u636e\u8303\u56f4\uff0c\u5c06\u6570\u636e\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff1b\u7136\u540e\uff0c\u5728\u6bcf\u4e2a\u6876\u5185\u90e8\u5206\u522b\u6267\u884c\u6392\u5e8f\uff1b\u6700\u7ec8\u6309\u7167\u6876\u7684\u987a\u5e8f\u5c06\u6240\u6709\u6570\u636e\u5408\u5e76\u3002</p>"},{"location":"chapter_sorting/bucket_sort/#1181","title":"11.8.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8003\u8651\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\uff0c\u5176\u5143\u7d20\u662f\u8303\u56f4 \\([0, 1)\\) \u5185\u7684\u6d6e\u70b9\u6570\u3002\u6876\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-13 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316 \\(k\\) \u4e2a\u6876\uff0c\u5c06 \\(n\\) \u4e2a\u5143\u7d20\u5206\u914d\u5230 \\(k\\) \u4e2a\u6876\u4e2d\u3002</li> <li>\u5bf9\u6bcf\u4e2a\u6876\u5206\u522b\u6267\u884c\u6392\u5e8f\uff08\u8fd9\u91cc\u91c7\u7528\u7f16\u7a0b\u8bed\u8a00\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff09\u3002</li> <li>\u6309\u7167\u6876\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u5408\u5e76\u7ed3\u679c\u3002</li> </ol> <p></p> <p> \u56fe 11-13 \u00a0 \u6876\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig bucket_sort.py<pre><code>def bucket_sort(nums: list[float]):\n    \"\"\"\u6876\u6392\u5e8f\"\"\"\n    # \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    k = len(nums) // 2\n    buckets = [[] for _ in range(k)]\n    # 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for num in nums:\n        # \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        i = int(num * k)\n        # \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].append(num)\n    # 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for bucket in buckets:\n        # \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort()\n    # 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    i = 0\n    for bucket in buckets:\n        for num in bucket:\n            nums[i] = num\n            i += 1\n</code></pre> bucket_sort.cpp<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(vector&lt;float&gt; &amp;nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.size() / 2;\n    vector&lt;vector&lt;float&gt;&gt; buckets(k);\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = num * k;\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n        buckets[i].push_back(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        sort(bucket.begin(), bucket.end());\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int i = 0;\n    for (vector&lt;float&gt; &amp;bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.java<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.length / 2;\n    List&lt;List&lt;Float&gt;&gt; buckets = new ArrayList&lt;&gt;();\n    for (int i = 0; i &lt; k; i++) {\n        buckets.add(new ArrayList&lt;&gt;());\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (float num : nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = (int) (num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets.get(i).add(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (List&lt;Float&gt; bucket : buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        Collections.sort(bucket);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int i = 0;\n    for (List&lt;Float&gt; bucket : buckets) {\n        for (float num : bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.cs<pre><code>/* \u6876\u6392\u5e8f */\nvoid BucketSort(float[] nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = nums.Length / 2;\n    List&lt;List&lt;float&gt;&gt; buckets = [];\n    for (int i = 0; i &lt; k; i++) {\n        buckets.Add([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    foreach (float num in nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int i = (int)(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].Add(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    foreach (List&lt;float&gt; bucket in buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.Sort();\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    int j = 0;\n    foreach (List&lt;float&gt; bucket in buckets) {\n        foreach (float num in bucket) {\n            nums[j++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.go<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums []float64) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    k := len(nums) / 2\n    buckets := make([][]float64, k)\n    for i := 0; i &lt; k; i++ {\n        buckets[i] = make([]float64, 0)\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for _, num := range nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        i := int(num * float64(k))\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i] = append(buckets[i], num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for i := 0; i &lt; k; i++ {\n        // \u4f7f\u7528\u5185\u7f6e\u5207\u7247\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        sort.Float64s(buckets[i])\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    i := 0\n    for _, bucket := range buckets {\n        for _, num := range bucket {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> bucket_sort.swift<pre><code>/* \u6876\u6392\u5e8f */\nfunc bucketSort(nums: inout [Double]) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    let k = nums.count / 2\n    var buckets = (0 ..&lt; k).map { _ in [Double]() }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for num in nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        let i = Int(num * Double(k))\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].append(num)\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for i in buckets.indices {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        buckets[i].sort()\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    var i = nums.startIndex\n    for bucket in buckets {\n        for num in bucket {\n            nums[i] = num\n            nums.formIndex(after: &amp;i)\n        }\n    }\n}\n</code></pre> bucket_sort.js<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.ts<pre><code>/* \u6876\u6392\u5e8f */\nfunction bucketSort(nums: number[]): void {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    const k = nums.length / 2;\n    const buckets: number[][] = [];\n    for (let i = 0; i &lt; k; i++) {\n        buckets.push([]);\n    }\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (const num of nums) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        const i = Math.floor(num * k);\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (const bucket of buckets) {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort((a, b) =&gt; a - b);\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let i = 0;\n    for (const bucket of buckets) {\n        for (const num of bucket) {\n            nums[i++] = num;\n        }\n    }\n}\n</code></pre> bucket_sort.dart<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(List&lt;double&gt; nums) {\n  // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n  int k = nums.length ~/ 2;\n  List&lt;List&lt;double&gt;&gt; buckets = List.generate(k, (index) =&gt; []);\n\n  // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n  for (double _num in nums) {\n    // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 _num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n    int i = (_num * k).toInt();\n    // \u5c06 _num \u6dfb\u52a0\u8fdb\u6876 bucket_idx\n    buckets[i].add(_num);\n  }\n  // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n  for (List&lt;double&gt; bucket in buckets) {\n    bucket.sort();\n  }\n  // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n  int i = 0;\n  for (List&lt;double&gt; bucket in buckets) {\n    for (double _num in bucket) {\n      nums[i++] = _num;\n    }\n  }\n}\n</code></pre> bucket_sort.rs<pre><code>/* \u6876\u6392\u5e8f */\nfn bucket_sort(nums: &amp;mut [f64]) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    let k = nums.len() / 2;\n    let mut buckets = vec![vec![]; k];\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for &amp;mut num in &amp;mut *nums {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        let i = (num * k as f64) as usize;\n        // \u5c06 num \u6dfb\u52a0\u8fdb\u6876 i\n        buckets[i].push(num);\n    }\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for bucket in &amp;mut buckets {\n        // \u4f7f\u7528\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u66ff\u6362\u6210\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\n        bucket.sort_by(|a, b| a.partial_cmp(b).unwrap());\n    }\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    let mut i = 0;\n    for bucket in &amp;mut buckets {\n        for &amp;mut num in bucket {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> bucket_sort.c<pre><code>/* \u6876\u6392\u5e8f */\nvoid bucketSort(float nums[], int size) {\n    // \u521d\u59cb\u5316 k = n/2 \u4e2a\u6876\uff0c\u9884\u671f\u5411\u6bcf\u4e2a\u6876\u5206\u914d 2 \u4e2a\u5143\u7d20\n    int k = size / 2;\n    float **buckets = calloc(k, sizeof(float *));\n    for (int i = 0; i &lt; k; i++) {\n        // \u6bcf\u4e2a\u6876\u6700\u591a\u53ef\u4ee5\u5206\u914d k \u4e2a\u5143\u7d20\n        buckets[i] = calloc(ARRAY_SIZE, sizeof(float));\n    }\n\n    // 1. \u5c06\u6570\u7ec4\u5143\u7d20\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\n    for (int i = 0; i &lt; size; i++) {\n        // \u8f93\u5165\u6570\u636e\u8303\u56f4\u4e3a [0, 1)\uff0c\u4f7f\u7528 num * k \u6620\u5c04\u5230\u7d22\u5f15\u8303\u56f4 [0, k-1]\n        int bucket_idx = nums[i] * k;\n        int j = 0;\n        // \u5982\u679c\u6876\u4e2d\u6709\u6570\u636e\u4e14\u6570\u636e\u5c0f\u4e8e\u5f53\u524d\u503c nums[i], \u8981\u5c06\u5176\u653e\u5230\u5f53\u524d\u6876\u7684\u540e\u9762\uff0c\u76f8\u5f53\u4e8e cpp \u4e2d\u7684 push_back\n        while (buckets[bucket_idx][j] &gt; 0 &amp;&amp; buckets[bucket_idx][j] &lt; nums[i]) {\n            j++;\n        }\n        float temp = nums[i];\n        while (j &lt; ARRAY_SIZE &amp;&amp; buckets[bucket_idx][j] &gt; 0) {\n            swap(&amp;temp, &amp;buckets[bucket_idx][j]);\n            j++;\n        }\n        buckets[bucket_idx][j] = temp;\n    }\n\n    // 2. \u5bf9\u5404\u4e2a\u6876\u6267\u884c\u6392\u5e8f\n    for (int i = 0; i &lt; k; i++) {\n        qsort(buckets[i], ARRAY_SIZE, sizeof(float), compare_float);\n    }\n\n    // 3. \u904d\u5386\u6876\u5408\u5e76\u7ed3\u679c\n    for (int i = 0, j = 0; j &lt; k; j++) {\n        for (int l = 0; l &lt; ARRAY_SIZE; l++) {\n            if (buckets[j][l] &gt; 0) {\n                nums[i++] = buckets[j][l];\n            }\n        }\n    }\n\n    // \u91ca\u653e\u4e0a\u8ff0\u5206\u914d\u7684\u5185\u5b58\n    for (int i = 0; i &lt; k; i++) {\n        free(buckets[i]);\n    }\n    free(buckets);\n}\n</code></pre> bucket_sort.zig<pre><code>[class]{}-[func]{bucketSort}\n</code></pre>"},{"location":"chapter_sorting/bucket_sort/#1182","title":"11.8.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u6876\u6392\u5e8f\u9002\u7528\u4e8e\u5904\u7406\u4f53\u91cf\u5f88\u5927\u7684\u6570\u636e\u3002\u4f8b\u5982\uff0c\u8f93\u5165\u6570\u636e\u5305\u542b 100 \u4e07\u4e2a\u5143\u7d20\uff0c\u7531\u4e8e\u7a7a\u95f4\u9650\u5236\uff0c\u7cfb\u7edf\u5185\u5b58\u65e0\u6cd5\u4e00\u6b21\u6027\u52a0\u8f7d\u6240\u6709\u6570\u636e\u3002\u6b64\u65f6\uff0c\u53ef\u4ee5\u5c06\u6570\u636e\u5206\u6210 1000 \u4e2a\u6876\uff0c\u7136\u540e\u5206\u522b\u5bf9\u6bcf\u4e2a\u6876\u8fdb\u884c\u6392\u5e8f\uff0c\u6700\u540e\u5c06\u7ed3\u679c\u5408\u5e76\u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\) \uff1a\u5047\u8bbe\u5143\u7d20\u5728\u5404\u4e2a\u6876\u5185\u5e73\u5747\u5206\u5e03\uff0c\u90a3\u4e48\u6bcf\u4e2a\u6876\u5185\u7684\u5143\u7d20\u6570\u91cf\u4e3a \\(\\frac{n}{k}\\) \u3002\u5047\u8bbe\u6392\u5e8f\u5355\u4e2a\u6876\u4f7f\u7528 \\(O(\\frac{n}{k} \\log\\frac{n}{k})\\) \u65f6\u95f4\uff0c\u5219\u6392\u5e8f\u6240\u6709\u6876\u4f7f\u7528 \\(O(n \\log\\frac{n}{k})\\) \u65f6\u95f4\u3002\u5f53\u6876\u6570\u91cf \\(k\\) \u6bd4\u8f83\u5927\u65f6\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5219\u8d8b\u5411\u4e8e \\(O(n)\\) \u3002\u5408\u5e76\u7ed3\u679c\u65f6\u9700\u8981\u904d\u5386\u6240\u6709\u6876\u548c\u5143\u7d20\uff0c\u82b1\u8d39 \\(O(n + k)\\) \u65f6\u95f4\u3002</li> <li>\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u6570\u636e\u88ab\u5206\u914d\u5230\u4e00\u4e2a\u6876\u4e2d\uff0c\u4e14\u6392\u5e8f\u8be5\u6876\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + k)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9700\u8981\u501f\u52a9 \\(k\\) \u4e2a\u6876\u548c\u603b\u5171 \\(n\\) \u4e2a\u5143\u7d20\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u6876\u6392\u5e8f\u662f\u5426\u7a33\u5b9a\u53d6\u51b3\u4e8e\u6392\u5e8f\u6876\u5185\u5143\u7d20\u7684\u7b97\u6cd5\u662f\u5426\u7a33\u5b9a\u3002</li> </ul>"},{"location":"chapter_sorting/bucket_sort/#1183","title":"11.8.3 \u00a0 \u5982\u4f55\u5b9e\u73b0\u5e73\u5747\u5206\u914d","text":"<p>\u6876\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7406\u8bba\u4e0a\u53ef\u4ee5\u8fbe\u5230 \\(O(n)\\) \uff0c\u5173\u952e\u5728\u4e8e\u5c06\u5143\u7d20\u5747\u5300\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\uff0c\u56e0\u4e3a\u5b9e\u9645\u6570\u636e\u5f80\u5f80\u4e0d\u662f\u5747\u5300\u5206\u5e03\u7684\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u60f3\u8981\u5c06\u6dd8\u5b9d\u4e0a\u7684\u6240\u6709\u5546\u54c1\u6309\u4ef7\u683c\u8303\u56f4\u5e73\u5747\u5206\u914d\u5230 10 \u4e2a\u6876\u4e2d\uff0c\u4f46\u5546\u54c1\u4ef7\u683c\u5206\u5e03\u4e0d\u5747\uff0c\u4f4e\u4e8e 100 \u5143\u7684\u975e\u5e38\u591a\uff0c\u9ad8\u4e8e 1000 \u5143\u7684\u975e\u5e38\u5c11\u3002\u82e5\u5c06\u4ef7\u683c\u533a\u95f4\u5e73\u5747\u5212\u5206\u4e3a 10 \u4e2a\uff0c\u5404\u4e2a\u6876\u4e2d\u7684\u5546\u54c1\u6570\u91cf\u5dee\u8ddd\u4f1a\u975e\u5e38\u5927\u3002</p> <p>\u4e3a\u5b9e\u73b0\u5e73\u5747\u5206\u914d\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u8bbe\u5b9a\u4e00\u6761\u5927\u81f4\u7684\u5206\u754c\u7ebf\uff0c\u5c06\u6570\u636e\u7c97\u7565\u5730\u5206\u5230 3 \u4e2a\u6876\u4e2d\u3002\u5206\u914d\u5b8c\u6bd5\u540e\uff0c\u518d\u5c06\u5546\u54c1\u8f83\u591a\u7684\u6876\u7ee7\u7eed\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u76f4\u81f3\u6240\u6709\u6876\u4e2d\u7684\u5143\u7d20\u6570\u91cf\u5927\u81f4\u76f8\u7b49\u3002</p> <p>\u5982\u56fe 11-14 \u6240\u793a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u672c\u8d28\u4e0a\u662f\u521b\u5efa\u4e00\u68f5\u9012\u5f52\u6811\uff0c\u76ee\u6807\u662f\u8ba9\u53f6\u8282\u70b9\u7684\u503c\u5c3d\u53ef\u80fd\u5e73\u5747\u3002\u5f53\u7136\uff0c\u4e0d\u4e00\u5b9a\u8981\u6bcf\u8f6e\u5c06\u6570\u636e\u5212\u5206\u4e3a 3 \u4e2a\u6876\uff0c\u5177\u4f53\u5212\u5206\u65b9\u5f0f\u53ef\u6839\u636e\u6570\u636e\u7279\u70b9\u7075\u6d3b\u9009\u62e9\u3002</p> <p></p> <p> \u56fe 11-14 \u00a0 \u9012\u5f52\u5212\u5206\u6876 </p> <p>\u5982\u679c\u6211\u4eec\u63d0\u524d\u77e5\u9053\u5546\u54c1\u4ef7\u683c\u7684\u6982\u7387\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u6839\u636e\u6570\u636e\u6982\u7387\u5206\u5e03\u8bbe\u7f6e\u6bcf\u4e2a\u6876\u7684\u4ef7\u683c\u5206\u754c\u7ebf\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6570\u636e\u5206\u5e03\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u7279\u610f\u7edf\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u6839\u636e\u6570\u636e\u7279\u70b9\u91c7\u7528\u67d0\u79cd\u6982\u7387\u6a21\u578b\u8fdb\u884c\u8fd1\u4f3c\u3002</p> <p>\u5982\u56fe 11-15 \u6240\u793a\uff0c\u6211\u4eec\u5047\u8bbe\u5546\u54c1\u4ef7\u683c\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5408\u7406\u5730\u8bbe\u5b9a\u4ef7\u683c\u533a\u95f4\uff0c\u4ece\u800c\u5c06\u5546\u54c1\u5e73\u5747\u5206\u914d\u5230\u5404\u4e2a\u6876\u4e2d\u3002</p> <p></p> <p> \u56fe 11-15 \u00a0 \u6839\u636e\u6982\u7387\u5206\u5e03\u5212\u5206\u6876 </p>"},{"location":"chapter_sorting/counting_sort/","title":"11.9 \u00a0 \u8ba1\u6570\u6392\u5e8f","text":"<p>\u300c\u8ba1\u6570\u6392\u5e8f counting sort\u300d\u901a\u8fc7\u7edf\u8ba1\u5143\u7d20\u6570\u91cf\u6765\u5b9e\u73b0\u6392\u5e8f\uff0c\u901a\u5e38\u5e94\u7528\u4e8e\u6574\u6570\u6570\u7ec4\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1191","title":"11.9.1 \u00a0 \u7b80\u5355\u5b9e\u73b0","text":"<p>\u5148\u6765\u770b\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u3002\u7ed9\u5b9a\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4 <code>nums</code> \uff0c\u5176\u4e2d\u7684\u5143\u7d20\u90fd\u662f\u201c\u975e\u8d1f\u6574\u6570\u201d\uff0c\u8ba1\u6570\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-16 \u6240\u793a\u3002</p> <ol> <li>\u904d\u5386\u6570\u7ec4\uff0c\u627e\u51fa\u5176\u4e2d\u7684\u6700\u5927\u6570\u5b57\uff0c\u8bb0\u4e3a \\(m\\) \uff0c\u7136\u540e\u521b\u5efa\u4e00\u4e2a\u957f\u5ea6\u4e3a \\(m + 1\\) \u7684\u8f85\u52a9\u6570\u7ec4 <code>counter</code> \u3002</li> <li>\u501f\u52a9 <code>counter</code> \u7edf\u8ba1 <code>nums</code> \u4e2d\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\uff0c\u5176\u4e2d <code>counter[num]</code> \u5bf9\u5e94\u6570\u5b57 <code>num</code> \u7684\u51fa\u73b0\u6b21\u6570\u3002\u7edf\u8ba1\u65b9\u6cd5\u5f88\u7b80\u5355\uff0c\u53ea\u9700\u904d\u5386 <code>nums</code>\uff08\u8bbe\u5f53\u524d\u6570\u5b57\u4e3a <code>num</code>\uff09\uff0c\u6bcf\u8f6e\u5c06 <code>counter[num]</code> \u589e\u52a0 \\(1\\) \u5373\u53ef\u3002</li> <li>\u7531\u4e8e <code>counter</code> \u7684\u5404\u4e2a\u7d22\u5f15\u5929\u7136\u6709\u5e8f\uff0c\u56e0\u6b64\u76f8\u5f53\u4e8e\u6240\u6709\u6570\u5b57\u5df2\u7ecf\u6392\u5e8f\u597d\u4e86\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u904d\u5386 <code>counter</code> \uff0c\u6839\u636e\u5404\u6570\u5b57\u51fa\u73b0\u6b21\u6570\u4ece\u5c0f\u5230\u5927\u7684\u987a\u5e8f\u586b\u5165 <code>nums</code> \u5373\u53ef\u3002</li> </ol> <p></p> <p> \u56fe 11-16 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b </p> <p>\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py<pre><code>def counting_sort_naive(nums: list[int]):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n    # \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\n    # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m = 0\n    for num in nums:\n        m = max(m, num)\n    # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    i = 0\n    for num in range(m + 1):\n        for _ in range(counter[num]):\n            nums[i] = num\n            i += 1\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid CountingSortNaive(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums []int) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    for i, num := 0, 0; num &lt; m+1; num++ {\n        for j := 0; j &lt; counter[num]; j++ {\n            nums[i] = num\n            i++\n        }\n    }\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunc countingSortNaive(nums: inout [Int]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    var i = 0\n    for num in stride(from: 0, to: m + 1, by: 1) {\n        for _ in stride(from: 0, to: counter[num], by: 1) {\n            nums[i] = num\n            i += 1\n        }\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfunction countingSortNaive(nums: number[]): void {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let i = 0;\n    for (let num = 0; num &lt; m + 1; num++) {\n        for (let j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(List&lt;int&gt; nums) {\n  // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n  int i = 0;\n  for (int _num = 0; _num &lt; m + 1; _num++) {\n    for (int j = 0; j &lt; counter[_num]; j++, i++) {\n      nums[i] = _num;\n    }\n  }\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nfn counting_sort_naive(nums: &amp;mut [i32]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = *nums.into_iter().max().unwrap();\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in &amp;*nums {\n        counter[num as usize] += 1;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    let mut i = 0;\n    for num in 0..m + 1 {\n        for _ in 0..counter[num as usize] {\n            nums[i] = num;\n            i += 1;\n        }\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u7b80\u5355\u5b9e\u73b0\uff0c\u65e0\u6cd5\u7528\u4e8e\u6392\u5e8f\u5bf9\u8c61\nvoid countingSortNaive(int nums[], int size) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u904d\u5386 counter \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u539f\u6570\u7ec4 nums\n    int i = 0;\n    for (int num = 0; num &lt; m + 1; num++) {\n        for (int j = 0; j &lt; counter[num]; j++, i++) {\n            nums[i] = num;\n        }\n    }\n    // 4. \u91ca\u653e\u5185\u5b58\n    free(counter);\n}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSortNaive}\n</code></pre> <p>\u8ba1\u6570\u6392\u5e8f\u4e0e\u6876\u6392\u5e8f\u7684\u8054\u7cfb</p> <p>\u4ece\u6876\u6392\u5e8f\u7684\u89d2\u5ea6\u770b\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u8ba1\u6570\u6392\u5e8f\u4e2d\u7684\u8ba1\u6570\u6570\u7ec4 <code>counter</code> \u7684\u6bcf\u4e2a\u7d22\u5f15\u89c6\u4e3a\u4e00\u4e2a\u6876\uff0c\u5c06\u7edf\u8ba1\u6570\u91cf\u7684\u8fc7\u7a0b\u770b\u4f5c\u5c06\u5404\u4e2a\u5143\u7d20\u5206\u914d\u5230\u5bf9\u5e94\u7684\u6876\u4e2d\u3002\u672c\u8d28\u4e0a\uff0c\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u5728\u6574\u578b\u6570\u636e\u4e0b\u7684\u4e00\u4e2a\u7279\u4f8b\u3002</p>"},{"location":"chapter_sorting/counting_sort/#1192","title":"11.9.2 \u00a0 \u5b8c\u6574\u5b9e\u73b0","text":"<p>\u7ec6\u5fc3\u7684\u8bfb\u8005\u53ef\u80fd\u53d1\u73b0\u4e86\uff0c\u5982\u679c\u8f93\u5165\u6570\u636e\u662f\u5bf9\u8c61\uff0c\u4e0a\u8ff0\u6b65\u9aa4 <code>3.</code> \u5c31\u5931\u6548\u4e86\u3002\u5047\u8bbe\u8f93\u5165\u6570\u636e\u662f\u5546\u54c1\u5bf9\u8c61\uff0c\u6211\u4eec\u60f3\u6309\u7167\u5546\u54c1\u4ef7\u683c\uff08\u7c7b\u7684\u6210\u5458\u53d8\u91cf\uff09\u5bf9\u5546\u54c1\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u4e0a\u8ff0\u7b97\u6cd5\u53ea\u80fd\u7ed9\u51fa\u4ef7\u683c\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p> <p>\u90a3\u4e48\u5982\u4f55\u624d\u80fd\u5f97\u5230\u539f\u6570\u636e\u7684\u6392\u5e8f\u7ed3\u679c\u5462\uff1f\u6211\u4eec\u9996\u5148\u8ba1\u7b97 <code>counter</code> \u7684\u201c\u524d\u7f00\u548c\u201d\u3002\u987e\u540d\u601d\u4e49\uff0c\u7d22\u5f15 <code>i</code> \u5904\u7684\u524d\u7f00\u548c <code>prefix[i]</code> \u7b49\u4e8e\u6570\u7ec4\u524d <code>i</code> \u4e2a\u5143\u7d20\u4e4b\u548c\uff1a</p> \\[ \\text{prefix}[i] = \\sum_{j=0}^i \\text{counter[j]} \\] <p>\u524d\u7f00\u548c\u5177\u6709\u660e\u786e\u7684\u610f\u4e49\uff0c<code>prefix[num] - 1</code> \u4ee3\u8868\u5143\u7d20 <code>num</code> \u5728\u7ed3\u679c\u6570\u7ec4 <code>res</code> \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\u3002\u8fd9\u4e2a\u4fe1\u606f\u975e\u5e38\u5173\u952e\uff0c\u56e0\u4e3a\u5b83\u544a\u8bc9\u6211\u4eec\u5404\u4e2a\u5143\u7d20\u5e94\u8be5\u51fa\u73b0\u5728\u7ed3\u679c\u6570\u7ec4\u7684\u54ea\u4e2a\u4f4d\u7f6e\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5012\u5e8f\u904d\u5386\u539f\u6570\u7ec4 <code>nums</code> \u7684\u6bcf\u4e2a\u5143\u7d20 <code>num</code> \uff0c\u5728\u6bcf\u8f6e\u8fed\u4ee3\u4e2d\u6267\u884c\u4ee5\u4e0b\u4e24\u6b65\u3002</p> <ol> <li>\u5c06 <code>num</code> \u586b\u5165\u6570\u7ec4 <code>res</code> \u7684\u7d22\u5f15 <code>prefix[num] - 1</code> \u5904\u3002</li> <li>\u4ee4\u524d\u7f00\u548c <code>prefix[num]</code> \u51cf\u5c0f \\(1\\) \uff0c\u4ece\u800c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e <code>num</code> \u7684\u7d22\u5f15\u3002</li> </ol> <p>\u904d\u5386\u5b8c\u6210\u540e\uff0c\u6570\u7ec4 <code>res</code> \u4e2d\u5c31\u662f\u6392\u5e8f\u597d\u7684\u7ed3\u679c\uff0c\u6700\u540e\u4f7f\u7528 <code>res</code> \u8986\u76d6\u539f\u6570\u7ec4 <code>nums</code> \u5373\u53ef\u3002\u56fe 11-17 \u5c55\u793a\u4e86\u5b8c\u6574\u7684\u8ba1\u6570\u6392\u5e8f\u6d41\u7a0b\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-17 \u00a0 \u8ba1\u6570\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u8ba1\u6570\u6392\u5e8f\u7684\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig counting_sort.py<pre><code>def counting_sort(nums: list[int]):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\"\"\"\n    # \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\n    # 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m = max(nums)\n    # 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    # counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter = [0] * (m + 1)\n    for num in nums:\n        counter[num] += 1\n    # 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    # \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in range(m):\n        counter[i + 1] += counter[i]\n    # 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    # \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    n = len(nums)\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        num = nums[i]\n        res[counter[num] - 1] = num  # \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num] -= 1  # \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    # \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in range(n):\n        nums[i] = res[i]\n</code></pre> counting_sort.cpp<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(vector&lt;int&gt; &amp;nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    vector&lt;int&gt; counter(m + 1, 0);\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.size();\n    vector&lt;int&gt; res(n);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    nums = res;\n}\n</code></pre> counting_sort.java<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int num : nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    for (int num : nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.cs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid CountingSort(int[] nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    foreach (int num in nums) {\n        m = Math.Max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int[] counter = new int[m + 1];\n    foreach (int num in nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int n = nums.Length;\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.go<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums []int) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    m := 0\n    for _, num := range nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    counter := make([]int, m+1)\n    for _, num := range nums {\n        counter[num]++\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i := 0; i &lt; m; i++ {\n        counter[i+1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    n := len(nums)\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        num := nums[i]\n        // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        res[counter[num]-1] = num\n        // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n        counter[num]--\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    copy(nums, res)\n}\n</code></pre> counting_sort.swift<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunc countingSort(nums: inout [Int]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = nums.max()!\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    var counter = Array(repeating: 0, count: m + 1)\n    for num in nums {\n        counter[num] += 1\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in stride(from: 0, to: m, by: 1) {\n        counter[i + 1] += counter[i]\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    var res = Array(repeating: 0, count: nums.count)\n    for i in stride(from: nums.count - 1, through: 0, by: -1) {\n        let num = nums[i]\n        res[counter[num] - 1] = num // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num] -= 1 // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in stride(from: 0, to: nums.count, by: 1) {\n        nums[i] = res[i]\n    }\n}\n</code></pre> counting_sort.js<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter = new Array(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    const n = nums.length;\n    const res = new Array(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.ts<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfunction countingSort(nums: number[]): void {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = 0;\n    for (const num of nums) {\n        m = Math.max(m, num);\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    const counter: number[] = new Array&lt;number&gt;(m + 1).fill(0);\n    for (const num of nums) {\n        counter[num]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (let i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    const n = nums.length;\n    const res: number[] = new Array&lt;number&gt;(n);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.dart<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(List&lt;int&gt; nums) {\n  // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n  int m = 0;\n  for (int _num in nums) {\n    m = max(m, _num);\n  }\n  // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  // counter[_num] \u4ee3\u8868 _num \u7684\u51fa\u73b0\u6b21\u6570\n  List&lt;int&gt; counter = List.filled(m + 1, 0);\n  for (int _num in nums) {\n    counter[_num]++;\n  }\n  // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n  // \u5373 counter[_num]-1 \u662f _num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n  for (int i = 0; i &lt; m; i++) {\n    counter[i + 1] += counter[i];\n  }\n  // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n  // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n  int n = nums.length;\n  List&lt;int&gt; res = List.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int _num = nums[i];\n    res[counter[_num] - 1] = _num; // \u5c06 _num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n    counter[_num]--; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e _num \u7684\u7d22\u5f15\n  }\n  // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n  nums.setAll(0, res);\n}\n</code></pre> counting_sort.rs<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nfn counting_sort(nums: &amp;mut [i32]) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    let m = *nums.into_iter().max().unwrap();\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    let mut counter = vec![0; m as usize + 1];\n    for &amp;num in &amp;*nums {\n        counter[num as usize] += 1;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for i in 0..m as usize {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    let n = nums.len();\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let num = nums[i];\n        res[counter[num as usize] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num as usize] -= 1; // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in 0..n {\n        nums[i] = res[i];\n    }\n}\n</code></pre> counting_sort.c<pre><code>/* \u8ba1\u6570\u6392\u5e8f */\n// \u5b8c\u6574\u5b9e\u73b0\uff0c\u53ef\u6392\u5e8f\u5bf9\u8c61\uff0c\u5e76\u4e14\u662f\u7a33\u5b9a\u6392\u5e8f\nvoid countingSort(int nums[], int size) {\n    // 1. \u7edf\u8ba1\u6570\u7ec4\u6700\u5927\u5143\u7d20 m\n    int m = 0;\n    for (int i = 0; i &lt; size; i++) {\n        if (nums[i] &gt; m) {\n            m = nums[i];\n        }\n    }\n    // 2. \u7edf\u8ba1\u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    // counter[num] \u4ee3\u8868 num \u7684\u51fa\u73b0\u6b21\u6570\n    int *counter = calloc(m, sizeof(int));\n    for (int i = 0; i &lt; size; i++) {\n        counter[nums[i]]++;\n    }\n    // 3. \u6c42 counter \u7684\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u6b21\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u5c3e\u7d22\u5f15\u201d\n    // \u5373 counter[num]-1 \u662f num \u5728 res \u4e2d\u6700\u540e\u4e00\u6b21\u51fa\u73b0\u7684\u7d22\u5f15\n    for (int i = 0; i &lt; m; i++) {\n        counter[i + 1] += counter[i];\n    }\n    // 4. \u5012\u5e8f\u904d\u5386 nums \uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165\u7ed3\u679c\u6570\u7ec4 res\n    // \u521d\u59cb\u5316\u6570\u7ec4 res \u7528\u4e8e\u8bb0\u5f55\u7ed3\u679c\n    int *res = malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int num = nums[i];\n        res[counter[num] - 1] = num; // \u5c06 num \u653e\u7f6e\u5230\u5bf9\u5e94\u7d22\u5f15\u5904\n        counter[num]--;              // \u4ee4\u524d\u7f00\u548c\u81ea\u51cf 1 \uff0c\u5f97\u5230\u4e0b\u6b21\u653e\u7f6e num \u7684\u7d22\u5f15\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u6570\u7ec4 res \u8986\u76d6\u539f\u6570\u7ec4 nums\n    memcpy(nums, res, size * sizeof(int));\n    // 5. \u91ca\u653e\u5185\u5b58\n    free(counter);\n}\n</code></pre> counting_sort.zig<pre><code>[class]{}-[func]{countingSort}\n</code></pre>"},{"location":"chapter_sorting/counting_sort/#1193","title":"11.9.3 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\) \uff1a\u6d89\u53ca\u904d\u5386 <code>nums</code> \u548c\u904d\u5386 <code>counter</code> \uff0c\u90fd\u4f7f\u7528\u7ebf\u6027\u65f6\u95f4\u3002\u4e00\u822c\u60c5\u51b5\u4e0b \\(n \\gg m\\) \uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u4e8e \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + m)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u501f\u52a9\u4e86\u957f\u5ea6\u5206\u522b\u4e3a \\(n\\) \u548c \\(m\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u7531\u4e8e\u5411 <code>res</code> \u4e2d\u586b\u5145\u5143\u7d20\u7684\u987a\u5e8f\u662f\u201c\u4ece\u53f3\u5411\u5de6\u201d\u7684\uff0c\u56e0\u6b64\u5012\u5e8f\u904d\u5386 <code>nums</code> \u53ef\u4ee5\u907f\u514d\u6539\u53d8\u76f8\u7b49\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b9e\u73b0\u7a33\u5b9a\u6392\u5e8f\u3002\u5b9e\u9645\u4e0a\uff0c\u6b63\u5e8f\u904d\u5386 <code>nums</code> \u4e5f\u53ef\u4ee5\u5f97\u5230\u6b63\u786e\u7684\u6392\u5e8f\u7ed3\u679c\uff0c\u4f46\u7ed3\u679c\u662f\u975e\u7a33\u5b9a\u7684\u3002</li> </ul>"},{"location":"chapter_sorting/counting_sort/#1194","title":"11.9.4 \u00a0 \u5c40\u9650\u6027","text":"<p>\u770b\u5230\u8fd9\u91cc\uff0c\u4f60\u4e5f\u8bb8\u4f1a\u89c9\u5f97\u8ba1\u6570\u6392\u5e8f\u975e\u5e38\u5de7\u5999\uff0c\u4ec5\u901a\u8fc7\u7edf\u8ba1\u6570\u91cf\u5c31\u53ef\u4ee5\u5b9e\u73b0\u9ad8\u6548\u7684\u6392\u5e8f\u3002\u7136\u800c\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u7684\u524d\u7f6e\u6761\u4ef6\u76f8\u5bf9\u8f83\u4e3a\u4e25\u683c\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u53ea\u9002\u7528\u4e8e\u975e\u8d1f\u6574\u6570\u3002\u82e5\u60f3\u5c06\u5176\u7528\u4e8e\u5176\u4ed6\u7c7b\u578b\u7684\u6570\u636e\uff0c\u9700\u8981\u786e\u4fdd\u8fd9\u4e9b\u6570\u636e\u53ef\u4ee5\u8f6c\u6362\u4e3a\u975e\u8d1f\u6574\u6570\uff0c\u5e76\u4e14\u5728\u8f6c\u6362\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u6539\u53d8\u5404\u4e2a\u5143\u7d20\u4e4b\u95f4\u7684\u76f8\u5bf9\u5927\u5c0f\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u5305\u542b\u8d1f\u6570\u7684\u6574\u6570\u6570\u7ec4\uff0c\u53ef\u4ee5\u5148\u7ed9\u6240\u6709\u6570\u5b57\u52a0\u4e0a\u4e00\u4e2a\u5e38\u6570\uff0c\u5c06\u5168\u90e8\u6570\u5b57\u8f6c\u5316\u4e3a\u6b63\u6570\uff0c\u6392\u5e8f\u5b8c\u6210\u540e\u518d\u8f6c\u6362\u56de\u53bb\u3002</p> <p>\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u6bd4\u5982\uff0c\u5728\u4e0a\u8ff0\u793a\u4f8b\u4e2d \\(m\\) \u4e0d\u80fd\u592a\u5927\uff0c\u5426\u5219\u4f1a\u5360\u7528\u8fc7\u591a\u7a7a\u95f4\u3002\u800c\u5f53 \\(n \\ll m\\) \u65f6\uff0c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(m)\\) \u65f6\u95f4\uff0c\u53ef\u80fd\u6bd4 \\(O(n \\log n)\\) \u7684\u6392\u5e8f\u7b97\u6cd5\u8fd8\u8981\u6162\u3002</p>"},{"location":"chapter_sorting/heap_sort/","title":"11.7 \u00a0 \u5806\u6392\u5e8f","text":"<p>Tip</p> <p>\u9605\u8bfb\u672c\u8282\u524d\uff0c\u8bf7\u786e\u4fdd\u5df2\u5b66\u5b8c\u201c\u5806\u201c\u7ae0\u8282\u3002</p> <p>\u300c\u5806\u6392\u5e8f heap sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5806\u6570\u636e\u7ed3\u6784\u5b9e\u73b0\u7684\u9ad8\u6548\u6392\u5e8f\u7b97\u6cd5\u3002\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5df2\u7ecf\u5b66\u8fc7\u7684\u201c\u5efa\u5806\u64cd\u4f5c\u201d\u548c\u201c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u201d\u5b9e\u73b0\u5806\u6392\u5e8f\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5c0f\u9876\u5806\uff0c\u6b64\u65f6\u6700\u5c0f\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u4e0d\u65ad\u6267\u884c\u51fa\u5806\u64cd\u4f5c\uff0c\u4f9d\u6b21\u8bb0\u5f55\u51fa\u5806\u5143\u7d20\uff0c\u5373\u53ef\u5f97\u5230\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u7684\u5e8f\u5217\u3002</li> </ol> <p>\u4ee5\u4e0a\u65b9\u6cd5\u867d\u7136\u53ef\u884c\uff0c\u4f46\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u989d\u5916\u6570\u7ec4\u6765\u4fdd\u5b58\u5f39\u51fa\u7684\u5143\u7d20\uff0c\u6bd4\u8f83\u6d6a\u8d39\u7a7a\u95f4\u3002\u5728\u5b9e\u9645\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u4e00\u79cd\u66f4\u52a0\u4f18\u96c5\u7684\u5b9e\u73b0\u65b9\u5f0f\u3002</p>"},{"location":"chapter_sorting/heap_sort/#1171","title":"11.7.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u5806\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-12 \u6240\u793a\u3002</p> <ol> <li>\u8f93\u5165\u6570\u7ec4\u5e76\u5efa\u7acb\u5927\u9876\u5806\u3002\u5b8c\u6210\u540e\uff0c\u6700\u5927\u5143\u7d20\u4f4d\u4e8e\u5806\u9876\u3002</li> <li>\u5c06\u5806\u9876\u5143\u7d20\uff08\u7b2c\u4e00\u4e2a\u5143\u7d20\uff09\u4e0e\u5806\u5e95\u5143\u7d20\uff08\u6700\u540e\u4e00\u4e2a\u5143\u7d20\uff09\u4ea4\u6362\u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u5806\u7684\u957f\u5ea6\u51cf \\(1\\) \uff0c\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u52a0 \\(1\\) \u3002</li> <li>\u4ece\u5806\u9876\u5143\u7d20\u5f00\u59cb\uff0c\u4ece\u9876\u5230\u5e95\u6267\u884c\u5806\u5316\u64cd\u4f5c\uff08Sift Down\uff09\u3002\u5b8c\u6210\u5806\u5316\u540e\uff0c\u5806\u7684\u6027\u8d28\u5f97\u5230\u4fee\u590d\u3002</li> <li>\u5faa\u73af\u6267\u884c\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\u3002\u5faa\u73af \\(n - 1\\) \u8f6e\u540e\uff0c\u5373\u53ef\u5b8c\u6210\u6570\u7ec4\u6392\u5e8f\u3002</li> </ol> <p>Tip</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5143\u7d20\u51fa\u5806\u64cd\u4f5c\u4e2d\u4e5f\u5305\u542b\u7b2c <code>2.</code> \u6b65\u548c\u7b2c <code>3.</code> \u6b65\uff0c\u53ea\u662f\u591a\u4e86\u4e00\u4e2a\u5f39\u51fa\u5143\u7d20\u7684\u6b65\u9aa4\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt;&lt;12&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-12 \u00a0 \u5806\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u6211\u4eec\u4f7f\u7528\u4e86\u4e0e\u201c\u5806\u201d\u7ae0\u8282\u76f8\u540c\u7684\u4ece\u9876\u81f3\u5e95\u5806\u5316 <code>sift_down()</code> \u51fd\u6570\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7531\u4e8e\u5806\u7684\u957f\u5ea6\u4f1a\u968f\u7740\u63d0\u53d6\u6700\u5927\u5143\u7d20\u800c\u51cf\u5c0f\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7ed9 <code>sift_down()</code> \u51fd\u6570\u6dfb\u52a0\u4e00\u4e2a\u957f\u5ea6\u53c2\u6570 \\(n\\) \uff0c\u7528\u4e8e\u6307\u5b9a\u5806\u7684\u5f53\u524d\u6709\u6548\u957f\u5ea6\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig heap_sort.py<pre><code>def sift_down(nums: list[int], n: int, i: int):\n    \"\"\"\u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316\"\"\"\n    while True:\n        # \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l = 2 * i + 1\n        r = 2 * i + 2\n        ma = i\n        if l &lt; n and nums[l] &gt; nums[ma]:\n            ma = l\n        if r &lt; n and nums[r] &gt; nums[ma]:\n            ma = r\n        # \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i:\n            break\n        # \u4ea4\u6362\u4e24\u8282\u70b9\n        nums[i], nums[ma] = nums[ma], nums[i]\n        # \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n\ndef heap_sort(nums: list[int]):\n    \"\"\"\u5806\u6392\u5e8f\"\"\"\n    # \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in range(len(nums) // 2 - 1, -1, -1):\n        sift_down(nums, len(nums), i)\n    # \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in range(len(nums) - 1, 0, -1):\n        # \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums[0], nums[i] = nums[i], nums[0]\n        # \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        sift_down(nums, i, 0)\n</code></pre> heap_sort.cpp<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(vector&lt;int&gt; &amp;nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        swap(nums[i], nums[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(vector&lt;int&gt; &amp;nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.size() / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, nums.size(), i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.size() - 1; i &gt; 0; --i) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        swap(nums[0], nums[i]);\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.java<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int[] nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.length / 2 - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.cs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid SiftDown(int[] nums, int n, int i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i)\n            break;\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        (nums[ma], nums[i]) = (nums[i], nums[ma]);\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid HeapSort(int[] nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = nums.Length / 2 - 1; i &gt;= 0; i--) {\n        SiftDown(nums, nums.Length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = nums.Length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        (nums[i], nums[0]) = (nums[0], nums[i]);\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        SiftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.go<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums *[]int, n, i int) {\n    for true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        l := 2*i + 1\n        r := 2*i + 2\n        ma := i\n        if l &lt; n &amp;&amp; (*nums)[l] &gt; (*nums)[ma] {\n            ma = l\n        }\n        if r &lt; n &amp;&amp; (*nums)[r] &gt; (*nums)[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        (*nums)[i], (*nums)[ma] = (*nums)[ma], (*nums)[i]\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums *[]int) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i := len(*nums)/2 - 1; i &gt;= 0; i-- {\n        siftDown(nums, len(*nums), i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i := len(*nums) - 1; i &gt; 0; i-- {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        (*nums)[0], (*nums)[i] = (*nums)[i], (*nums)[0]\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0)\n    }\n}\n</code></pre> heap_sort.swift<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunc siftDown(nums: inout [Int], n: Int, i: Int) {\n    var i = i\n    while true {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1\n        let r = 2 * i + 2\n        var ma = i\n        if l &lt; n, nums[l] &gt; nums[ma] {\n            ma = l\n        }\n        if r &lt; n, nums[r] &gt; nums[ma] {\n            ma = r\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        nums.swapAt(i, ma)\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunc heapSort(nums: inout [Int]) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in stride(from: nums.count / 2 - 1, through: 0, by: -1) {\n        siftDown(nums: &amp;nums, n: nums.count, i: i)\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in stride(from: nums.count - 1, to: 0, by: -1) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        nums.swapAt(0, i)\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums: &amp;nums, n: i, i: 0)\n    }\n}\n</code></pre> heap_sort.js<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums, n, i) {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.ts<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfunction siftDown(nums: number[], n: number, i: number): void {\n    while (true) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) {\n            ma = l;\n        }\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma === i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        [nums[i], nums[ma]] = [nums[ma], nums[i]];\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfunction heapSort(nums: number[]): void {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (let i = Math.floor(nums.length / 2) - 1; i &gt;= 0; i--) {\n        siftDown(nums, nums.length, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (let i = nums.length - 1; i &gt; 0; i--) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        [nums[0], nums[i]] = [nums[i], nums[0]];\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.dart<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(List&lt;int&gt; nums, int n, int i) {\n  while (true) {\n    // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n    int l = 2 * i + 1;\n    int r = 2 * i + 2;\n    int ma = i;\n    if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma]) ma = l;\n    if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma]) ma = r;\n    // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n    if (ma == i) break;\n    // \u4ea4\u6362\u4e24\u8282\u70b9\n    int temp = nums[i];\n    nums[i] = nums[ma];\n    nums[ma] = temp;\n    // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n    i = ma;\n  }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(List&lt;int&gt; nums) {\n  // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n  for (int i = nums.length ~/ 2 - 1; i &gt;= 0; i--) {\n    siftDown(nums, nums.length, i);\n  }\n  // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n  for (int i = nums.length - 1; i &gt; 0; i--) {\n    // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n    int tmp = nums[0];\n    nums[0] = nums[i];\n    nums[i] = tmp;\n    // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n    siftDown(nums, i, 0);\n  }\n}\n</code></pre> heap_sort.rs<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nfn sift_down(nums: &amp;mut [i32], n: usize, mut i: usize) {\n    loop {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        let l = 2 * i + 1;\n        let r = 2 * i + 2;\n        let mut ma = i;\n        if l &lt; n &amp;&amp; nums[l] &gt; nums[ma] {\n            ma = l;\n        }\n        if r &lt; n &amp;&amp; nums[r] &gt; nums[ma] {\n            ma = r;\n        }\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if ma == i {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        let temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nfn heap_sort(nums: &amp;mut [i32]) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for i in (0..=nums.len() / 2 - 1).rev() {\n        sift_down(nums, nums.len(), i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for i in (1..=nums.len() - 1).rev() {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        let tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        sift_down(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.c<pre><code>/* \u5806\u7684\u957f\u5ea6\u4e3a n \uff0c\u4ece\u8282\u70b9 i \u5f00\u59cb\uff0c\u4ece\u9876\u81f3\u5e95\u5806\u5316 */\nvoid siftDown(int nums[], int n, int i) {\n    while (1) {\n        // \u5224\u65ad\u8282\u70b9 i, l, r \u4e2d\u503c\u6700\u5927\u7684\u8282\u70b9\uff0c\u8bb0\u4e3a ma\n        int l = 2 * i + 1;\n        int r = 2 * i + 2;\n        int ma = i;\n        if (l &lt; n &amp;&amp; nums[l] &gt; nums[ma])\n            ma = l;\n        if (r &lt; n &amp;&amp; nums[r] &gt; nums[ma])\n            ma = r;\n        // \u82e5\u8282\u70b9 i \u6700\u5927\u6216\u7d22\u5f15 l, r \u8d8a\u754c\uff0c\u5219\u65e0\u987b\u7ee7\u7eed\u5806\u5316\uff0c\u8df3\u51fa\n        if (ma == i) {\n            break;\n        }\n        // \u4ea4\u6362\u4e24\u8282\u70b9\n        int temp = nums[i];\n        nums[i] = nums[ma];\n        nums[ma] = temp;\n        // \u5faa\u73af\u5411\u4e0b\u5806\u5316\n        i = ma;\n    }\n}\n\n/* \u5806\u6392\u5e8f */\nvoid heapSort(int nums[], int n) {\n    // \u5efa\u5806\u64cd\u4f5c\uff1a\u5806\u5316\u9664\u53f6\u8282\u70b9\u4ee5\u5916\u7684\u5176\u4ed6\u6240\u6709\u8282\u70b9\n    for (int i = n / 2 - 1; i &gt;= 0; --i) {\n        siftDown(nums, n, i);\n    }\n    // \u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\uff0c\u5faa\u73af n-1 \u8f6e\n    for (int i = n - 1; i &gt; 0; --i) {\n        // \u4ea4\u6362\u6839\u8282\u70b9\u4e0e\u6700\u53f3\u53f6\u8282\u70b9\uff08\u4ea4\u6362\u9996\u5143\u7d20\u4e0e\u5c3e\u5143\u7d20\uff09\n        int tmp = nums[0];\n        nums[0] = nums[i];\n        nums[i] = tmp;\n        // \u4ee5\u6839\u8282\u70b9\u4e3a\u8d77\u70b9\uff0c\u4ece\u9876\u81f3\u5e95\u8fdb\u884c\u5806\u5316\n        siftDown(nums, i, 0);\n    }\n}\n</code></pre> heap_sort.zig<pre><code>[class]{}-[func]{siftDown}\n\n[class]{}-[func]{heapSort}\n</code></pre>"},{"location":"chapter_sorting/heap_sort/#1172","title":"11.7.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5efa\u5806\u64cd\u4f5c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002\u4ece\u5806\u4e2d\u63d0\u53d6\u6700\u5927\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\) \uff0c\u5171\u5faa\u73af \\(n - 1\\) \u8f6e\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u51e0\u4e2a\u6307\u9488\u53d8\u91cf\u4f7f\u7528 \\(O(1)\\) \u7a7a\u95f4\u3002\u5143\u7d20\u4ea4\u6362\u548c\u5806\u5316\u64cd\u4f5c\u90fd\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u4ea4\u6362\u5806\u9876\u5143\u7d20\u548c\u5806\u5e95\u5143\u7d20\u65f6\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u76f8\u5bf9\u4f4d\u7f6e\u53ef\u80fd\u53d1\u751f\u53d8\u5316\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/","title":"11.4 \u00a0 \u63d2\u5165\u6392\u5e8f","text":"<p>\u300c\u63d2\u5165\u6392\u5e8f insertion sort\u300d\u662f\u4e00\u79cd\u7b80\u5355\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u624b\u52a8\u6574\u7406\u4e00\u526f\u724c\u7684\u8fc7\u7a0b\u975e\u5e38\u76f8\u4f3c\u3002</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u5728\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u4e00\u4e2a\u57fa\u51c6\u5143\u7d20\uff0c\u5c06\u8be5\u5143\u7d20\u4e0e\u5176\u5de6\u4fa7\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u5143\u7d20\u9010\u4e00\u6bd4\u8f83\u5927\u5c0f\uff0c\u5e76\u5c06\u8be5\u5143\u7d20\u63d2\u5165\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u3002</p> <p>\u56fe 11-6 \u5c55\u793a\u4e86\u6570\u7ec4\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u6d41\u7a0b\u3002\u8bbe\u57fa\u51c6\u5143\u7d20\u4e3a <code>base</code> \uff0c\u6211\u4eec\u9700\u8981\u5c06\u4ece\u76ee\u6807\u7d22\u5f15\u5230 <code>base</code> \u4e4b\u95f4\u7684\u6240\u6709\u5143\u7d20\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\uff0c\u7136\u540e\u5c06 <code>base</code> \u8d4b\u503c\u7ed9\u76ee\u6807\u7d22\u5f15\u3002</p> <p></p> <p> \u56fe 11-6 \u00a0 \u5355\u6b21\u63d2\u5165\u64cd\u4f5c </p>"},{"location":"chapter_sorting/insertion_sort/#1141","title":"11.4.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-7 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6570\u7ec4\u7684\u7b2c 1 \u4e2a\u5143\u7d20\u5df2\u5b8c\u6210\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u6570\u7ec4\u7684\u7b2c 2 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u7b2c 3 \u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6570\u7ec4\u7684\u524d 3 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u6700\u540e\u4e00\u8f6e\u4e2d\uff0c\u9009\u53d6\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a <code>base</code> \uff0c\u5c06\u5176\u63d2\u5165\u5230\u6b63\u786e\u4f4d\u7f6e\u540e\uff0c\u6240\u6709\u5143\u7d20\u5747\u5df2\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-7 \u00a0 \u63d2\u5165\u6392\u5e8f\u6d41\u7a0b </p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig insertion_sort.py<pre><code>def insertion_sort(nums: list[int]):\n    \"\"\"\u63d2\u5165\u6392\u5e8f\"\"\"\n    # \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u533a\u95f4\u4e3a [0, i-1]\n    for i in range(1, len(nums)):\n        base = nums[i]\n        j = i - 1\n        # \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4 [0, i-1] \u4e2d\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0 and nums[j] &gt; base:\n            nums[j + 1] = nums[j]  # \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1\n        nums[j + 1] = base  # \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n</code></pre> insertion_sort.cpp<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(vector&lt;int&gt; &amp;nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for (int i = 1; i &lt; nums.size(); i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.java<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for (int i = 1; i &lt; nums.length; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base;        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.cs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid InsertionSort(int[] nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for (int i = 1; i &lt; nums.Length; i++) {\n        int bas = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; bas) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = bas;         // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.go<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums []int) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [0, i]\n    for i := 1; i &lt; len(nums); i++ {\n        base := nums[i]\n        j := i - 1\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        for j &gt;= 0 &amp;&amp; nums[j] &gt; base {\n            nums[j+1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--\n        }\n        nums[j+1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.swift<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunc insertionSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for i in stride(from: 1, to: nums.count, by: 1) {\n        let base = nums[i]\n        var j = i - 1\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0, nums[j] &gt; base {\n            nums[j + 1] = nums[j] // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1\n        }\n        nums[j + 1] = base // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.js<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for (let i = 1; i &lt; nums.length; i++) {\n        let base = nums[i],\n            j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.ts<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfunction insertionSort(nums: number[]): void {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for (let i = 1; i &lt; nums.length; i++) {\n        const base = nums[i];\n        let j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j--;\n        }\n        nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.dart<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(List&lt;int&gt; nums) {\n  // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n  for (int i = 1; i &lt; nums.length; i++) {\n    int base = nums[i], j = i - 1;\n    // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n    while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n      nums[j + 1] = nums[j]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n      j--;\n    }\n    nums[j + 1] = base; // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n  }\n}\n</code></pre> insertion_sort.rs<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nfn insertion_sort(nums: &amp;mut [i32]) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for i in 1..nums.len() {\n        let (base, mut j) = (nums[i],  (i - 1) as i32);\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while j &gt;= 0 &amp;&amp; nums[j as usize] &gt; base {\n            nums[(j + 1) as usize] = nums[j as usize]; // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            j -= 1;\n        }\n        nums[(j + 1) as usize] = base;  // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre> insertion_sort.c<pre><code>/* \u63d2\u5165\u6392\u5e8f */\nvoid insertionSort(int nums[], int size) {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    for (int i = 1; i &lt; size; i++) {\n        int base = nums[i], j = i - 1;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 0 &amp;&amp; nums[j] &gt; base) {\n            // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n            nums[j + 1] = nums[j];\n            j--;\n        }\n        // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n        nums[j + 1] = base;\n    }\n}\n</code></pre> insertion_sort.zig<pre><code>// \u63d2\u5165\u6392\u5e8f\nfn insertionSort(nums: []i32) void {\n    // \u5916\u5faa\u73af\uff1a\u5df2\u6392\u5e8f\u5143\u7d20\u6570\u91cf\u4e3a 1, 2, ..., n\n    var i: usize = 1;\n    while (i &lt; nums.len) : (i += 1) {\n        var base = nums[i];\n        var j: usize = i;\n        // \u5185\u5faa\u73af\uff1a\u5c06 base \u63d2\u5165\u5230\u5df2\u6392\u5e8f\u90e8\u5206\u7684\u6b63\u786e\u4f4d\u7f6e\n        while (j &gt;= 1 and nums[j - 1] &gt; base) : (j -= 1) {\n            nums[j] = nums[j - 1];  // \u5c06 nums[j] \u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n        }\n        nums[j] = base;             // \u5c06 base \u8d4b\u503c\u5230\u6b63\u786e\u4f4d\u7f6e\n    }\n}\n</code></pre>"},{"location":"chapter_sorting/insertion_sort/#1142","title":"11.4.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n^2)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u6b21\u63d2\u5165\u64cd\u4f5c\u5206\u522b\u9700\u8981\u5faa\u73af \\(n - 1\\)\u3001\\(n-2\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \u6b21\uff0c\u6c42\u548c\u5f97\u5230 \\((n - 1) n / 2\\) \uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002\u5728\u9047\u5230\u6709\u5e8f\u6570\u636e\u65f6\uff0c\u63d2\u5165\u64cd\u4f5c\u4f1a\u63d0\u524d\u7ec8\u6b62\u3002\u5f53\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u6709\u5e8f\u65f6\uff0c\u63d2\u5165\u6392\u5e8f\u8fbe\u5230\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u63d2\u5165\u64cd\u4f5c\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4f1a\u5c06\u5143\u7d20\u63d2\u5165\u5230\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\uff0c\u4e0d\u4f1a\u6539\u53d8\u5b83\u4eec\u7684\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/insertion_sort/#1143","title":"11.4.3 \u00a0 \u63d2\u5165\u6392\u5e8f\u4f18\u52bf","text":"<p>\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u800c\u6211\u4eec\u5373\u5c06\u5b66\u4e60\u7684\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u5c3d\u7ba1\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u66f4\u9ad8\uff0c\u4f46\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u7684\u60c5\u51b5\u4e0b\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u66f4\u5feb\u3002</p> <p>\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u7ebf\u6027\u67e5\u627e\u548c\u4e8c\u5206\u67e5\u627e\u7684\u9002\u7528\u60c5\u51b5\u7684\u7ed3\u8bba\u7c7b\u4f3c\u3002\u5feb\u901f\u6392\u5e8f\u8fd9\u7c7b \\(O(n \\log n)\\) \u7684\u7b97\u6cd5\u5c5e\u4e8e\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5f80\u5f80\u5305\u542b\u66f4\u591a\u5355\u5143\u8ba1\u7b97\u64cd\u4f5c\u3002\u800c\u5728\u6570\u636e\u91cf\u8f83\u5c0f\u65f6\uff0c\\(n^2\\) \u548c \\(n \\log n\\) \u7684\u6570\u503c\u6bd4\u8f83\u63a5\u8fd1\uff0c\u590d\u6742\u5ea6\u4e0d\u5360\u4e3b\u5bfc\u5730\u4f4d\uff1b\u6bcf\u8f6e\u4e2d\u7684\u5355\u5143\u64cd\u4f5c\u6570\u91cf\u8d77\u5230\u51b3\u5b9a\u6027\u4f5c\u7528\u3002</p> <p>\u5b9e\u9645\u4e0a\uff0c\u8bb8\u591a\u7f16\u7a0b\u8bed\u8a00\uff08\u4f8b\u5982 Java\uff09\u7684\u5185\u7f6e\u6392\u5e8f\u51fd\u6570\u91c7\u7528\u4e86\u63d2\u5165\u6392\u5e8f\uff0c\u5927\u81f4\u601d\u8def\u4e3a\uff1a\u5bf9\u4e8e\u957f\u6570\u7ec4\uff0c\u91c7\u7528\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u4f8b\u5982\u5feb\u901f\u6392\u5e8f\uff1b\u5bf9\u4e8e\u77ed\u6570\u7ec4\uff0c\u76f4\u63a5\u4f7f\u7528\u63d2\u5165\u6392\u5e8f\u3002</p> <p>\u867d\u7136\u5192\u6ce1\u6392\u5e8f\u3001\u9009\u62e9\u6392\u5e8f\u548c\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u5728\u5b9e\u9645\u60c5\u51b5\u4e2d\uff0c\u63d2\u5165\u6392\u5e8f\u7684\u4f7f\u7528\u9891\u7387\u663e\u8457\u9ad8\u4e8e\u5192\u6ce1\u6392\u5e8f\u548c\u9009\u62e9\u6392\u5e8f\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u5192\u6ce1\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u4ea4\u6362\u5b9e\u73b0\uff0c\u9700\u8981\u501f\u52a9\u4e00\u4e2a\u4e34\u65f6\u53d8\u91cf\uff0c\u5171\u6d89\u53ca 3 \u4e2a\u5355\u5143\u64cd\u4f5c\uff1b\u63d2\u5165\u6392\u5e8f\u57fa\u4e8e\u5143\u7d20\u8d4b\u503c\u5b9e\u73b0\uff0c\u4ec5\u9700 1 \u4e2a\u5355\u5143\u64cd\u4f5c\u3002\u56e0\u6b64\uff0c\u5192\u6ce1\u6392\u5e8f\u7684\u8ba1\u7b97\u5f00\u9500\u901a\u5e38\u6bd4\u63d2\u5165\u6392\u5e8f\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u5728\u4efb\u4f55\u60c5\u51b5\u4e0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(n^2)\\) \u3002\u5982\u679c\u7ed9\u5b9a\u4e00\u7ec4\u90e8\u5206\u6709\u5e8f\u7684\u6570\u636e\uff0c\u63d2\u5165\u6392\u5e8f\u901a\u5e38\u6bd4\u9009\u62e9\u6392\u5e8f\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u9009\u62e9\u6392\u5e8f\u4e0d\u7a33\u5b9a\uff0c\u65e0\u6cd5\u5e94\u7528\u4e8e\u591a\u7ea7\u6392\u5e8f\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/","title":"11.6 \u00a0 \u5f52\u5e76\u6392\u5e8f","text":"<p>\u300c\u5f52\u5e76\u6392\u5e8f merge sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u5305\u542b\u56fe 11-10 \u6240\u793a\u7684\u201c\u5212\u5206\u201d\u548c\u201c\u5408\u5e76\u201d\u9636\u6bb5\u3002</p> <ol> <li>\u5212\u5206\u9636\u6bb5\uff1a\u901a\u8fc7\u9012\u5f52\u4e0d\u65ad\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5904\u5206\u5f00\uff0c\u5c06\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u8f6c\u6362\u4e3a\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u5212\u5206\uff0c\u5f00\u59cb\u5408\u5e76\uff0c\u6301\u7eed\u5730\u5c06\u5de6\u53f3\u4e24\u4e2a\u8f83\u77ed\u7684\u6709\u5e8f\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u8f83\u957f\u7684\u6709\u5e8f\u6570\u7ec4\uff0c\u76f4\u81f3\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-10 \u00a0 \u5f52\u5e76\u6392\u5e8f\u7684\u5212\u5206\u4e0e\u5408\u5e76\u9636\u6bb5 </p>"},{"location":"chapter_sorting/merge_sort/#1161","title":"11.6.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5982\u56fe 11-11 \u6240\u793a\uff0c\u201c\u5212\u5206\u9636\u6bb5\u201d\u4ece\u9876\u81f3\u5e95\u9012\u5f52\u5730\u5c06\u6570\u7ec4\u4ece\u4e2d\u70b9\u5207\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002</p> <ol> <li>\u8ba1\u7b97\u6570\u7ec4\u4e2d\u70b9 <code>mid</code> \uff0c\u9012\u5f52\u5212\u5206\u5de6\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[left, mid]</code> \uff09\u548c\u53f3\u5b50\u6570\u7ec4\uff08\u533a\u95f4 <code>[mid + 1, right]</code> \uff09\u3002</li> <li>\u9012\u5f52\u6267\u884c\u6b65\u9aa4 <code>1.</code> \uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u533a\u95f4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u3002</li> </ol> <p>\u201c\u5408\u5e76\u9636\u6bb5\u201d\u4ece\u5e95\u81f3\u9876\u5730\u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5408\u5e76\u4e3a\u4e00\u4e2a\u6709\u5e8f\u6570\u7ec4\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4ece\u957f\u5ea6\u4e3a 1 \u7684\u5b50\u6570\u7ec4\u5f00\u59cb\u5408\u5e76\uff0c\u5408\u5e76\u9636\u6bb5\u4e2d\u7684\u6bcf\u4e2a\u5b50\u6570\u7ec4\u90fd\u662f\u6709\u5e8f\u7684\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-11 \u00a0 \u5f52\u5e76\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u89c2\u5bdf\u53d1\u73b0\uff0c\u5f52\u5e76\u6392\u5e8f\u4e0e\u4e8c\u53c9\u6811\u540e\u5e8f\u904d\u5386\u7684\u9012\u5f52\u987a\u5e8f\u662f\u4e00\u81f4\u7684\u3002</p> <ul> <li>\u540e\u5e8f\u904d\u5386\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6811\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6811\uff0c\u6700\u540e\u5904\u7406\u6839\u8282\u70b9\u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\uff1a\u5148\u9012\u5f52\u5de6\u5b50\u6570\u7ec4\uff0c\u518d\u9012\u5f52\u53f3\u5b50\u6570\u7ec4\uff0c\u6700\u540e\u5904\u7406\u5408\u5e76\u3002</li> </ul> <p>\u5f52\u5e76\u6392\u5e8f\u7684\u5b9e\u73b0\u5982\u4ee5\u4e0b\u4ee3\u7801\u6240\u793a\u3002\u8bf7\u6ce8\u610f\uff0c<code>nums</code> \u7684\u5f85\u5408\u5e76\u533a\u95f4\u4e3a <code>[left, right]</code> \uff0c\u800c <code>tmp</code> \u7684\u5bf9\u5e94\u533a\u95f4\u4e3a <code>[0, right - left]</code> \u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig merge_sort.py<pre><code>def merge(nums: list[int], left: int, mid: int, right: int):\n    \"\"\"\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\"\"\"\n    # \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    # \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    tmp = [0] * (right - left + 1)\n    # \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k = left, mid + 1, 0\n    # \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid and j &lt;= right:\n        if nums[i] &lt;= nums[j]:\n            tmp[k] = nums[i]\n            i += 1\n        else:\n            tmp[k] = nums[j]\n            j += 1\n        k += 1\n    # \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid:\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    while j &lt;= right:\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    # \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in range(0, len(tmp)):\n        nums[left + k] = tmp[k]\n\ndef merge_sort(nums: list[int], left: int, right: int):\n    \"\"\"\u5f52\u5e76\u6392\u5e8f\"\"\"\n    # \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right:\n        return  # \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    # \u5212\u5206\u9636\u6bb5\n    mid = (left + right) // 2  # \u8ba1\u7b97\u4e2d\u70b9\n    merge_sort(nums, left, mid)  # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    merge_sort(nums, mid + 1, right)  # \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    # \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n</code></pre> merge_sort.cpp<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    vector&lt;int&gt; tmp(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.size(); k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.java<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int[] nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2; // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.cs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid Merge(int[] nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int[] tmp = new int[right - left + 1];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j])\n            tmp[k++] = nums[i++];\n        else\n            tmp[k++] = nums[j++];\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.Length; ++k) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid MergeSort(int[] nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return;       // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    MergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    MergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    Merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.go<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums []int, left, mid, right int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    tmp := make([]int, right-left+1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    i, j, k := left, mid+1, 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    for i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i++\n        } else {\n            tmp[k] = nums[j]\n            j++\n        }\n        k++\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    for i &lt;= mid {\n        tmp[k] = nums[i]\n        i++\n        k++\n    }\n    for j &lt;= right {\n        tmp[k] = nums[j]\n        j++\n        k++\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k := 0; k &lt; len(tmp); k++ {\n        nums[left+k] = tmp[k]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums []int, left, right int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right {\n        return\n    }\n    // \u5212\u5206\u9636\u6bb5\n    mid := (left + right) / 2\n    mergeSort(nums, left, mid)\n    mergeSort(nums, mid+1, right)\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right)\n}\n</code></pre> merge_sort.swift<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunc merge(nums: inout [Int], left: Int, mid: Int, right: Int) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    var tmp = Array(repeating: 0, count: right - left + 1)\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    var i = left, j = mid + 1, k = 0\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid, j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[i]\n            i += 1\n            k += 1\n        } else {\n            tmp[k] = nums[j]\n            j += 1\n            k += 1\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i]\n        i += 1\n        k += 1\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j]\n        j += 1\n        k += 1\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in tmp.indices {\n        nums[left + k] = tmp[k]\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunc mergeSort(nums: inout [Int], left: Int, right: Int) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right { // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n        return\n    }\n    // \u5212\u5206\u9636\u6bb5\n    let mid = (left + right) / 2 // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums: &amp;nums, left: left, right: mid) // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums: &amp;nums, left: mid + 1, right: right) // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums: &amp;nums, left: left, mid: mid, right: right)\n}\n</code></pre> merge_sort.js<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums, left, mid, right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums, left, right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.ts<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfunction merge(nums: number[], left: number, mid: number, right: number): void {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    const tmp = new Array(right - left + 1);\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let i = left,\n        j = mid + 1,\n        k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmp.length; k++) {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfunction mergeSort(nums: number[], left: number, right: number): void {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    let mid = Math.floor((left + right) / 2); // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.dart<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(List&lt;int&gt; nums, int left, int mid, int right) {\n  // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n  // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n  List&lt;int&gt; tmp = List.filled(right - left + 1, 0);\n  // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n  int i = left, j = mid + 1, k = 0;\n  // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n  while (i &lt;= mid &amp;&amp; j &lt;= right) {\n    if (nums[i] &lt;= nums[j])\n      tmp[k++] = nums[i++];\n    else\n      tmp[k++] = nums[j++];\n  }\n  // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n  while (i &lt;= mid) {\n    tmp[k++] = nums[i++];\n  }\n  while (j &lt;= right) {\n    tmp[k++] = nums[j++];\n  }\n  // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n  for (k = 0; k &lt; tmp.length; k++) {\n    nums[left + k] = tmp[k];\n  }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(List&lt;int&gt; nums, int left, int right) {\n  // \u7ec8\u6b62\u6761\u4ef6\n  if (left &gt;= right) return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n  // \u5212\u5206\u9636\u6bb5\n  int mid = (left + right) ~/ 2; // \u8ba1\u7b97\u4e2d\u70b9\n  mergeSort(nums, left, mid); // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n  mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n  // \u5408\u5e76\u9636\u6bb5\n  merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.rs<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nfn merge(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    let tmp_size = right - left + 1;\n    let mut tmp = vec![0; tmp_size];\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    let (mut i, mut j, mut k) = (left, mid + 1, 0);\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid &amp;&amp; j &lt;= right {\n        if nums[i] &lt;= nums[j] {\n            tmp[k] = nums[j];\n            i += 1;\n        } else {\n            tmp[k] = nums[j];\n            j += 1;\n        }\n        k += 1;\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while i &lt;= mid {\n        tmp[k] = nums[i];\n        k += 1;\n        i += 1;\n    }\n    while j &lt;= right {\n        tmp[k] = nums[j];\n        k += 1;\n        j += 1;\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for k in 0..tmp_size {\n        nums[left + k] = tmp[k];\n    }\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nfn merge_sort(nums: &amp;mut [i32], left: usize, right: usize) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if left &gt;= right { return; }             // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    let mid = (left + right) / 2;     // \u8ba1\u7b97\u4e2d\u70b9\n    merge_sort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    merge_sort(nums, mid + 1, right);  // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.c<pre><code>/* \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4 */\nvoid merge(int *nums, int left, int mid, int right) {\n    // \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid], \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid+1, right]\n    // \u521b\u5efa\u4e00\u4e2a\u4e34\u65f6\u6570\u7ec4 tmp \uff0c\u7528\u4e8e\u5b58\u653e\u5408\u5e76\u540e\u7684\u7ed3\u679c\n    int tmpSize = right - left + 1;\n    int *tmp = (int *)malloc(tmpSize * sizeof(int));\n    // \u521d\u59cb\u5316\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\n    int i = left, j = mid + 1, k = 0;\n    // \u5f53\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u8fd8\u6709\u5143\u7d20\u65f6\uff0c\u6bd4\u8f83\u5e76\u5c06\u8f83\u5c0f\u7684\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid &amp;&amp; j &lt;= right) {\n        if (nums[i] &lt;= nums[j]) {\n            tmp[k++] = nums[i++];\n        } else {\n            tmp[k++] = nums[j++];\n        }\n    }\n    // \u5c06\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u7684\u5269\u4f59\u5143\u7d20\u590d\u5236\u5230\u4e34\u65f6\u6570\u7ec4\u4e2d\n    while (i &lt;= mid) {\n        tmp[k++] = nums[i++];\n    }\n    while (j &lt;= right) {\n        tmp[k++] = nums[j++];\n    }\n    // \u5c06\u4e34\u65f6\u6570\u7ec4 tmp \u4e2d\u7684\u5143\u7d20\u590d\u5236\u56de\u539f\u6570\u7ec4 nums \u7684\u5bf9\u5e94\u533a\u95f4\n    for (k = 0; k &lt; tmpSize; ++k) {\n        nums[left + k] = tmp[k];\n    }\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n}\n\n/* \u5f52\u5e76\u6392\u5e8f */\nvoid mergeSort(int *nums, int left, int right) {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right)\n        return; // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    int mid = (left + right) / 2;    // \u8ba1\u7b97\u4e2d\u70b9\n    mergeSort(nums, left, mid);      // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    mergeSort(nums, mid + 1, right); // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    merge(nums, left, mid, right);\n}\n</code></pre> merge_sort.zig<pre><code>// \u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n// \u5de6\u5b50\u6570\u7ec4\u533a\u95f4 [left, mid]\n// \u53f3\u5b50\u6570\u7ec4\u533a\u95f4 [mid + 1, right]\nfn merge(nums: []i32, left: usize, mid: usize, right: usize) !void {\n    // \u521d\u59cb\u5316\u8f85\u52a9\u6570\u7ec4\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var tmp = try mem_allocator.alloc(i32, right + 1 - left);\n    std.mem.copy(i32, tmp, nums[left..right+1]);\n    // \u5de6\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15  \n    var leftStart = left - left;\n    var leftEnd = mid - left;\n    // \u53f3\u5b50\u6570\u7ec4\u7684\u8d77\u59cb\u7d22\u5f15\u548c\u7ed3\u675f\u7d22\u5f15       \n    var rightStart = mid + 1 - left;\n    var rightEnd = right - left;\n    // i, j \u5206\u522b\u6307\u5411\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\u7684\u9996\u5143\u7d20\n    var i = leftStart;\n    var j = rightStart;\n    // \u901a\u8fc7\u8986\u76d6\u539f\u6570\u7ec4 nums \u6765\u5408\u5e76\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\n    var k = left;\n    while (k &lt;= right) : (k += 1) {\n        // \u82e5\u201c\u5de6\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n        if (i &gt; leftEnd) {\n            nums[k] = tmp[j];\n            j += 1;\n        // \u5426\u5219\uff0c\u82e5\u201c\u53f3\u5b50\u6570\u7ec4\u5df2\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u6216\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &lt;= \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u5de6\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 i++\n        } else if  (j &gt; rightEnd or tmp[i] &lt;= tmp[j]) {\n            nums[k] = tmp[i];\n            i += 1;\n        // \u5426\u5219\uff0c\u82e5\u201c\u5de6\u53f3\u5b50\u6570\u7ec4\u90fd\u672a\u5168\u90e8\u5408\u5e76\u5b8c\u201d\u4e14\u201c\u5de6\u5b50\u6570\u7ec4\u5143\u7d20 &gt; \u53f3\u5b50\u6570\u7ec4\u5143\u7d20\u201d\uff0c\u5219\u9009\u53d6\u53f3\u5b50\u6570\u7ec4\u5143\u7d20\uff0c\u5e76\u4e14 j++\n        } else {\n            nums[k] = tmp[j];\n            j += 1;\n        }\n    }\n}\n\n// \u5f52\u5e76\u6392\u5e8f\nfn mergeSort(nums: []i32, left: usize, right: usize) !void {\n    // \u7ec8\u6b62\u6761\u4ef6\n    if (left &gt;= right) return;              // \u5f53\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    // \u5212\u5206\u9636\u6bb5\n    var mid = (left + right) / 2;           // \u8ba1\u7b97\u4e2d\u70b9\n    try mergeSort(nums, left, mid);         // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\n    try mergeSort(nums, mid + 1, right);    // \u9012\u5f52\u53f3\u5b50\u6570\u7ec4\n    // \u5408\u5e76\u9636\u6bb5\n    try merge(nums, left, mid, right);\n}\n</code></pre>"},{"location":"chapter_sorting/merge_sort/#1162","title":"11.6.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5212\u5206\u4ea7\u751f\u9ad8\u5ea6\u4e3a \\(\\log n\\) \u7684\u9012\u5f52\u6811\uff0c\u6bcf\u5c42\u5408\u5e76\u7684\u603b\u64cd\u4f5c\u6570\u91cf\u4e3a \\(n\\) \uff0c\u56e0\u6b64\u603b\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u9012\u5f52\u6df1\u5ea6\u4e3a \\(\\log n\\) \uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002\u5408\u5e76\u64cd\u4f5c\u9700\u8981\u501f\u52a9\u8f85\u52a9\u6570\u7ec4\u5b9e\u73b0\uff0c\u4f7f\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u76f8\u7b49\u5143\u7d20\u7684\u6b21\u5e8f\u4fdd\u6301\u4e0d\u53d8\u3002</li> </ul>"},{"location":"chapter_sorting/merge_sort/#1163","title":"11.6.3 \u00a0 \u94fe\u8868\u6392\u5e8f","text":"<p>\u5bf9\u4e8e\u94fe\u8868\uff0c\u5f52\u5e76\u6392\u5e8f\u76f8\u8f83\u4e8e\u5176\u4ed6\u6392\u5e8f\u7b97\u6cd5\u5177\u6709\u663e\u8457\u4f18\u52bf\uff0c\u53ef\u4ee5\u5c06\u94fe\u8868\u6392\u5e8f\u4efb\u52a1\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</p> <ul> <li>\u5212\u5206\u9636\u6bb5\uff1a\u53ef\u4ee5\u4f7f\u7528\u201c\u8fed\u4ee3\u201d\u66ff\u4ee3\u201c\u9012\u5f52\u201d\u6765\u5b9e\u73b0\u94fe\u8868\u5212\u5206\u5de5\u4f5c\uff0c\u4ece\u800c\u7701\u53bb\u9012\u5f52\u4f7f\u7528\u7684\u6808\u5e27\u7a7a\u95f4\u3002</li> <li>\u5408\u5e76\u9636\u6bb5\uff1a\u5728\u94fe\u8868\u4e2d\uff0c\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u4ec5\u9700\u6539\u53d8\u5f15\u7528\uff08\u6307\u9488\uff09\u5373\u53ef\u5b9e\u73b0\uff0c\u56e0\u6b64\u5408\u5e76\u9636\u6bb5\uff08\u5c06\u4e24\u4e2a\u77ed\u6709\u5e8f\u94fe\u8868\u5408\u5e76\u4e3a\u4e00\u4e2a\u957f\u6709\u5e8f\u94fe\u8868\uff09\u65e0\u987b\u521b\u5efa\u989d\u5916\u94fe\u8868\u3002</li> </ul> <p>\u5177\u4f53\u5b9e\u73b0\u7ec6\u8282\u6bd4\u8f83\u590d\u6742\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u67e5\u9605\u76f8\u5173\u8d44\u6599\u8fdb\u884c\u5b66\u4e60\u3002</p>"},{"location":"chapter_sorting/quick_sort/","title":"11.5 \u00a0 \u5feb\u901f\u6392\u5e8f","text":"<p>\u300c\u5feb\u901f\u6392\u5e8f quick sort\u300d\u662f\u4e00\u79cd\u57fa\u4e8e\u5206\u6cbb\u7b56\u7565\u7684\u6392\u5e8f\u7b97\u6cd5\uff0c\u8fd0\u884c\u9ad8\u6548\uff0c\u5e94\u7528\u5e7f\u6cdb\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u6838\u5fc3\u64cd\u4f5c\u662f\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5176\u76ee\u6807\u662f\uff1a\u9009\u62e9\u6570\u7ec4\u4e2d\u7684\u67d0\u4e2a\u5143\u7d20\u4f5c\u4e3a\u201c\u57fa\u51c6\u6570\u201d\uff0c\u5c06\u6240\u6709\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u5de6\u4fa7\uff0c\u800c\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\u79fb\u5230\u5176\u53f3\u4fa7\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u54e8\u5175\u5212\u5206\u7684\u6d41\u7a0b\u5982\u56fe 11-8 \u6240\u793a\u3002</p> <ol> <li>\u9009\u53d6\u6570\u7ec4\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u521d\u59cb\u5316\u4e24\u4e2a\u6307\u9488 <code>i</code> \u548c <code>j</code> \u5206\u522b\u6307\u5411\u6570\u7ec4\u7684\u4e24\u7aef\u3002</li> <li>\u8bbe\u7f6e\u4e00\u4e2a\u5faa\u73af\uff0c\u5728\u6bcf\u8f6e\u4e2d\u4f7f\u7528 <code>i</code>\uff08<code>j</code>\uff09\u5206\u522b\u5bfb\u627e\u7b2c\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u5927\uff08\u5c0f\uff09\u7684\u5143\u7d20\uff0c\u7136\u540e\u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\u3002</li> <li>\u5faa\u73af\u6267\u884c\u6b65\u9aa4 <code>2.</code> \uff0c\u76f4\u5230 <code>i</code> \u548c <code>j</code> \u76f8\u9047\u65f6\u505c\u6b62\uff0c\u6700\u540e\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-8 \u00a0 \u54e8\u5175\u5212\u5206\u6b65\u9aa4 </p> <p>\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u539f\u6570\u7ec4\u88ab\u5212\u5206\u6210\u4e09\u90e8\u5206\uff1a\u5de6\u5b50\u6570\u7ec4\u3001\u57fa\u51c6\u6570\u3001\u53f3\u5b50\u6570\u7ec4\uff0c\u4e14\u6ee1\u8db3\u201c\u5de6\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20 \\(\\leq\\) \u57fa\u51c6\u6570 \\(\\leq\\) \u53f3\u5b50\u6570\u7ec4\u4efb\u610f\u5143\u7d20\u201d\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u63a5\u4e0b\u6765\u53ea\u9700\u5bf9\u8fd9\u4e24\u4e2a\u5b50\u6570\u7ec4\u8fdb\u884c\u6392\u5e8f\u3002</p> <p>\u5feb\u901f\u6392\u5e8f\u7684\u5206\u6cbb\u7b56\u7565</p> <p>\u54e8\u5175\u5212\u5206\u7684\u5b9e\u8d28\u662f\u5c06\u4e00\u4e2a\u8f83\u957f\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u7b80\u5316\u4e3a\u4e24\u4e2a\u8f83\u77ed\u6570\u7ec4\u7684\u6392\u5e8f\u95ee\u9898\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5212\u5206\"\"\"\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(vector&lt;int&gt; &amp;nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int[] nums, int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid Swap(int[] nums, int i, int j) {\n    (nums[j], nums[i]) = (nums[i], nums[j]);\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint Partition(int[] nums, int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfunc (q *quickSort) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nfunc swap(nums: inout [Int], i: Int, j: Int) {\n    let tmp = nums[i]\n    nums[i] = nums[j]\n    nums[j] = tmp\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nfunc partition(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left\n    var j = right\n    while i &lt; j {\n        while i &lt; j, nums[j] &gt;= nums[left] {\n            j -= 1 // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j, nums[i] &lt;= nums[left] {\n            i += 1 // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        swap(nums: &amp;nums, i: i, j: j) // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums: &amp;nums, i: i, j: left) // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.js<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums, i, j) {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums, left, right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nswap(nums: number[], i: number, j: number): void {\n    let tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\npartition(nums: number[], left: number, right: number): number {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        // \u5143\u7d20\u4ea4\u6362\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid _swap(List&lt;int&gt; nums, int i, int j) {\n  int tmp = nums[i];\n  nums[i] = nums[j];\n  nums[j] = tmp;\n}\n\n/* \u54e8\u5175\u5212\u5206 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n  return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u54e8\u5175\u5212\u5206 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1;      // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1;      // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    nums.swap(i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    i                    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5143\u7d20\u4ea4\u6362 */\nvoid swap(int nums[], int i, int j) {\n    int tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            j--;\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            i++;\n        }\n        // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n    return i;\n}\n</code></pre> quick_sort.zig<pre><code>// \u5143\u7d20\u4ea4\u6362\nfn swap(nums: []i32, i: usize, j: usize) void {\n    var tmp = nums[i];\n    nums[i] = nums[j];\n    nums[j] = tmp;\n}\n\n// \u54e8\u5175\u5212\u5206\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre>"},{"location":"chapter_sorting/quick_sort/#1151","title":"11.5.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u5feb\u901f\u6392\u5e8f\u7684\u6574\u4f53\u6d41\u7a0b\u5982\u56fe 11-9 \u6240\u793a\u3002</p> <ol> <li>\u9996\u5148\uff0c\u5bf9\u539f\u6570\u7ec4\u6267\u884c\u4e00\u6b21\u201c\u54e8\u5175\u5212\u5206\u201d\uff0c\u5f97\u5230\u672a\u6392\u5e8f\u7684\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u3002</li> <li>\u7136\u540e\uff0c\u5bf9\u5de6\u5b50\u6570\u7ec4\u548c\u53f3\u5b50\u6570\u7ec4\u5206\u522b\u9012\u5f52\u6267\u884c\u201c\u54e8\u5175\u5212\u5206\u201d\u3002</li> <li>\u6301\u7eed\u9012\u5f52\uff0c\u76f4\u81f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\uff0c\u4ece\u800c\u5b8c\u6210\u6574\u4e2a\u6570\u7ec4\u7684\u6392\u5e8f\u3002</li> </ol> <p></p> <p> \u56fe 11-9 \u00a0 \u5feb\u901f\u6392\u5e8f\u6d41\u7a0b </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\"\"\"\n    # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right:\n        return\n    # \u54e8\u5175\u5212\u5206\n    pivot = self.partition(nums, left, right)\n    # \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    self.quick_sort(nums, left, pivot - 1)\n    self.quick_sort(nums, pivot + 1, right)\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right)\n        return;\n    // \u54e8\u5175\u5212\u5206\n    int pivot = Partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    QuickSort(nums, left, pivot - 1);\n    QuickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc (q *quickSort) quickSort(nums []int, left, right int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5212\u5206\n    pivot := q.partition(nums, left, right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    q.quickSort(nums, left, pivot-1)\n    q.quickSort(nums, pivot+1, right)\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nfunc quickSort(nums: inout [Int], left: Int, right: Int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return\n    }\n    // \u54e8\u5175\u5212\u5206\n    let pivot = partition(nums: &amp;nums, left: left, right: right)\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums: &amp;nums, left: left, right: pivot - 1)\n    quickSort(nums: &amp;nums, left: pivot + 1, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums, left, right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5212\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    const pivot = this.partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    this.quickSort(nums, left, pivot - 1);\n    this.quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n  if (left &gt;= right) return;\n  // \u54e8\u5175\u5212\u5206\n  int pivot = _partition(nums, left, right);\n  // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n  quickSort(nums, left, pivot - 1);\n  quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f */\npub fn quick_sort(left: i32, right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if left &gt;= right {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    Self::quick_sort(left, pivot - 1, nums);\n    Self::quick_sort(pivot + 1, right, nums);\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b */\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u54e8\u5175\u5212\u5206\nint partition(int nums[], int left, int right) {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            j--;\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n            i++;\n        }\n        // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n        swap(nums, i, j);\n    }\n    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    swap(nums, i, left);\n    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n    return i;\n}\n\n// \u5feb\u901f\u6392\u5e8f\u7c7b-\u5feb\u901f\u6392\u5e8f\nvoid quickSort(int nums[], int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) {\n        return;\n    }\n    // \u54e8\u5175\u5212\u5206\n    int pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\nfn quickSort(nums: []i32, left: usize, right: usize) void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    if (left &gt;= right) return;\n    // \u54e8\u5175\u5212\u5206\n    var pivot = partition(nums, left, right);\n    // \u9012\u5f52\u5de6\u5b50\u6570\u7ec4\u3001\u53f3\u5b50\u6570\u7ec4\n    quickSort(nums, left, pivot - 1);\n    quickSort(nums, pivot + 1, right);\n}\n</code></pre>"},{"location":"chapter_sorting/quick_sort/#1152","title":"11.5.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n \\log n)\\)\u3001\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5728\u5e73\u5747\u60c5\u51b5\u4e0b\uff0c\u54e8\u5175\u5212\u5206\u7684\u9012\u5f52\u5c42\u6570\u4e3a \\(\\log n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u603b\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n \\log n)\\) \u65f6\u95f4\u3002\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u957f\u5ea6\u4e3a \\(n\\) \u7684\u6570\u7ec4\u5212\u5206\u4e3a\u957f\u5ea6\u4e3a \\(0\\) \u548c \\(n - 1\\) \u7684\u4e24\u4e2a\u5b50\u6570\u7ec4\uff0c\u6b64\u65f6\u9012\u5f52\u5c42\u6570\u8fbe\u5230 \\(n\\) \uff0c\u6bcf\u5c42\u4e2d\u7684\u5faa\u73af\u6570\u4e3a \\(n\\) \uff0c\u603b\u4f53\u4f7f\u7528 \\(O(n^2)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u5728\u8f93\u5165\u6570\u7ec4\u5b8c\u5168\u5012\u5e8f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u6700\u5dee\u9012\u5f52\u6df1\u5ea6 \\(n\\) \uff0c\u4f7f\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002\u6392\u5e8f\u64cd\u4f5c\u662f\u5728\u539f\u6570\u7ec4\u4e0a\u8fdb\u884c\u7684\uff0c\u672a\u501f\u52a9\u989d\u5916\u6570\u7ec4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5728\u54e8\u5175\u5212\u5206\u7684\u6700\u540e\u4e00\u6b65\uff0c\u57fa\u51c6\u6570\u53ef\u80fd\u4f1a\u88ab\u4ea4\u6362\u81f3\u76f8\u7b49\u5143\u7d20\u7684\u53f3\u4fa7\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1153","title":"11.5.3 \u00a0 \u5feb\u901f\u6392\u5e8f\u4e3a\u4ec0\u4e48\u5feb","text":"<p>\u4ece\u540d\u79f0\u4e0a\u5c31\u80fd\u770b\u51fa\uff0c\u5feb\u901f\u6392\u5e8f\u5728\u6548\u7387\u65b9\u9762\u5e94\u8be5\u5177\u6709\u4e00\u5b9a\u7684\u4f18\u52bf\u3002\u5c3d\u7ba1\u5feb\u901f\u6392\u5e8f\u7684\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u201c\u5f52\u5e76\u6392\u5e8f\u201d\u548c\u201c\u5806\u6392\u5e8f\u201d\u76f8\u540c\uff0c\u4f46\u901a\u5e38\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u66f4\u9ad8\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u539f\u56e0\u3002</p> <ul> <li>\u51fa\u73b0\u6700\u5dee\u60c5\u51b5\u7684\u6982\u7387\u5f88\u4f4e\uff1a\u867d\u7136\u5feb\u901f\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u6ca1\u6709\u5f52\u5e76\u6392\u5e8f\u7a33\u5b9a\uff0c\u4f46\u5728\u7edd\u5927\u591a\u6570\u60c5\u51b5\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u80fd\u5728 \\(O(n \\log n)\\) \u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0b\u8fd0\u884c\u3002</li> <li>\u7f13\u5b58\u4f7f\u7528\u6548\u7387\u9ad8\uff1a\u5728\u6267\u884c\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u65f6\uff0c\u7cfb\u7edf\u53ef\u5c06\u6574\u4e2a\u5b50\u6570\u7ec4\u52a0\u8f7d\u5230\u7f13\u5b58\uff0c\u56e0\u6b64\u8bbf\u95ee\u5143\u7d20\u7684\u6548\u7387\u8f83\u9ad8\u3002\u800c\u50cf\u201c\u5806\u6392\u5e8f\u201d\u8fd9\u7c7b\u7b97\u6cd5\u9700\u8981\u8df3\u8dc3\u5f0f\u8bbf\u95ee\u5143\u7d20\uff0c\u4ece\u800c\u7f3a\u4e4f\u8fd9\u4e00\u7279\u6027\u3002</li> <li>\u590d\u6742\u5ea6\u7684\u5e38\u6570\u7cfb\u6570\u5c0f\uff1a\u5728\u4e0a\u8ff0\u4e09\u79cd\u7b97\u6cd5\u4e2d\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u6bd4\u8f83\u3001\u8d4b\u503c\u3001\u4ea4\u6362\u7b49\u64cd\u4f5c\u7684\u603b\u6570\u91cf\u6700\u5c11\u3002\u8fd9\u4e0e\u201c\u63d2\u5165\u6392\u5e8f\u201d\u6bd4\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u66f4\u5feb\u7684\u539f\u56e0\u7c7b\u4f3c\u3002</li> </ul>"},{"location":"chapter_sorting/quick_sort/#1154","title":"11.5.4 \u00a0 \u57fa\u51c6\u6570\u4f18\u5316","text":"<p>\u5feb\u901f\u6392\u5e8f\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\u7684\u65f6\u95f4\u6548\u7387\u53ef\u80fd\u964d\u4f4e\u3002\u4e3e\u4e00\u4e2a\u6781\u7aef\u4f8b\u5b50\uff0c\u5047\u8bbe\u8f93\u5165\u6570\u7ec4\u662f\u5b8c\u5168\u5012\u5e8f\u7684\uff0c\u7531\u4e8e\u6211\u4eec\u9009\u62e9\u6700\u5de6\u7aef\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u5728\u54e8\u5175\u5212\u5206\u5b8c\u6210\u540e\uff0c\u57fa\u51c6\u6570\u88ab\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u53f3\u7aef\uff0c\u5bfc\u81f4\u5de6\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(n - 1\\)\u3001\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(0\\) \u3002\u5982\u6b64\u9012\u5f52\u4e0b\u53bb\uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u540e\u7684\u53f3\u5b50\u6570\u7ec4\u957f\u5ea6\u90fd\u4e3a \\(0\\) \uff0c\u5206\u6cbb\u7b56\u7565\u5931\u6548\uff0c\u5feb\u901f\u6392\u5e8f\u9000\u5316\u4e3a\u201c\u5192\u6ce1\u6392\u5e8f\u201d\u3002</p> <p>\u4e3a\u4e86\u5c3d\u91cf\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u53d1\u751f\uff0c\u6211\u4eec\u53ef\u4ee5\u4f18\u5316\u54e8\u5175\u5212\u5206\u4e2d\u7684\u57fa\u51c6\u6570\u7684\u9009\u53d6\u7b56\u7565\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u968f\u673a\u9009\u53d6\u4e00\u4e2a\u5143\u7d20\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u7136\u800c\uff0c\u5982\u679c\u8fd0\u6c14\u4e0d\u4f73\uff0c\u6bcf\u6b21\u90fd\u9009\u5230\u4e0d\u7406\u60f3\u7684\u57fa\u51c6\u6570\uff0c\u6548\u7387\u4ecd\u7136\u4e0d\u5c3d\u5982\u4eba\u610f\u3002</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u7f16\u7a0b\u8bed\u8a00\u901a\u5e38\u751f\u6210\u7684\u662f\u201c\u4f2a\u968f\u673a\u6570\u201d\u3002\u5982\u679c\u6211\u4eec\u9488\u5bf9\u4f2a\u968f\u673a\u6570\u5e8f\u5217\u6784\u5efa\u4e00\u4e2a\u7279\u5b9a\u7684\u6d4b\u8bd5\u6837\u4f8b\uff0c\u90a3\u4e48\u5feb\u901f\u6392\u5e8f\u7684\u6548\u7387\u4ecd\u7136\u53ef\u80fd\u52a3\u5316\u3002</p> <p>\u4e3a\u4e86\u8fdb\u4e00\u6b65\u6539\u8fdb\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6570\u7ec4\u4e2d\u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\uff08\u901a\u5e38\u4e3a\u6570\u7ec4\u7684\u9996\u3001\u5c3e\u3001\u4e2d\u70b9\u5143\u7d20\uff09\uff0c\u5e76\u5c06\u8fd9\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\u4f5c\u4e3a\u57fa\u51c6\u6570\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u57fa\u51c6\u6570\u201c\u65e2\u4e0d\u592a\u5c0f\u4e5f\u4e0d\u592a\u5927\u201d\u7684\u6982\u7387\u5c06\u5927\u5e45\u63d0\u5347\u3002\u5f53\u7136\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u9009\u53d6\u66f4\u591a\u5019\u9009\u5143\u7d20\uff0c\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7b97\u6cd5\u7684\u7a33\u5065\u6027\u3002\u91c7\u7528\u8fd9\u79cd\u65b9\u6cd5\u540e\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u7684\u6982\u7387\u5927\u5927\u964d\u4f4e\u3002</p> <p>\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def median_three(self, nums: list[int], left: int, mid: int, right: int) -&gt; int:\n    \"\"\"\u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\"\"\"\n    # \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    # \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if (nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]):\n        return left\n    elif (nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]):\n        return mid\n    return right\n\ndef partition(self, nums: list[int], left: int, right: int) -&gt; int:\n    \"\"\"\u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\"\"\"\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    med = self.median_three(nums, left, (left + right) // 2, right)\n    # \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    # \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j = left, right\n    while i &lt; j:\n        while i &lt; j and nums[j] &gt;= nums[left]:\n            j -= 1  # \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while i &lt; j and nums[i] &lt;= nums[left]:\n            i += 1  # \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        # \u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    # \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i  # \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n</code></pre> quick_sort.cpp<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(vector&lt;int&gt; &amp;nums, int left, int mid, int right) {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\n        return left;\n    else if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\n        return mid;\n    else\n        return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.java<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint medianThree(int[] nums, int left, int mid, int right) {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\n        return left;\n    else if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\n        return mid;\n    else\n        return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint partition(int[] nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.cs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint MedianThree(int[] nums, int left, int mid, int right) {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\n        return left;\n    else if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\n        return mid;\n    else\n        return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint Partition(int[] nums, int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = MedianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    Swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--;          // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        Swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    Swap(nums, i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;             // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.go<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc (q *quickSortMedian) medianThree(nums []int, left, mid, right int) int {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\uff08!= \u5728\u8fd9\u91cc\u8d77\u5230\u5f02\u6216\u7684\u4f5c\u7528\uff09\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if (nums[left] &lt; nums[mid]) != (nums[left] &lt; nums[right]) {\n        return left\n    } else if (nums[mid] &lt; nums[left]) != (nums[mid] &lt; nums[right]) {\n        return mid\n    }\n    return right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09*/\nfunc (q *quickSortMedian) partition(nums []int, left, right int) int {\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    med := q.medianThree(nums, left, (left+right)/2, right)\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums[left], nums[med] = nums[med], nums[left]\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    i, j := left, right\n    for i &lt; j {\n        for i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j-- //\u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        for i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i++ //\u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        //\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[j] = nums[j], nums[i]\n    }\n    //\u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    nums[i], nums[left] = nums[left], nums[i]\n    return i //\u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.swift<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfunc medianThree(nums: [Int], left: Int, mid: Int, right: Int) -&gt; Int {\n    if (nums[left] &lt; nums[mid]) != (nums[left] &lt; nums[right]) {\n        return left\n    } else if (nums[mid] &lt; nums[left]) != (nums[mid] &lt; nums[right]) {\n        return mid\n    } else {\n        return right\n    }\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfunc partitionMedian(nums: inout [Int], left: Int, right: Int) -&gt; Int {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = medianThree(nums: nums, left: left, mid: (left + right) / 2, right: right)\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums: &amp;nums, i: left, j: med)\n    return partition(nums: &amp;nums, left: left, right: right)\n}\n</code></pre> quick_sort.js<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(nums, left, mid, right) {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right])) return left;\n    else if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\n        return mid;\n    else return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums, left, right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.ts<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nmedianThree(\n    nums: number[],\n    left: number,\n    mid: number,\n    right: number\n): number {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if (Number(nums[left] &lt; nums[mid]) ^ Number(nums[left] &lt; nums[right])) {\n        return left;\n    } else if (\n        Number(nums[mid] &lt; nums[left]) ^ Number(nums[mid] &lt; nums[right])\n    ) {\n        return mid;\n    } else {\n        return right;\n    }\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\npartition(nums: number[], left: number, right: number): number {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = this.medianThree(\n        nums,\n        left,\n        Math.floor((left + right) / 2),\n        right\n    );\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    this.swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let i = left,\n        j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) {\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) {\n            i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        this.swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    this.swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.dart<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nint _medianThree(List&lt;int&gt; nums, int left, int mid, int right) {\n  // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n  // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n  if ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\n    return left;\n  else if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\n    return mid;\n  else\n    return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nint _partition(List&lt;int&gt; nums, int left, int right) {\n  // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n  int med = _medianThree(nums, left, (left + right) ~/ 2, right);\n  // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n  _swap(nums, left, med);\n  // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n  int i = left, j = right;\n  while (i &lt; j) {\n    while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left]) j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left]) i++; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n    _swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n  }\n  _swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n  return i; // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.rs<pre><code>/* \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570 */\nfn median_three(nums: &amp;mut [i32], left: usize, mid: usize, right: usize) -&gt; usize {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if (nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]) {\n        return left;\n    } else if (nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]) {\n        return mid;\n    } \n    right\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */\nfn partition(nums: &amp;mut [i32], left: usize, right: usize) -&gt; usize {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    let med = Self::median_three(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    nums.swap(left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    let (mut i, mut j) = (left, right);\n    while i &lt; j {\n        while i &lt; j &amp;&amp; nums[j] &gt;= nums[left] {\n            j -= 1;      // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        while i &lt; j &amp;&amp; nums[i] &lt;= nums[left] {\n            i += 1;      // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        }\n        nums.swap(i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n     nums.swap(i, left);  // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n     i                    // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u4e2d\u4f4d\u57fa\u51c6\u6570\u4f18\u5316\uff09 */\n// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nint medianThree(int nums[], int left, int mid, int right) {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if ((nums[left] &lt; nums[mid]) ^ (nums[left] &lt; nums[right]))\n        return left;\n    else if ((nums[mid] &lt; nums[left]) ^ (nums[mid] &lt; nums[right]))\n        return mid;\n    else\n        return right;\n}\n\n/* \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09 */ \nint partitionMedian(int nums[], int left, int right) {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    int med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    int i = left, j = right;\n    while (i &lt; j) {\n        while (i &lt; j &amp;&amp; nums[j] &gt;= nums[left])\n            j--; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j &amp;&amp; nums[i] &lt;= nums[left])\n            i++;          // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j); // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left); // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;            // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre> quick_sort.zig<pre><code>// \u9009\u53d6\u4e09\u4e2a\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\nfn medianThree(nums: []i32, left: usize, mid: usize, right: usize) usize {\n    // \u6b64\u5904\u4f7f\u7528\u5f02\u6216\u8fd0\u7b97\u6765\u7b80\u5316\u4ee3\u7801\n    // \u5f02\u6216\u89c4\u5219\u4e3a 0 ^ 0 = 1 ^ 1 = 0, 0 ^ 1 = 1 ^ 0 = 1\n    if ((nums[left] &lt; nums[mid]) != (nums[left] &lt; nums[right])) {\n        return left;\n    } else if ((nums[mid] &lt; nums[left]) != (nums[mid] &lt; nums[right])) {\n        return mid;\n    } else {\n        return right;\n    }\n}\n\n// \u54e8\u5175\u5212\u5206\uff08\u4e09\u6570\u53d6\u4e2d\u503c\uff09\nfn partition(nums: []i32, left: usize, right: usize) usize {\n    // \u9009\u53d6\u4e09\u4e2a\u5019\u9009\u5143\u7d20\u7684\u4e2d\u4f4d\u6570\n    var med = medianThree(nums, left, (left + right) / 2, right);\n    // \u5c06\u4e2d\u4f4d\u6570\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\n    swap(nums, left, med);\n    // \u4ee5 nums[left] \u4e3a\u57fa\u51c6\u6570\n    var i = left;\n    var j = right;\n    while (i &lt; j) {\n        while (i &lt; j and nums[j] &gt;= nums[left]) j -= 1; // \u4ece\u53f3\u5411\u5de6\u627e\u9996\u4e2a\u5c0f\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        while (i &lt; j and nums[i] &lt;= nums[left]) i += 1; // \u4ece\u5de6\u5411\u53f3\u627e\u9996\u4e2a\u5927\u4e8e\u57fa\u51c6\u6570\u7684\u5143\u7d20\n        swap(nums, i, j);   // \u4ea4\u6362\u8fd9\u4e24\u4e2a\u5143\u7d20\n    }\n    swap(nums, i, left);    // \u5c06\u57fa\u51c6\u6570\u4ea4\u6362\u81f3\u4e24\u5b50\u6570\u7ec4\u7684\u5206\u754c\u7ebf\n    return i;               // \u8fd4\u56de\u57fa\u51c6\u6570\u7684\u7d22\u5f15\n}\n</code></pre>"},{"location":"chapter_sorting/quick_sort/#1155","title":"11.5.5 \u00a0 \u5c3e\u9012\u5f52\u4f18\u5316","text":"<p>\u5728\u67d0\u4e9b\u8f93\u5165\u4e0b\uff0c\u5feb\u901f\u6392\u5e8f\u53ef\u80fd\u5360\u7528\u7a7a\u95f4\u8f83\u591a\u3002\u4ee5\u5b8c\u5168\u6709\u5e8f\u7684\u8f93\u5165\u6570\u7ec4\u4e3a\u4f8b\uff0c\u8bbe\u9012\u5f52\u4e2d\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a \\(m\\) \uff0c\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u90fd\u5c06\u4ea7\u751f\u957f\u5ea6\u4e3a \\(0\\) \u7684\u5de6\u5b50\u6570\u7ec4\u548c\u957f\u5ea6\u4e3a \\(m - 1\\) \u7684\u53f3\u5b50\u6570\u7ec4\uff0c\u8fd9\u610f\u5473\u7740\u6bcf\u4e00\u5c42\u9012\u5f52\u8c03\u7528\u51cf\u5c11\u7684\u95ee\u9898\u89c4\u6a21\u975e\u5e38\u5c0f\uff08\u53ea\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff09\uff0c\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4f1a\u8fbe\u5230 \\(n - 1\\) \uff0c\u6b64\u65f6\u9700\u8981\u5360\u7528 \\(O(n)\\) \u5927\u5c0f\u7684\u6808\u5e27\u7a7a\u95f4\u3002</p> <p>\u4e3a\u4e86\u9632\u6b62\u6808\u5e27\u7a7a\u95f4\u7684\u7d2f\u79ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6bcf\u8f6e\u54e8\u5175\u6392\u5e8f\u5b8c\u6210\u540e\uff0c\u6bd4\u8f83\u4e24\u4e2a\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\uff0c\u4ec5\u5bf9\u8f83\u77ed\u7684\u5b50\u6570\u7ec4\u8fdb\u884c\u9012\u5f52\u3002\u7531\u4e8e\u8f83\u77ed\u5b50\u6570\u7ec4\u7684\u957f\u5ea6\u4e0d\u4f1a\u8d85\u8fc7 \\(n / 2\\) \uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u80fd\u786e\u4fdd\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff0c\u4ece\u800c\u5c06\u6700\u5dee\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u81f3 \\(O(\\log n)\\) \u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig quick_sort.py<pre><code>def quick_sort(self, nums: list[int], left: int, right: int):\n    \"\"\"\u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\"\"\"\n    # \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right:\n        # \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        pivot = self.partition(nums, left, right)\n        # \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot - left &lt; right - pivot:\n            self.quick_sort(nums, left, pivot - 1)  # \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        else:\n            self.quick_sort(nums, pivot + 1, right)  # \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1  # \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n</code></pre> quick_sort.cpp<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(vector&lt;int&gt; &amp;nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.java<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.cs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid QuickSort(int[] nums, int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = Partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            QuickSort(nums, left, pivot - 1);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            QuickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.go<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09*/\nfunc (q *quickSortTailCall) quickSort(nums []int, left, right int) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    for left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        pivot := q.partition(nums, left, right)\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if pivot-left &lt; right-pivot {\n            q.quickSort(nums, left, pivot-1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            q.quickSort(nums, pivot+1, right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1                 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.swift<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nfunc quickSortTailCall(nums: inout [Int], left: Int, right: Int) {\n    var left = left\n    var right = right\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = partition(nums: &amp;nums, left: left, right: right)\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left) &lt; (right - pivot) {\n            quickSortTailCall(nums: &amp;nums, left: left, right: pivot - 1) // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums: &amp;nums, left: pivot + 1, right: right) // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1 // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.js<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums, left, right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.ts<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nquickSort(nums: number[], left: number, right: number): void {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = this.partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            this.quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            this.quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.dart<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\nvoid quickSort(List&lt;int&gt; nums, int left, int right) {\n  // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n  while (left &lt; right) {\n    // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n    int pivot = _partition(nums, left, right);\n    // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n    if (pivot - left &lt; right - pivot) {\n      quickSort(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n      left = pivot + 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n    } else {\n      quickSort(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n      right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n    }\n  }\n}\n</code></pre> quick_sort.rs<pre><code>/* \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\npub fn quick_sort(mut left: i32, mut right: i32, nums: &amp;mut [i32]) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while left &lt; right {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        let pivot = Self::partition(nums, left as usize, right as usize) as i32;\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if  pivot - left &lt; right - pivot {\n            Self::quick_sort(left, pivot - 1, nums);  // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            Self::quick_sort(pivot + 1, right, nums); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1; // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.c<pre><code>/* \u5feb\u901f\u6392\u5e8f\u7c7b\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09 */\n// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nvoid quickSortTailCall(int nums[], int left, int right) {\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        int pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSortTailCall(nums, left, pivot - 1); // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSortTailCall(nums, pivot + 1, right); // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                         // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre> quick_sort.zig<pre><code>// \u5feb\u901f\u6392\u5e8f\uff08\u5c3e\u9012\u5f52\u4f18\u5316\uff09\nfn quickSort(nums: []i32, left_: usize, right_: usize) void {\n    var left = left_;\n    var right = right_;\n    // \u5b50\u6570\u7ec4\u957f\u5ea6\u4e3a 1 \u65f6\u7ec8\u6b62\u9012\u5f52\n    while (left &lt; right) {\n        // \u54e8\u5175\u5212\u5206\u64cd\u4f5c\n        var pivot = partition(nums, left, right);\n        // \u5bf9\u4e24\u4e2a\u5b50\u6570\u7ec4\u4e2d\u8f83\u77ed\u7684\u90a3\u4e2a\u6267\u884c\u5feb\u901f\u6392\u5e8f\n        if (pivot - left &lt; right - pivot) {\n            quickSort(nums, left, pivot - 1);   // \u9012\u5f52\u6392\u5e8f\u5de6\u5b50\u6570\u7ec4\n            left = pivot + 1;                   // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [pivot + 1, right]\n        } else {\n            quickSort(nums, pivot + 1, right);  // \u9012\u5f52\u6392\u5e8f\u53f3\u5b50\u6570\u7ec4\n            right = pivot - 1;                  // \u5269\u4f59\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [left, pivot - 1]\n        }\n    }\n}\n</code></pre>"},{"location":"chapter_sorting/radix_sort/","title":"11.10 \u00a0 \u57fa\u6570\u6392\u5e8f","text":"<p>\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u8ba1\u6570\u6392\u5e8f\uff0c\u5b83\u9002\u7528\u4e8e\u6570\u636e\u91cf \\(n\\) \u8f83\u5927\u4f46\u6570\u636e\u8303\u56f4 \\(m\\) \u8f83\u5c0f\u7684\u60c5\u51b5\u3002\u5047\u8bbe\u6211\u4eec\u9700\u8981\u5bf9 \\(n = 10^6\\) \u4e2a\u5b66\u53f7\u8fdb\u884c\u6392\u5e8f\uff0c\u800c\u5b66\u53f7\u662f\u4e00\u4e2a \\(8\\) \u4f4d\u6570\u5b57\uff0c\u8fd9\u610f\u5473\u7740\u6570\u636e\u8303\u56f4 \\(m = 10^8\\) \u975e\u5e38\u5927\uff0c\u4f7f\u7528\u8ba1\u6570\u6392\u5e8f\u9700\u8981\u5206\u914d\u5927\u91cf\u5185\u5b58\u7a7a\u95f4\uff0c\u800c\u57fa\u6570\u6392\u5e8f\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u3002</p> <p>\u300c\u57fa\u6570\u6392\u5e8f radix sort\u300d\u7684\u6838\u5fc3\u601d\u60f3\u4e0e\u8ba1\u6570\u6392\u5e8f\u4e00\u81f4\uff0c\u4e5f\u901a\u8fc7\u7edf\u8ba1\u4e2a\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u6b64\u57fa\u7840\u4e0a\uff0c\u57fa\u6570\u6392\u5e8f\u5229\u7528\u6570\u5b57\u5404\u4f4d\u4e4b\u95f4\u7684\u9012\u8fdb\u5173\u7cfb\uff0c\u4f9d\u6b21\u5bf9\u6bcf\u4e00\u4f4d\u8fdb\u884c\u6392\u5e8f\uff0c\u4ece\u800c\u5f97\u5230\u6700\u7ec8\u7684\u6392\u5e8f\u7ed3\u679c\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11101","title":"11.10.1 \u00a0 \u7b97\u6cd5\u6d41\u7a0b","text":"<p>\u4ee5\u5b66\u53f7\u6570\u636e\u4e3a\u4f8b\uff0c\u5047\u8bbe\u6570\u5b57\u7684\u6700\u4f4e\u4f4d\u662f\u7b2c \\(1\\) \u4f4d\uff0c\u6700\u9ad8\u4f4d\u662f\u7b2c \\(8\\) \u4f4d\uff0c\u57fa\u6570\u6392\u5e8f\u7684\u6d41\u7a0b\u5982\u56fe 11-18 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u5316\u4f4d\u6570 \\(k = 1\\) \u3002</li> <li>\u5bf9\u5b66\u53f7\u7684\u7b2c \\(k\\) \u4f4d\u6267\u884c\u201c\u8ba1\u6570\u6392\u5e8f\u201d\u3002\u5b8c\u6210\u540e\uff0c\u6570\u636e\u4f1a\u6839\u636e\u7b2c \\(k\\) \u4f4d\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002</li> <li>\u5c06 \\(k\\) \u589e\u52a0 \\(1\\) \uff0c\u7136\u540e\u8fd4\u56de\u6b65\u9aa4 <code>2.</code> \u7ee7\u7eed\u8fed\u4ee3\uff0c\u76f4\u5230\u6240\u6709\u4f4d\u90fd\u6392\u5e8f\u5b8c\u6210\u540e\u7ed3\u675f\u3002</li> </ol> <p></p> <p> \u56fe 11-18 \u00a0 \u57fa\u6570\u6392\u5e8f\u7b97\u6cd5\u6d41\u7a0b </p> <p>\u4e0b\u9762\u5256\u6790\u4ee3\u7801\u5b9e\u73b0\u3002\u5bf9\u4e8e\u4e00\u4e2a \\(d\\) \u8fdb\u5236\u7684\u6570\u5b57 \\(x\\) \uff0c\u8981\u83b7\u53d6\u5176\u7b2c \\(k\\) \u4f4d \\(x_k\\) \uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ x_k = \\lfloor\\frac{x}{d^{k-1}}\\rfloor \\bmod d \\] <p>\u5176\u4e2d \\(\\lfloor a \\rfloor\\) \u8868\u793a\u5bf9\u6d6e\u70b9\u6570 \\(a\\) \u5411\u4e0b\u53d6\u6574\uff0c\u800c \\(\\bmod \\: d\\) \u8868\u793a\u5bf9 \\(d\\) \u53d6\u4f59\u3002\u5bf9\u4e8e\u5b66\u53f7\u6570\u636e\uff0c\\(d = 10\\) \u4e14 \\(k \\in [1, 8]\\) \u3002</p> <p>\u6b64\u5916\uff0c\u6211\u4eec\u9700\u8981\u5c0f\u5e45\u6539\u52a8\u8ba1\u6570\u6392\u5e8f\u4ee3\u7801\uff0c\u4f7f\u4e4b\u53ef\u4ee5\u6839\u636e\u6570\u5b57\u7684\u7b2c \\(k\\) \u4f4d\u8fdb\u884c\u6392\u5e8f\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig radix_sort.py<pre><code>def digit(num: int, exp: int) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\"\"\"\n    # \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num // exp) % 10\n\ndef counting_sort_digit(nums: list[int], exp: int):\n    \"\"\"\u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\"\"\"\n    # \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    counter = [0] * 10\n    n = len(nums)\n    # \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in range(n):\n        d = digit(nums[i], exp)  # \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1  # \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    # \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in range(1, 10):\n        counter[i] += counter[i - 1]\n    # \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    res = [0] * n\n    for i in range(n - 1, -1, -1):\n        d = digit(nums[i], exp)\n        j = counter[d] - 1  # \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]  # \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1  # \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    # \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in range(n):\n        nums[i] = res[i]\n\ndef radix_sort(nums: list[int]):\n    \"\"\"\u57fa\u6570\u6392\u5e8f\"\"\"\n    # \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    m = max(nums)\n    # \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    exp = 1\n    while exp &lt;= m:\n        # \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        # k = 1 -&gt; exp = 1\n        # k = 2 -&gt; exp = 10\n        # \u5373 exp = 10^(k-1)\n        counting_sort_digit(nums, exp)\n        exp *= 10\n</code></pre> radix_sort.cpp<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(vector&lt;int&gt; &amp;nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    vector&lt;int&gt; counter(10, 0);\n    int n = nums.size();\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    vector&lt;int&gt; res(n, 0);\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(vector&lt;int&gt; &amp;nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = *max_element(nums.begin(), nums.end());\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.java<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int[] nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int[] counter = new int[10];\n    int n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++)\n        nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int[] nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = Integer.MIN_VALUE;\n    for (int num : nums)\n        if (num &gt; m)\n            m = num;\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.cs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint Digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid CountingSortDigit(int[] nums, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int[] counter = new int[10];\n    int n = nums.Length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; n; i++) {\n        int d = Digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++;                // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int[] res = new int[n];\n    for (int i = n - 1; i &gt;= 0; i--) {\n        int d = Digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid RadixSort(int[] nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int m = int.MinValue;\n    foreach (int num in nums) {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        CountingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.go<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num, exp int) int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums []int, exp int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    counter := make([]int, 10)\n    n := len(nums)\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i := 0; i &lt; n; i++ {\n        d := digit(nums[i], exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++             // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i := 1; i &lt; 10; i++ {\n        counter[i] += counter[i-1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    res := make([]int, n)\n    for i := n - 1; i &gt;= 0; i-- {\n        d := digit(nums[i], exp)\n        j := counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]    // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i := 0; i &lt; n; i++ {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums []int) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    max := math.MinInt\n    for _, num := range nums {\n        if num &gt; max {\n            max = num\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for exp := 1; max &gt;= exp; exp *= 10 {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp)\n    }\n}\n</code></pre> radix_sort.swift<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunc digit(num: Int, exp: Int) -&gt; Int {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    (num / exp) % 10\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunc countingSortDigit(nums: inout [Int], exp: Int) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    var counter = Array(repeating: 0, count: 10)\n    let n = nums.count\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in nums.indices {\n        let d = digit(num: nums[i], exp: exp) // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1 // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in 1 ..&lt; 10 {\n        counter[i] += counter[i - 1]\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    var res = Array(repeating: 0, count: n)\n    for i in stride(from: n - 1, through: 0, by: -1) {\n        let d = digit(num: nums[i], exp: exp)\n        let j = counter[d] - 1 // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i] // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1 // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in nums.indices {\n        nums[i] = res[i]\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunc radixSort(nums: inout [Int]) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m = Int.min\n    for num in nums {\n        if num &gt; m {\n            m = num\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for exp in sequence(first: 1, next: { m &gt;= ($0 * 10) ? $0 * 10 : nil }) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums: &amp;nums, exp: exp)\n    }\n}\n</code></pre> radix_sort.js<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num, exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums, exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.ts<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfunction digit(num: number, exp: number): number {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return Math.floor(num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfunction countingSortDigit(nums: number[], exp: number): void {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    const counter = new Array(10).fill(0);\n    const n = nums.length;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (let i = 0; i &lt; n; i++) {\n        const d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (let i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    const res = new Array(n).fill(0);\n    for (let i = n - 1; i &gt;= 0; i--) {\n        const d = digit(nums[i], exp);\n        const j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (let i = 0; i &lt; n; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfunction radixSort(nums: number[]): void {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = Number.MIN_VALUE;\n    for (const num of nums) {\n        if (num &gt; m) {\n            m = num;\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (let exp = 1; exp &lt;= m; exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, exp);\n    }\n}\n</code></pre> radix_sort.dart<pre><code>/* \u83b7\u53d6\u5143\u7d20 _num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int _num, int exp) {\n  // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n  return (_num ~/ exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(List&lt;int&gt; nums, int exp) {\n  // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n  List&lt;int&gt; counter = List&lt;int&gt;.filled(10, 0);\n  int n = nums.length;\n  // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n  for (int i = 0; i &lt; n; i++) {\n    int d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n    counter[d]++; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n  }\n  // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n  for (int i = 1; i &lt; 10; i++) {\n    counter[i] += counter[i - 1];\n  }\n  // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n  List&lt;int&gt; res = List&lt;int&gt;.filled(n, 0);\n  for (int i = n - 1; i &gt;= 0; i--) {\n    int d = digit(nums[i], exp);\n    int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n    res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n    counter[d]--; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n  }\n  // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n  for (int i = 0; i &lt; n; i++) nums[i] = res[i];\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(List&lt;int&gt; nums) {\n  // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n  // dart \u4e2d int \u7684\u957f\u5ea6\u662f 64 \u4f4d\u7684\n  int m = -1 &lt;&lt; 63;\n  for (int _num in nums) if (_num &gt; m) m = _num;\n  // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n  for (int exp = 1; exp &lt;= m; exp *= 10)\n    // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n    // k = 1 -&gt; exp = 1\n    // k = 2 -&gt; exp = 10\n    // \u5373 exp = 10^(k-1)\n    countingSortDigit(nums, exp);\n}\n</code></pre> radix_sort.rs<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nfn digit(num: i32, exp: i32) -&gt; usize {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return ((num / exp) % 10) as usize;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nfn counting_sort_digit(nums: &amp;mut [i32], exp: i32) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    let mut counter = [0; 10];\n    let n = nums.len();\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for i in 0..n {\n        let d = digit(nums[i], exp); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for i in 1..10 {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    let mut res = vec![0; n];\n    for i in (0..n).rev() {\n        let d = digit(nums[i], exp);\n        let j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i]; // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1; // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for i in 0..n {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nfn radix_sort(nums: &amp;mut [i32]) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    let m = *nums.into_iter().max().unwrap();\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    let mut exp = 1;\n    while exp &lt;= m {\n        counting_sort_digit(nums, exp);\n        exp *= 10;\n    }\n}\n</code></pre> radix_sort.c<pre><code>/* \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1) */\nint digit(int num, int exp) {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return (num / exp) % 10;\n}\n\n/* \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09 */\nvoid countingSortDigit(int nums[], int size, int exp) {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    int *counter = (int *)malloc((sizeof(int) * 10));\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (int i = 0; i &lt; size; i++) {\n        // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        int d = digit(nums[i], exp);\n        // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n        counter[d]++;\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    for (int i = 1; i &lt; 10; i++) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    int *res = (int *)malloc(sizeof(int) * size);\n    for (int i = size - 1; i &gt;= 0; i--) {\n        int d = digit(nums[i], exp);\n        int j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d]--;           // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    for (int i = 0; i &lt; size; i++) {\n        nums[i] = res[i];\n    }\n}\n\n/* \u57fa\u6570\u6392\u5e8f */\nvoid radixSort(int nums[], int size) {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    int max = INT32_MIN;\n    for (size_t i = 0; i &lt; size - 1; i++) {\n        if (nums[i] &gt; max) {\n            max = nums[i];\n        }\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    for (int exp = 1; max &gt;= exp; exp *= 10)\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        countingSortDigit(nums, size, exp);\n}\n</code></pre> radix_sort.zig<pre><code>// \u83b7\u53d6\u5143\u7d20 num \u7684\u7b2c k \u4f4d\uff0c\u5176\u4e2d exp = 10^(k-1)\nfn digit(num: i32, exp: i32) i32 {\n    // \u4f20\u5165 exp \u800c\u975e k \u53ef\u4ee5\u907f\u514d\u5728\u6b64\u91cd\u590d\u6267\u884c\u6602\u8d35\u7684\u6b21\u65b9\u8ba1\u7b97\n    return @mod(@divFloor(num, exp), 10);\n}\n\n// \u8ba1\u6570\u6392\u5e8f\uff08\u6839\u636e nums \u7b2c k \u4f4d\u6392\u5e8f\uff09\nfn countingSortDigit(nums: []i32, exp: i32) !void {\n    // \u5341\u8fdb\u5236\u7684\u4f4d\u8303\u56f4\u4e3a 0~9 \uff0c\u56e0\u6b64\u9700\u8981\u957f\u5ea6\u4e3a 10 \u7684\u6876\u6570\u7ec4\n    var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator);\n    // defer mem_arena.deinit();\n    const mem_allocator = mem_arena.allocator();\n    var counter = try mem_allocator.alloc(usize, 10);\n    @memset(counter, 0);\n    var n = nums.len;\n    // \u7edf\u8ba1 0~9 \u5404\u6570\u5b57\u7684\u51fa\u73b0\u6b21\u6570\n    for (nums) |num| {\n        var d: u32 = @bitCast(digit(num, exp)); // \u83b7\u53d6 nums[i] \u7b2c k \u4f4d\uff0c\u8bb0\u4e3a d\n        counter[d] += 1; // \u7edf\u8ba1\u6570\u5b57 d \u7684\u51fa\u73b0\u6b21\u6570\n    }\n    // \u6c42\u524d\u7f00\u548c\uff0c\u5c06\u201c\u51fa\u73b0\u4e2a\u6570\u201d\u8f6c\u6362\u4e3a\u201c\u6570\u7ec4\u7d22\u5f15\u201d\n    var i: usize = 1;\n    while (i &lt; 10) : (i += 1) {\n        counter[i] += counter[i - 1];\n    }\n    // \u5012\u5e8f\u904d\u5386\uff0c\u6839\u636e\u6876\u5185\u7edf\u8ba1\u7ed3\u679c\uff0c\u5c06\u5404\u5143\u7d20\u586b\u5165 res\n    var res = try mem_allocator.alloc(i32, n);\n    i = n - 1;\n    while (i &gt;= 0) : (i -= 1) {\n        var d: u32 = @bitCast(digit(nums[i], exp));\n        var j = counter[d] - 1; // \u83b7\u53d6 d \u5728\u6570\u7ec4\u4e2d\u7684\u7d22\u5f15 j\n        res[j] = nums[i];       // \u5c06\u5f53\u524d\u5143\u7d20\u586b\u5165\u7d22\u5f15 j\n        counter[d] -= 1;        // \u5c06 d \u7684\u6570\u91cf\u51cf 1\n        if (i == 0) break;\n    }\n    // \u4f7f\u7528\u7ed3\u679c\u8986\u76d6\u539f\u6570\u7ec4 nums\n    i = 0;\n    while (i &lt; n) : (i += 1) {\n        nums[i] = res[i];\n    }\n}\n\n// \u57fa\u6570\u6392\u5e8f\nfn radixSort(nums: []i32) !void {\n    // \u83b7\u53d6\u6570\u7ec4\u7684\u6700\u5927\u5143\u7d20\uff0c\u7528\u4e8e\u5224\u65ad\u6700\u5927\u4f4d\u6570\n    var m: i32 = std.math.minInt(i32);\n    for (nums) |num| {\n        if (num &gt; m) m = num;\n    }\n    // \u6309\u7167\u4ece\u4f4e\u4f4d\u5230\u9ad8\u4f4d\u7684\u987a\u5e8f\u904d\u5386\n    var exp: i32 = 1;\n    while (exp &lt;= m) : (exp *= 10) {\n        // \u5bf9\u6570\u7ec4\u5143\u7d20\u7684\u7b2c k \u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\n        // k = 1 -&gt; exp = 1\n        // k = 2 -&gt; exp = 10\n        // \u5373 exp = 10^(k-1)\n        try countingSortDigit(nums, exp);    \n    }\n} \n</code></pre> <p>\u4e3a\u4ec0\u4e48\u4ece\u6700\u4f4e\u4f4d\u5f00\u59cb\u6392\u5e8f\uff1f</p> <p>\u5728\u8fde\u7eed\u7684\u6392\u5e8f\u8f6e\u6b21\u4e2d\uff0c\u540e\u4e00\u8f6e\u6392\u5e8f\u4f1a\u8986\u76d6\u524d\u4e00\u8f6e\u6392\u5e8f\u7684\u7ed3\u679c\u3002\u4e3e\u4f8b\u6765\u8bf4\uff0c\u5982\u679c\u7b2c\u4e00\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &lt; b\\) \uff0c\u800c\u7b2c\u4e8c\u8f6e\u6392\u5e8f\u7ed3\u679c \\(a &gt; b\\) \uff0c\u90a3\u4e48\u7b2c\u4e8c\u8f6e\u7684\u7ed3\u679c\u5c06\u53d6\u4ee3\u7b2c\u4e00\u8f6e\u7684\u7ed3\u679c\u3002\u7531\u4e8e\u6570\u5b57\u7684\u9ad8\u4f4d\u4f18\u5148\u7ea7\u9ad8\u4e8e\u4f4e\u4f4d\uff0c\u56e0\u6b64\u5e94\u8be5\u5148\u6392\u5e8f\u4f4e\u4f4d\u518d\u6392\u5e8f\u9ad8\u4f4d\u3002</p>"},{"location":"chapter_sorting/radix_sort/#11102","title":"11.10.2 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<p>\u76f8\u8f83\u4e8e\u8ba1\u6570\u6392\u5e8f\uff0c\u57fa\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u503c\u8303\u56f4\u8f83\u5927\u7684\u60c5\u51b5\uff0c\u4f46\u524d\u63d0\u662f\u6570\u636e\u5fc5\u987b\u53ef\u4ee5\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u683c\u5f0f\uff0c\u4e14\u4f4d\u6570\u4e0d\u80fd\u8fc7\u5927\u3002\u4f8b\u5982\uff0c\u6d6e\u70b9\u6570\u4e0d\u9002\u5408\u4f7f\u7528\u57fa\u6570\u6392\u5e8f\uff0c\u56e0\u4e3a\u5176\u4f4d\u6570 \\(k\\) \u8fc7\u5927\uff0c\u53ef\u80fd\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk) \\gg O(n^2)\\) \u3002</p> <ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(nk)\\)\uff1a\u8bbe\u6570\u636e\u91cf\u4e3a \\(n\\)\u3001\u6570\u636e\u4e3a \\(d\\) \u8fdb\u5236\u3001\u6700\u5927\u4f4d\u6570\u4e3a \\(k\\) \uff0c\u5219\u5bf9\u67d0\u4e00\u4f4d\u6267\u884c\u8ba1\u6570\u6392\u5e8f\u4f7f\u7528 \\(O(n + d)\\) \u65f6\u95f4\uff0c\u6392\u5e8f\u6240\u6709 \\(k\\) \u4f4d\u4f7f\u7528 \\(O((n + d)k)\\) \u65f6\u95f4\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\\(d\\) \u548c \\(k\\) \u90fd\u76f8\u5bf9\u8f83\u5c0f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u8d8b\u5411 \\(O(n)\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n + d)\\)\u3001\u975e\u539f\u5730\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\uff0c\u57fa\u6570\u6392\u5e8f\u9700\u8981\u501f\u52a9\u957f\u5ea6\u4e3a \\(n\\) \u548c \\(d\\) \u7684\u6570\u7ec4 <code>res</code> \u548c <code>counter</code> \u3002</li> <li>\u7a33\u5b9a\u6392\u5e8f\uff1a\u4e0e\u8ba1\u6570\u6392\u5e8f\u76f8\u540c\u3002</li> </ul>"},{"location":"chapter_sorting/selection_sort/","title":"11.2 \u00a0 \u9009\u62e9\u6392\u5e8f","text":"<p>\u300c\u9009\u62e9\u6392\u5e8f selection sort\u300d\u7684\u5de5\u4f5c\u539f\u7406\u975e\u5e38\u7b80\u5355\uff1a\u5f00\u542f\u4e00\u4e2a\u5faa\u73af\uff0c\u6bcf\u8f6e\u4ece\u672a\u6392\u5e8f\u533a\u95f4\u9009\u62e9\u6700\u5c0f\u7684\u5143\u7d20\uff0c\u5c06\u5176\u653e\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u672b\u5c3e\u3002</p> <p>\u8bbe\u6570\u7ec4\u7684\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u9009\u62e9\u6392\u5e8f\u7684\u7b97\u6cd5\u6d41\u7a0b\u5982\u56fe 11-2 \u6240\u793a\u3002</p> <ol> <li>\u521d\u59cb\u72b6\u6001\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u672a\u6392\u5e8f\uff0c\u5373\u672a\u6392\u5e8f\uff08\u7d22\u5f15\uff09\u533a\u95f4\u4e3a \\([0, n-1]\\) \u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([0, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(0\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 1 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u9009\u53d6\u533a\u95f4 \\([1, n-1]\\) \u4e2d\u7684\u6700\u5c0f\u5143\u7d20\uff0c\u5c06\u5176\u4e0e\u7d22\u5f15 \\(1\\) \u5904\u7684\u5143\u7d20\u4ea4\u6362\u3002\u5b8c\u6210\u540e\uff0c\u6570\u7ec4\u524d 2 \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ee5\u6b64\u7c7b\u63a8\u3002\u7ecf\u8fc7 \\(n - 1\\) \u8f6e\u9009\u62e9\u4e0e\u4ea4\u6362\u540e\uff0c\u6570\u7ec4\u524d \\(n - 1\\) \u4e2a\u5143\u7d20\u5df2\u6392\u5e8f\u3002</li> <li>\u4ec5\u5269\u7684\u4e00\u4e2a\u5143\u7d20\u5fc5\u5b9a\u662f\u6700\u5927\u5143\u7d20\uff0c\u65e0\u987b\u6392\u5e8f\uff0c\u56e0\u6b64\u6570\u7ec4\u6392\u5e8f\u5b8c\u6210\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 11-2 \u00a0 \u9009\u62e9\u6392\u5e8f\u6b65\u9aa4 </p> <p>\u5728\u4ee3\u7801\u4e2d\uff0c\u6211\u4eec\u7528 \\(k\\) \u6765\u8bb0\u5f55\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig selection_sort.py<pre><code>def selection_sort(nums: list[int]):\n    \"\"\"\u9009\u62e9\u6392\u5e8f\"\"\"\n    n = len(nums)\n    # \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in range(n - 1):\n        # \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        k = i\n        for j in range(i + 1, n):\n            if nums[j] &lt; nums[k]:\n                k = j  # \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        # \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[k] = nums[k], nums[i]\n</code></pre> selection_sort.cpp<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(vector&lt;int&gt; &amp;nums) {\n    int n = nums.size();\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        swap(nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.java<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int[] nums) {\n    int n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.cs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid SelectionSort(int[] nums) {\n    int n = nums.Length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        (nums[k], nums[i]) = (nums[i], nums[k]);\n    }\n}\n</code></pre> selection_sort.go<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums []int) {\n    n := len(nums)\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i := 0; i &lt; n-1; i++ {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        k := i\n        for j := i + 1; j &lt; n; j++ {\n            if nums[j] &lt; nums[k] {\n                // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n                k = j\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums[i], nums[k] = nums[k], nums[i]\n\n    }\n}\n</code></pre> selection_sort.swift<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunc selectionSort(nums: inout [Int]) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in nums.indices.dropLast() {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        var k = i\n        for j in nums.indices.dropFirst(i + 1) {\n            if nums[j] &lt; nums[k] {\n                k = j // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums.swapAt(i, k)\n    }\n}\n</code></pre> selection_sort.js<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums) {\n    let n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.ts<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfunction selectionSort(nums: number[]): void {\n    let n = nums.length;\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (let i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let k = i;\n        for (let j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k]) {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        [nums[i], nums[k]] = [nums[k], nums[i]];\n    }\n}\n</code></pre> selection_sort.dart<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(List&lt;int&gt; nums) {\n  int n = nums.length;\n  // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n  for (int i = 0; i &lt; n - 1; i++) {\n    // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n    int k = i;\n    for (int j = i + 1; j &lt; n; j++) {\n      if (nums[j] &lt; nums[k]) k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n    }\n    // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n    int temp = nums[i];\n    nums[i] = nums[k];\n    nums[k] = temp;\n  }\n}\n</code></pre> selection_sort.rs<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nfn selection_sort(nums: &amp;mut [i32]) {\n    let n = nums.len();\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for i in 0..n-1 {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        let mut k = i;\n        for j in i+1..n {\n            if nums[j] &lt; nums[k] {\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n            }\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        nums.swap(i, k);\n    }\n}\n</code></pre> selection_sort.c<pre><code>/* \u9009\u62e9\u6392\u5e8f */\nvoid selectionSort(int nums[], int n) {\n    // \u5916\u5faa\u73af\uff1a\u672a\u6392\u5e8f\u533a\u95f4\u4e3a [i, n-1]\n    for (int i = 0; i &lt; n - 1; i++) {\n        // \u5185\u5faa\u73af\uff1a\u627e\u5230\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u6700\u5c0f\u5143\u7d20\n        int k = i;\n        for (int j = i + 1; j &lt; n; j++) {\n            if (nums[j] &lt; nums[k])\n                k = j; // \u8bb0\u5f55\u6700\u5c0f\u5143\u7d20\u7684\u7d22\u5f15\n        }\n        // \u5c06\u8be5\u6700\u5c0f\u5143\u7d20\u4e0e\u672a\u6392\u5e8f\u533a\u95f4\u7684\u9996\u4e2a\u5143\u7d20\u4ea4\u6362\n        int temp = nums[i];\n        nums[i] = nums[k];\n        nums[k] = temp;\n    }\n}\n</code></pre> selection_sort.zig<pre><code>[class]{}-[func]{selectionSort}\n</code></pre>"},{"location":"chapter_sorting/selection_sort/#1121","title":"11.2.1 \u00a0 \u7b97\u6cd5\u7279\u6027","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\)\u3001\u975e\u81ea\u9002\u5e94\u6392\u5e8f\uff1a\u5916\u5faa\u73af\u5171 \\(n - 1\\) \u8f6e\uff0c\u7b2c\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(n\\) \uff0c\u6700\u540e\u4e00\u8f6e\u7684\u672a\u6392\u5e8f\u533a\u95f4\u957f\u5ea6\u4e3a \\(2\\) \uff0c\u5373\u5404\u8f6e\u5916\u5faa\u73af\u5206\u522b\u5305\u542b \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(3\\)\u3001\\(2\\) \u8f6e\u5185\u5faa\u73af\uff0c\u6c42\u548c\u4e3a \\(\\frac{(n - 1)(n + 2)}{2}\\) \u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(1)\\)\u3001\u539f\u5730\u6392\u5e8f\uff1a\u6307\u9488 \\(i\\) \u548c \\(j\\) \u4f7f\u7528\u5e38\u6570\u5927\u5c0f\u7684\u989d\u5916\u7a7a\u95f4\u3002</li> <li>\u975e\u7a33\u5b9a\u6392\u5e8f\uff1a\u5982\u56fe 11-3 \u6240\u793a\uff0c\u5143\u7d20 <code>nums[i]</code> \u6709\u53ef\u80fd\u88ab\u4ea4\u6362\u81f3\u4e0e\u5176\u76f8\u7b49\u7684\u5143\u7d20\u7684\u53f3\u8fb9\uff0c\u5bfc\u81f4\u4e24\u8005\u7684\u76f8\u5bf9\u987a\u5e8f\u53d1\u751f\u6539\u53d8\u3002</li> </ul> <p> \u56fe 11-3 \u00a0 \u9009\u62e9\u6392\u5e8f\u975e\u7a33\u5b9a\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/","title":"11.1 \u00a0 \u6392\u5e8f\u7b97\u6cd5","text":"<p>\u300c\u6392\u5e8f\u7b97\u6cd5 sorting algorithm\u300d\u7528\u4e8e\u5bf9\u4e00\u7ec4\u6570\u636e\u6309\u7167\u7279\u5b9a\u987a\u5e8f\u8fdb\u884c\u6392\u5217\u3002\u6392\u5e8f\u7b97\u6cd5\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u56e0\u4e3a\u6709\u5e8f\u6570\u636e\u901a\u5e38\u80fd\u591f\u88ab\u66f4\u9ad8\u6548\u5730\u67e5\u627e\u3001\u5206\u6790\u548c\u5904\u7406\u3002</p> <p>\u5982\u56fe 11-1 \u6240\u793a\uff0c\u6392\u5e8f\u7b97\u6cd5\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u53ef\u4ee5\u662f\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7b49\u3002\u6392\u5e8f\u7684\u5224\u65ad\u89c4\u5219\u53ef\u6839\u636e\u9700\u6c42\u8bbe\u5b9a\uff0c\u5982\u6570\u5b57\u5927\u5c0f\u3001\u5b57\u7b26 ASCII \u7801\u987a\u5e8f\u6216\u81ea\u5b9a\u4e49\u89c4\u5219\u3002</p> <p></p> <p> \u56fe 11-1 \u00a0 \u6570\u636e\u7c7b\u578b\u548c\u5224\u65ad\u89c4\u5219\u793a\u4f8b </p>"},{"location":"chapter_sorting/sorting_algorithm/#1111","title":"11.1.1 \u00a0 \u8bc4\u4ef7\u7ef4\u5ea6","text":"<p>\u8fd0\u884c\u6548\u7387\uff1a\u6211\u4eec\u671f\u671b\u6392\u5e8f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c3d\u91cf\u4f4e\uff0c\u4e14\u603b\u4f53\u64cd\u4f5c\u6570\u91cf\u8f83\u5c11\uff08\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u5e38\u6570\u9879\u53d8\u5c0f\uff09\u3002\u5bf9\u4e8e\u5927\u6570\u636e\u91cf\u7684\u60c5\u51b5\uff0c\u8fd0\u884c\u6548\u7387\u663e\u5f97\u5c24\u4e3a\u91cd\u8981\u3002</p> <p>\u5c31\u5730\u6027\uff1a\u987e\u540d\u601d\u4e49\uff0c\u300c\u539f\u5730\u6392\u5e8f\u300d\u901a\u8fc7\u5728\u539f\u6570\u7ec4\u4e0a\u76f4\u63a5\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\uff0c\u65e0\u987b\u501f\u52a9\u989d\u5916\u7684\u8f85\u52a9\u6570\u7ec4\uff0c\u4ece\u800c\u8282\u7701\u5185\u5b58\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u539f\u5730\u6392\u5e8f\u7684\u6570\u636e\u642c\u8fd0\u64cd\u4f5c\u8f83\u5c11\uff0c\u8fd0\u884c\u901f\u5ea6\u4e5f\u66f4\u5feb\u3002</p> <p>\u7a33\u5b9a\u6027\uff1a\u300c\u7a33\u5b9a\u6392\u5e8f\u300d\u5728\u5b8c\u6210\u6392\u5e8f\u540e\uff0c\u76f8\u7b49\u5143\u7d20\u5728\u6570\u7ec4\u4e2d\u7684\u76f8\u5bf9\u987a\u5e8f\u4e0d\u53d1\u751f\u6539\u53d8\u3002</p> <p>\u7a33\u5b9a\u6392\u5e8f\u662f\u591a\u7ea7\u6392\u5e8f\u573a\u666f\u7684\u5fc5\u8981\u6761\u4ef6\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5b58\u50a8\u5b66\u751f\u4fe1\u606f\u7684\u8868\u683c\uff0c\u7b2c 1 \u5217\u548c\u7b2c 2 \u5217\u5206\u522b\u662f\u59d3\u540d\u548c\u5e74\u9f84\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u300c\u975e\u7a33\u5b9a\u6392\u5e8f\u300d\u53ef\u80fd\u5bfc\u81f4\u8f93\u5165\u6570\u636e\u7684\u6709\u5e8f\u6027\u4e27\u5931\uff1a</p> <pre><code># \u8f93\u5165\u6570\u636e\u662f\u6309\u7167\u59d3\u540d\u6392\u5e8f\u597d\u7684\n# (name, age)\n  ('A', 19)\n  ('B', 18)\n  ('C', 21)\n  ('D', 19)\n  ('E', 23)\n\n# \u5047\u8bbe\u4f7f\u7528\u975e\u7a33\u5b9a\u6392\u5e8f\u7b97\u6cd5\u6309\u5e74\u9f84\u6392\u5e8f\u5217\u8868\uff0c\n# \u7ed3\u679c\u4e2d ('D', 19) \u548c ('A', 19) \u7684\u76f8\u5bf9\u4f4d\u7f6e\u6539\u53d8\uff0c\n# \u8f93\u5165\u6570\u636e\u6309\u59d3\u540d\u6392\u5e8f\u7684\u6027\u8d28\u4e22\u5931\n  ('B', 18)\n  ('D', 19)\n  ('A', 19)\n  ('C', 21)\n  ('E', 23)\n</code></pre> <p>\u81ea\u9002\u5e94\u6027\uff1a\u300c\u81ea\u9002\u5e94\u6392\u5e8f\u300d\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u53d7\u8f93\u5165\u6570\u636e\u7684\u5f71\u54cd\uff0c\u5373\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u3001\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u5e76\u4e0d\u5b8c\u5168\u76f8\u7b49\u3002</p> <p>\u81ea\u9002\u5e94\u6027\u9700\u8981\u6839\u636e\u5177\u4f53\u60c5\u51b5\u6765\u8bc4\u4f30\u3002\u5982\u679c\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u5dee\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u8bf4\u660e\u6392\u5e8f\u7b97\u6cd5\u5728\u67d0\u4e9b\u6570\u636e\u4e0b\u6027\u80fd\u53ef\u80fd\u52a3\u5316\uff0c\u56e0\u6b64\u88ab\u89c6\u4e3a\u8d1f\u9762\u5c5e\u6027\uff1b\u800c\u5982\u679c\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u4e8e\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5219\u88ab\u89c6\u4e3a\u6b63\u9762\u5c5e\u6027\u3002</p> <p>\u662f\u5426\u57fa\u4e8e\u6bd4\u8f83\uff1a\u300c\u57fa\u4e8e\u6bd4\u8f83\u7684\u6392\u5e8f\u300d\u4f9d\u8d56\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff08\\(&lt;\\)\u3001\\(=\\)\u3001\\(&gt;\\)\uff09\u6765\u5224\u65ad\u5143\u7d20\u7684\u76f8\u5bf9\u987a\u5e8f\uff0c\u4ece\u800c\u6392\u5e8f\u6574\u4e2a\u6570\u7ec4\uff0c\u7406\u8bba\u6700\u4f18\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n \\log n)\\) \u3002\u800c\u300c\u975e\u6bd4\u8f83\u6392\u5e8f\u300d\u4e0d\u4f7f\u7528\u6bd4\u8f83\u8fd0\u7b97\u7b26\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u53ef\u8fbe \\(O(n)\\) \uff0c\u4f46\u5176\u901a\u7528\u6027\u76f8\u5bf9\u8f83\u5dee\u3002</p>"},{"location":"chapter_sorting/sorting_algorithm/#1112","title":"11.1.2 \u00a0 \u7406\u60f3\u6392\u5e8f\u7b97\u6cd5","text":"<p>\u8fd0\u884c\u5feb\u3001\u539f\u5730\u3001\u7a33\u5b9a\u3001\u6b63\u5411\u81ea\u9002\u5e94\u3001\u901a\u7528\u6027\u597d\u3002\u663e\u7136\uff0c\u8fc4\u4eca\u4e3a\u6b62\u5c1a\u672a\u53d1\u73b0\u517c\u5177\u4ee5\u4e0a\u6240\u6709\u7279\u6027\u7684\u6392\u5e8f\u7b97\u6cd5\u3002\u56e0\u6b64\uff0c\u5728\u9009\u62e9\u6392\u5e8f\u7b97\u6cd5\u65f6\uff0c\u9700\u8981\u6839\u636e\u5177\u4f53\u7684\u6570\u636e\u7279\u70b9\u548c\u95ee\u9898\u9700\u6c42\u6765\u51b3\u5b9a\u3002</p> <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u5171\u540c\u5b66\u4e60\u5404\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5e76\u57fa\u4e8e\u4e0a\u8ff0\u8bc4\u4ef7\u7ef4\u5ea6\u5bf9\u5404\u4e2a\u6392\u5e8f\u7b97\u6cd5\u7684\u4f18\u7f3a\u70b9\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_sorting/summary/","title":"11.11 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_sorting/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u5192\u6ce1\u6392\u5e8f\u901a\u8fc7\u4ea4\u6362\u76f8\u90bb\u5143\u7d20\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u6807\u5fd7\u4f4d\u6765\u5b9e\u73b0\u63d0\u524d\u8fd4\u56de\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5192\u6ce1\u6392\u5e8f\u7684\u6700\u4f73\u65f6\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(n)\\) \u3002</li> <li>\u63d2\u5165\u6392\u5e8f\u6bcf\u8f6e\u5c06\u672a\u6392\u5e8f\u533a\u95f4\u5185\u7684\u5143\u7d20\u63d2\u5165\u5230\u5df2\u6392\u5e8f\u533a\u95f4\u7684\u6b63\u786e\u4f4d\u7f6e\uff0c\u4ece\u800c\u5b8c\u6210\u6392\u5e8f\u3002\u867d\u7136\u63d2\u5165\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \uff0c\u4f46\u7531\u4e8e\u5355\u5143\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u56e0\u6b64\u5728\u5c0f\u6570\u636e\u91cf\u7684\u6392\u5e8f\u4efb\u52a1\u4e2d\u975e\u5e38\u53d7\u6b22\u8fce\u3002</li> <li>\u5feb\u901f\u6392\u5e8f\u57fa\u4e8e\u54e8\u5175\u5212\u5206\u64cd\u4f5c\u5b9e\u73b0\u6392\u5e8f\u3002\u5728\u54e8\u5175\u5212\u5206\u4e2d\uff0c\u6709\u53ef\u80fd\u6bcf\u6b21\u90fd\u9009\u53d6\u5230\u6700\u5dee\u7684\u57fa\u51c6\u6570\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u52a3\u5316\u81f3 \\(O(n^2)\\) \u3002\u5f15\u5165\u4e2d\u4f4d\u6570\u57fa\u51c6\u6570\u6216\u968f\u673a\u57fa\u51c6\u6570\u53ef\u4ee5\u964d\u4f4e\u8fd9\u79cd\u52a3\u5316\u7684\u6982\u7387\u3002\u5c3e\u9012\u5f52\u65b9\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u51cf\u5c11\u9012\u5f52\u6df1\u5ea6\uff0c\u5c06\u7a7a\u95f4\u590d\u6742\u5ea6\u4f18\u5316\u5230 \\(O(\\log n)\\) \u3002</li> <li>\u5f52\u5e76\u6392\u5e8f\u5305\u62ec\u5212\u5206\u548c\u5408\u5e76\u4e24\u4e2a\u9636\u6bb5\uff0c\u5178\u578b\u5730\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\u3002\u5728\u5f52\u5e76\u6392\u5e8f\u4e2d\uff0c\u6392\u5e8f\u6570\u7ec4\u9700\u8981\u521b\u5efa\u8f85\u52a9\u6570\u7ec4\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff1b\u7136\u800c\u6392\u5e8f\u94fe\u8868\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u53ef\u4ee5\u4f18\u5316\u81f3 \\(O(1)\\) \u3002</li> <li>\u6876\u6392\u5e8f\u5305\u542b\u4e09\u4e2a\u6b65\u9aa4\uff1a\u6570\u636e\u5206\u6876\u3001\u6876\u5185\u6392\u5e8f\u548c\u5408\u5e76\u7ed3\u679c\u3002\u5b83\u540c\u6837\u4f53\u73b0\u4e86\u5206\u6cbb\u7b56\u7565\uff0c\u9002\u7528\u4e8e\u6570\u636e\u4f53\u91cf\u5f88\u5927\u7684\u60c5\u51b5\u3002\u6876\u6392\u5e8f\u7684\u5173\u952e\u5728\u4e8e\u5bf9\u6570\u636e\u8fdb\u884c\u5e73\u5747\u5206\u914d\u3002</li> <li>\u8ba1\u6570\u6392\u5e8f\u662f\u6876\u6392\u5e8f\u7684\u4e00\u4e2a\u7279\u4f8b\uff0c\u5b83\u901a\u8fc7\u7edf\u8ba1\u6570\u636e\u51fa\u73b0\u7684\u6b21\u6570\u6765\u5b9e\u73b0\u6392\u5e8f\u3002\u8ba1\u6570\u6392\u5e8f\u9002\u7528\u4e8e\u6570\u636e\u91cf\u5927\u4f46\u6570\u636e\u8303\u56f4\u6709\u9650\u7684\u60c5\u51b5\uff0c\u5e76\u4e14\u8981\u6c42\u6570\u636e\u80fd\u591f\u8f6c\u6362\u4e3a\u6b63\u6574\u6570\u3002</li> <li>\u57fa\u6570\u6392\u5e8f\u901a\u8fc7\u9010\u4f4d\u6392\u5e8f\u6765\u5b9e\u73b0\u6570\u636e\u6392\u5e8f\uff0c\u8981\u6c42\u6570\u636e\u80fd\u591f\u8868\u793a\u4e3a\u56fa\u5b9a\u4f4d\u6570\u7684\u6570\u5b57\u3002</li> <li>\u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\uff0c\u5177\u6709\u9ad8\u6548\u7387\u3001\u7a33\u5b9a\u3001\u539f\u5730\u4ee5\u53ca\u6b63\u5411\u81ea\u9002\u5e94\u6027\u7b49\u4f18\u70b9\u3002\u7136\u800c\uff0c\u6b63\u5982\u5176\u4ed6\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u4e00\u6837\uff0c\u6ca1\u6709\u4e00\u79cd\u6392\u5e8f\u7b97\u6cd5\u80fd\u591f\u540c\u65f6\u6ee1\u8db3\u6240\u6709\u8fd9\u4e9b\u6761\u4ef6\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u9700\u8981\u6839\u636e\u6570\u636e\u7684\u7279\u6027\u6765\u9009\u62e9\u5408\u9002\u7684\u6392\u5e8f\u7b97\u6cd5\u3002</li> <li>\u56fe 11-19 \u5bf9\u6bd4\u4e86\u4e3b\u6d41\u6392\u5e8f\u7b97\u6cd5\u7684\u6548\u7387\u3001\u7a33\u5b9a\u6027\u3001\u5c31\u5730\u6027\u548c\u81ea\u9002\u5e94\u6027\u7b49\u3002</li> </ul> <p> \u56fe 11-19 \u00a0 \u6392\u5e8f\u7b97\u6cd5\u5bf9\u6bd4 </p>"},{"location":"chapter_sorting/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6392\u5e8f\u7b97\u6cd5\u7a33\u5b9a\u6027\u5728\u4ec0\u4e48\u60c5\u51b5\u4e0b\u662f\u5fc5\u9700\u7684\uff1f</p> <p>\u5728\u73b0\u5b9e\u4e2d\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u662f\u57fa\u4e8e\u5bf9\u8c61\u7684\u67d0\u4e2a\u5c5e\u6027\u8fdb\u884c\u6392\u5e8f\u3002\u4f8b\u5982\uff0c\u5b66\u751f\u6709\u59d3\u540d\u548c\u8eab\u9ad8\u4e24\u4e2a\u5c5e\u6027\uff0c\u6211\u4eec\u5e0c\u671b\u5b9e\u73b0\u4e00\u4e2a\u591a\u7ea7\u6392\u5e8f\uff1a</p> <p>\u5148\u6309\u7167\u59d3\u540d\u8fdb\u884c\u6392\u5e8f\uff0c\u5f97\u5230 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> \uff1b\u518d\u5bf9\u8eab\u9ad8\u8fdb\u884c\u6392\u5e8f\u3002\u7531\u4e8e\u6392\u5e8f\u7b97\u6cd5\u4e0d\u7a33\u5b9a\uff0c\u56e0\u6b64\u53ef\u80fd\u5f97\u5230 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> \u3002</p> <p>\u53ef\u4ee5\u53d1\u73b0\uff0c\u5b66\u751f D \u548c C \u7684\u4f4d\u7f6e\u53d1\u751f\u4e86\u4ea4\u6362\uff0c\u59d3\u540d\u7684\u6709\u5e8f\u6027\u88ab\u7834\u574f\u4e86\uff0c\u800c\u8fd9\u662f\u6211\u4eec\u4e0d\u5e0c\u671b\u770b\u5230\u7684\u3002</p> <p>\u54e8\u5175\u5212\u5206\u4e2d\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u4e0e\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u7684\u987a\u5e8f\u53ef\u4ee5\u4ea4\u6362\u5417\uff1f</p> <p>\u4e0d\u884c\uff0c\u5f53\u6211\u4eec\u4ee5\u6700\u5de6\u7aef\u5143\u7d20\u4e3a\u57fa\u51c6\u6570\u65f6\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u53f3\u5f80\u5de6\u67e5\u627e\u201d\u518d\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002\u8fd9\u4e2a\u7ed3\u8bba\u6709\u4e9b\u53cd\u76f4\u89c9\uff0c\u6211\u4eec\u6765\u5256\u6790\u4e00\u4e0b\u539f\u56e0\u3002</p> <p>\u54e8\u5175\u5212\u5206 <code>partition()</code> \u7684\u6700\u540e\u4e00\u6b65\u662f\u4ea4\u6362 <code>nums[left]</code> \u548c <code>nums[i]</code> \u3002\u5b8c\u6210\u4ea4\u6362\u540e\uff0c\u57fa\u51c6\u6570\u5de6\u8fb9\u7684\u5143\u7d20\u90fd <code>&lt;=</code> \u57fa\u51c6\u6570\uff0c\u8fd9\u5c31\u8981\u6c42\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u524d <code>nums[left] &gt;= nums[i]</code> \u5fc5\u987b\u6210\u7acb\u3002\u5047\u8bbe\u6211\u4eec\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\uff0c\u90a3\u4e48\u5982\u679c\u627e\u4e0d\u5230\u6bd4\u57fa\u51c6\u6570\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u4f1a\u5728 <code>i == j</code> \u65f6\u8df3\u51fa\u5faa\u73af\uff0c\u6b64\u65f6\u53ef\u80fd <code>nums[j] == nums[i] &gt; nums[left]</code>\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u6b64\u65f6\u6700\u540e\u4e00\u6b65\u4ea4\u6362\u64cd\u4f5c\u4f1a\u628a\u4e00\u4e2a\u6bd4\u57fa\u51c6\u6570\u66f4\u5927\u7684\u5143\u7d20\u4ea4\u6362\u81f3\u6570\u7ec4\u6700\u5de6\u7aef\uff0c\u5bfc\u81f4\u54e8\u5175\u5212\u5206\u5931\u8d25\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u7ed9\u5b9a\u6570\u7ec4 <code>[0, 0, 0, 0, 1]</code> \uff0c\u5982\u679c\u5148\u201c\u4ece\u5de6\u5411\u53f3\u67e5\u627e\u201d\uff0c\u54e8\u5175\u5212\u5206\u540e\u6570\u7ec4\u4e3a <code>[1, 0, 0, 0, 0]</code> \uff0c\u8fd9\u4e2a\u7ed3\u679c\u662f\u4e0d\u6b63\u786e\u7684\u3002</p> <p>\u518d\u6df1\u5165\u601d\u8003\u4e00\u4e0b\uff0c\u5982\u679c\u6211\u4eec\u9009\u62e9 <code>nums[right]</code> \u4e3a\u57fa\u51c6\u6570\uff0c\u90a3\u4e48\u6b63\u597d\u53cd\u8fc7\u6765\uff0c\u5fc5\u987b\u5148\u201c\u4ece\u5de6\u5f80\u53f3\u67e5\u627e\u201d\u3002</p> <p>\u5173\u4e8e\u5c3e\u9012\u5f52\u4f18\u5316\uff0c\u4e3a\u4ec0\u4e48\u9009\u77ed\u7684\u6570\u7ec4\u80fd\u4fdd\u8bc1\u9012\u5f52\u6df1\u5ea6\u4e0d\u8d85\u8fc7 \\(\\log n\\) \uff1f</p> <p>\u9012\u5f52\u6df1\u5ea6\u5c31\u662f\u5f53\u524d\u672a\u8fd4\u56de\u7684\u9012\u5f52\u65b9\u6cd5\u7684\u6570\u91cf\u3002\u6bcf\u8f6e\u54e8\u5175\u5212\u5206\u6211\u4eec\u5c06\u539f\u6570\u7ec4\u5212\u5206\u4e3a\u4e24\u4e2a\u5b50\u6570\u7ec4\u3002\u5728\u5c3e\u9012\u5f52\u4f18\u5316\u540e\uff0c\u5411\u4e0b\u9012\u5f52\u7684\u5b50\u6570\u7ec4\u957f\u5ea6\u6700\u5927\u4e3a\u539f\u6570\u7ec4\u957f\u5ea6\u7684\u4e00\u534a\u3002\u5047\u8bbe\u6700\u5dee\u60c5\u51b5\uff0c\u4e00\u76f4\u4e3a\u4e00\u534a\u957f\u5ea6\uff0c\u90a3\u4e48\u6700\u7ec8\u7684\u9012\u5f52\u6df1\u5ea6\u5c31\u662f \\(\\log n\\) \u3002</p> <p>\u56de\u987e\u539f\u59cb\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u4f1a\u8fde\u7eed\u5730\u9012\u5f52\u957f\u5ea6\u8f83\u5927\u7684\u6570\u7ec4\uff0c\u6700\u5dee\u60c5\u51b5\u4e0b\u4e3a \\(n\\)\u3001\\(n - 1\\)\u3001\\(\\dots\\)\u3001\\(2\\)\u3001\\(1\\) \uff0c\u9012\u5f52\u6df1\u5ea6\u4e3a \\(n\\) \u3002\u5c3e\u9012\u5f52\u4f18\u5316\u53ef\u4ee5\u907f\u514d\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002</p> <p>\u5f53\u6570\u7ec4\u4e2d\u6240\u6709\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n^2)\\) \u5417\uff1f\u8be5\u5982\u4f55\u5904\u7406\u8fd9\u79cd\u9000\u5316\u60c5\u51b5\uff1f</p> <p>\u662f\u7684\u3002\u5bf9\u4e8e\u8fd9\u79cd\u60c5\u51b5\uff0c\u53ef\u4ee5\u8003\u8651\u901a\u8fc7\u54e8\u5175\u5212\u5206\u5c06\u6570\u7ec4\u5212\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff1a\u5c0f\u4e8e\u3001\u7b49\u4e8e\u3001\u5927\u4e8e\u57fa\u51c6\u6570\u3002\u4ec5\u5411\u4e0b\u9012\u5f52\u5c0f\u4e8e\u548c\u5927\u4e8e\u7684\u4e24\u90e8\u5206\u3002\u5728\u8be5\u65b9\u6cd5\u4e0b\uff0c\u8f93\u5165\u5143\u7d20\u5168\u90e8\u76f8\u7b49\u7684\u6570\u7ec4\uff0c\u4ec5\u4e00\u8f6e\u54e8\u5175\u5212\u5206\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\u3002</p> <p>\u6876\u6392\u5e8f\u7684\u6700\u5dee\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\u4ec0\u4e48\u662f \\(O(n^2)\\) \uff1f</p> <p>\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u5143\u7d20\u88ab\u5206\u81f3\u540c\u4e00\u4e2a\u6876\u4e2d\u3002\u5982\u679c\u6211\u4eec\u91c7\u7528\u4e00\u4e2a \\(O(n^2)\\) \u7b97\u6cd5\u6765\u6392\u5e8f\u8fd9\u4e9b\u5143\u7d20\uff0c\u5219\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n^2)\\) \u3002</p>"},{"location":"chapter_stack_and_queue/","title":"\u7b2c 5 \u7ae0 \u00a0 \u6808\u4e0e\u961f\u5217","text":"<p>Abstract</p> <p>\u6808\u5982\u540c\u53e0\u732b\u732b\uff0c\u800c\u961f\u5217\u5c31\u50cf\u732b\u732b\u6392\u961f\u3002</p> <p>\u4e24\u8005\u5206\u522b\u4ee3\u8868\u5148\u5165\u540e\u51fa\u548c\u5148\u5165\u5148\u51fa\u7684\u903b\u8f91\u5173\u7cfb\u3002</p>"},{"location":"chapter_stack_and_queue/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>5.1 \u00a0 \u6808</li> <li>5.2 \u00a0 \u961f\u5217</li> <li>5.3 \u00a0 \u53cc\u5411\u961f\u5217</li> <li>5.4 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_stack_and_queue/deque/","title":"5.3 \u00a0 \u53cc\u5411\u961f\u5217","text":"<p>\u5728\u961f\u5217\u4e2d\uff0c\u6211\u4eec\u4ec5\u80fd\u5220\u9664\u5934\u90e8\u5143\u7d20\u6216\u5728\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u3002\u5982\u56fe 5-7 \u6240\u793a\uff0c\u300c\u53cc\u5411\u961f\u5217 double-ended queue\u300d\u63d0\u4f9b\u4e86\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\uff0c\u5141\u8bb8\u5728\u5934\u90e8\u548c\u5c3e\u90e8\u6267\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u6216\u5220\u9664\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 5-7 \u00a0 \u53cc\u5411\u961f\u5217\u7684\u64cd\u4f5c </p>"},{"location":"chapter_stack_and_queue/deque/#531","title":"5.3.1 \u00a0 \u53cc\u5411\u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-3 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u79f0\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002</p> <p> \u8868 5-3 \u00a0 \u53cc\u5411\u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 pushFirst() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u9996 \\(O(1)\\) pushLast() \u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) popFirst() \u5220\u9664\u961f\u9996\u5143\u7d20 \\(O(1)\\) popLast() \u5220\u9664\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) peekFirst() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) peekLast() \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 \\(O(1)\\) <p>\u540c\u6837\u5730\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u5df2\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig deque.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217\ndeque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\ndeque.append(2)      # \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.appendleft(3)  # \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.appendleft(1)\n\n# \u8bbf\u95ee\u5143\u7d20\nfront: int = deque[0]  # \u961f\u9996\u5143\u7d20\nrear: int = deque[-1]  # \u961f\u5c3e\u5143\u7d20\n\n# \u5143\u7d20\u51fa\u961f\npop_front: int = deque.popleft()  # \u961f\u9996\u5143\u7d20\u51fa\u961f\npop_rear: int = deque.pop()       # \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n# \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(deque)\n\n# \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(deque) == 0\n</code></pre> deque.cpp<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\ndeque&lt;int&gt; deque;\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint front = deque.front(); // \u961f\u9996\u5143\u7d20\nint back = deque.back();   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.pop_front();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.pop_back();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = deque.empty();\n</code></pre> deque.java<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nDeque&lt;Integer&gt; deque = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.offerLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.offerLast(5);\ndeque.offerLast(4);\ndeque.offerFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.offerFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.peekFirst();  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.peekLast();    // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.pollFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.pollLast();    // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.size();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = deque.isEmpty();\n</code></pre> deque.cs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 C# \u4e2d\uff0c\u5c06\u94fe\u8868 LinkedList \u770b\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nLinkedList&lt;int&gt; deque = new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.AddLast(2);   // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.AddLast(5);\ndeque.AddLast(4);\ndeque.AddFirst(3);  // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.AddFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.First.Value;  // \u961f\u9996\u5143\u7d20\nint peekLast = deque.Last.Value;    // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.RemoveFirst();  // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.RemoveLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.Count;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.Count == 0;\n</code></pre> deque_test.go<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u4f7f\u7528\ndeque := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.PushBack(2)      // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.PushBack(5)\ndeque.PushBack(4)\ndeque.PushFront(3)     // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.PushFront(1)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nfront := deque.Front() // \u961f\u9996\u5143\u7d20\nrear := deque.Back()   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\ndeque.Remove(front)    // \u961f\u9996\u5143\u7d20\u51fa\u961f\ndeque.Remove(rear)     // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nsize := deque.Len()\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := deque.Len() == 0\n</code></pre> deque.swift<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u5411\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u53cc\u5411\u961f\u5217\u6765\u4f7f\u7528\nvar deque: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.append(2) // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.append(5)\ndeque.append(4)\ndeque.insert(3, at: 0) // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.insert(1, at: 0)\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nlet peekFirst = deque.first! // \u961f\u9996\u5143\u7d20\nlet peekLast = deque.last! // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u4f7f\u7528 Array \u6a21\u62df\u65f6 popFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet popFirst = deque.removeFirst() // \u961f\u9996\u5143\u7d20\u51fa\u961f\nlet popLast = deque.removeLast() // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.count\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = deque.isEmpty\n</code></pre> deque.js<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront = deque.shift();\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack = deque.pop();\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n</code></pre> deque.ts<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u53cc\u7aef\u961f\u5217\uff0c\u53ea\u80fd\u628a Array \u5f53\u4f5c\u53cc\u7aef\u961f\u5217\u6765\u4f7f\u7528\nconst deque: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push(2);\ndeque.push(5);\ndeque.push(4);\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cunshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\ndeque.unshift(3);\ndeque.unshift(1);\nconsole.log(\"\u53cc\u5411\u961f\u5217 deque = \", deque);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nconst peekFirst: number = deque[0];\nconsole.log(\"\u961f\u9996\u5143\u7d20 peekFirst = \" + peekFirst);\nconst peekLast: number = deque[deque.length - 1];\nconsole.log(\"\u961f\u5c3e\u5143\u7d20 peekLast = \" + peekLast);\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u8bf7\u6ce8\u610f\uff0c\u7531\u4e8e\u662f\u6570\u7ec4\uff0cshift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst popFront: number = deque.shift() as number;\nconsole.log(\"\u961f\u9996\u51fa\u961f\u5143\u7d20 popFront = \" + popFront + \"\uff0c\u961f\u9996\u51fa\u961f\u540e deque = \" + deque);\nconst popBack: number = deque.pop() as number;\nconsole.log(\"\u961f\u5c3e\u51fa\u961f\u5143\u7d20 popBack = \" + popBack + \"\uff0c\u961f\u5c3e\u51fa\u961f\u540e deque = \" + deque);\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nconst size: number = deque.length;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u957f\u5ea6 size = \" + size);\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst isEmpty: boolean = size === 0;\nconsole.log(\"\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a = \" + isEmpty);\n</code></pre> deque.dart<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0cQueue \u88ab\u5b9a\u4e49\u4e3a\u53cc\u5411\u961f\u5217\nQueue&lt;int&gt; deque = Queue&lt;int&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.addLast(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.addLast(5);\ndeque.addLast(4);\ndeque.addFirst(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.addFirst(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nint peekFirst = deque.first; // \u961f\u9996\u5143\u7d20\nint peekLast = deque.last;   // \u961f\u5c3e\u5143\u7d20\n\n/* \u5143\u7d20\u51fa\u961f */\nint popFirst = deque.removeFirst(); // \u961f\u9996\u5143\u7d20\u51fa\u961f\nint popLast = deque.removeLast();   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size = deque.length;\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = deque.isEmpty;W\n</code></pre> deque.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(2);  // \u6dfb\u52a0\u81f3\u961f\u5c3e\ndeque.push_back(5);\ndeque.push_back(4);\ndeque.push_front(3); // \u6dfb\u52a0\u81f3\u961f\u9996\ndeque.push_front(1);\n\n/* \u8bbf\u95ee\u5143\u7d20 */\nif let Some(front) = deque.front() { // \u961f\u9996\u5143\u7d20\n}\nif let Some(rear) = deque.back() {   // \u961f\u5c3e\u5143\u7d20\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop_front) = deque.pop_front() { // \u961f\u9996\u5143\u7d20\u51fa\u961f\n}\nif let Some(pop_rear) = deque.pop_back() {   // \u961f\u5c3e\u5143\u7d20\u51fa\u961f\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> deque.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u53cc\u5411\u961f\u5217\n</code></pre> deque.zig<pre><code>\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#532","title":"5.3.2 \u00a0 \u53cc\u5411\u961f\u5217\u5b9e\u73b0 *","text":"<p>\u53cc\u5411\u961f\u5217\u7684\u5b9e\u73b0\u4e0e\u961f\u5217\u7c7b\u4f3c\uff0c\u53ef\u4ee5\u9009\u62e9\u94fe\u8868\u6216\u6570\u7ec4\u4f5c\u4e3a\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p>"},{"location":"chapter_stack_and_queue/deque/#1","title":"1. \u00a0 \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u56de\u987e\u4e0a\u4e00\u8282\u5185\u5bb9\uff0c\u6211\u4eec\u4f7f\u7528\u666e\u901a\u5355\u5411\u94fe\u8868\u6765\u5b9e\u73b0\u961f\u5217\uff0c\u56e0\u4e3a\u5b83\u53ef\u4ee5\u65b9\u4fbf\u5730\u5220\u9664\u5934\u8282\u70b9\uff08\u5bf9\u5e94\u51fa\u961f\u64cd\u4f5c\uff09\u548c\u5728\u5c3e\u8282\u70b9\u540e\u6dfb\u52a0\u65b0\u8282\u70b9\uff08\u5bf9\u5e94\u5165\u961f\u64cd\u4f5c\uff09\u3002</p> <p>\u5bf9\u4e8e\u53cc\u5411\u961f\u5217\u800c\u8a00\uff0c\u5934\u90e8\u548c\u5c3e\u90e8\u90fd\u53ef\u4ee5\u6267\u884c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u53cc\u5411\u961f\u5217\u9700\u8981\u5b9e\u73b0\u53e6\u4e00\u4e2a\u5bf9\u79f0\u65b9\u5411\u7684\u64cd\u4f5c\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u91c7\u7528\u201c\u53cc\u5411\u94fe\u8868\u201d\u4f5c\u4e3a\u53cc\u5411\u961f\u5217\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-8 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u53cc\u5411\u94fe\u8868\u7684\u5934\u8282\u70b9\u548c\u5c3e\u8282\u70b9\u89c6\u4e3a\u53cc\u5411\u961f\u5217\u7684\u961f\u9996\u548c\u961f\u5c3e\uff0c\u540c\u65f6\u5b9e\u73b0\u5728\u4e24\u7aef\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u7684\u529f\u80fd\u3002</p> LinkedListDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-8 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_deque.py<pre><code>class ListNode:\n    \"\"\"\u53cc\u5411\u94fe\u8868\u8282\u70b9\"\"\"\n\n    def __init__(self, val: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self.val: int = val\n        self.next: ListNode | None = None  # \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n        self.prev: ListNode | None = None  # \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\nclass LinkedListDeque:\n    \"\"\"\u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0  # \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self.size() == 0\n\n    def push(self, num: int, is_front: bool):\n        \"\"\"\u5165\u961f\u64cd\u4f5c\"\"\"\n        node = ListNode(num)\n        # \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if self.is_empty():\n            self._front = self._rear = node\n        # \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        elif is_front:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            self._front.prev = node\n            node.next = self._front\n            self._front = node  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else:\n            # \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            self._rear.next = node\n            node.prev = self._rear\n            self._rear = node  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size += 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        self.push(num, True)\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        self.push(num, False)\n\n    def pop(self, is_front: bool) -&gt; int:\n        \"\"\"\u51fa\u961f\u64cd\u4f5c\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front:\n            val: int = self._front.val  # \u6682\u5b58\u5934\u8282\u70b9\u503c\n            # \u5220\u9664\u5934\u8282\u70b9\n            fnext: ListNode | None = self._front.next\n            if fnext != None:\n                fnext.prev = None\n                self._front.next = None\n            self._front = fnext  # \u66f4\u65b0\u5934\u8282\u70b9\n        # \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else:\n            val: int = self._rear.val  # \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            # \u5220\u9664\u5c3e\u8282\u70b9\n            rprev: ListNode | None = self._rear.prev\n            if rprev != None:\n                rprev.next = None\n                self._rear.prev = None\n            self._rear = rprev  # \u66f4\u65b0\u5c3e\u8282\u70b9\n        self._size -= 1  # \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        return self.pop(True)\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        return self.pop(False)\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._rear.val\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        node = self._front\n        res = [0] * self.size()\n        for i in range(self.size()):\n            res[i] = node.val\n            node = node.next\n        return res\n</code></pre> linkedlist_deque.cpp<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nstruct DoublyListNode {\n    int val;              // \u8282\u70b9\u503c\n    DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\u6307\u9488\n    DoublyListNode(int val) : val(val), prev(nullptr), next(nullptr) {\n    }\n};\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n  private:\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize = 0;              // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    LinkedListDeque() : front(nullptr), rear(nullptr) {\n    }\n\n    /* \u6790\u6784\u65b9\u6cd5 */\n    ~LinkedListDeque() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        DoublyListNode *pre, *cur = front;\n        while (cur != nullptr) {\n            pre = cur;\n            cur = cur-&gt;next;\n            delete pre;\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void push(int num, bool isFront) {\n        DoublyListNode *node = new DoublyListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front-&gt;prev = node;\n            node-&gt;next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear-&gt;next = node;\n            node-&gt;prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int pop(bool isFront) {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front-&gt;val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            DoublyListNode *fNext = front-&gt;next;\n            if (fNext != nullptr) {\n                fNext-&gt;prev = nullptr;\n                front-&gt;next = nullptr;\n                delete front;\n            }\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear-&gt;val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            DoublyListNode *rPrev = rear-&gt;prev;\n            if (rPrev != nullptr) {\n                rPrev-&gt;next = nullptr;\n                rear-&gt;prev = nullptr;\n                delete rear;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return rear-&gt;val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        DoublyListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_deque.java<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    int val; // \u8282\u70b9\u503c\n    ListNode next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    ListNode prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    ListNode(int val) {\n        this.val = val;\n        prev = next = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private void push(int num, boolean isFront) {\n        ListNode node = new ListNode(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (isEmpty())\n            front = rear = node;\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear.next = node;\n            node.prev = rear;\n            rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private int pop(boolean isFront) {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        int val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode fNext = front.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front.next = null;\n            }\n            front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        } else {\n            val = rear.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode rPrev = rear.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear.prev = null;\n            }\n            rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        return pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        return pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return rear.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.cs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode(int val) {\n    public int val = val;       // \u8282\u70b9\u503c\n    public ListNode? next = null; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    public ListNode? prev = null; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    ListNode? front, rear; // \u5934\u8282\u70b9 front, \u5c3e\u8282\u70b9 rear\n    int queSize = 0;      // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    public LinkedListDeque() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    void Push(int num, bool isFront) {\n        ListNode node = new(num);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (IsEmpty()) {\n            front = node;\n            rear = node;\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if (isFront) {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front!.prev = node;\n            node.next = front;\n            front = node; // \u66f4\u65b0\u5934\u8282\u70b9                           \n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear!.next = node;\n            node.prev = rear;\n            rear = node;  // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        Push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        Push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    int? Pop(bool isFront) {\n        if (IsEmpty())\n            throw new Exception();\n        int? val;\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if (isFront) {\n            val = front?.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            ListNode? fNext = front?.next;\n            if (fNext != null) {\n                fNext.prev = null;\n                front!.next = null;\n            }\n            front = fNext;   // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear?.val;  // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            ListNode? rPrev = rear?.prev;\n            if (rPrev != null) {\n                rPrev.next = null;\n                rear!.prev = null;\n            }\n            rear = rPrev;    // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n\n        queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int? PopFirst() {\n        return Pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int? PopLast() {\n        return Pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int? PeekFirst() {\n        if (IsEmpty())\n            throw new Exception();\n        return front?.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int? PeekLast() {\n        if (IsEmpty())\n            throw new Exception();\n        return rear?.val;\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int?[] ToArray() {\n        ListNode? node = front;\n        int?[] res = new int?[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node?.val;\n            node = node?.next;\n        }\n\n        return res;\n    }\n}\n</code></pre> linkedlist_deque.go<pre><code>/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype linkedListDeque struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u53cc\u7aef\u961f\u5217 */\nfunc newLinkedListDeque() *linkedListDeque {\n    return &amp;linkedListDeque{\n        data: list.New(),\n    }\n}\n\n/* \u961f\u9996\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushFirst(value any) {\n    s.data.PushFront(value)\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u5165\u961f */\nfunc (s *linkedListDeque) pushLast(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u961f\u9996\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u961f\u5c3e\u5143\u7d20\u51fa\u961f */\nfunc (s *linkedListDeque) popLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListDeque) peekFirst() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (s *linkedListDeque) peekLast() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListDeque) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListDeque) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListDeque) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_deque.swift<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    var val: Int // \u8282\u70b9\u503c\n    var next: ListNode? // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n    weak var prev: ListNode? // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n    init(val: Int) {\n        self.val = val\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private var front: ListNode? // \u5934\u8282\u70b9 front\n    private var rear: ListNode? // \u5c3e\u8282\u70b9 rear\n    private var queSize: Int // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    init() {\n        queSize = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    private func push(num: Int, isFront: Bool) {\n        let node = ListNode(val: num)\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if isEmpty() {\n            front = node\n            rear = node\n        }\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        else if isFront {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            front?.prev = node\n            node.next = front\n            front = node // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            rear?.next = node\n            node.prev = rear\n            rear = node // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize += 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        push(num: num, isFront: true)\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        push(num: num, isFront: false)\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    private func pop(isFront: Bool) -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        let val: Int\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if isFront {\n            val = front!.val // \u6682\u5b58\u5934\u8282\u70b9\u503c\n            // \u5220\u9664\u5934\u8282\u70b9\n            let fNext = front?.next\n            if fNext != nil {\n                fNext?.prev = nil\n                front?.next = nil\n            }\n            front = fNext // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            val = rear!.val // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n            // \u5220\u9664\u5c3e\u8282\u70b9\n            let rPrev = rear?.prev\n            if rPrev != nil {\n                rPrev?.next = nil\n                rear?.prev = nil\n            }\n            rear = rPrev // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        queSize -= 1 // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        return val\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        pop(isFront: true)\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        pop(isFront: false)\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int? {\n        isEmpty() ? nil : front?.val\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int? {\n        isEmpty() ? nil : rear?.val\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_deque.js<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val; // \u8282\u70b9\u503c\n\n    constructor(val) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    #front; // \u5934\u8282\u70b9 front\n    #rear; // \u5c3e\u8282\u70b9 rear\n    #queSize; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n        this.#queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.#rear.next = node;\n            node.prev = this.#rear;\n            this.#rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val) {\n        const node = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.#queSize === 0) {\n            this.#front = node;\n            this.#rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.#front.prev = node;\n            node.next = this.#front;\n            this.#front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp = this.#rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.#rear.prev = null;\n        }\n        this.#rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst() {\n        if (this.#queSize === 0) {\n            return null;\n        }\n        const value = this.#front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp = this.#front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.#front.next = null;\n        }\n        this.#front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.#queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        return this.#queSize === 0 ? null : this.#rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        return this.#queSize === 0 ? null : this.#front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print() {\n        const arr = [];\n        let temp = this.#front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.ts<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n    prev: ListNode; // \u524d\u9a71\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    next: ListNode; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528 (\u6307\u9488)\n    val: number; // \u8282\u70b9\u503c\n\n    constructor(val: number) {\n        this.val = val;\n        this.next = null;\n        this.prev = null;\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass LinkedListDeque {\n    private front: ListNode; // \u5934\u8282\u70b9 front\n    private rear: ListNode; // \u5c3e\u8282\u70b9 rear\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n        this.queSize = 0;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f\u64cd\u4f5c */\n    pushLast(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n            this.rear.next = node;\n            node.prev = this.rear;\n            this.rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u5165\u961f\u64cd\u4f5c */\n    pushFirst(val: number): void {\n        const node: ListNode = new ListNode(val);\n        // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n        if (this.queSize === 0) {\n            this.front = node;\n            this.rear = node;\n        } else {\n            // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n            this.front.prev = node;\n            node.next = this.front;\n            this.front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n        }\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c */\n    popLast(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.rear.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5c3e\u8282\u70b9\n        let temp: ListNode = this.rear.prev;\n        if (temp !== null) {\n            temp.next = null;\n            this.rear.prev = null;\n        }\n        this.rear = temp; // \u66f4\u65b0\u5c3e\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u961f\u9996\u51fa\u961f\u64cd\u4f5c */\n    popFirst(): number {\n        if (this.queSize === 0) {\n            return null;\n        }\n        const value: number = this.front.val; // \u5b58\u50a8\u5c3e\u8282\u70b9\u503c\n        // \u5220\u9664\u5934\u8282\u70b9\n        let temp: ListNode = this.front.next;\n        if (temp !== null) {\n            temp.prev = null;\n            this.front.next = null;\n        }\n        this.front = temp; // \u66f4\u65b0\u5934\u8282\u70b9\n        this.queSize--;\n        return value;\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        return this.queSize === 0 ? null : this.rear.val;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        return this.queSize === 0 ? null : this.front.val;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u6253\u5370\u53cc\u5411\u961f\u5217 */\n    print(): void {\n        const arr: number[] = [];\n        let temp: ListNode = this.front;\n        while (temp !== null) {\n            arr.push(temp.val);\n            temp = temp.next;\n        }\n        console.log('[' + arr.join(', ') + ']');\n    }\n}\n</code></pre> linkedlist_deque.dart<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\nclass ListNode {\n  int val; // \u8282\u70b9\u503c\n  ListNode? next; // \u540e\u7ee7\u8282\u70b9\u5f15\u7528\n  ListNode? prev; // \u524d\u9a71\u8282\u70b9\u5f15\u7528\n\n  ListNode(this.val, {this.next, this.prev});\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u5bf9\u5217 */\nclass LinkedListDeque {\n  late ListNode? _front; // \u5934\u8282\u70b9 _front\n  late ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n\n  LinkedListDeque() {\n    this._front = null;\n    this._rear = null;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u957f\u5ea6 */\n  int size() {\n    return this._queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return size() == 0;\n  }\n\n  /* \u5165\u961f\u64cd\u4f5c */\n  void push(int _num, bool isFront) {\n    final ListNode node = ListNode(_num);\n    if (isEmpty()) {\n      // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 _front \u548c _rear \u90fd\u6307\u5411 node\n      _front = _rear = node;\n    } else if (isFront) {\n      // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n      _front!.prev = node;\n      node.next = _front;\n      _front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n      // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n      _rear!.next = node;\n      node.prev = _rear;\n      _rear = node; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    push(_num, true);\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    push(_num, false);\n  }\n\n  /* \u51fa\u961f\u64cd\u4f5c */\n  int? pop(bool isFront) {\n    // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de null\n    if (isEmpty()) {\n      return null;\n    }\n    final int val;\n    if (isFront) {\n      // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n      val = _front!.val; // \u6682\u5b58\u5934\u8282\u70b9\u503c\n      // \u5220\u9664\u5934\u8282\u70b9\n      ListNode? fNext = _front!.next;\n      if (fNext != null) {\n        fNext.prev = null;\n        _front!.next = null;\n      }\n      _front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    } else {\n      // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n      val = _rear!.val; // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n      // \u5220\u9664\u5c3e\u8282\u70b9\n      ListNode? rPrev = _rear!.prev;\n      if (rPrev != null) {\n        rPrev.next = null;\n        _rear!.prev = null;\n      }\n      _rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    _queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int? popFirst() {\n    return pop(true);\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int? popLast() {\n    return pop(false);\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int? peekFirst() {\n    return _front?.val;\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int? peekLast() {\n    return _rear?.val;\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; _queSize; i++) {\n      res.add(node!.val);\n      node = node.next;\n    }\n    return res;\n  }\n}\n</code></pre> linkedlist_deque.rs<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\npub struct ListNode&lt;T&gt; {\n    pub val: T,                                 // \u8282\u70b9\u503c\n    pub next: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n    pub prev: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;, // \u524d\u9a71\u8282\u70b9\u6307\u9488\n}\n\nimpl&lt;T&gt; ListNode&lt;T&gt; {\n    pub fn new(val: T) -&gt; Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt; {\n        Rc::new(RefCell::new(ListNode {\n            val,\n            next: None,\n            prev: None,\n        }))\n    }\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListDeque&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,    // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,     // \u5c3e\u8282\u70b9 rear \n    que_size: usize,                            // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListDeque&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0, \n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f\u64cd\u4f5c */\n    pub fn push(&amp;mut self, num: T, is_front: bool) {\n        let node = ListNode::new(num);\n        // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n        if is_front {\n            match self.front.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.rear = Some(node.clone());\n                    self.front = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                Some(old_front) =&gt; {\n                    old_front.borrow_mut().prev = Some(node.clone());\n                    node.borrow_mut().next = Some(old_front);\n                    self.front = Some(node); // \u66f4\u65b0\u5934\u8282\u70b9\n                }\n            }\n        } \n        // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n        else {\n            match self.rear.take() {\n                // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n                None =&gt; {\n                    self.front = Some(node.clone());\n                    self.rear = Some(node);\n                }\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                Some(old_rear) =&gt; {\n                    old_rear.borrow_mut().next = Some(node.clone());\n                    node.borrow_mut().prev = Some(old_rear);\n                    self.rear = Some(node); // \u66f4\u65b0\u5c3e\u8282\u70b9\n                }\n            }\n        }\n        self.que_size += 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: T) {\n        self.push(num, true);\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: T) {\n        self.push(num, false);\n    }\n\n    /* \u51fa\u961f\u64cd\u4f5c */\n    pub fn pop(&amp;mut self, is_front: bool) -&gt; Option&lt;T&gt; {\n        // \u82e5\u961f\u5217\u4e3a\u7a7a\uff0c\u76f4\u63a5\u8fd4\u56de None\n        if self.is_empty() { \n            return None \n        };\n        // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n        if is_front {\n            self.front.take().map(|old_front| {\n                match old_front.borrow_mut().next.take() {\n                    Some(new_front) =&gt; {\n                        new_front.borrow_mut().prev.take();\n                        self.front = Some(new_front);   // \u66f4\u65b0\u5934\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.rear.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n            })\n\n        } \n        // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n        else {\n            self.rear.take().map(|old_rear| {\n                match old_rear.borrow_mut().prev.take() {\n                    Some(new_rear) =&gt; {\n                        new_rear.borrow_mut().next.take();\n                        self.rear = Some(new_rear);     // \u66f4\u65b0\u5c3e\u8282\u70b9\n                    }\n                    None =&gt; {\n                        self.front.take();\n                    }\n                }\n                self.que_size -= 1; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n                Rc::try_unwrap(old_rear).ok().unwrap().into_inner().val\n            })\n        }\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    pub fn pop_first(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(true);\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    pub fn pop_last(&amp;mut self) -&gt; Option&lt;T&gt; {\n        return self.pop(false);\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek_first(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    pub fn peek_last(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.rear.as_ref()\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_deque.c<pre><code>/* \u53cc\u5411\u94fe\u8868\u8282\u70b9 */\ntypedef struct DoublyListNode {\n    int val;                     // \u8282\u70b9\u503c\n    struct DoublyListNode *next; // \u540e\u7ee7\u8282\u70b9\n    struct DoublyListNode *prev; // \u524d\u9a71\u8282\u70b9\n} DoublyListNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nDoublyListNode *newDoublyListNode(int num) {\n    DoublyListNode *new = (DoublyListNode *)malloc(sizeof(DoublyListNode));\n    new-&gt;val = num;\n    new-&gt;next = NULL;\n    new-&gt;prev = NULL;\n    return new;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delDoublyListNode(DoublyListNode *node) {\n    free(node);\n}\n\n/* \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    DoublyListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;                  // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n} LinkedListDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListDeque *newLinkedListDeque() {\n    LinkedListDeque *deque = (LinkedListDeque *)malloc(sizeof(LinkedListDeque));\n    deque-&gt;front = NULL;\n    deque-&gt;rear = NULL;\n    deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListdeque(LinkedListDeque *deque) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    for (int i = 0; i &lt; deque-&gt;queSize &amp;&amp; deque-&gt;front != NULL; i++) {\n        DoublyListNode *tmp = deque-&gt;front;\n        deque-&gt;front = deque-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e deque \u7ed3\u6784\u4f53\n    free(deque);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListDeque *deque) {\n    return (size(deque) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListDeque *deque, int num, bool isFront) {\n    DoublyListNode *node = newDoublyListNode(num);\n    // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411node\n    if (empty(deque)) {\n        deque-&gt;front = deque-&gt;rear = node;\n    }\n    // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n    else if (isFront) {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n        deque-&gt;front-&gt;prev = node;\n        node-&gt;next = deque-&gt;front;\n        deque-&gt;front = node; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n    else {\n        // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n        deque-&gt;rear-&gt;next = node;\n        node-&gt;prev = deque-&gt;rear;\n        deque-&gt;rear = node;\n    }\n    deque-&gt;queSize++; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(LinkedListDeque *deque, int num) {\n    push(deque, num, true);\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(LinkedListDeque *deque, int num) {\n    push(deque, num, false);\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;front);\n    return deque-&gt;front-&gt;val;\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(LinkedListDeque *deque) {\n    assert(size(deque) &amp;&amp; deque-&gt;rear);\n    return deque-&gt;rear-&gt;val;\n}\n\n/* \u51fa\u961f */\nint pop(LinkedListDeque *deque, bool isFront) {\n    if (empty(deque))\n        return -1;\n    int val;\n    // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n    if (isFront) {\n        val = peekFirst(deque); // \u6682\u5b58\u5934\u8282\u70b9\u503c\n        DoublyListNode *fNext = deque-&gt;front-&gt;next;\n        if (fNext) {\n            fNext-&gt;prev = NULL;\n            deque-&gt;front-&gt;next = NULL;\n            delDoublyListNode(deque-&gt;front);\n        }\n        deque-&gt;front = fNext; // \u66f4\u65b0\u5934\u8282\u70b9\n    }\n    // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n    else {\n        val = peekLast(deque); // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n        DoublyListNode *rPrev = deque-&gt;rear-&gt;prev;\n        if (rPrev) {\n            rPrev-&gt;next = NULL;\n            deque-&gt;rear-&gt;prev = NULL;\n            delDoublyListNode(deque-&gt;rear);\n        }\n        deque-&gt;rear = rPrev; // \u66f4\u65b0\u5c3e\u8282\u70b9\n    }\n    deque-&gt;queSize--; // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n    return val;\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(LinkedListDeque *deque) {\n    return pop(deque, true);\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(LinkedListDeque *deque) {\n    return pop(deque, false);\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListDeque(LinkedListDeque *deque) {\n    int *arr = malloc(sizeof(int) * deque-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    DoublyListNode *node;\n    for (i = 0, node = deque-&gt;front; i &lt; deque-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, deque-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_deque.zig<pre><code>// \u53cc\u5411\u94fe\u8868\u8282\u70b9\nfn ListNode(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        val: T = undefined,     // \u8282\u70b9\u503c\n        next: ?*Self = null,    // \u540e\u7ee7\u8282\u70b9\u6307\u9488\n        prev: ?*Self = null,    // \u524d\u9a71\u8282\u70b9\u6307\u9488\n\n        // Initialize a list node with specific value\n        pub fn init(self: *Self, x: i32) void {\n            self.val = x;\n            self.next = null;\n            self.prev = null;\n        }\n    };\n}\n\n// \u57fa\u4e8e\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\nfn LinkedListDeque(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*ListNode(T) = null,                    // \u5934\u8282\u70b9 front\n        rear: ?*ListNode(T) = null,                     // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                             // \u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u5165\u961f\u64cd\u4f5c\n        pub fn push(self: *Self, num: T, is_front: bool) !void {\n            var node = try self.mem_allocator.create(ListNode(T));\n            node.init(num);\n            // \u82e5\u94fe\u8868\u4e3a\u7a7a\uff0c\u5219\u4ee4 front \u548c rear \u90fd\u6307\u5411 node\n            if (self.isEmpty()) {\n                self.front = node;\n                self.rear = node;\n            // \u961f\u9996\u5165\u961f\u64cd\u4f5c\n            } else if (is_front) {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5934\u90e8\n                self.front.?.prev = node;\n                node.next = self.front;\n                self.front = node;  // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u5165\u961f\u64cd\u4f5c\n            } else {\n                // \u5c06 node \u6dfb\u52a0\u81f3\u94fe\u8868\u5c3e\u90e8\n                self.rear.?.next = node;\n                node.prev = self.rear;\n                self.rear = node;   // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size += 1;      // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n        } \n\n        // \u961f\u9996\u5165\u961f\n        pub fn pushFirst(self: *Self, num: T) !void {\n            try self.push(num, true);\n        } \n\n        // \u961f\u5c3e\u5165\u961f\n        pub fn pushLast(self: *Self, num: T) !void {\n            try self.push(num, false);\n        } \n\n        // \u51fa\u961f\u64cd\u4f5c\n        pub fn pop(self: *Self, is_front: bool) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            var val: T = undefined;\n            // \u961f\u9996\u51fa\u961f\u64cd\u4f5c\n            if (is_front) {\n                val = self.front.?.val;     // \u6682\u5b58\u5934\u8282\u70b9\u503c\n                // \u5220\u9664\u5934\u8282\u70b9\n                var fNext = self.front.?.next;\n                if (fNext != null) {\n                    fNext.?.prev = null;\n                    self.front.?.next = null;\n                }\n                self.front = fNext;         // \u66f4\u65b0\u5934\u8282\u70b9\n            // \u961f\u5c3e\u51fa\u961f\u64cd\u4f5c\n            } else {\n                val = self.rear.?.val;      // \u6682\u5b58\u5c3e\u8282\u70b9\u503c\n                // \u5220\u9664\u5c3e\u8282\u70b9\n                var rPrev = self.rear.?.prev;\n                if (rPrev != null) {\n                    rPrev.?.next = null;\n                    self.rear.?.prev = null;\n                }\n                self.rear = rPrev;          // \u66f4\u65b0\u5c3e\u8282\u70b9\n            }\n            self.que_size -= 1;              // \u66f4\u65b0\u961f\u5217\u957f\u5ea6\n            return val;\n        } \n\n        // \u961f\u9996\u51fa\u961f\n        pub fn popFirst(self: *Self) T {\n            return self.pop(true);\n        } \n\n        // \u961f\u5c3e\u51fa\u961f\n        pub fn popLast(self: *Self) T {\n            return self.pop(false);\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peekFirst(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\n        pub fn peekLast(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n            return self.rear.?.val;\n        }\n\n        // \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-9 \u6240\u793a\uff0c\u4e0e\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7c7b\u4f3c\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u4f7f\u7528\u73af\u5f62\u6570\u7ec4\u6765\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u3002</p> ArrayDequepushLast()pushFirst()popLast()popFirst() <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 5-9 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u53cc\u5411\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u5728\u961f\u5217\u7684\u5b9e\u73b0\u57fa\u7840\u4e0a\uff0c\u4ec5\u9700\u589e\u52a0\u201c\u961f\u9996\u5165\u961f\u201d\u548c\u201c\u961f\u5c3e\u51fa\u961f\u201d\u7684\u65b9\u6cd5\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_deque.py<pre><code>class ArrayDeque:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217\"\"\"\n\n    def __init__(self, capacity: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * capacity\n        self._front: int = 0\n        self._size: int = 0\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def index(self, i: int) -&gt; int:\n        \"\"\"\u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15\"\"\"\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        # \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + self.capacity()) % self.capacity()\n\n    def push_first(self, num: int):\n        \"\"\"\u961f\u9996\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self._front = self.index(self._front - 1)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self._nums[self._front] = num\n        self._size += 1\n\n    def push_last(self, num: int):\n        \"\"\"\u961f\u5c3e\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        # \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        rear = self.index(self._front + self._size)\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop_first(self) -&gt; int:\n        \"\"\"\u961f\u9996\u51fa\u961f\"\"\"\n        num = self.peek_first()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self._front = self.index(self._front + 1)\n        self._size -= 1\n        return num\n\n    def pop_last(self) -&gt; int:\n        \"\"\"\u961f\u5c3e\u51fa\u961f\"\"\"\n        num = self.peek_last()\n        self._size -= 1\n        return num\n\n    def peek_first(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def peek_last(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u5c3e\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        # \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        last = self.index(self._front + self._size - 1)\n        return self._nums[last]\n\n    def to_array(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370\"\"\"\n        # \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        res = []\n        for i in range(self._size):\n            res.append(self._nums[self.index(self._front + i)])\n        return res\n</code></pre> array_deque.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  private:\n    vector&lt;int&gt; nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;        // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;      // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayDeque(int capacity) {\n        nums.resize(capacity);\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return nums.size();\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    void pushFirst(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    void pushLast(int num) {\n        if (queSize == capacity()) {\n            cout &lt;&lt; \"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peekFirst() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    int peekLast() {\n        if (isEmpty())\n            throw out_of_range(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; res(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n};\n</code></pre> array_deque.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        this.nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private int index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + capacity()) % capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void pushFirst(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void pushLast(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int popFirst() {\n        int num = peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int popLast() {\n        int num = peekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peekFirst() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int peekLast() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayDeque(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    int Index(int i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + Capacity()) % Capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    public void PushFirst(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = Index(front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num;\n        queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    public void PushLast(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        int rear = Index(front + queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    public int PopFirst() {\n        int num = PeekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = Index(front + 1);\n        queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    public int PopLast() {\n        int num = PeekLast();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int PeekFirst() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        return nums[front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    public int PeekLast() {\n        if (IsEmpty()) {\n            throw new InvalidOperationException();\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        int last = Index(front + queSize - 1);\n        return nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[Index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntype arrayDeque struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayDeque(queCapacity int) *arrayDeque {\n    return &amp;arrayDeque{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayDeque) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayDeque) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nfunc (q *arrayDeque) index(i int) int {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + q.queCapacity) % q.queCapacity\n}\n\n/* \u961f\u9996\u5165\u961f */\nfunc (q *arrayDeque) pushFirst(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    q.front = q.index(q.front - 1)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n    q.nums[q.front] = num\n    q.queSize++\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nfunc (q *arrayDeque) pushLast(num int) {\n    if q.queSize == q.queCapacity {\n        fmt.Println(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n        return\n    }\n    // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    rear := q.index(q.front + q.queSize)\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u961f\u9996\u51fa\u961f */\nfunc (q *arrayDeque) popFirst() any {\n    num := q.peekFirst()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    q.front = q.index(q.front + 1)\n    q.queSize--\n    return num\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nfunc (q *arrayDeque) popLast() any {\n    num := q.peekLast()\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayDeque) peekFirst() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nfunc (q *arrayDeque) peekLast() any {\n    if q.isEmpty() {\n        return nil\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    last := q.index(q.front + q.queSize - 1)\n    return q.nums[last]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayDeque) toSlice() []int {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    res := make([]int, q.queSize)\n    for i, j := 0, q.front; i &lt; q.queSize; i++ {\n        res[i] = q.nums[q.index(j)]\n        j++\n    }\n    return res\n}\n</code></pre> array_deque.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front: Int // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize: Int // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(capacity: Int) {\n        nums = Array(repeating: 0, count: capacity)\n        front = 0\n        queSize = 0\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        queSize\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    private func index(i: Int) -&gt; Int {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        (i + capacity()) % capacity()\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    func pushFirst(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        front = index(i: front - 1)\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        nums[front] = num\n        queSize += 1\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    func pushLast(num: Int) {\n        if size() == capacity() {\n            print(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = index(i: front + size())\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize += 1\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    func popFirst() -&gt; Int {\n        let num = peekFirst()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        front = index(i: front + 1)\n        queSize -= 1\n        return num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    func popLast() -&gt; Int {\n        let num = peekLast()\n        queSize -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peekFirst() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    func peekLast() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\")\n        }\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = index(i: front + size() - 1)\n        return nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        var res = Array(repeating: 0, count: size())\n        for (i, j) in sequence(first: (0, front), next: { $0 &lt; self.size() - 1 ? ($0 + 1, $1 + 1) : nil }) {\n            res[i] = nums[index(i: j)]\n        }\n        return res\n    }\n}\n</code></pre> array_deque.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n        this.#front = 0;\n        this.#queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i) {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.#front = this.index(this.#front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.#nums[this.#front] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num) {\n        if (this.#queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear = this.index(this.#front + this.#queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst() {\n        const num = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.#front = this.index(this.#front + 1);\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast() {\n        const num = this.peekLast();\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast() {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.#front + this.#queSize - 1);\n        return this.#nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res = [];\n        for (let i = 0, j = this.#front; i &lt; this.#queSize; i++, j++) {\n            res[i] = this.#nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = 0;\n        this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    index(i: number): number {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return (i + this.capacity()) % this.capacity();\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pushFirst(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        this.front = this.index(this.front - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        this.nums[this.front] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pushLast(num: number): void {\n        if (this.queSize === this.capacity()) {\n            console.log('\u53cc\u5411\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        const rear: number = this.index(this.front + this.queSize);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    popFirst(): number {\n        const num: number = this.peekFirst();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        this.front = this.index(this.front + 1);\n        this.queSize--;\n        return num;\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    popLast(): number {\n        const num: number = this.peekLast();\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peekFirst(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        return this.nums[this.front];\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    peekLast(): number {\n        if (this.isEmpty()) throw new Error('The Deque Is Empty.');\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        const last = this.index(this.front + this.queSize - 1);\n        return this.nums[last];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const res: number[] = [];\n        for (let i = 0, j = this.front; i &lt; this.queSize; i++, j++) {\n            res[i] = this.nums[this.index(j)];\n        }\n        return res;\n    }\n}\n</code></pre> array_deque.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nclass ArrayDeque {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayDeque(int capacity) {\n    this._nums = List.filled(capacity, 0);\n    this._front = this._queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n  int capacity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n  int index(int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return (i + capacity()) % capacity();\n  }\n\n  /* \u961f\u9996\u5165\u961f */\n  void pushFirst(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 _front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n    _front = index(_front - 1);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u9996\n    _nums[_front] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u5c3e\u5165\u961f */\n  void pushLast(int _num) {\n    if (_queSize == capacity()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = index(_front + _queSize);\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u961f\u9996\u51fa\u961f */\n  int popFirst() {\n    int _num = peekFirst();\n    // \u961f\u9996\u6307\u9488\u5411\u53f3\u79fb\u52a8\u4e00\u4f4d\n    _front = index(_front + 1);\n    _queSize--;\n    return _num;\n  }\n\n  /* \u961f\u5c3e\u51fa\u961f */\n  int popLast() {\n    int _num = peekLast();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peekFirst() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n  int peekLast() {\n    if (isEmpty()) {\n      throw Exception(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\");\n    }\n    // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n    int last = index(_front + _queSize - 1);\n    return _nums[last];\n  }\n\n  /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[index(j)];\n    }\n    return res;\n  }\n}\n</code></pre> array_deque.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\nstruct ArrayDeque {\n    nums: Vec&lt;i32&gt;,     // \u7528\u4e8e\u5b58\u50a8\u53cc\u5411\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: usize,       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: usize,    // \u53cc\u5411\u961f\u5217\u957f\u5ea6\n}\n\nimpl ArrayDeque {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    pub fn new(capacity: usize) -&gt; Self {\n        Self {\n            nums: vec![0; capacity],\n            front: 0,\n            que_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\n    pub fn capacity(&amp;self) -&gt; usize {\n        self.nums.len()\n    }\n\n    /* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\n    fn index(&amp;self, i: i32) -&gt; usize {\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\uff0c\u56de\u5230\u5934\u90e8\n        // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n        return ((i + self.capacity() as i32) % self.capacity() as i32) as usize;\n    }\n\n    /* \u961f\u9996\u5165\u961f */\n    pub fn push_first(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return\n        }\n        // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\u56de\u5230\u5c3e\u90e8\n        self.front = self.index(self.front as i32 - 1);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u9996\n        self.nums[self.front] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u5c3e\u5165\u961f */\n    pub fn push_last(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\");\n            return\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        let rear = self.index(self.front as i32 + self.que_size as i32);\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear] = num;\n        self.que_size += 1;\n    }\n\n    /* \u961f\u9996\u51fa\u961f */\n    fn pop_first(&amp;mut self) -&gt; i32 {\n        let num = self.peek_first();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n        self.front = self.index(self.front as i32 + 1);\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u961f\u5c3e\u51fa\u961f */\n    fn pop_last(&amp;mut self) -&gt; i32 {\n        let num = self.peek_last();\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek_first(&amp;self) -&gt; i32 {\n        if self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\n        self.nums[self.front]\n    }\n\n    /* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\n    fn peek_last(&amp;self) -&gt; i32 {\n        if self.is_empty() { panic!(\"\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\") };\n        // \u8ba1\u7b97\u5c3e\u5143\u7d20\u7d22\u5f15\n        let last = self.index(self.front as i32 + self.que_size as i32 - 1);\n        self.nums[last]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4\u7528\u4e8e\u6253\u5370 */\n    fn to_array(&amp;self) -&gt; Vec&lt;i32&gt; {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        let mut res = vec![0; self.que_size];\n        let mut j = self.front;\n        for i in 0..self.que_size {\n            res[i] = self.nums[self.index(j as i32)];\n            j += 1;\n        }\n        res\n    }\n}\n</code></pre> array_deque.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u53cc\u5411\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayDeque;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayDeque *newArrayDeque(int capacity) {\n    ArrayDeque *deque = (ArrayDeque *)malloc(sizeof(ArrayDeque));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    deque-&gt;queCapacity = capacity;\n    deque-&gt;nums = (int *)malloc(sizeof(int) * deque-&gt;queCapacity);\n    deque-&gt;front = deque-&gt;queSize = 0;\n    return deque;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayDeque(ArrayDeque *deque) {\n    free(deque-&gt;nums);\n    free(deque);\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayDeque *deque) {\n    return deque-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u53cc\u5411\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayDeque *deque) {\n    return deque-&gt;queSize;\n}\n\n/* \u5224\u65ad\u53cc\u5411\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayDeque *deque) {\n    return deque-&gt;queSize == 0;\n}\n\n/* \u8ba1\u7b97\u73af\u5f62\u6570\u7ec4\u7d22\u5f15 */\nint dequeIndex(ArrayDeque *deque, int i) {\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\u5b9e\u73b0\u6570\u7ec4\u9996\u5c3e\u76f8\u8fde\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u56de\u5230\u5934\u90e8\n    // \u5f53 i \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u540e\uff0c\u56de\u5230\u5c3e\u90e8\n    return ((i + capacity(deque)) % capacity(deque));\n}\n\n/* \u961f\u9996\u5165\u961f */\nvoid pushFirst(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u961f\u9996\u6307\u9488\u5411\u5de6\u79fb\u52a8\u4e00\u4f4d\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 front \u8d8a\u8fc7\u6570\u7ec4\u5934\u90e8\u56de\u5230\u5c3e\u90e8\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front - 1);\n    // \u5c06 num \u6dfb\u52a0\u5230\u961f\u9996\n    deque-&gt;nums[deque-&gt;front] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u961f\u5c3e\u5165\u961f */\nvoid pushLast(ArrayDeque *deque, int num) {\n    if (deque-&gt;queSize == capacity(deque)) {\n        printf(\"\u53cc\u5411\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    int rear = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize);\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    deque-&gt;nums[rear] = num;\n    deque-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peekFirst(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    return deque-&gt;nums[deque-&gt;front];\n}\n\n/* \u8bbf\u95ee\u961f\u5c3e\u5143\u7d20 */\nint peekLast(ArrayDeque *deque) {\n    // \u8bbf\u95ee\u5f02\u5e38\uff1a\u53cc\u5411\u961f\u5217\u4e3a\u7a7a\n    assert(empty(deque) == 0);\n    int last = dequeIndex(deque, deque-&gt;front + deque-&gt;queSize - 1);\n    return deque-&gt;nums[last];\n}\n\n/* \u961f\u9996\u51fa\u961f */\nint popFirst(ArrayDeque *deque) {\n    int num = peekFirst(deque);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\n    deque-&gt;front = dequeIndex(deque, deque-&gt;front + 1);\n    deque-&gt;queSize--;\n    return num;\n}\n\n/* \u961f\u5c3e\u51fa\u961f */\nint popLast(ArrayDeque *deque) {\n    int num = peekLast(deque);\n    deque-&gt;queSize--;\n    return num;\n}\n</code></pre> array_deque.zig<pre><code>[class]{ArrayDeque}-[func]{}\n</code></pre>"},{"location":"chapter_stack_and_queue/deque/#533","title":"5.3.3 \u00a0 \u53cc\u5411\u961f\u5217\u5e94\u7528","text":"<p>\u53cc\u5411\u961f\u5217\u517c\u5177\u6808\u4e0e\u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u5b9e\u73b0\u8fd9\u4e24\u8005\u7684\u6240\u6709\u5e94\u7528\u573a\u666f\uff0c\u540c\u65f6\u63d0\u4f9b\u66f4\u9ad8\u7684\u81ea\u7531\u5ea6\u3002</p> <p>\u6211\u4eec\u77e5\u9053\uff0c\u8f6f\u4ef6\u7684\u201c\u64a4\u9500\u201d\u529f\u80fd\u901a\u5e38\u4f7f\u7528\u6808\u6765\u5b9e\u73b0\uff1a\u7cfb\u7edf\u5c06\u6bcf\u6b21\u66f4\u6539\u64cd\u4f5c <code>push</code> \u5230\u6808\u4e2d\uff0c\u7136\u540e\u901a\u8fc7 <code>pop</code> \u5b9e\u73b0\u64a4\u9500\u3002\u7136\u800c\uff0c\u8003\u8651\u5230\u7cfb\u7edf\u8d44\u6e90\u7684\u9650\u5236\uff0c\u8f6f\u4ef6\u901a\u5e38\u4f1a\u9650\u5236\u64a4\u9500\u7684\u6b65\u6570\uff08\u4f8b\u5982\u4ec5\u5141\u8bb8\u4fdd\u5b58 \\(50\\) \u6b65\uff09\u3002\u5f53\u6808\u7684\u957f\u5ea6\u8d85\u8fc7 \\(50\\) \u65f6\uff0c\u8f6f\u4ef6\u9700\u8981\u5728\u6808\u5e95\uff08\u961f\u9996\uff09\u6267\u884c\u5220\u9664\u64cd\u4f5c\u3002\u4f46\u6808\u65e0\u6cd5\u5b9e\u73b0\u8be5\u529f\u80fd\uff0c\u6b64\u65f6\u5c31\u9700\u8981\u4f7f\u7528\u53cc\u5411\u961f\u5217\u6765\u66ff\u4ee3\u6808\u3002\u8bf7\u6ce8\u610f\uff0c\u201c\u64a4\u9500\u201d\u7684\u6838\u5fc3\u903b\u8f91\u4ecd\u7136\u9075\u5faa\u6808\u7684\u5148\u5165\u540e\u51fa\u539f\u5219\uff0c\u53ea\u662f\u53cc\u5411\u961f\u5217\u80fd\u591f\u66f4\u52a0\u7075\u6d3b\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u903b\u8f91\u3002</p>"},{"location":"chapter_stack_and_queue/queue/","title":"5.2 \u00a0 \u961f\u5217","text":"<p>\u300c\u961f\u5217 queue\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u89c4\u5219\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u961f\u5217\u6a21\u62df\u4e86\u6392\u961f\u73b0\u8c61\uff0c\u5373\u65b0\u6765\u7684\u4eba\u4e0d\u65ad\u52a0\u5165\u961f\u5217\u5c3e\u90e8\uff0c\u800c\u4f4d\u4e8e\u961f\u5217\u5934\u90e8\u7684\u4eba\u9010\u4e2a\u79bb\u5f00\u3002</p> <p>\u5982\u56fe 5-4 \u6240\u793a\uff0c\u6211\u4eec\u5c06\u961f\u5217\u5934\u90e8\u79f0\u4e3a\u201c\u961f\u9996\u201d\uff0c\u5c3e\u90e8\u79f0\u4e3a\u201c\u961f\u5c3e\u201d\uff0c\u5c06\u628a\u5143\u7d20\u52a0\u5165\u961f\u5c3e\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u5165\u961f\u201d\uff0c\u5220\u9664\u961f\u9996\u5143\u7d20\u7684\u64cd\u4f5c\u79f0\u4e3a\u201c\u51fa\u961f\u201d\u3002</p> <p></p> <p> \u56fe 5-4 \u00a0 \u961f\u5217\u7684\u5148\u5165\u5148\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/queue/#521","title":"5.2.1 \u00a0 \u961f\u5217\u5e38\u7528\u64cd\u4f5c","text":"<p>\u961f\u5217\u7684\u5e38\u89c1\u64cd\u4f5c\u5982\u8868 5-2 \u6240\u793a\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0d\u540c\u7f16\u7a0b\u8bed\u8a00\u7684\u65b9\u6cd5\u540d\u79f0\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u6211\u4eec\u5728\u6b64\u91c7\u7528\u4e0e\u6808\u76f8\u540c\u7684\u65b9\u6cd5\u547d\u540d\u3002</p> <p> \u8868 5-2 \u00a0 \u961f\u5217\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5\u540d \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u961f\uff0c\u5373\u5c06\u5143\u7d20\u6dfb\u52a0\u81f3\u961f\u5c3e \\(O(1)\\) pop() \u961f\u9996\u5143\u7d20\u51fa\u961f \\(O(1)\\) peek() \u8bbf\u95ee\u961f\u9996\u5143\u7d20 \\(O(1)\\) <p>\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u4e2d\u73b0\u6210\u7684\u961f\u5217\u7c7b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig queue.py<pre><code>from collections import deque\n\n# \u521d\u59cb\u5316\u961f\u5217\n# \u5728 Python \u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u5c06\u53cc\u5411\u961f\u5217\u7c7b deque \u5f53\u4f5c\u961f\u5217\u4f7f\u7528\n# \u867d\u7136 queue.Queue() \u662f\u7eaf\u6b63\u7684\u961f\u5217\u7c7b\uff0c\u4f46\u4e0d\u592a\u597d\u7528\uff0c\u56e0\u6b64\u4e0d\u63a8\u8350\nque: deque[int] = deque()\n\n# \u5143\u7d20\u5165\u961f\nque.append(1)\nque.append(3)\nque.append(2)\nque.append(5)\nque.append(4)\n\n# \u8bbf\u95ee\u961f\u9996\u5143\u7d20\nfront: int = que[0];\n\n# \u5143\u7d20\u51fa\u961f\npop: int = que.popleft()\n\n# \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\nsize: int = len(que)\n\n# \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(que) == 0\n</code></pre> queue.cpp<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nqueue&lt;int&gt; queue;\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint front = queue.front();\n\n/* \u5143\u7d20\u51fa\u961f */\nqueue.pop();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty = queue.empty();\n</code></pre> queue.java<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;Integer&gt; queue = new LinkedList&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.offer(1);\nqueue.offer(3);\nqueue.offer(2);\nqueue.offer(5);\nqueue.offer(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.poll();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.size();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = queue.isEmpty();\n</code></pre> queue.cs<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\nQueue&lt;int&gt; queue = new();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.Enqueue(1);\nqueue.Enqueue(3);\nqueue.Enqueue(2);\nqueue.Enqueue(5);\nqueue.Enqueue(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.Peek();\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.Dequeue();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.Count;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.Count == 0;\n</code></pre> queue_test.go<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Go \u4e2d\uff0c\u5c06 list \u4f5c\u4e3a\u961f\u5217\u6765\u4f7f\u7528\nqueue := list.New()\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.PushBack(1)\nqueue.PushBack(3)\nqueue.PushBack(2)\nqueue.PushBack(5)\nqueue.PushBack(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\npeek := queue.Front()\n\n/* \u5143\u7d20\u51fa\u961f */\npop := queue.Front()\nqueue.Remove(pop)\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nsize := queue.Len()\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nisEmpty := queue.Len() == 0\n</code></pre> queue.swift<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nvar queue: [Int] = []\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.append(1)\nqueue.append(3)\nqueue.append(2)\nqueue.append(5)\nqueue.append(4)\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nlet peek = queue.first!\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u7531\u4e8e\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 removeFirst \u7684\u590d\u6742\u5ea6\u4e3a O(n)\nlet pool = queue.removeFirst()\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = queue.count\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = queue.isEmpty\n</code></pre> queue.js<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// JavaScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528\nconst queue = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.ts<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// TypeScript \u6ca1\u6709\u5185\u7f6e\u7684\u961f\u5217\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u961f\u5217\u6765\u4f7f\u7528 \nconst queue: number[] = [];\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.push(1);\nqueue.push(3);\nqueue.push(2);\nqueue.push(5);\nqueue.push(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nconst peek = queue[0];\n\n/* \u5143\u7d20\u51fa\u961f */\n// \u5e95\u5c42\u662f\u6570\u7ec4\uff0c\u56e0\u6b64 shift() \u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n)\nconst pop = queue.shift();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nconst size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nconst empty = queue.length === 0;\n</code></pre> queue.dart<pre><code>/* \u521d\u59cb\u5316\u961f\u5217 */\n// \u5728 Dart \u4e2d\uff0c\u961f\u5217\u7c7b Qeque \u662f\u53cc\u5411\u961f\u5217\uff0c\u4e5f\u53ef\u4f5c\u4e3a\u961f\u5217\u4f7f\u7528\nQueue&lt;int&gt; queue = Queue();\n\n/* \u5143\u7d20\u5165\u961f */\nqueue.add(1);\nqueue.add(3);\nqueue.add(2);\nqueue.add(5);\nqueue.add(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek = queue.first;\n\n/* \u5143\u7d20\u51fa\u961f */\nint pop = queue.removeFirst();\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size = queue.length;\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = queue.isEmpty;\n</code></pre> queue.rs<pre><code>/* \u521d\u59cb\u5316\u53cc\u5411\u961f\u5217 */\n// \u5728 Rust \u4e2d\u4f7f\u7528\u53cc\u5411\u961f\u5217\u4f5c\u4e3a\u666e\u901a\u961f\u5217\u6765\u4f7f\u7528\nlet mut deque: VecDeque&lt;u32&gt; = VecDeque::new();\n\n/* \u5143\u7d20\u5165\u961f */\ndeque.push_back(1);\ndeque.push_back(3);\ndeque.push_back(2);\ndeque.push_back(5);\ndeque.push_back(4);\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nif let Some(front) = deque.front() {\n}\n\n/* \u5143\u7d20\u51fa\u961f */\nif let Some(pop) = deque.pop_front() {\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nlet size = deque.len();\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = deque.is_empty();\n</code></pre> queue.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u961f\u5217\n</code></pre> queue.zig<pre><code>\n</code></pre>"},{"location":"chapter_stack_and_queue/queue/#522","title":"5.2.2 \u00a0 \u961f\u5217\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u5b9e\u73b0\u961f\u5217\uff0c\u6211\u4eec\u9700\u8981\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u4ee5\u5728\u4e00\u7aef\u6dfb\u52a0\u5143\u7d20\uff0c\u5e76\u5728\u53e6\u4e00\u7aef\u5220\u9664\u5143\u7d20\u3002\u94fe\u8868\u548c\u6570\u7ec4\u90fd\u7b26\u5408\u8981\u6c42\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u5982\u56fe 5-5 \u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u201c\u5934\u8282\u70b9\u201d\u548c\u201c\u5c3e\u8282\u70b9\u201d\u5206\u522b\u89c6\u4e3a\u201c\u961f\u9996\u201d\u548c\u201c\u961f\u5c3e\u201d\uff0c\u89c4\u5b9a\u961f\u5c3e\u4ec5\u53ef\u6dfb\u52a0\u8282\u70b9\uff0c\u961f\u9996\u4ec5\u53ef\u5220\u9664\u8282\u70b9\u3002</p> LinkedListQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-5 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u7528\u94fe\u8868\u5b9e\u73b0\u961f\u5217\u7684\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_queue.py<pre><code>class LinkedListQueue:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._front: ListNode | None = None  # \u5934\u8282\u70b9 front\n        self._rear: ListNode | None = None  # \u5c3e\u8282\u70b9 rear\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._front\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        # \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        node = ListNode(num)\n        # \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if self._front is None:\n            self._front = node\n            self._rear = node\n        # \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else:\n            self._rear.next = node\n            self._rear = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num = self.peek()\n        # \u5220\u9664\u5934\u8282\u70b9\n        self._front = self._front.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._front.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        queue = []\n        temp = self._front\n        while temp:\n            queue.append(temp.val)\n            temp = temp.next\n        return queue\n</code></pre> linkedlist_queue.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  private:\n    ListNode *front, *rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    int queSize;\n\n  public:\n    LinkedListQueue() {\n        front = nullptr;\n        rear = nullptr;\n        queSize = 0;\n    }\n\n    ~LinkedListQueue() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(front);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode *node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == nullptr) {\n            front = node;\n            rear = node;\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear-&gt;next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    void pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        ListNode *tmp = front;\n        front = front-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        queSize--;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (size() == 0)\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return front-&gt;val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = front;\n        vector&lt;int&gt; res(size());\n        for (int i = 0; i &lt; res.size(); i++) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_queue.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private ListNode front, rear; // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear\n    private int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = front;\n        int[] res = new int[size()];\n        for (int i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    ListNode? front, rear;  // \u5934\u8282\u70b9 front \uff0c\u5c3e\u8282\u70b9 rear \n    int queSize = 0;\n\n    public LinkedListQueue() {\n        front = null;\n        rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        ListNode node = new(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (front == null) {\n            front = node;\n            rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else if (rear != null) {\n            rear.next = node;\n            rear = node;\n        }\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next;\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (front == null)\n            return [];\n\n        ListNode? node = front;\n        int[] res = new int[Size()];\n        for (int i = 0; i &lt; res.Length; i++) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntype linkedListQueue struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u961f\u5217\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newLinkedListQueue() *linkedListQueue {\n    return &amp;linkedListQueue{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u961f */\nfunc (s *linkedListQueue) push(value any) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u961f */\nfunc (s *linkedListQueue) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (s *linkedListQueue) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Front()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (s *linkedListQueue) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListQueue) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListQueue) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_queue.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private var front: ListNode? // \u5934\u8282\u70b9\n    private var rear: ListNode? // \u5c3e\u8282\u70b9\n    private var _size = 0\n\n    init() {}\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let node = ListNode(x: num)\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if front == nil {\n            front = node\n            rear = node\n        }\n        // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        else {\n            rear?.next = node\n            rear = node\n        }\n        _size += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u5220\u9664\u5934\u8282\u70b9\n        front = front?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return front!.val\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = front\n        var res = Array(repeating: 0, count: size())\n        for i in res.indices {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_queue.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    #front; // \u5934\u8282\u70b9 #front\n    #rear; // \u5c3e\u8282\u70b9 #rear\n    #queSize = 0;\n\n    constructor() {\n        this.#front = null;\n        this.#rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.#front) {\n            this.#front = node;\n            this.#rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.#rear.next = node;\n            this.#rear = node;\n        }\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.#front = this.#front.next;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#front.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#front;\n        const res = new Array(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n    private front: ListNode | null; // \u5934\u8282\u70b9 front\n    private rear: ListNode | null; // \u5c3e\u8282\u70b9 rear\n    private queSize: number = 0;\n\n    constructor() {\n        this.front = null;\n        this.rear = null;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        const node = new ListNode(num);\n        // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n        if (!this.front) {\n            this.front = node;\n            this.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n        } else {\n            this.rear!.next = node;\n            this.rear = node;\n        }\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        if (!this.front) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        // \u5220\u9664\u5934\u8282\u70b9\n        this.front = this.front.next;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.size === 0) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.front!.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.front;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = 0; i &lt; res.length; i++) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_queue.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\nclass LinkedListQueue {\n  ListNode? _front; // \u5934\u8282\u70b9 _front\n  ListNode? _rear; // \u5c3e\u8282\u70b9 _rear\n  int _queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n  LinkedListQueue() {\n    _front = null;\n    _rear = null;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 _num\n    final node = ListNode(_num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (_front == null) {\n      _front = node;\n      _rear = node;\n    } else {\n      // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n      _rear!.next = node;\n      _rear = node;\n    }\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    final int _num = peek();\n    // \u5220\u9664\u5934\u8282\u70b9\n    _front = _front!.next;\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (_queSize == 0) {\n      throw Exception('\u961f\u5217\u4e3a\u7a7a');\n    }\n    return _front!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() {\n    ListNode? node = _front;\n    final List&lt;int&gt; queue = [];\n    while (node != null) {\n      queue.add(node.val);\n      node = node.next;\n    }\n    return queue;\n  }\n}\n</code></pre> linkedlist_queue.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\n#[allow(dead_code)]\npub struct LinkedListQueue&lt;T&gt; {\n    front: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,    // \u5934\u8282\u70b9 front\n    rear: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,     // \u5c3e\u8282\u70b9 rear \n    que_size: usize,                            // \u961f\u5217\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListQueue&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            front: None,\n            rear: None,\n            que_size: 0, \n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.que_size;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u961f */\n    pub fn push(&amp;mut self, num: T) {\n        // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n        let new_rear = ListNode::new(num);\n        match self.rear.take() {\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            Some(old_rear) =&gt; {\n                old_rear.borrow_mut().next = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            None =&gt; {\n                self.front = Some(new_rear.clone());\n                self.rear = Some(new_rear);\n            }\n        }\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.front.take().map(|old_front| {\n            match old_front.borrow_mut().next.take() {\n                Some(new_front) =&gt; {\n                    self.front = Some(new_front);\n                }\n                None =&gt; {\n                    self.rear.take();\n                }\n            }\n            self.que_size -= 1;\n            Rc::try_unwrap(old_front).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.front.as_ref()\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.insert(0, node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_queue.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    ListNode *front, *rear;\n    int queSize;\n} LinkedListQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListQueue *newLinkedListQueue() {\n    LinkedListQueue *queue = (LinkedListQueue *)malloc(sizeof(LinkedListQueue));\n    queue-&gt;front = NULL;\n    queue-&gt;rear = NULL;\n    queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListQueue(LinkedListQueue *queue) {\n    // \u91ca\u653e\u6240\u6709\u8282\u70b9\n    for (int i = 0; i &lt; queue-&gt;queSize &amp;&amp; queue-&gt;front != NULL; i++) {\n        ListNode *tmp = queue-&gt;front;\n        queue-&gt;front = queue-&gt;front-&gt;next;\n        free(tmp);\n    }\n    // \u91ca\u653e queue \u7ed3\u6784\u4f53\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(LinkedListQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(LinkedListQueue *queue) {\n    return (size(queue) == 0);\n}\n\n/* \u5165\u961f */\nvoid push(LinkedListQueue *queue, int num) {\n    // \u5c3e\u8282\u70b9\u5904\u6dfb\u52a0 node\n    ListNode *node = newListNode(num);\n    // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n    if (queue-&gt;front == NULL) {\n        queue-&gt;front = node;\n        queue-&gt;rear = node;\n    }\n    // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n    else {\n        queue-&gt;rear-&gt;next = node;\n        queue-&gt;rear = node;\n    }\n    queue-&gt;queSize++;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(LinkedListQueue *queue) {\n    assert(size(queue) &amp;&amp; queue-&gt;front);\n    return queue-&gt;front-&gt;val;\n}\n\n/* \u51fa\u961f */\nvoid pop(LinkedListQueue *queue) {\n    int num = peek(queue);\n    ListNode *tmp = queue-&gt;front;\n    queue-&gt;front = queue-&gt;front-&gt;next;\n    free(tmp);\n    queue-&gt;queSize--;\n}\n\n/* \u6253\u5370\u961f\u5217 */\nvoid printLinkedListQueue(LinkedListQueue *queue) {\n    int *arr = malloc(sizeof(int) * queue-&gt;queSize);\n    // \u62f7\u8d1d\u94fe\u8868\u4e2d\u7684\u6570\u636e\u5230\u6570\u7ec4\n    int i;\n    ListNode *node;\n    for (i = 0, node = queue-&gt;front; i &lt; queue-&gt;queSize; i++) {\n        arr[i] = node-&gt;val;\n        node = node-&gt;next;\n    }\n    printArray(arr, queue-&gt;queSize);\n    free(arr);\n}\n</code></pre> linkedlist_queue.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u961f\u5217\nfn LinkedListQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        front: ?*inc.ListNode(T) = null,                // \u5934\u8282\u70b9 front\n        rear: ?*inc.ListNode(T) = null,                 // \u5c3e\u8282\u70b9 rear\n        que_size: usize = 0,                            // \u961f\u5217\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u961f\u5217\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.front = null;\n            self.rear = null;\n            self.que_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.que_size;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.front.?.val;\n        }  \n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            // \u5982\u679c\u961f\u5217\u4e3a\u7a7a\uff0c\u5219\u4ee4\u5934\u3001\u5c3e\u8282\u70b9\u90fd\u6307\u5411\u8be5\u8282\u70b9\n            if (self.front == null) {\n                self.front = node;\n                self.rear = node;\n            // \u5982\u679c\u961f\u5217\u4e0d\u4e3a\u7a7a\uff0c\u5219\u5c06\u8be5\u8282\u70b9\u6dfb\u52a0\u5230\u5c3e\u8282\u70b9\u540e\n            } else {\n                self.rear.?.next = node;\n                self.rear = node;\n            }\n            self.que_size += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u5220\u9664\u5934\u8282\u70b9\n            self.front = self.front.?.next;\n            self.que_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u94fe\u8868\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.front;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[i] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/queue/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u5728\u6570\u7ec4\u4e2d\u5220\u9664\u9996\u5143\u7d20\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(n)\\) \uff0c\u8fd9\u4f1a\u5bfc\u81f4\u51fa\u961f\u64cd\u4f5c\u6548\u7387\u8f83\u4f4e\u3002\u7136\u800c\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u4ee5\u4e0b\u5de7\u5999\u65b9\u6cd5\u6765\u907f\u514d\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u53d8\u91cf <code>front</code> \u6307\u5411\u961f\u9996\u5143\u7d20\u7684\u7d22\u5f15\uff0c\u5e76\u7ef4\u62a4\u4e00\u4e2a\u53d8\u91cf <code>size</code> \u7528\u4e8e\u8bb0\u5f55\u961f\u5217\u957f\u5ea6\u3002\u5b9a\u4e49 <code>rear = front + size</code> \uff0c\u8fd9\u4e2a\u516c\u5f0f\u8ba1\u7b97\u51fa\u7684 <code>rear</code> \u6307\u5411\u961f\u5c3e\u5143\u7d20\u4e4b\u540e\u7684\u4e0b\u4e00\u4e2a\u4f4d\u7f6e\u3002</p> <p>\u57fa\u4e8e\u6b64\u8bbe\u8ba1\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u5143\u7d20\u7684\u6709\u6548\u533a\u95f4\u4e3a <code>[front, rear - 1]</code>\uff0c\u5404\u79cd\u64cd\u4f5c\u7684\u5b9e\u73b0\u65b9\u6cd5\u5982\u56fe 5-6 \u6240\u793a\u3002</p> <ul> <li>\u5165\u961f\u64cd\u4f5c\uff1a\u5c06\u8f93\u5165\u5143\u7d20\u8d4b\u503c\u7ed9 <code>rear</code> \u7d22\u5f15\u5904\uff0c\u5e76\u5c06 <code>size</code> \u589e\u52a0 1 \u3002</li> <li>\u51fa\u961f\u64cd\u4f5c\uff1a\u53ea\u9700\u5c06 <code>front</code> \u589e\u52a0 1 \uff0c\u5e76\u5c06 <code>size</code> \u51cf\u5c11 1 \u3002</li> </ul> <p>\u53ef\u4ee5\u770b\u5230\uff0c\u5165\u961f\u548c\u51fa\u961f\u64cd\u4f5c\u90fd\u53ea\u9700\u8fdb\u884c\u4e00\u6b21\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(1)\\) \u3002</p> ArrayQueuepush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-6 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u961f\u5217\u7684\u5165\u961f\u51fa\u961f\u64cd\u4f5c </p> <p>\u4f60\u53ef\u80fd\u4f1a\u53d1\u73b0\u4e00\u4e2a\u95ee\u9898\uff1a\u5728\u4e0d\u65ad\u8fdb\u884c\u5165\u961f\u548c\u51fa\u961f\u7684\u8fc7\u7a0b\u4e2d\uff0c<code>front</code> \u548c <code>rear</code> \u90fd\u5728\u5411\u53f3\u79fb\u52a8\uff0c\u5f53\u5b83\u4eec\u5230\u8fbe\u6570\u7ec4\u5c3e\u90e8\u65f6\u5c31\u65e0\u6cd5\u7ee7\u7eed\u79fb\u52a8\u4e86\u3002\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u89c6\u4e3a\u9996\u5c3e\u76f8\u63a5\u7684\u201c\u73af\u5f62\u6570\u7ec4\u201d\u3002</p> <p>\u5bf9\u4e8e\u73af\u5f62\u6570\u7ec4\uff0c\u6211\u4eec\u9700\u8981\u8ba9 <code>front</code> \u6216 <code>rear</code> \u5728\u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u65f6\uff0c\u76f4\u63a5\u56de\u5230\u6570\u7ec4\u5934\u90e8\u7ee7\u7eed\u904d\u5386\u3002\u8fd9\u79cd\u5468\u671f\u6027\u89c4\u5f8b\u53ef\u4ee5\u901a\u8fc7\u201c\u53d6\u4f59\u64cd\u4f5c\u201d\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_queue.py<pre><code>class ArrayQueue:\n    \"\"\"\u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\"\"\"\n\n    def __init__(self, size: int):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._nums: list[int] = [0] * size  # \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n        self._front: int = 0  # \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        self._size: int = 0  # \u961f\u5217\u957f\u5ea6\n\n    def capacity(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\"\"\"\n        return len(self._nums)\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._size == 0\n\n    def push(self, num: int):\n        \"\"\"\u5165\u961f\"\"\"\n        if self._size == self.capacity():\n            raise IndexError(\"\u961f\u5217\u5df2\u6ee1\")\n        # \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        # \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        rear: int = (self._front + self._size) % self.capacity()\n        # \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self._nums[rear] = num\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u961f\"\"\"\n        num: int = self.peek()\n        # \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self._front = (self._front + 1) % self.capacity()\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u961f\u9996\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u961f\u5217\u4e3a\u7a7a\")\n        return self._nums[self._front]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        res = [0] * self.size()\n        j: int = self._front\n        for i in range(self.size()):\n            res[i] = self._nums[(j % self.capacity())]\n            j += 1\n        return res\n</code></pre> array_queue.cpp<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  private:\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u961f\u5217\u957f\u5ea6\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n\n  public:\n    ArrayQueue(int capacity) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = new int[capacity];\n        queCapacity = capacity;\n        front = queSize = 0;\n    }\n\n    ~ArrayQueue() {\n        delete[] nums;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int capacity() {\n        return queCapacity;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u961f */\n    void push(int num) {\n        if (queSize == queCapacity) {\n            cout &lt;&lt; \"\u961f\u5217\u5df2\u6ee1\" &lt;&lt; endl;\n            return;\n        }\n        // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % queCapacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    void pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % queCapacity;\n        queSize--;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    int peek() {\n        if (isEmpty())\n            throw out_of_range(\"\u961f\u5217\u4e3a\u7a7a\");\n        return nums[front];\n    }\n\n    /* \u5c06\u6570\u7ec4\u8f6c\u5316\u4e3a Vector \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        vector&lt;int&gt; arr(queSize);\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            arr[i] = nums[j % queCapacity];\n        }\n        return arr;\n    }\n};\n</code></pre> array_queue.java<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private int[] nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private int front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    public int capacity() {\n        return nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void push(int num) {\n        if (queSize == capacity()) {\n            System.out.println(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int pop() {\n        int num = peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.cs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    int[] nums;  // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize; // \u961f\u5217\u957f\u5ea6\n\n    public ArrayQueue(int capacity) {\n        nums = new int[capacity];\n        front = queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    int Capacity() {\n        return nums.Length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    public int Size() {\n        return queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return queSize == 0;\n    }\n\n    /* \u5165\u961f */\n    public void Push(int num) {\n        if (queSize == Capacity()) {\n            Console.WriteLine(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        int rear = (front + queSize) % Capacity();\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num;\n        queSize++;\n    }\n\n    /* \u51fa\u961f */\n    public int Pop() {\n        int num = Peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % Capacity();\n        queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return nums[front];\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    public int[] ToArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        int[] res = new int[queSize];\n        for (int i = 0, j = front; i &lt; queSize; i++, j++) {\n            res[i] = nums[j % this.Capacity()];\n        }\n        return res;\n    }\n}\n</code></pre> array_queue.go<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntype arrayQueue struct {\n    nums        []int // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front       int   // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    queSize     int   // \u961f\u5217\u957f\u5ea6\n    queCapacity int   // \u961f\u5217\u5bb9\u91cf\uff08\u5373\u6700\u5927\u5bb9\u7eb3\u5143\u7d20\u6570\u91cf\uff09\n}\n\n/* \u521d\u59cb\u5316\u961f\u5217 */\nfunc newArrayQueue(queCapacity int) *arrayQueue {\n    return &amp;arrayQueue{\n        nums:        make([]int, queCapacity),\n        queCapacity: queCapacity,\n        front:       0,\n        queSize:     0,\n    }\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nfunc (q *arrayQueue) size() int {\n    return q.queSize\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nfunc (q *arrayQueue) isEmpty() bool {\n    return q.queSize == 0\n}\n\n/* \u5165\u961f */\nfunc (q *arrayQueue) push(num int) {\n    // \u5f53 rear == queCapacity \u8868\u793a\u961f\u5217\u5df2\u6ee1\n    if q.queSize == q.queCapacity {\n        return\n    }\n    // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    rear := (q.front + q.queSize) % q.queCapacity\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    q.nums[rear] = num\n    q.queSize++\n}\n\n/* \u51fa\u961f */\nfunc (q *arrayQueue) pop() any {\n    num := q.peek()\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    q.front = (q.front + 1) % q.queCapacity\n    q.queSize--\n    return num\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nfunc (q *arrayQueue) peek() any {\n    if q.isEmpty() {\n        return nil\n    }\n    return q.nums[q.front]\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (q *arrayQueue) toSlice() []int {\n    rear := (q.front + q.queSize)\n    if rear &gt;= q.queCapacity {\n        rear %= q.queCapacity\n        return append(q.nums[q.front:], q.nums[:rear]...)\n    }\n    return q.nums[q.front:rear]\n}\n</code></pre> array_queue.swift<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private var nums: [Int] // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private var front = 0 // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private var queSize = 0 // \u961f\u5217\u957f\u5ea6\n\n    init(capacity: Int) {\n        // \u521d\u59cb\u5316\u6570\u7ec4\n        nums = Array(repeating: 0, count: capacity)\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    func capacity() -&gt; Int {\n        nums.count\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        queSize\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        queSize == 0\n    }\n\n    /* \u5165\u961f */\n    func push(num: Int) {\n        if size() == capacity() {\n            print(\"\u961f\u5217\u5df2\u6ee1\")\n            return\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (front + queSize) % capacity()\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        nums[rear] = num\n        queSize += 1\n    }\n\n    /* \u51fa\u961f */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        front = (front + 1) % capacity()\n        queSize -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u961f\u5217\u4e3a\u7a7a\")\n        }\n        return nums[front]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    func toArray() -&gt; [Int] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        var res = Array(repeating: 0, count: queSize)\n        for (i, j) in sequence(first: (0, front), next: { $0 &lt; self.queSize - 1 ? ($0 + 1, $1 + 1) : nil }) {\n            res[i] = nums[j % capacity()]\n        }\n        return res\n    }\n}\n</code></pre> array_queue.js<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    #nums; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    #front = 0; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    #queSize = 0; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity) {\n        this.#nums = new Array(capacity);\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity() {\n        return this.#nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num) {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.#front + this.size) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.#nums[rear] = num;\n        this.#queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop() {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.#front = (this.#front + 1) % this.capacity;\n        this.#queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek() {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.#nums[this.#front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.#front; i &lt; this.size; i++, j++) {\n            arr[i] = this.#nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.ts<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n    private nums: number[]; // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    private front: number; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    private queSize: number; // \u961f\u5217\u957f\u5ea6\n\n    constructor(capacity: number) {\n        this.nums = new Array(capacity);\n        this.front = this.queSize = 0;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    get capacity(): number {\n        return this.nums.length;\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.queSize;\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.queSize === 0;\n    }\n\n    /* \u5165\u961f */\n    push(num: number): void {\n        if (this.size === this.capacity) {\n            console.log('\u961f\u5217\u5df2\u6ee1');\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        const rear = (this.front + this.queSize) % this.capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        this.nums[rear] = num;\n        this.queSize++;\n    }\n\n    /* \u51fa\u961f */\n    pop(): number {\n        const num = this.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        this.front = (this.front + 1) % this.capacity;\n        this.queSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    peek(): number {\n        if (this.isEmpty()) throw new Error('\u961f\u5217\u4e3a\u7a7a');\n        return this.nums[this.front];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray(): number[] {\n        // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n        const arr = new Array(this.size);\n        for (let i = 0, j = this.front; i &lt; this.size; i++, j++) {\n            arr[i] = this.nums[j % this.capacity];\n        }\n        return arr;\n    }\n}\n</code></pre> array_queue.dart<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nclass ArrayQueue {\n  late List&lt;int&gt; _nums; // \u7528\u4e8e\u50a8\u5b58\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n  late int _front; // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n  late int _queSize; // \u961f\u5217\u957f\u5ea6\n\n  ArrayQueue(int capacity) {\n    _nums = List.filled(capacity, 0);\n    _front = _queSize = 0;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n  int capaCity() {\n    return _nums.length;\n  }\n\n  /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n  int size() {\n    return _queSize;\n  }\n\n  /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _queSize == 0;\n  }\n\n  /* \u5165\u961f */\n  void push(int _num) {\n    if (_queSize == capaCity()) {\n      throw Exception(\"\u961f\u5217\u5df2\u6ee1\");\n    }\n    // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (_front + _queSize) % capaCity();\n    // \u5c06 _num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    _nums[rear] = _num;\n    _queSize++;\n  }\n\n  /* \u51fa\u961f */\n  int pop() {\n    int _num = peek();\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    _front = (_front + 1) % capaCity();\n    _queSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u961f\u5217\u4e3a\u7a7a\");\n    }\n    return _nums[_front];\n  }\n\n  /* \u8fd4\u56de Array */\n  List&lt;int&gt; toArray() {\n    // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n    final List&lt;int&gt; res = List.filled(_queSize, 0);\n    for (int i = 0, j = _front; i &lt; _queSize; i++, j++) {\n      res[i] = _nums[j % capaCity()];\n    }\n    return res;\n  }\n}\n</code></pre> array_queue.rs<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\nstruct ArrayQueue {\n    nums: Vec&lt;i32&gt;,     // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    front: i32,         // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    que_size: i32,      // \u961f\u5217\u957f\u5ea6\n    que_capacity: i32,  // \u961f\u5217\u5bb9\u91cf\n}\n\nimpl ArrayQueue {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(capacity: i32) -&gt; ArrayQueue {\n        ArrayQueue {\n            nums: vec![0; capacity as usize],\n            front: 0,\n            que_size: 0,\n            que_capacity: capacity,\n        }\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\n    fn capacity(&amp;self) -&gt; i32 {\n        self.que_capacity\n    }\n\n    /* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; i32 {\n        self.que_size\n    }\n\n    /* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.que_size == 0\n    }\n\n    /* \u5165\u961f */\n    fn push(&amp;mut self, num: i32) {\n        if self.que_size == self.capacity() {\n            println!(\"\u961f\u5217\u5df2\u6ee1\");\n            return;\n        }\n        // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n        // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n        let rear = (self.front + self.que_size) % self.que_capacity;\n        // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n        self.nums[rear as usize] = num;\n        self.que_size += 1;\n    }\n\n    /* \u51fa\u961f */\n    fn pop(&amp;mut self) -&gt; i32 {\n        let num = self.peek();\n        // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n        self.front = (self.front + 1) % self.que_capacity;\n        self.que_size -= 1;\n        num\n    }\n\n    /* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; i32 {\n        if self.is_empty() {\n            panic!(\"index out of bounds\");\n        }\n        self.nums[self.front as usize]\n    }\n\n    /* \u8fd4\u56de\u6570\u7ec4 */\n    fn to_vector(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let cap = self.que_capacity;\n        let mut j = self.front;\n        let mut arr = vec![0; self.que_size as usize];\n        for i in 0..self.que_size {\n            arr[i as usize] = self.nums[(j % cap) as usize];\n            j += 1;\n        }\n        arr\n    }\n}\n</code></pre> array_queue.c<pre><code>/* \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217 */\ntypedef struct {\n    int *nums;       // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4\n    int front;       // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n    int queSize;     // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n    int queCapacity; // \u961f\u5217\u5bb9\u91cf\n} ArrayQueue;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayQueue *newArrayQueue(int capacity) {\n    ArrayQueue *queue = (ArrayQueue *)malloc(sizeof(ArrayQueue));\n    // \u521d\u59cb\u5316\u6570\u7ec4\n    queue-&gt;queCapacity = capacity;\n    queue-&gt;nums = (int *)malloc(sizeof(int) * queue-&gt;queCapacity);\n    queue-&gt;front = queue-&gt;queSize = 0;\n    return queue;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayQueue(ArrayQueue *queue) {\n    free(queue-&gt;nums);\n    free(queue);\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf */\nint capacity(ArrayQueue *queue) {\n    return queue-&gt;queCapacity;\n}\n\n/* \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6 */\nint size(ArrayQueue *queue) {\n    return queue-&gt;queSize;\n}\n\n/* \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a */\nbool empty(ArrayQueue *queue) {\n    return queue-&gt;queSize == 0;\n}\n\n/* \u8bbf\u95ee\u961f\u9996\u5143\u7d20 */\nint peek(ArrayQueue *queue) {\n    assert(size(queue) != 0);\n    return queue-&gt;nums[queue-&gt;front];\n}\n\n/* \u5165\u961f */\nvoid push(ArrayQueue *queue, int num) {\n    if (size(queue) == capacity(queue)) {\n        printf(\"\u961f\u5217\u5df2\u6ee1\\r\\n\");\n        return;\n    }\n    // \u8ba1\u7b97\u961f\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n    // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n    int rear = (queue-&gt;front + queue-&gt;queSize) % queue-&gt;queCapacity;\n    // \u5c06 num \u6dfb\u52a0\u81f3\u961f\u5c3e\n    queue-&gt;nums[rear] = num;\n    queue-&gt;queSize++;\n}\n\n/* \u51fa\u961f */\nvoid pop(ArrayQueue *queue) {\n    int num = peek(queue);\n    // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n    queue-&gt;front = (queue-&gt;front + 1) % queue-&gt;queCapacity;\n    queue-&gt;queSize--;\n}\n</code></pre> array_queue.zig<pre><code>// \u57fa\u4e8e\u73af\u5f62\u6570\u7ec4\u5b9e\u73b0\u7684\u961f\u5217\nfn ArrayQueue(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        nums: []T = undefined,                          // \u7528\u4e8e\u5b58\u50a8\u961f\u5217\u5143\u7d20\u7684\u6570\u7ec4     \n        cap: usize = 0,                                 // \u961f\u5217\u5bb9\u91cf\n        front: usize = 0,                               // \u961f\u9996\u6307\u9488\uff0c\u6307\u5411\u961f\u9996\u5143\u7d20\n        queSize: usize = 0,                             // \u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e + 1\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,   // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6570\u7ec4\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator, cap: usize) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.cap = cap;\n            self.nums = try self.mem_allocator.alloc(T, self.cap);\n            @memset(self.nums, @as(T, 0));\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u5bb9\u91cf\n        pub fn capacity(self: *Self) usize {\n            return self.cap;\n        }\n\n        // \u83b7\u53d6\u961f\u5217\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.queSize;\n        }\n\n        // \u5224\u65ad\u961f\u5217\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.queSize == 0;\n        }\n\n        // \u5165\u961f\n        pub fn push(self: *Self, num: T) !void {\n            if (self.size() == self.capacity()) {\n                std.debug.print(\"\u961f\u5217\u5df2\u6ee1\\n\", .{});\n                return;\n            }\n            // \u8ba1\u7b97\u5c3e\u6307\u9488\uff0c\u6307\u5411\u961f\u5c3e\u7d22\u5f15 + 1\n            // \u901a\u8fc7\u53d6\u4f59\u64cd\u4f5c\uff0c\u5b9e\u73b0 rear \u8d8a\u8fc7\u6570\u7ec4\u5c3e\u90e8\u540e\u56de\u5230\u5934\u90e8\n            var rear = (self.front + self.queSize) % self.capacity();\n            // \u5c3e\u8282\u70b9\u540e\u6dfb\u52a0 num\n            self.nums[rear] = num;\n            self.queSize += 1;\n        } \n\n        // \u51fa\u961f\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            // \u961f\u9996\u6307\u9488\u5411\u540e\u79fb\u52a8\u4e00\u4f4d\uff0c\u82e5\u8d8a\u8fc7\u5c3e\u90e8\u5219\u8fd4\u56de\u5230\u6570\u7ec4\u5934\u90e8\n            self.front = (self.front + 1) % self.capacity();\n            self.queSize -= 1;\n            return num;\n        } \n\n        // \u8bbf\u95ee\u961f\u9996\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u961f\u5217\u4e3a\u7a7a\");\n            return self.nums[self.front];\n        } \n\n        // \u8fd4\u56de\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            // \u4ec5\u8f6c\u6362\u6709\u6548\u957f\u5ea6\u8303\u56f4\u5185\u7684\u5217\u8868\u5143\u7d20\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            var j: usize = self.front;\n            while (i &lt; self.size()) : ({ i += 1; j += 1; }) {\n                res[i] = self.nums[j % self.capacity()];\n            }\n            return res;\n        }\n    };\n}\n</code></pre> <p>\u4ee5\u4e0a\u5b9e\u73b0\u7684\u961f\u5217\u4ecd\u7136\u5177\u6709\u5c40\u9650\u6027\uff1a\u5176\u957f\u5ea6\u4e0d\u53ef\u53d8\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u4e0d\u96be\u89e3\u51b3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u66ff\u6362\u4e3a\u52a8\u6001\u6570\u7ec4\uff0c\u4ece\u800c\u5f15\u5165\u6269\u5bb9\u673a\u5236\u3002\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u5c1d\u8bd5\u81ea\u884c\u5b9e\u73b0\u3002</p> <p>\u4e24\u79cd\u5b9e\u73b0\u7684\u5bf9\u6bd4\u7ed3\u8bba\u4e0e\u6808\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_stack_and_queue/queue/#523","title":"5.2.3 \u00a0 \u961f\u5217\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6dd8\u5b9d\u8ba2\u5355\u3002\u8d2d\u7269\u8005\u4e0b\u5355\u540e\uff0c\u8ba2\u5355\u5c06\u52a0\u5165\u961f\u5217\u4e2d\uff0c\u7cfb\u7edf\u968f\u540e\u4f1a\u6839\u636e\u987a\u5e8f\u5904\u7406\u961f\u5217\u4e2d\u7684\u8ba2\u5355\u3002\u5728\u53cc\u5341\u4e00\u671f\u95f4\uff0c\u77ed\u65f6\u95f4\u5185\u4f1a\u4ea7\u751f\u6d77\u91cf\u8ba2\u5355\uff0c\u9ad8\u5e76\u53d1\u6210\u4e3a\u5de5\u7a0b\u5e08\u4eec\u9700\u8981\u91cd\u70b9\u653b\u514b\u7684\u95ee\u9898\u3002</li> <li>\u5404\u7c7b\u5f85\u529e\u4e8b\u9879\u3002\u4efb\u4f55\u9700\u8981\u5b9e\u73b0\u201c\u5148\u6765\u540e\u5230\u201d\u529f\u80fd\u7684\u573a\u666f\uff0c\u4f8b\u5982\u6253\u5370\u673a\u7684\u4efb\u52a1\u961f\u5217\u3001\u9910\u5385\u7684\u51fa\u9910\u961f\u5217\u7b49\uff0c\u961f\u5217\u5728\u8fd9\u4e9b\u573a\u666f\u4e2d\u53ef\u4ee5\u6709\u6548\u5730\u7ef4\u62a4\u5904\u7406\u987a\u5e8f\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/stack/","title":"5.1 \u00a0 \u6808","text":"<p>\u300c\u6808 stack\u300d\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u903b\u8f91\u7684\u7ebf\u6027\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u5c06\u6808\u7c7b\u6bd4\u4e3a\u684c\u9762\u4e0a\u7684\u4e00\u645e\u76d8\u5b50\uff0c\u5982\u679c\u60f3\u53d6\u51fa\u5e95\u90e8\u7684\u76d8\u5b50\uff0c\u5219\u9700\u8981\u5148\u5c06\u4e0a\u9762\u7684\u76d8\u5b50\u4f9d\u6b21\u79fb\u8d70\u3002\u6211\u4eec\u5c06\u76d8\u5b50\u66ff\u6362\u4e3a\u5404\u79cd\u7c7b\u578b\u7684\u5143\u7d20\uff08\u5982\u6574\u6570\u3001\u5b57\u7b26\u3001\u5bf9\u8c61\u7b49\uff09\uff0c\u5c31\u5f97\u5230\u4e86\u6808\u8fd9\u79cd\u6570\u636e\u7ed3\u6784\u3002</p> <p>\u5982\u56fe 5-1 \u6240\u793a\uff0c\u6211\u4eec\u628a\u5806\u53e0\u5143\u7d20\u7684\u9876\u90e8\u79f0\u4e3a\u201c\u6808\u9876\u201d\uff0c\u5e95\u90e8\u79f0\u4e3a\u201c\u6808\u5e95\u201d\u3002\u5c06\u628a\u5143\u7d20\u6dfb\u52a0\u5230\u6808\u9876\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u5165\u6808\u201d\uff0c\u5220\u9664\u6808\u9876\u5143\u7d20\u7684\u64cd\u4f5c\u53eb\u4f5c\u201c\u51fa\u6808\u201d\u3002</p> <p></p> <p> \u56fe 5-1 \u00a0 \u6808\u7684\u5148\u5165\u540e\u51fa\u89c4\u5219 </p>"},{"location":"chapter_stack_and_queue/stack/#511","title":"5.1.1 \u00a0 \u6808\u5e38\u7528\u64cd\u4f5c","text":"<p>\u6808\u7684\u5e38\u7528\u64cd\u4f5c\u5982\u8868 5-1 \u6240\u793a\uff0c\u5177\u4f53\u7684\u65b9\u6cd5\u540d\u9700\u8981\u6839\u636e\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u6765\u786e\u5b9a\u3002\u5728\u6b64\uff0c\u6211\u4eec\u4ee5\u5e38\u89c1\u7684 <code>push()</code>\u3001<code>pop()</code>\u3001<code>peek()</code> \u547d\u540d\u4e3a\u4f8b\u3002</p> <p> \u8868 5-1 \u00a0 \u6808\u7684\u64cd\u4f5c\u6548\u7387 </p> \u65b9\u6cd5 \u63cf\u8ff0 \u65f6\u95f4\u590d\u6742\u5ea6 push() \u5143\u7d20\u5165\u6808\uff08\u6dfb\u52a0\u81f3\u6808\u9876\uff09 \\(O(1)\\) pop() \u6808\u9876\u5143\u7d20\u51fa\u6808 \\(O(1)\\) peek() \u8bbf\u95ee\u6808\u9876\u5143\u7d20 \\(O(1)\\) <p>\u901a\u5e38\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u7f16\u7a0b\u8bed\u8a00\u5185\u7f6e\u7684\u6808\u7c7b\u3002\u7136\u800c\uff0c\u67d0\u4e9b\u8bed\u8a00\u53ef\u80fd\u6ca1\u6709\u4e13\u95e8\u63d0\u4f9b\u6808\u7c7b\uff0c\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u5c06\u8be5\u8bed\u8a00\u7684\u201c\u6570\u7ec4\u201d\u6216\u201c\u94fe\u8868\u201d\u5f53\u4f5c\u6808\u6765\u4f7f\u7528\uff0c\u5e76\u5728\u7a0b\u5e8f\u903b\u8f91\u4e0a\u5ffd\u7565\u4e0e\u6808\u65e0\u5173\u7684\u64cd\u4f5c\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig stack.py<pre><code># \u521d\u59cb\u5316\u6808\n# Python \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nstack: list[int] = []\n\n# \u5143\u7d20\u5165\u6808\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n# \u8bbf\u95ee\u6808\u9876\u5143\u7d20\npeek: int = stack[-1]\n\n# \u5143\u7d20\u51fa\u6808\npop: int = stack.pop()\n\n# \u83b7\u53d6\u6808\u7684\u957f\u5ea6\nsize: int = len(stack)\n\n# \u5224\u65ad\u662f\u5426\u4e3a\u7a7a\nis_empty: bool = len(stack) == 0\n</code></pre> stack.cpp<pre><code>/* \u521d\u59cb\u5316\u6808 */\nstack&lt;int&gt; stack;\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint top = stack.top();\n\n/* \u5143\u7d20\u51fa\u6808 */\nstack.pop(); // \u65e0\u8fd4\u56de\u503c\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool empty = stack.empty();\n</code></pre> stack.java<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;Integer&gt; stack = new Stack&lt;&gt;();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.size();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nboolean isEmpty = stack.isEmpty();\n</code></pre> stack.cs<pre><code>/* \u521d\u59cb\u5316\u6808 */\nStack&lt;int&gt; stack = new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.Push(1);\nstack.Push(3);\nstack.Push(2);\nstack.Push(5);\nstack.Push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.Peek();\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.Pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.Count;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.Count == 0;\n</code></pre> stack_test.go<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u5728 Go \u4e2d\uff0c\u63a8\u8350\u5c06 Slice \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack []int\n\n/* \u5143\u7d20\u5165\u6808 */\nstack = append(stack, 1)\nstack = append(stack, 3)\nstack = append(stack, 2)\nstack = append(stack, 5)\nstack = append(stack, 4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\npeek := stack[len(stack)-1]\n\n/* \u5143\u7d20\u51fa\u6808 */\npop := stack[len(stack)-1]\nstack = stack[:len(stack)-1]\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nsize := len(stack)\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nisEmpty := len(stack) == 0\n</code></pre> stack.swift<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Swift \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nvar stack: [Int] = []\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.append(1)\nstack.append(3)\nstack.append(2)\nstack.append(5)\nstack.append(4)\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet peek = stack.last!\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.removeLast()\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.count\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet isEmpty = stack.isEmpty\n</code></pre> stack.js<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Javascript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length-1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.ts<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Typescript \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a Array \u5f53\u4f5c\u6808\u6765\u4f7f\u7528 \nconst stack: number[] = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nconst peek = stack[stack.length - 1];\n\n/* \u5143\u7d20\u51fa\u6808 */\nconst pop = stack.pop();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nconst size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nconst is_empty = stack.length === 0;\n</code></pre> stack.dart<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// Dart \u6ca1\u6709\u5185\u7f6e\u7684\u6808\u7c7b\uff0c\u53ef\u4ee5\u628a List \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nList&lt;int&gt; stack = [];\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.add(1);\nstack.add(3);\nstack.add(2);\nstack.add(5);\nstack.add(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek = stack.last;\n\n/* \u5143\u7d20\u51fa\u6808 */\nint pop = stack.removeLast();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size = stack.length;\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty = stack.isEmpty;\n</code></pre> stack.rs<pre><code>/* \u521d\u59cb\u5316\u6808 */\n// \u628a Vec \u5f53\u4f5c\u6808\u6765\u4f7f\u7528\nlet mut stack: Vec&lt;i32&gt; = Vec::new();\n\n/* \u5143\u7d20\u5165\u6808 */\nstack.push(1);\nstack.push(3);\nstack.push(2);\nstack.push(5);\nstack.push(4);\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nlet top = stack.last().unwrap();\n\n/* \u5143\u7d20\u51fa\u6808 */\nlet pop = stack.pop().unwrap();\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nlet size = stack.len();\n\n/* \u5224\u65ad\u662f\u5426\u4e3a\u7a7a */\nlet is_empty = stack.is_empty();\n</code></pre> stack.c<pre><code>// C \u672a\u63d0\u4f9b\u5185\u7f6e\u6808\n</code></pre> stack.zig<pre><code>\n</code></pre>"},{"location":"chapter_stack_and_queue/stack/#512","title":"5.1.2 \u00a0 \u6808\u7684\u5b9e\u73b0","text":"<p>\u4e3a\u4e86\u6df1\u5165\u4e86\u89e3\u6808\u7684\u8fd0\u884c\u673a\u5236\uff0c\u6211\u4eec\u6765\u5c1d\u8bd5\u81ea\u5df1\u5b9e\u73b0\u4e00\u4e2a\u6808\u7c7b\u3002</p> <p>\u6808\u9075\u5faa\u5148\u5165\u540e\u51fa\u7684\u539f\u5219\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u80fd\u5728\u6808\u9876\u6dfb\u52a0\u6216\u5220\u9664\u5143\u7d20\u3002\u7136\u800c\uff0c\u6570\u7ec4\u548c\u94fe\u8868\u90fd\u53ef\u4ee5\u5728\u4efb\u610f\u4f4d\u7f6e\u6dfb\u52a0\u548c\u5220\u9664\u5143\u7d20\uff0c\u56e0\u6b64\u6808\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u53d7\u9650\u5236\u7684\u6570\u7ec4\u6216\u94fe\u8868\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u201c\u5c4f\u853d\u201d\u6570\u7ec4\u6216\u94fe\u8868\u7684\u90e8\u5206\u65e0\u5173\u64cd\u4f5c\uff0c\u4f7f\u5176\u5bf9\u5916\u8868\u73b0\u7684\u903b\u8f91\u7b26\u5408\u6808\u7684\u7279\u6027\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#1","title":"1. \u00a0 \u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u94fe\u8868\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u94fe\u8868\u7684\u5934\u8282\u70b9\u89c6\u4e3a\u6808\u9876\uff0c\u5c3e\u8282\u70b9\u89c6\u4e3a\u6808\u5e95\u3002</p> <p>\u5982\u56fe 5-2 \u6240\u793a\uff0c\u5bf9\u4e8e\u5165\u6808\u64cd\u4f5c\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u5143\u7d20\u63d2\u5165\u94fe\u8868\u5934\u90e8\uff0c\u8fd9\u79cd\u8282\u70b9\u63d2\u5165\u65b9\u6cd5\u88ab\u79f0\u4e3a\u201c\u5934\u63d2\u6cd5\u201d\u3002\u800c\u5bf9\u4e8e\u51fa\u6808\u64cd\u4f5c\uff0c\u53ea\u9700\u5c06\u5934\u8282\u70b9\u4ece\u94fe\u8868\u4e2d\u5220\u9664\u5373\u53ef\u3002</p> LinkedListStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-2 \u00a0 \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u4ee5\u4e0b\u662f\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u6808\u7684\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig linkedlist_stack.py<pre><code>class LinkedListStack:\n    \"\"\"\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._peek: ListNode | None = None\n        self._size: int = 0\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return self._size\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return not self._peek\n\n    def push(self, val: int):\n        \"\"\"\u5165\u6808\"\"\"\n        node = ListNode(val)\n        node.next = self._peek\n        self._peek = node\n        self._size += 1\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        num = self.peek()\n        self._peek = self._peek.next\n        self._size -= 1\n        return num\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._peek.val\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8f6c\u5316\u4e3a\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        arr = []\n        node = self._peek\n        while node:\n            arr.append(node.val)\n            node = node.next\n        arr.reverse()\n        return arr\n</code></pre> linkedlist_stack.cpp<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  private:\n    ListNode *stackTop; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize;        // \u6808\u7684\u957f\u5ea6\n\n  public:\n    LinkedListStack() {\n        stackTop = nullptr;\n        stkSize = 0;\n    }\n\n    ~LinkedListStack() {\n        // \u904d\u5386\u94fe\u8868\u5220\u9664\u8282\u70b9\uff0c\u91ca\u653e\u5185\u5b58\n        freeMemoryLinkedList(stackTop);\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        ListNode *node = new ListNode(num);\n        node-&gt;next = stackTop;\n        stackTop = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    void pop() {\n        int num = top();\n        ListNode *tmp = stackTop;\n        stackTop = stackTop-&gt;next;\n        // \u91ca\u653e\u5185\u5b58\n        delete tmp;\n        stkSize--;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stackTop-&gt;val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    vector&lt;int&gt; toVector() {\n        ListNode *node = stackTop;\n        vector&lt;int&gt; res(size());\n        for (int i = res.size() - 1; i &gt;= 0; i--) {\n            res[i] = node-&gt;val;\n            node = node-&gt;next;\n        }\n        return res;\n    }\n};\n</code></pre> linkedlist_stack.java<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private ListNode stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private int stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        ListNode node = new ListNode(num);\n        node.next = stackPeek;\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        int num = peek();\n        stackPeek = stackPeek.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stackPeek.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] toArray() {\n        ListNode node = stackPeek;\n        int[] res = new int[size()];\n        for (int i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.cs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    ListNode? stackPeek;  // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int stkSize = 0;   // \u6808\u7684\u957f\u5ea6\n\n    public LinkedListStack() {\n        stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        ListNode node = new(num) {\n            next = stackPeek\n        };\n        stackPeek = node;\n        stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        int num = Peek();\n        stackPeek = stackPeek!.next;\n        stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stackPeek!.val;\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        if (stackPeek == null)\n            return [];\n\n        ListNode? node = stackPeek;\n        int[] res = new int[Size()];\n        for (int i = res.Length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.go<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntype linkedListStack struct {\n    // \u4f7f\u7528\u5185\u7f6e\u5305 list \u6765\u5b9e\u73b0\u6808\n    data *list.List\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newLinkedListStack() *linkedListStack {\n    return &amp;linkedListStack{\n        data: list.New(),\n    }\n}\n\n/* \u5165\u6808 */\nfunc (s *linkedListStack) push(value int) {\n    s.data.PushBack(value)\n}\n\n/* \u51fa\u6808 */\nfunc (s *linkedListStack) pop() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    s.data.Remove(e)\n    return e.Value\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nfunc (s *linkedListStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    e := s.data.Back()\n    return e.Value\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nfunc (s *linkedListStack) size() int {\n    return s.data.Len()\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *linkedListStack) isEmpty() bool {\n    return s.data.Len() == 0\n}\n\n/* \u83b7\u53d6 List \u7528\u4e8e\u6253\u5370 */\nfunc (s *linkedListStack) toList() *list.List {\n    return s.data\n}\n</code></pre> linkedlist_stack.swift<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private var _peek: ListNode? // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private var _size = 0 // \u6808\u7684\u957f\u5ea6\n\n    init() {}\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        _size\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        size() == 0\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        let node = ListNode(x: num)\n        node.next = _peek\n        _peek = node\n        _size += 1\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        let num = peek()\n        _peek = _peek?.next\n        _size -= 1\n        return num\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return _peek!.val\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        var node = _peek\n        var res = Array(repeating: 0, count: _size)\n        for i in sequence(first: res.count - 1, next: { $0 &gt;= 0 + 1 ? $0 - 1 : nil }) {\n            res[i] = node!.val\n            node = node?.next\n        }\n        return res\n    }\n}\n</code></pre> linkedlist_stack.js<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    #stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    #stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.#stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        const node = new ListNode(num);\n        node.next = this.#stackPeek;\n        this.#stackPeek = node;\n        this.#stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        const num = this.peek();\n        this.#stackPeek = this.#stackPeek.next;\n        this.#stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek() {\n        if (!this.#stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray() {\n        let node = this.#stackPeek;\n        const res = new Array(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node.val;\n            node = node.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.ts<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n    private stackPeek: ListNode | null; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    private stkSize: number = 0; // \u6808\u7684\u957f\u5ea6\n\n    constructor() {\n        this.stackPeek = null;\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stkSize;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.size === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        const node = new ListNode(num);\n        node.next = this.stackPeek;\n        this.stackPeek = node;\n        this.stkSize++;\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number {\n        const num = this.peek();\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        this.stackPeek = this.stackPeek.next;\n        this.stkSize--;\n        return num;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    peek(): number {\n        if (!this.stackPeek) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stackPeek.val;\n    }\n\n    /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    toArray(): number[] {\n        let node = this.stackPeek;\n        const res = new Array&lt;number&gt;(this.size);\n        for (let i = res.length - 1; i &gt;= 0; i--) {\n            res[i] = node!.val;\n            node = node!.next;\n        }\n        return res;\n    }\n}\n</code></pre> linkedlist_stack.dart<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u7c7b\u5b9e\u73b0\u7684\u6808 */\nclass LinkedListStack {\n  ListNode? _stackPeek; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n  int _stkSize = 0; // \u6808\u7684\u957f\u5ea6\n\n  LinkedListStack() {\n    _stackPeek = null;\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stkSize;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stkSize == 0;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    final ListNode node = ListNode(_num);\n    node.next = _stackPeek;\n    _stackPeek = node;\n    _stkSize++;\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    final int _num = peek();\n    _stackPeek = _stackPeek!.next;\n    _stkSize--;\n    return _num;\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (_stackPeek == null) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stackPeek!.val;\n  }\n\n  /* \u5c06\u94fe\u8868\u8f6c\u5316\u4e3a List \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toList() {\n    ListNode? node = _stackPeek;\n    List&lt;int&gt; list = [];\n    while (node != null) {\n      list.add(node.val);\n      node = node.next;\n    }\n    list = list.reversed.toList();\n    return list;\n  }\n}\n</code></pre> linkedlist_stack.rs<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\n#[allow(dead_code)]\npub struct LinkedListStack&lt;T&gt; {\n    stack_peek: Option&lt;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;,   // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    stk_size: usize,                                // \u6808\u7684\u957f\u5ea6\n}\n\nimpl&lt;T: Copy&gt; LinkedListStack&lt;T&gt; {\n    pub fn new() -&gt; Self {\n        Self {\n            stack_peek: None,\n            stk_size: 0,\n        }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    pub fn size(&amp;self) -&gt; usize {\n        return self.stk_size;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    pub fn is_empty(&amp;self) -&gt; bool {\n        return self.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    pub fn push(&amp;mut self, num: T) {\n        let node = ListNode::new(num);\n        node.borrow_mut().next = self.stack_peek.take();\n        self.stack_peek = Some(node);\n        self.stk_size += 1;\n    }\n\n    /* \u51fa\u6808 */\n    pub fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        self.stack_peek.take().map(|old_head| {\n            match old_head.borrow_mut().next.take() {\n                Some(new_head) =&gt; {\n                    self.stack_peek = Some(new_head);\n                }\n                None =&gt; {\n                    self.stack_peek = None;\n                }\n            }\n            self.stk_size -= 1;\n            Rc::try_unwrap(old_head).ok().unwrap().into_inner().val\n        })\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    pub fn peek(&amp;self) -&gt; Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt; {\n        self.stack_peek.as_ref()\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    pub fn to_array(&amp;self, head: Option&lt;&amp;Rc&lt;RefCell&lt;ListNode&lt;T&gt;&gt;&gt;&gt;) -&gt; Vec&lt;T&gt; {\n        if let Some(node) = head {\n            let mut nums = self.to_array(node.borrow().next.as_ref());\n            nums.push(node.borrow().val);\n            return nums;\n        }\n        return Vec::new();\n    }\n}\n</code></pre> linkedlist_stack.c<pre><code>/* \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    ListNode *top; // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n    int size;      // \u6808\u7684\u957f\u5ea6\n} LinkedListStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nLinkedListStack *newLinkedListStack() {\n    LinkedListStack *s = malloc(sizeof(LinkedListStack));\n    s-&gt;top = NULL;\n    s-&gt;size = 0;\n    return s;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delLinkedListStack(LinkedListStack *s) {\n    while (s-&gt;top) {\n        ListNode *n = s-&gt;top-&gt;next;\n        free(s-&gt;top);\n        s-&gt;top = n;\n    }\n    free(s);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(LinkedListStack *s) {\n    assert(s);\n    return s-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(LinkedListStack *s) {\n    assert(s);\n    return size(s) == 0;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(LinkedListStack *s) {\n    assert(s);\n    assert(size(s) != 0);\n    return s-&gt;top-&gt;val;\n}\n\n/* \u5165\u6808 */\nvoid push(LinkedListStack *s, int num) {\n    assert(s);\n    ListNode *node = (ListNode *)malloc(sizeof(ListNode));\n    node-&gt;next = s-&gt;top; // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6307\u9488\u57df\n    node-&gt;val = num;     // \u66f4\u65b0\u65b0\u52a0\u8282\u70b9\u6570\u636e\u57df\n    s-&gt;top = node;       // \u66f4\u65b0\u6808\u9876\n    s-&gt;size++;           // \u66f4\u65b0\u6808\u5927\u5c0f\n}\n\n/* \u51fa\u6808 */\nint pop(LinkedListStack *s) {\n    if (s-&gt;size == 0) {\n        printf(\"stack is empty.\\n\");\n        return INT_MAX;\n    }\n    assert(s);\n    int val = peek(s);\n    ListNode *tmp = s-&gt;top;\n    s-&gt;top = s-&gt;top-&gt;next;\n    // \u91ca\u653e\u5185\u5b58\n    free(tmp);\n    s-&gt;size--;\n    return val;\n}\n</code></pre> linkedlist_stack.zig<pre><code>// \u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\nfn LinkedListStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack_top: ?*inc.ListNode(T) = null,             // \u5c06\u5934\u8282\u70b9\u4f5c\u4e3a\u6808\u9876\n        stk_size: usize = 0,                             // \u6808\u7684\u957f\u5ea6\n        mem_arena: ?std.heap.ArenaAllocator = null,\n        mem_allocator: std.mem.Allocator = undefined,    // \u5185\u5b58\u5206\u914d\u5668\n\n        // \u6784\u9020\u51fd\u6570\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) !void {\n            if (self.mem_arena == null) {\n                self.mem_arena = std.heap.ArenaAllocator.init(allocator);\n                self.mem_allocator = self.mem_arena.?.allocator();\n            }\n            self.stack_top = null;\n            self.stk_size = 0;\n        }\n\n        // \u6790\u6784\u51fd\u6570\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.mem_arena == null) return;\n            self.mem_arena.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stk_size;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.size() == 0) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack_top.?.val;\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            var node = try self.mem_allocator.create(inc.ListNode(T));\n            node.init(num);\n            node.next = self.stack_top;\n            self.stack_top = node;\n            self.stk_size += 1;\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.peek();\n            self.stack_top = self.stack_top.?.next;\n            self.stk_size -= 1;\n            return num;\n        } \n\n        // \u5c06\u6808\u8f6c\u6362\u4e3a\u6570\u7ec4\n        pub fn toArray(self: *Self) ![]T {\n            var node = self.stack_top;\n            var res = try self.mem_allocator.alloc(T, self.size());\n            @memset(res, @as(T, 0));\n            var i: usize = 0;\n            while (i &lt; res.len) : (i += 1) {\n                res[res.len - i - 1] = node.?.val;\n                node = node.?.next;\n            }\n            return res;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/stack/#2","title":"2. \u00a0 \u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0","text":"<p>\u4f7f\u7528\u6570\u7ec4\u5b9e\u73b0\u6808\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6570\u7ec4\u7684\u5c3e\u90e8\u4f5c\u4e3a\u6808\u9876\u3002\u5982\u56fe 5-3 \u6240\u793a\uff0c\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u5206\u522b\u5bf9\u5e94\u5728\u6570\u7ec4\u5c3e\u90e8\u6dfb\u52a0\u5143\u7d20\u4e0e\u5220\u9664\u5143\u7d20\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u4e3a \\(O(1)\\) \u3002</p> ArrayStackpush()pop() <p></p> <p></p> <p></p> <p> \u56fe 5-3 \u00a0 \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u6808\u7684\u5165\u6808\u51fa\u6808\u64cd\u4f5c </p> <p>\u7531\u4e8e\u5165\u6808\u7684\u5143\u7d20\u53ef\u80fd\u4f1a\u6e90\u6e90\u4e0d\u65ad\u5730\u589e\u52a0\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u6570\u7ec4\uff0c\u8fd9\u6837\u5c31\u65e0\u987b\u81ea\u884c\u5904\u7406\u6570\u7ec4\u6269\u5bb9\u95ee\u9898\u3002\u4ee5\u4e0b\u4e3a\u793a\u4f8b\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_stack.py<pre><code>class ArrayStack:\n    \"\"\"\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\"\"\"\n\n    def __init__(self):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._stack: list[int] = []\n\n    def size(self) -&gt; int:\n        \"\"\"\u83b7\u53d6\u6808\u7684\u957f\u5ea6\"\"\"\n        return len(self._stack)\n\n    def is_empty(self) -&gt; bool:\n        \"\"\"\u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\"\"\"\n        return self._stack == []\n\n    def push(self, item: int):\n        \"\"\"\u5165\u6808\"\"\"\n        self._stack.append(item)\n\n    def pop(self) -&gt; int:\n        \"\"\"\u51fa\u6808\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack.pop()\n\n    def peek(self) -&gt; int:\n        \"\"\"\u8bbf\u95ee\u6808\u9876\u5143\u7d20\"\"\"\n        if self.is_empty():\n            raise IndexError(\"\u6808\u4e3a\u7a7a\")\n        return self._stack[-1]\n\n    def to_list(self) -&gt; list[int]:\n        \"\"\"\u8fd4\u56de\u5217\u8868\u7528\u4e8e\u6253\u5370\"\"\"\n        return self._stack\n</code></pre> array_stack.cpp<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  private:\n    vector&lt;int&gt; stack;\n\n  public:\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    bool isEmpty() {\n        return stack.size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    void push(int num) {\n        stack.push_back(num);\n    }\n\n    /* \u51fa\u6808 */\n    void pop() {\n        int oldTop = top();\n        stack.pop_back();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    int top() {\n        if (isEmpty())\n            throw out_of_range(\"\u6808\u4e3a\u7a7a\");\n        return stack.back();\n    }\n\n    /* \u8fd4\u56de Vector */\n    vector&lt;int&gt; toVector() {\n        return stack;\n    }\n};\n</code></pre> array_stack.java<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private ArrayList&lt;Integer&gt; stack;\n\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = new ArrayList&lt;&gt;();\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int size() {\n        return stack.size();\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public boolean isEmpty() {\n        return size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void push(int num) {\n        stack.add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int pop() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.remove(size() - 1);\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int peek() {\n        if (isEmpty())\n            throw new IndexOutOfBoundsException();\n        return stack.get(size() - 1);\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public Object[] toArray() {\n        return stack.toArray();\n    }\n}\n</code></pre> array_stack.cs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    List&lt;int&gt; stack;\n    public ArrayStack() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    public int Size() {\n        return stack.Count;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    public bool IsEmpty() {\n        return Size() == 0;\n    }\n\n    /* \u5165\u6808 */\n    public void Push(int num) {\n        stack.Add(num);\n    }\n\n    /* \u51fa\u6808 */\n    public int Pop() {\n        if (IsEmpty())\n            throw new Exception();\n        var val = Peek();\n        stack.RemoveAt(Size() - 1);\n        return val;\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    public int Peek() {\n        if (IsEmpty())\n            throw new Exception();\n        return stack[Size() - 1];\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    public int[] ToArray() {\n        return [.. stack];\n    }\n}\n</code></pre> array_stack.go<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntype arrayStack struct {\n    data []int // \u6570\u636e\n}\n\n/* \u521d\u59cb\u5316\u6808 */\nfunc newArrayStack() *arrayStack {\n    return &amp;arrayStack{\n        // \u8bbe\u7f6e\u6808\u7684\u957f\u5ea6\u4e3a 0\uff0c\u5bb9\u91cf\u4e3a 16\n        data: make([]int, 0, 16),\n    }\n}\n\n/* \u6808\u7684\u957f\u5ea6 */\nfunc (s *arrayStack) size() int {\n    return len(s.data)\n}\n\n/* \u6808\u662f\u5426\u4e3a\u7a7a */\nfunc (s *arrayStack) isEmpty() bool {\n    return s.size() == 0\n}\n\n/* \u5165\u6808 */\nfunc (s *arrayStack) push(v int) {\n    // \u5207\u7247\u4f1a\u81ea\u52a8\u6269\u5bb9\n    s.data = append(s.data, v)\n}\n\n/* \u51fa\u6808 */\nfunc (s *arrayStack) pop() any {\n    val := s.peek()\n    s.data = s.data[:len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6\u6808\u9876\u5143\u7d20 */\nfunc (s *arrayStack) peek() any {\n    if s.isEmpty() {\n        return nil\n    }\n    val := s.data[len(s.data)-1]\n    return val\n}\n\n/* \u83b7\u53d6 Slice \u7528\u4e8e\u6253\u5370 */\nfunc (s *arrayStack) toSlice() []int {\n    return s.data\n}\n</code></pre> array_stack.swift<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private var stack: [Int]\n\n    init() {\n        // \u521d\u59cb\u5316\u5217\u8868\uff08\u52a8\u6001\u6570\u7ec4\uff09\n        stack = []\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    func size() -&gt; Int {\n        stack.count\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    func isEmpty() -&gt; Bool {\n        stack.isEmpty\n    }\n\n    /* \u5165\u6808 */\n    func push(num: Int) {\n        stack.append(num)\n    }\n\n    /* \u51fa\u6808 */\n    @discardableResult\n    func pop() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.removeLast()\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    func peek() -&gt; Int {\n        if isEmpty() {\n            fatalError(\"\u6808\u4e3a\u7a7a\")\n        }\n        return stack.last!\n    }\n\n    /* \u5c06 List \u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n    func toArray() -&gt; [Int] {\n        stack\n    }\n}\n</code></pre> array_stack.js<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    #stack;\n    constructor() {\n        this.#stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size() {\n        return this.#stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty() {\n        return this.#stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num) {\n        this.#stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top() {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.#stack[this.#stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.#stack;\n    }\n}\n</code></pre> array_stack.ts<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n    private stack: number[];\n    constructor() {\n        this.stack = [];\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    get size(): number {\n        return this.stack.length;\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    isEmpty(): boolean {\n        return this.stack.length === 0;\n    }\n\n    /* \u5165\u6808 */\n    push(num: number): void {\n        this.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    pop(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack.pop();\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    top(): number | undefined {\n        if (this.isEmpty()) throw new Error('\u6808\u4e3a\u7a7a');\n        return this.stack[this.stack.length - 1];\n    }\n\n    /* \u8fd4\u56de Array */\n    toArray() {\n        return this.stack;\n    }\n}\n</code></pre> array_stack.dart<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nclass ArrayStack {\n  late List&lt;int&gt; _stack;\n  ArrayStack() {\n    _stack = [];\n  }\n\n  /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n  int size() {\n    return _stack.length;\n  }\n\n  /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n  bool isEmpty() {\n    return _stack.isEmpty;\n  }\n\n  /* \u5165\u6808 */\n  void push(int _num) {\n    _stack.add(_num);\n  }\n\n  /* \u51fa\u6808 */\n  int pop() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.removeLast();\n  }\n\n  /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n  int peek() {\n    if (isEmpty()) {\n      throw Exception(\"\u6808\u4e3a\u7a7a\");\n    }\n    return _stack.last;\n  }\n\n  /* \u5c06\u6808\u8f6c\u5316\u4e3a Array \u5e76\u8fd4\u56de */\n  List&lt;int&gt; toArray() =&gt; _stack;\n}\n</code></pre> array_stack.rs<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\nstruct ArrayStack&lt;T&gt; {\n    stack: Vec&lt;T&gt;,\n}\n\nimpl&lt;T&gt; ArrayStack&lt;T&gt; {\n    /* \u521d\u59cb\u5316\u6808 */\n    fn new() -&gt; ArrayStack&lt;T&gt; {\n        ArrayStack::&lt;T&gt; { stack: Vec::&lt;T&gt;::new() }\n    }\n\n    /* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\n    fn size(&amp;self) -&gt; usize {\n        self.stack.len()\n    }\n\n    /* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\n    fn is_empty(&amp;self) -&gt; bool {\n        self.size() == 0\n    }\n\n    /* \u5165\u6808 */\n    fn push(&amp;mut self, num: T) {\n        self.stack.push(num);\n    }\n\n    /* \u51fa\u6808 */\n    fn pop(&amp;mut self) -&gt; Option&lt;T&gt; {\n        match self.stack.pop() {\n            Some(num) =&gt; Some(num),\n            None =&gt; None,\n        }\n    }\n\n    /* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\n    fn peek(&amp;self) -&gt; Option&lt;&amp;T&gt; {\n        if self.is_empty() { panic!(\"\u6808\u4e3a\u7a7a\") };\n        self.stack.last()\n    }\n\n    /* \u8fd4\u56de &amp;Vec */\n    fn to_array(&amp;self) -&gt; &amp;Vec&lt;T&gt; {\n        &amp;self.stack\n    }\n}\n</code></pre> array_stack.c<pre><code>/* \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808 */\ntypedef struct {\n    int *data;\n    int size;\n} ArrayStack;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayStack *newArrayStack() {\n    ArrayStack *stack = malloc(sizeof(ArrayStack));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5927\u5bb9\u91cf\uff0c\u907f\u514d\u6269\u5bb9\n    stack-&gt;data = malloc(sizeof(int) * MAX_SIZE);\n    stack-&gt;size = 0;\n    return stack;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayStack(ArrayStack *stack) {\n    free(stack-&gt;data);\n    free(stack);\n}\n\n/* \u83b7\u53d6\u6808\u7684\u957f\u5ea6 */\nint size(ArrayStack *stack) {\n    return stack-&gt;size;\n}\n\n/* \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a */\nbool isEmpty(ArrayStack *stack) {\n    return stack-&gt;size == 0;\n}\n\n/* \u5165\u6808 */\nvoid push(ArrayStack *stack, int num) {\n    if (stack-&gt;size == MAX_SIZE) {\n        printf(\"stack is full.\\n\");\n        return;\n    }\n    stack-&gt;data[stack-&gt;size] = num;\n    stack-&gt;size++;\n}\n\n/* \u8bbf\u95ee\u6808\u9876\u5143\u7d20 */\nint peek(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"stack is empty.\\n\");\n        return INT_MAX;\n    }\n    return stack-&gt;data[stack-&gt;size - 1];\n}\n\n/* \u51fa\u6808 */\nint pop(ArrayStack *stack) {\n    if (stack-&gt;size == 0) {\n        printf(\"stack is empty.\\n\");\n        return INT_MAX;\n    }\n    int val = peek(stack);\n    stack-&gt;size--;\n    return val;\n}\n</code></pre> array_stack.zig<pre><code>// \u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\nfn ArrayStack(comptime T: type) type {\n    return struct {\n        const Self = @This();\n\n        stack: ?std.ArrayList(T) = null,     \n\n        // \u6784\u9020\u65b9\u6cd5\uff08\u5206\u914d\u5185\u5b58+\u521d\u59cb\u5316\u6808\uff09\n        pub fn init(self: *Self, allocator: std.mem.Allocator) void {\n            if (self.stack == null) {\n                self.stack = std.ArrayList(T).init(allocator);\n            }\n        }\n\n        // \u6790\u6784\u65b9\u6cd5\uff08\u91ca\u653e\u5185\u5b58\uff09\n        pub fn deinit(self: *Self) void {\n            if (self.stack == null) return;\n            self.stack.?.deinit();\n        }\n\n        // \u83b7\u53d6\u6808\u7684\u957f\u5ea6\n        pub fn size(self: *Self) usize {\n            return self.stack.?.items.len;\n        }\n\n        // \u5224\u65ad\u6808\u662f\u5426\u4e3a\u7a7a\n        pub fn isEmpty(self: *Self) bool {\n            return self.size() == 0;\n        }\n\n        // \u8bbf\u95ee\u6808\u9876\u5143\u7d20\n        pub fn peek(self: *Self) T {\n            if (self.isEmpty()) @panic(\"\u6808\u4e3a\u7a7a\");\n            return self.stack.?.items[self.size() - 1];\n        }  \n\n        // \u5165\u6808\n        pub fn push(self: *Self, num: T) !void {\n            try self.stack.?.append(num);\n        } \n\n        // \u51fa\u6808\n        pub fn pop(self: *Self) T {\n            var num = self.stack.?.pop();\n            return num;\n        } \n\n        // \u8fd4\u56de ArrayList\n        pub fn toList(self: *Self) std.ArrayList(T) {\n            return self.stack.?;\n        }\n    };\n}\n</code></pre>"},{"location":"chapter_stack_and_queue/stack/#513","title":"5.1.3 \u00a0 \u4e24\u79cd\u5b9e\u73b0\u5bf9\u6bd4","text":"<p>\u652f\u6301\u64cd\u4f5c</p> <p>\u4e24\u79cd\u5b9e\u73b0\u90fd\u652f\u6301\u6808\u5b9a\u4e49\u4e2d\u7684\u5404\u9879\u64cd\u4f5c\u3002\u6570\u7ec4\u5b9e\u73b0\u989d\u5916\u652f\u6301\u968f\u673a\u8bbf\u95ee\uff0c\u4f46\u8fd9\u5df2\u8d85\u51fa\u4e86\u6808\u7684\u5b9a\u4e49\u8303\u7574\uff0c\u56e0\u6b64\u4e00\u822c\u4e0d\u4f1a\u7528\u5230\u3002</p> <p>\u65f6\u95f4\u6548\u7387</p> <p>\u5728\u57fa\u4e8e\u6570\u7ec4\u7684\u5b9e\u73b0\u4e2d\uff0c\u5165\u6808\u548c\u51fa\u6808\u64cd\u4f5c\u90fd\u5728\u9884\u5148\u5206\u914d\u597d\u7684\u8fde\u7eed\u5185\u5b58\u4e2d\u8fdb\u884c\uff0c\u5177\u6709\u5f88\u597d\u7684\u7f13\u5b58\u672c\u5730\u6027\uff0c\u56e0\u6b64\u6548\u7387\u8f83\u9ad8\u3002\u7136\u800c\uff0c\u5982\u679c\u5165\u6808\u65f6\u8d85\u51fa\u6570\u7ec4\u5bb9\u91cf\uff0c\u4f1a\u89e6\u53d1\u6269\u5bb9\u673a\u5236\uff0c\u5bfc\u81f4\u8be5\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u53d8\u4e3a \\(O(n)\\) \u3002</p> <p>\u5728\u57fa\u4e8e\u94fe\u8868\u7684\u5b9e\u73b0\u4e2d\uff0c\u94fe\u8868\u7684\u6269\u5bb9\u975e\u5e38\u7075\u6d3b\uff0c\u4e0d\u5b58\u5728\u4e0a\u8ff0\u6570\u7ec4\u6269\u5bb9\u65f6\u6548\u7387\u964d\u4f4e\u7684\u95ee\u9898\u3002\u4f46\u662f\uff0c\u5165\u6808\u64cd\u4f5c\u9700\u8981\u521d\u59cb\u5316\u8282\u70b9\u5bf9\u8c61\u5e76\u4fee\u6539\u6307\u9488\uff0c\u56e0\u6b64\u6548\u7387\u76f8\u5bf9\u8f83\u4f4e\u3002\u4e0d\u8fc7\uff0c\u5982\u679c\u5165\u6808\u5143\u7d20\u672c\u8eab\u5c31\u662f\u8282\u70b9\u5bf9\u8c61\uff0c\u90a3\u4e48\u53ef\u4ee5\u7701\u53bb\u521d\u59cb\u5316\u6b65\u9aa4\uff0c\u4ece\u800c\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53\u5165\u6808\u4e0e\u51fa\u6808\u64cd\u4f5c\u7684\u5143\u7d20\u662f\u57fa\u672c\u6570\u636e\u7c7b\u578b\u65f6\uff0c\u4f8b\u5982 <code>int</code> \u6216 <code>double</code> \uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\u4ee5\u4e0b\u7ed3\u8bba\u3002</p> <ul> <li>\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u5728\u89e6\u53d1\u6269\u5bb9\u65f6\u6548\u7387\u4f1a\u964d\u4f4e\uff0c\u4f46\u7531\u4e8e\u6269\u5bb9\u662f\u4f4e\u9891\u64cd\u4f5c\uff0c\u56e0\u6b64\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> <li>\u57fa\u4e8e\u94fe\u8868\u5b9e\u73b0\u7684\u6808\u53ef\u4ee5\u63d0\u4f9b\u66f4\u52a0\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> </ul> <p>\u7a7a\u95f4\u6548\u7387</p> <p>\u5728\u521d\u59cb\u5316\u5217\u8868\u65f6\uff0c\u7cfb\u7edf\u4f1a\u4e3a\u5217\u8868\u5206\u914d\u201c\u521d\u59cb\u5bb9\u91cf\u201d\uff0c\u8be5\u5bb9\u91cf\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\uff1b\u5e76\u4e14\uff0c\u6269\u5bb9\u673a\u5236\u901a\u5e38\u662f\u6309\u7167\u7279\u5b9a\u500d\u7387\uff08\u4f8b\u5982 2 \u500d\uff09\u8fdb\u884c\u6269\u5bb9\u7684\uff0c\u6269\u5bb9\u540e\u7684\u5bb9\u91cf\u4e5f\u53ef\u80fd\u8d85\u51fa\u5b9e\u9645\u9700\u6c42\u3002\u56e0\u6b64\uff0c\u57fa\u4e8e\u6570\u7ec4\u5b9e\u73b0\u7684\u6808\u53ef\u80fd\u9020\u6210\u4e00\u5b9a\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002</p> <p>\u7136\u800c\uff0c\u7531\u4e8e\u94fe\u8868\u8282\u70b9\u9700\u8981\u989d\u5916\u5b58\u50a8\u6307\u9488\uff0c\u56e0\u6b64\u94fe\u8868\u8282\u70b9\u5360\u7528\u7684\u7a7a\u95f4\u76f8\u5bf9\u8f83\u5927\u3002</p> <p>\u7efc\u4e0a\uff0c\u6211\u4eec\u4e0d\u80fd\u7b80\u5355\u5730\u786e\u5b9a\u54ea\u79cd\u5b9e\u73b0\u66f4\u52a0\u8282\u7701\u5185\u5b58\uff0c\u9700\u8981\u9488\u5bf9\u5177\u4f53\u60c5\u51b5\u8fdb\u884c\u5206\u6790\u3002</p>"},{"location":"chapter_stack_and_queue/stack/#514","title":"5.1.4 \u00a0 \u6808\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u6d4f\u89c8\u5668\u4e2d\u7684\u540e\u9000\u4e0e\u524d\u8fdb\u3001\u8f6f\u4ef6\u4e2d\u7684\u64a4\u9500\u4e0e\u53cd\u64a4\u9500\u3002\u6bcf\u5f53\u6211\u4eec\u6253\u5f00\u65b0\u7684\u7f51\u9875\uff0c\u6d4f\u89c8\u5668\u5c31\u4f1a\u5bf9\u4e0a\u4e00\u4e2a\u7f51\u9875\u6267\u884c\u5165\u6808\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u540e\u9000\u64cd\u4f5c\u56de\u5230\u4e0a\u4e00\u4e2a\u7f51\u9875\u3002\u540e\u9000\u64cd\u4f5c\u5b9e\u9645\u4e0a\u662f\u5728\u6267\u884c\u51fa\u6808\u3002\u5982\u679c\u8981\u540c\u65f6\u652f\u6301\u540e\u9000\u548c\u524d\u8fdb\uff0c\u90a3\u4e48\u9700\u8981\u4e24\u4e2a\u6808\u6765\u914d\u5408\u5b9e\u73b0\u3002</li> <li>\u7a0b\u5e8f\u5185\u5b58\u7ba1\u7406\u3002\u6bcf\u6b21\u8c03\u7528\u51fd\u6570\u65f6\uff0c\u7cfb\u7edf\u90fd\u4f1a\u5728\u6808\u9876\u6dfb\u52a0\u4e00\u4e2a\u6808\u5e27\uff0c\u7528\u4e8e\u8bb0\u5f55\u51fd\u6570\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002\u5728\u9012\u5f52\u51fd\u6570\u4e2d\uff0c\u5411\u4e0b\u9012\u63a8\u9636\u6bb5\u4f1a\u4e0d\u65ad\u6267\u884c\u5165\u6808\u64cd\u4f5c\uff0c\u800c\u5411\u4e0a\u56de\u6eaf\u9636\u6bb5\u5219\u4f1a\u4e0d\u65ad\u6267\u884c\u51fa\u6808\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/","title":"5.4 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_stack_and_queue/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u6808\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u540e\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u53ef\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002</li> <li>\u5728\u65f6\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u5177\u6709\u8f83\u9ad8\u7684\u5e73\u5747\u6548\u7387\uff0c\u4f46\u5728\u6269\u5bb9\u8fc7\u7a0b\u4e2d\uff0c\u5355\u6b21\u5165\u6808\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6808\u7684\u94fe\u8868\u5b9e\u73b0\u5177\u6709\u66f4\u4e3a\u7a33\u5b9a\u7684\u6548\u7387\u8868\u73b0\u3002</li> <li>\u5728\u7a7a\u95f4\u6548\u7387\u65b9\u9762\uff0c\u6808\u7684\u6570\u7ec4\u5b9e\u73b0\u53ef\u80fd\u5bfc\u81f4\u4e00\u5b9a\u7a0b\u5ea6\u7684\u7a7a\u95f4\u6d6a\u8d39\u3002\u4f46\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u94fe\u8868\u8282\u70b9\u6240\u5360\u7528\u7684\u5185\u5b58\u7a7a\u95f4\u6bd4\u6570\u7ec4\u5143\u7d20\u66f4\u5927\u3002</li> <li>\u961f\u5217\u662f\u4e00\u79cd\u9075\u5faa\u5148\u5165\u5148\u51fa\u539f\u5219\u7684\u6570\u636e\u7ed3\u6784\uff0c\u540c\u6837\u53ef\u4ee5\u901a\u8fc7\u6570\u7ec4\u6216\u94fe\u8868\u6765\u5b9e\u73b0\u3002\u5728\u65f6\u95f4\u6548\u7387\u548c\u7a7a\u95f4\u6548\u7387\u7684\u5bf9\u6bd4\u4e0a\uff0c\u961f\u5217\u7684\u7ed3\u8bba\u4e0e\u524d\u8ff0\u6808\u7684\u7ed3\u8bba\u76f8\u4f3c\u3002</li> <li>\u53cc\u5411\u961f\u5217\u662f\u4e00\u79cd\u5177\u6709\u66f4\u9ad8\u81ea\u7531\u5ea6\u7684\u961f\u5217\uff0c\u5b83\u5141\u8bb8\u5728\u4e24\u7aef\u8fdb\u884c\u5143\u7d20\u7684\u6dfb\u52a0\u548c\u5220\u9664\u64cd\u4f5c\u3002</li> </ul>"},{"location":"chapter_stack_and_queue/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u662f\u5426\u662f\u53cc\u5411\u94fe\u8868\u5b9e\u73b0\uff1f</p> <p>\u6d4f\u89c8\u5668\u7684\u524d\u8fdb\u540e\u9000\u529f\u80fd\u672c\u8d28\u4e0a\u662f\u201c\u6808\u201d\u7684\u4f53\u73b0\u3002\u5f53\u7528\u6237\u8bbf\u95ee\u4e00\u4e2a\u65b0\u9875\u9762\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u88ab\u6dfb\u52a0\u5230\u6808\u9876\uff1b\u5f53\u7528\u6237\u70b9\u51fb\u540e\u9000\u6309\u94ae\u65f6\uff0c\u8be5\u9875\u9762\u4f1a\u4ece\u6808\u9876\u5f39\u51fa\u3002\u4f7f\u7528\u53cc\u5411\u961f\u5217\u53ef\u4ee5\u65b9\u4fbf\u5730\u5b9e\u73b0\u4e00\u4e9b\u989d\u5916\u64cd\u4f5c\uff0c\u8fd9\u4e2a\u5728\u201c\u53cc\u5411\u961f\u5217\u201d\u7ae0\u8282\u6709\u63d0\u5230\u3002</p> <p>\u5728\u51fa\u6808\u540e\uff0c\u662f\u5426\u9700\u8981\u91ca\u653e\u51fa\u6808\u8282\u70b9\u7684\u5185\u5b58\uff1f</p> <p>\u5982\u679c\u540e\u7eed\u4ecd\u9700\u8981\u4f7f\u7528\u5f39\u51fa\u8282\u70b9\uff0c\u5219\u4e0d\u9700\u8981\u91ca\u653e\u5185\u5b58\u3002\u82e5\u4e4b\u540e\u4e0d\u9700\u8981\u7528\u5230\uff0c<code>Java</code> \u548c <code>Python</code> \u7b49\u8bed\u8a00\u62e5\u6709\u81ea\u52a8\u5783\u573e\u56de\u6536\u673a\u5236\uff0c\u56e0\u6b64\u4e0d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\uff1b\u5728 <code>C</code> \u548c <code>C++</code> \u4e2d\u9700\u8981\u624b\u52a8\u91ca\u653e\u5185\u5b58\u3002</p> <p>\u53cc\u5411\u961f\u5217\u50cf\u662f\u4e24\u4e2a\u6808\u62fc\u63a5\u5728\u4e86\u4e00\u8d77\uff0c\u5b83\u7684\u7528\u9014\u662f\u4ec0\u4e48\uff1f</p> <p>\u53cc\u5411\u961f\u5217\u5c31\u50cf\u662f\u6808\u548c\u961f\u5217\u7684\u7ec4\u5408\uff0c\u6216\u4e24\u4e2a\u6808\u62fc\u5728\u4e86\u4e00\u8d77\u3002\u5b83\u8868\u73b0\u7684\u662f\u6808 + \u961f\u5217\u7684\u903b\u8f91\uff0c\u56e0\u6b64\u53ef\u4ee5\u5b9e\u73b0\u6808\u4e0e\u961f\u5217\u7684\u6240\u6709\u5e94\u7528\uff0c\u5e76\u4e14\u66f4\u52a0\u7075\u6d3b\u3002</p> <p>\u64a4\u9500\uff08undo\uff09\u548c\u53cd\u64a4\u9500\uff08redo\uff09\u5177\u4f53\u662f\u5982\u4f55\u5b9e\u73b0\u7684\uff1f</p> <p>\u4f7f\u7528\u4e24\u4e2a\u5806\u6808\uff0c\u6808 <code>A</code> \u7528\u4e8e\u64a4\u9500\uff0c\u6808 <code>B</code> \u7528\u4e8e\u53cd\u64a4\u9500\u3002</p> <ol> <li>\u6bcf\u5f53\u7528\u6237\u6267\u884c\u4e00\u4e2a\u64cd\u4f5c\uff0c\u5c06\u8fd9\u4e2a\u64cd\u4f5c\u538b\u5165\u6808 <code>A</code> \uff0c\u5e76\u6e05\u7a7a\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>A</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>B</code> \u3002</li> <li>\u5f53\u7528\u6237\u6267\u884c\u201c\u53cd\u64a4\u9500\u201d\u65f6\uff0c\u4ece\u6808 <code>B</code> \u4e2d\u5f39\u51fa\u6700\u8fd1\u7684\u64cd\u4f5c\uff0c\u5e76\u5c06\u5176\u538b\u5165\u6808 <code>A</code> \u3002</li> </ol>"},{"location":"chapter_tree/","title":"\u7b2c 7 \u7ae0 \u00a0 \u6811","text":"<p>Abstract</p> <p>\u53c2\u5929\u5927\u6811\u5145\u6ee1\u751f\u547d\u529b\uff0c\u6839\u6df1\u53f6\u8302\uff0c\u5206\u679d\u6276\u758f\u3002</p> <p>\u5b83\u4e3a\u6211\u4eec\u5c55\u73b0\u4e86\u6570\u636e\u5206\u6cbb\u7684\u751f\u52a8\u5f62\u6001\u3002</p>"},{"location":"chapter_tree/#_1","title":"\u672c\u7ae0\u5185\u5bb9","text":"<ul> <li>7.1 \u00a0 \u4e8c\u53c9\u6811</li> <li>7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386</li> <li>7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a</li> <li>7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811</li> <li>7.5 \u00a0 AVL \u6811 *</li> <li>7.6 \u00a0 \u5c0f\u7ed3</li> </ul>"},{"location":"chapter_tree/array_representation_of_tree/","title":"7.3 \u00a0 \u4e8c\u53c9\u6811\u6570\u7ec4\u8868\u793a","text":"<p>\u5728\u94fe\u8868\u8868\u793a\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u5b58\u50a8\u5355\u5143\u4e3a\u8282\u70b9 <code>TreeNode</code> \uff0c\u8282\u70b9\u4e4b\u95f4\u901a\u8fc7\u6307\u9488\u76f8\u8fde\u63a5\u3002\u4e0a\u4e00\u8282\u4ecb\u7ecd\u4e86\u94fe\u8868\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7684\u5404\u9879\u57fa\u672c\u64cd\u4f5c\u3002</p> <p>\u90a3\u4e48\uff0c\u6211\u4eec\u80fd\u5426\u7528\u6570\u7ec4\u6765\u8868\u793a\u4e8c\u53c9\u6811\u5462\uff1f\u7b54\u6848\u662f\u80af\u5b9a\u7684\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#731","title":"7.3.1 \u00a0 \u8868\u793a\u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5148\u5206\u6790\u4e00\u4e2a\u7b80\u5355\u6848\u4f8b\u3002\u7ed9\u5b9a\u4e00\u68f5\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u6211\u4eec\u5c06\u6240\u6709\u8282\u70b9\u6309\u7167\u5c42\u5e8f\u904d\u5386\u7684\u987a\u5e8f\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u5219\u6bcf\u4e2a\u8282\u70b9\u90fd\u5bf9\u5e94\u552f\u4e00\u7684\u6570\u7ec4\u7d22\u5f15\u3002</p> <p>\u6839\u636e\u5c42\u5e8f\u904d\u5386\u7684\u7279\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7236\u8282\u70b9\u7d22\u5f15\u4e0e\u5b50\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u201c\u6620\u5c04\u516c\u5f0f\u201d\uff1a\u82e5\u67d0\u8282\u70b9\u7684\u7d22\u5f15\u4e3a \\(i\\) \uff0c\u5219\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 1\\) \uff0c\u53f3\u5b50\u8282\u70b9\u7d22\u5f15\u4e3a \\(2i + 2\\) \u3002\u56fe 7-12 \u5c55\u793a\u4e86\u5404\u4e2a\u8282\u70b9\u7d22\u5f15\u4e4b\u95f4\u7684\u6620\u5c04\u5173\u7cfb\u3002</p> <p></p> <p> \u56fe 7-12 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u6620\u5c04\u516c\u5f0f\u7684\u89d2\u8272\u76f8\u5f53\u4e8e\u94fe\u8868\u4e2d\u7684\u6307\u9488\u3002\u7ed9\u5b9a\u6570\u7ec4\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u8282\u70b9\uff0c\u6211\u4eec\u90fd\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\u516c\u5f0f\u6765\u8bbf\u95ee\u5b83\u7684\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/array_representation_of_tree/#732","title":"7.3.2 \u00a0 \u8868\u793a\u4efb\u610f\u4e8c\u53c9\u6811","text":"<p>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u4e00\u4e2a\u7279\u4f8b\uff0c\u5728\u4e8c\u53c9\u6811\u7684\u4e2d\u95f4\u5c42\u901a\u5e38\u5b58\u5728\u8bb8\u591a \\(\\text{None}\\) \u3002\u7531\u4e8e\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5e76\u4e0d\u5305\u542b\u8fd9\u4e9b \\(\\text{None}\\) \uff0c\u56e0\u6b64\u6211\u4eec\u65e0\u6cd5\u4ec5\u51ed\u8be5\u5e8f\u5217\u6765\u63a8\u6d4b \\(\\text{None}\\) \u7684\u6570\u91cf\u548c\u5206\u5e03\u4f4d\u7f6e\u3002\u8fd9\u610f\u5473\u7740\u5b58\u5728\u591a\u79cd\u4e8c\u53c9\u6811\u7ed3\u6784\u90fd\u7b26\u5408\u8be5\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</p> <p>\u5982\u56fe 7-13 \u6240\u793a\uff0c\u7ed9\u5b9a\u4e00\u68f5\u975e\u5b8c\u7f8e\u4e8c\u53c9\u6811\uff0c\u4e0a\u8ff0\u6570\u7ec4\u8868\u793a\u65b9\u6cd5\u5df2\u7ecf\u5931\u6548\u3002</p> <p></p> <p> \u56fe 7-13 \u00a0 \u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5bf9\u5e94\u591a\u79cd\u4e8c\u53c9\u6811\u53ef\u80fd\u6027 </p> <p>\u4e3a\u4e86\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u4e2d\u663e\u5f0f\u5730\u5199\u51fa\u6240\u6709 \\(\\text{None}\\) \u3002\u5982\u56fe 7-14 \u6240\u793a\uff0c\u8fd9\u6837\u5904\u7406\u540e\uff0c\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u5c31\u53ef\u4ee5\u552f\u4e00\u8868\u793a\u4e8c\u53c9\u6811\u4e86\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code># \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\n# \u4f7f\u7528 None \u6765\u8868\u793a\u7a7a\u4f4d\ntree = [1, 2, 3, 4, None, 6, 7, 8, 9, None, None, 12, None, None, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c INT_MAX \u6807\u8bb0\u7a7a\u4f4d\nvector&lt;int&gt; tree = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u7684\u5305\u88c5\u7c7b Integer \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nInteger[] tree = { 1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15 };\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nint?[] tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 any \u7c7b\u578b\u7684\u5207\u7247, \u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\ntree := []any{1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 Int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 nil \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree: [Int?] = [1, 2, 3, 4, nil, 6, 7, 8, 9, nil, nil, 12, nil, nil, 15]\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 null \u6765\u8868\u793a\u7a7a\u4f4d\nlet tree: (number | null)[] = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int? \u53ef\u7a7a\u7c7b\u578b \uff0c\u5c31\u53ef\u4ee5\u4f7f\u7528 null \u6765\u6807\u8bb0\u7a7a\u4f4d\nList&lt;int?&gt; tree = [1, 2, 3, 4, null, 6, 7, 8, 9, null, null, 12, null, null, 15];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 None \u6765\u6807\u8bb0\u7a7a\u4f4d\nlet tree = [Some(1), Some(2), Some(3), Some(4), None, Some(6), Some(7), Some(8), Some(9), None, None, Some(12), None, None, Some(15)];\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a */\n// \u4f7f\u7528 int \u6700\u5927\u503c\u6807\u8bb0\u7a7a\u4f4d\uff0c\u56e0\u6b64\u8981\u6c42\u8282\u70b9\u503c\u4e0d\u80fd\u4e3a INT_MAX\nint tree[] = {1, 2, 3, 4, INT_MAX, 6, 7, 8, 9, INT_MAX, INT_MAX, 12, INT_MAX, INT_MAX, 15};\n</code></pre> <pre><code>\n</code></pre> <p></p> <p> \u56fe 7-14 \u00a0 \u4efb\u610f\u7c7b\u578b\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u503c\u5f97\u8bf4\u660e\u7684\u662f\uff0c\u5b8c\u5168\u4e8c\u53c9\u6811\u975e\u5e38\u9002\u5408\u4f7f\u7528\u6570\u7ec4\u6765\u8868\u793a\u3002\u56de\u987e\u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u5b9a\u4e49\uff0c\\(\\text{None}\\) \u53ea\u51fa\u73b0\u5728\u6700\u5e95\u5c42\u4e14\u9760\u53f3\u7684\u4f4d\u7f6e\uff0c\u56e0\u6b64\u6240\u6709 \\(\\text{None}\\) \u4e00\u5b9a\u51fa\u73b0\u5728\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u4f7f\u7528\u6570\u7ec4\u8868\u793a\u5b8c\u5168\u4e8c\u53c9\u6811\u65f6\uff0c\u53ef\u4ee5\u7701\u7565\u5b58\u50a8\u6240\u6709 \\(\\text{None}\\) \uff0c\u975e\u5e38\u65b9\u4fbf\u3002\u56fe 7-15 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f8b\u5b50\u3002</p> <p></p> <p> \u56fe 7-15 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a </p> <p>\u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u4e00\u68f5\u57fa\u4e8e\u6570\u7ec4\u8868\u793a\u7684\u4e8c\u53c9\u6811\uff0c\u5305\u62ec\u4ee5\u4e0b\u51e0\u79cd\u64cd\u4f5c\u3002</p> <ul> <li>\u7ed9\u5b9a\u67d0\u8282\u70b9\uff0c\u83b7\u53d6\u5b83\u7684\u503c\u3001\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u3001\u7236\u8282\u70b9\u3002</li> <li>\u83b7\u53d6\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u3001\u540e\u5e8f\u904d\u5386\u3001\u5c42\u5e8f\u904d\u5386\u5e8f\u5217\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig array_binary_tree.py<pre><code>class ArrayBinaryTree:\n    \"\"\"\u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b\"\"\"\n\n    def __init__(self, arr: list[int | None]):\n        \"\"\"\u6784\u9020\u65b9\u6cd5\"\"\"\n        self._tree = list(arr)\n\n    def size(self):\n        \"\"\"\u8282\u70b9\u6570\u91cf\"\"\"\n        return len(self._tree)\n\n    def val(self, i: int) -&gt; int:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c\"\"\"\n        # \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 or i &gt;= self.size():\n            return None\n        return self._tree[i]\n\n    def left(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 1\n\n    def right(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return 2 * i + 2\n\n    def parent(self, i: int) -&gt; int | None:\n        \"\"\"\u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15\"\"\"\n        return (i - 1) // 2\n\n    def level_order(self) -&gt; list[int]:\n        \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        # \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in range(self.size()):\n            if self.val(i) is not None:\n                self.res.append(self.val(i))\n        return self.res\n\n    def dfs(self, i: int, order: str):\n        \"\"\"\u6df1\u5ea6\u4f18\u5148\u904d\u5386\"\"\"\n        if self.val(i) is None:\n            return\n        # \u524d\u5e8f\u904d\u5386\n        if order == \"pre\":\n            self.res.append(self.val(i))\n        self.dfs(self.left(i), order)\n        # \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\":\n            self.res.append(self.val(i))\n        self.dfs(self.right(i), order)\n        # \u540e\u5e8f\u904d\u5386\n        if order == \"post\":\n            self.res.append(self.val(i))\n\n    def pre_order(self) -&gt; list[int]:\n        \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"pre\")\n        return self.res\n\n    def in_order(self) -&gt; list[int]:\n        \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"in\")\n        return self.res\n\n    def post_order(self) -&gt; list[int]:\n        \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n        self.res = []\n        self.dfs(0, order=\"post\")\n        return self.res\n</code></pre> array_binary_tree.cpp<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  public:\n    /* \u6784\u9020\u65b9\u6cd5 */\n    ArrayBinaryTree(vector&lt;int&gt; arr) {\n        tree = arr;\n    }\n\n    /* \u8282\u70b9\u6570\u91cf */\n    int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    int val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return INT_MAX;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    int right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    int parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; levelOrder() {\n        vector&lt;int&gt; res;\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != INT_MAX)\n                res.push_back(val(i));\n        }\n        return res;\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; preOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; inOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    vector&lt;int&gt; postOrder() {\n        vector&lt;int&gt; res;\n        dfs(0, \"post\", res);\n        return res;\n    }\n\n  private:\n    vector&lt;int&gt; tree;\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void dfs(int i, string order, vector&lt;int&gt; &amp;res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == INT_MAX)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.push_back(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.push_back(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.push_back(val(i));\n    }\n};\n</code></pre> array_binary_tree.java<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private List&lt;Integer&gt; tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    public ArrayBinaryTree(List&lt;Integer&gt; arr) {\n        tree = new ArrayList&lt;&gt;(arr);\n    }\n\n    /* \u8282\u70b9\u6570\u91cf */\n    public int size() {\n        return tree.size();\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public Integer val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= size())\n            return null;\n        return tree.get(i);\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public Integer parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; levelOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; size(); i++) {\n            if (val(i) != null)\n                res.add(val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private void dfs(Integer i, String order, List&lt;Integer&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (val(i) == null)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (\"pre\".equals(order))\n            res.add(val(i));\n        dfs(left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (\"in\".equals(order))\n            res.add(val(i));\n        dfs(right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (\"post\".equals(order))\n            res.add(val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; preOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; inOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;Integer&gt; postOrder() {\n        List&lt;Integer&gt; res = new ArrayList&lt;&gt;();\n        dfs(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.cs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree(List&lt;int?&gt; arr) {\n    List&lt;int?&gt; tree = new(arr);\n\n    /* \u8282\u70b9\u6570\u91cf */\n    public int Size() {\n        return tree.Count;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    public int? Val(int i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= Size())\n            return null;\n        return tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Left(int i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Right(int i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    public int Parent(int i) {\n        return (i - 1) / 2;\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; LevelOrder() {\n        List&lt;int&gt; res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (int i = 0; i &lt; Size(); i++) {\n            if (Val(i).HasValue)\n                res.Add(Val(i)!.Value);\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    void DFS(int i, string order, List&lt;int&gt; res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (!Val(i).HasValue)\n            return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order == \"pre\")\n            res.Add(Val(i)!.Value);\n        DFS(Left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order == \"in\")\n            res.Add(Val(i)!.Value);\n        DFS(Right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order == \"post\")\n            res.Add(Val(i)!.Value);\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PreOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"pre\", res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; InOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"in\", res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    public List&lt;int&gt; PostOrder() {\n        List&lt;int&gt; res = [];\n        DFS(0, \"post\", res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.go<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\ntype arrayBinaryTree struct {\n    tree []any\n}\n\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc newArrayBinaryTree(arr []any) *arrayBinaryTree {\n    return &amp;arrayBinaryTree{\n        tree: arr,\n    }\n}\n\n/* \u8282\u70b9\u6570\u91cf */\nfunc (abt *arrayBinaryTree) size() int {\n    return len(abt.tree)\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nfunc (abt *arrayBinaryTree) val(i int) any {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if i &lt; 0 || i &gt;= abt.size() {\n        return nil\n    }\n    return abt.tree[i]\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) left(i int) int {\n    return 2*i + 1\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) right(i int) int {\n    return 2*i + 2\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\nfunc (abt *arrayBinaryTree) parent(i int) int {\n    return (i - 1) / 2\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) levelOrder() []any {\n    var res []any\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for i := 0; i &lt; abt.size(); i++ {\n        if abt.val(i) != nil {\n            res = append(res, abt.val(i))\n        }\n    }\n    return res\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nfunc (abt *arrayBinaryTree) dfs(i int, order string, res *[]any) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if abt.val(i) == nil {\n        return\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if order == \"pre\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.left(i), order, res)\n    // \u4e2d\u5e8f\u904d\u5386\n    if order == \"in\" {\n        *res = append(*res, abt.val(i))\n    }\n    abt.dfs(abt.right(i), order, res)\n    // \u540e\u5e8f\u904d\u5386\n    if order == \"post\" {\n        *res = append(*res, abt.val(i))\n    }\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) preOrder() []any {\n    var res []any\n    abt.dfs(0, \"pre\", &amp;res)\n    return res\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) inOrder() []any {\n    var res []any\n    abt.dfs(0, \"in\", &amp;res)\n    return res\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc (abt *arrayBinaryTree) postOrder() []any {\n    var res []any\n    abt.dfs(0, \"post\", &amp;res)\n    return res\n}\n</code></pre> array_binary_tree.swift<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    private var tree: [Int?]\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    init(arr: [Int?]) {\n        tree = arr\n    }\n\n    /* \u8282\u70b9\u6570\u91cf */\n    func size() -&gt; Int {\n        tree.count\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    func val(i: Int) -&gt; Int? {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= size() {\n            return nil\n        }\n        return tree[i]\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func left(i: Int) -&gt; Int {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    func right(i: Int) -&gt; Int {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    func parent(i: Int) -&gt; Int {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    func levelOrder() -&gt; [Int] {\n        var res: [Int] = []\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in stride(from: 0, to: size(), by: 1) {\n            if let val = val(i: i) {\n                res.append(val)\n            }\n        }\n        return res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    private func dfs(i: Int, order: String, res: inout [Int]) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        guard let val = val(i: i) else {\n            return\n        }\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.append(val)\n        }\n        dfs(i: left(i: i), order: order, res: &amp;res)\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.append(val)\n        }\n        dfs(i: right(i: i), order: order, res: &amp;res)\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.append(val)\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    func preOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"pre\", res: &amp;res)\n        return res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    func inOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"in\", res: &amp;res)\n        return res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    func postOrder() -&gt; [Int] {\n        var res: [Int] = []\n        dfs(i: 0, order: \"post\", res: &amp;res)\n        return res\n    }\n}\n</code></pre> array_binary_tree.js<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree;\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr) {\n        this.#tree = arr;\n    }\n\n    /* \u8282\u70b9\u6570\u91cf */\n    size() {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i) {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i) {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i) {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i) {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder() {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i, order, res) {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder() {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder() {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder() {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.ts<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n    #tree: (number | null)[];\n\n    /* \u6784\u9020\u65b9\u6cd5 */\n    constructor(arr: (number | null)[]) {\n        this.#tree = arr;\n    }\n\n    /* \u8282\u70b9\u6570\u91cf */\n    size(): number {\n        return this.#tree.length;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    val(i: number): number | null {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if (i &lt; 0 || i &gt;= this.size()) return null;\n        return this.#tree[i];\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    left(i: number): number {\n        return 2 * i + 1;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    right(i: number): number {\n        return 2 * i + 2;\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    parent(i: number): number {\n        return Math.floor((i - 1) / 2); // \u5411\u4e0b\u6574\u9664\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    levelOrder(): number[] {\n        let res = [];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for (let i = 0; i &lt; this.size(); i++) {\n            if (this.val(i) !== null) res.push(this.val(i));\n        }\n        return res;\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    #dfs(i: number, order: Order, res: (number | null)[]): void {\n        // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n        if (this.val(i) === null) return;\n        // \u524d\u5e8f\u904d\u5386\n        if (order === 'pre') res.push(this.val(i));\n        this.#dfs(this.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if (order === 'in') res.push(this.val(i));\n        this.#dfs(this.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if (order === 'post') res.push(this.val(i));\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    preOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'pre', res);\n        return res;\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    inOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'in', res);\n        return res;\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    postOrder(): (number | null)[] {\n        const res = [];\n        this.#dfs(0, 'post', res);\n        return res;\n    }\n}\n</code></pre> array_binary_tree.dart<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nclass ArrayBinaryTree {\n  late List&lt;int?&gt; _tree;\n\n  /* \u6784\u9020\u65b9\u6cd5 */\n  ArrayBinaryTree(this._tree);\n\n  /* \u8282\u70b9\u6570\u91cf */\n  int size() {\n    return _tree.length;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n  int? val(int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de null \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size()) {\n      return null;\n    }\n    return _tree[i];\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? left(int i) {\n    return 2 * i + 1;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? right(int i) {\n    return 2 * i + 2;\n  }\n\n  /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n  int? parent(int i) {\n    return (i - 1) ~/ 2;\n  }\n\n  /* \u5c42\u5e8f\u904d\u5386 */\n  List&lt;int&gt; levelOrder() {\n    List&lt;int&gt; res = [];\n    for (int i = 0; i &lt; size(); i++) {\n      if (val(i) != null) {\n        res.add(val(i)!);\n      }\n    }\n    return res;\n  }\n\n  /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n  void dfs(int i, String order, List&lt;int?&gt; res) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(i) == null) {\n      return;\n    }\n    // \u524d\u5e8f\u904d\u5386\n    if (order == 'pre') {\n      res.add(val(i));\n    }\n    dfs(left(i)!, order, res);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (order == 'in') {\n      res.add(val(i));\n    }\n    dfs(right(i)!, order, res);\n    // \u540e\u5e8f\u904d\u5386\n    if (order == 'post') {\n      res.add(val(i));\n    }\n  }\n\n  /* \u524d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; preOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'pre', res);\n    return res;\n  }\n\n  /* \u4e2d\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; inOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'in', res);\n    return res;\n  }\n\n  /* \u540e\u5e8f\u904d\u5386 */\n  List&lt;int?&gt; postOrder() {\n    List&lt;int?&gt; res = [];\n    dfs(0, 'post', res);\n    return res;\n  }\n}\n</code></pre> array_binary_tree.rs<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7c7b */\nstruct ArrayBinaryTree {\n    tree: Vec&lt;Option&lt;i32&gt;&gt;,\n}\n\nimpl ArrayBinaryTree {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(arr: Vec&lt;Option&lt;i32&gt;&gt;) -&gt; Self {\n        Self { tree: arr }\n    }\n\n    /* \u8282\u70b9\u6570\u91cf */\n    fn size(&amp;self) -&gt; i32 {\n        self.tree.len() as i32\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\n    fn val(&amp;self, i: i32) -&gt; Option&lt;i32&gt; {\n        // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de None \uff0c\u4ee3\u8868\u7a7a\u4f4d\n        if i &lt; 0 || i &gt;= self.size() {\n            None\n        } else {\n            self.tree[i as usize]\n        }\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn left(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 1\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u53f3\u5b50\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn right(&amp;self, i: i32) -&gt; i32 {\n        2 * i + 2\n    }\n\n    /* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u7236\u8282\u70b9\u7684\u7d22\u5f15 */\n    fn parent(&amp;self, i: i32) -&gt; i32 {\n        (i - 1) / 2\n    }\n\n    /* \u5c42\u5e8f\u904d\u5386 */\n    fn level_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n        for i in 0..self.size() {\n            if let Some(val) = self.val(i) {\n                res.push(val)\n            }\n        }\n        res\n    }\n\n    /* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\n    fn dfs(&amp;self, i: i32, order: &amp;str, res: &amp;mut Vec&lt;i32&gt;) {\n        if self.val(i).is_none() {\n            return;\n        }\n        let val = self.val(i).unwrap();\n        // \u524d\u5e8f\u904d\u5386\n        if order == \"pre\" {\n            res.push(val);\n        }\n        self.dfs(self.left(i), order, res);\n        // \u4e2d\u5e8f\u904d\u5386\n        if order == \"in\" {\n            res.push(val);\n        }\n        self.dfs(self.right(i), order, res);\n        // \u540e\u5e8f\u904d\u5386\n        if order == \"post\" {\n            res.push(val);\n        }\n    }\n\n    /* \u524d\u5e8f\u904d\u5386 */\n    fn pre_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"pre\", &amp;mut res);\n        res\n    }\n\n    /* \u4e2d\u5e8f\u904d\u5386 */\n    fn in_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"in\", &amp;mut res);\n        res\n    }\n\n    /* \u540e\u5e8f\u904d\u5386 */\n    fn post_order(&amp;self) -&gt; Vec&lt;i32&gt; {\n        let mut res = vec![];\n        self.dfs(0, \"post\", &amp;mut res);\n        res\n    }\n}\n</code></pre> array_binary_tree.c<pre><code>/* \u6570\u7ec4\u8868\u793a\u4e0b\u7684\u4e8c\u53c9\u6811\u7ed3\u6784\u4f53 */\ntypedef struct {\n    int *tree;\n    int size;\n} ArrayBinaryTree;\n\n/* \u6784\u9020\u51fd\u6570 */\nArrayBinaryTree *newArrayBinaryTree(int *arr, int arrSize) {\n    ArrayBinaryTree *abt = (ArrayBinaryTree *)malloc(sizeof(ArrayBinaryTree));\n    abt-&gt;tree = malloc(sizeof(int) * arrSize);\n    memcpy(abt-&gt;tree, arr, sizeof(int) * arrSize);\n    abt-&gt;size = arrSize;\n    return abt;\n}\n\n/* \u6790\u6784\u51fd\u6570 */\nvoid delArrayBinaryTree(ArrayBinaryTree *abt) {\n    free(abt-&gt;tree);\n    free(abt);\n}\n\n/* \u8282\u70b9\u6570\u91cf */\nint size(ArrayBinaryTree *abt) {\n    return abt-&gt;size;\n}\n\n/* \u83b7\u53d6\u7d22\u5f15\u4e3a i \u8282\u70b9\u7684\u503c */\nint val(ArrayBinaryTree *abt, int i) {\n    // \u82e5\u7d22\u5f15\u8d8a\u754c\uff0c\u5219\u8fd4\u56de INT_MAX \uff0c\u4ee3\u8868\u7a7a\u4f4d\n    if (i &lt; 0 || i &gt;= size(abt))\n        return INT_MAX;\n    return abt-&gt;tree[i];\n}\n\n/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    // \u76f4\u63a5\u904d\u5386\u6570\u7ec4\n    for (int i = 0; i &lt; size(abt); i++) {\n        if (val(abt, i) != INT_MAX)\n            res[index++] = val(abt, i);\n    }\n    *returnSize = index;\n    return res;\n}\n\n/* \u6df1\u5ea6\u4f18\u5148\u904d\u5386 */\nvoid dfs(ArrayBinaryTree *abt, int i, char *order, int *res, int *index) {\n    // \u82e5\u4e3a\u7a7a\u4f4d\uff0c\u5219\u8fd4\u56de\n    if (val(abt, i) == INT_MAX)\n        return;\n    // \u524d\u5e8f\u904d\u5386\n    if (strcmp(order, \"pre\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, left(i), order, res, index);\n    // \u4e2d\u5e8f\u904d\u5386\n    if (strcmp(order, \"in\") == 0)\n        res[(*index)++] = val(abt, i);\n    dfs(abt, right(i), order, res, index);\n    // \u540e\u5e8f\u904d\u5386\n    if (strcmp(order, \"post\") == 0)\n        res[(*index)++] = val(abt, i);\n}\n\n/* \u524d\u5e8f\u904d\u5386 */\nint *preOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"pre\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nint *inOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"in\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nint *postOrder(ArrayBinaryTree *abt, int *returnSize) {\n    int *res = (int *)malloc(sizeof(int) * size(abt));\n    int index = 0;\n    dfs(abt, 0, \"post\", res, &amp;index);\n    *returnSize = index;\n    return res;\n}\n</code></pre> array_binary_tree.zig<pre><code>[class]{ArrayBinaryTree}-[func]{}\n</code></pre>"},{"location":"chapter_tree/array_representation_of_tree/#733","title":"7.3.3 \u00a0 \u4f18\u70b9\u4e0e\u5c40\u9650\u6027","text":"<p>\u4e8c\u53c9\u6811\u7684\u6570\u7ec4\u8868\u793a\u4e3b\u8981\u6709\u4ee5\u4e0b\u4f18\u70b9\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u5728\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4\u4e2d\uff0c\u5bf9\u7f13\u5b58\u53cb\u597d\uff0c\u8bbf\u95ee\u4e0e\u904d\u5386\u901f\u5ea6\u8f83\u5feb\u3002</li> <li>\u4e0d\u9700\u8981\u5b58\u50a8\u6307\u9488\uff0c\u6bd4\u8f83\u8282\u7701\u7a7a\u95f4\u3002</li> <li>\u5141\u8bb8\u968f\u673a\u8bbf\u95ee\u8282\u70b9\u3002</li> </ul> <p>\u7136\u800c\uff0c\u6570\u7ec4\u8868\u793a\u4e5f\u5b58\u5728\u4e00\u4e9b\u5c40\u9650\u6027\u3002</p> <ul> <li>\u6570\u7ec4\u5b58\u50a8\u9700\u8981\u8fde\u7eed\u5185\u5b58\u7a7a\u95f4\uff0c\u56e0\u6b64\u4e0d\u9002\u5408\u5b58\u50a8\u6570\u636e\u91cf\u8fc7\u5927\u7684\u6811\u3002</li> <li>\u589e\u5220\u8282\u70b9\u9700\u8981\u901a\u8fc7\u6570\u7ec4\u63d2\u5165\u4e0e\u5220\u9664\u64cd\u4f5c\u5b9e\u73b0\uff0c\u6548\u7387\u8f83\u4f4e\u3002</li> <li>\u5f53\u4e8c\u53c9\u6811\u4e2d\u5b58\u5728\u5927\u91cf \\(\\text{None}\\) \u65f6\uff0c\u6570\u7ec4\u4e2d\u5305\u542b\u7684\u8282\u70b9\u6570\u636e\u6bd4\u91cd\u8f83\u4f4e\uff0c\u7a7a\u95f4\u5229\u7528\u7387\u8f83\u4f4e\u3002</li> </ul>"},{"location":"chapter_tree/avl_tree/","title":"7.5 \u00a0 AVL \u6811 *","text":"<p>\u5728\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u63d0\u5230\uff0c\u5728\u591a\u6b21\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u53ef\u80fd\u9000\u5316\u4e3a\u94fe\u8868\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u6240\u6709\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c06\u4ece \\(O(\\log n)\\) \u6076\u5316\u4e3a \\(O(n)\\) \u3002</p> <p>\u5982\u56fe 7-24 \u6240\u793a\uff0c\u7ecf\u8fc7\u4e24\u6b21\u5220\u9664\u8282\u70b9\u64cd\u4f5c\uff0c\u8fd9\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\u4fbf\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u3002</p> <p></p> <p> \u56fe 7-24 \u00a0 AVL \u6811\u5728\u5220\u9664\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>\u518d\u4f8b\u5982\uff0c\u5728\u56fe 7-25 \u6240\u793a\u7684\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e24\u4e2a\u8282\u70b9\u540e\uff0c\u6811\u5c06\u4e25\u91cd\u5411\u5de6\u503e\u659c\uff0c\u67e5\u627e\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u968f\u4e4b\u6076\u5316\u3002</p> <p></p> <p> \u56fe 7-25 \u00a0 AVL \u6811\u5728\u63d2\u5165\u8282\u70b9\u540e\u53d1\u751f\u9000\u5316 </p> <p>1962 \u5e74 G. M. Adelson-Velsky \u548c E. M. Landis \u5728\u8bba\u6587 \"An algorithm for the organization of information\" \u4e2d\u63d0\u51fa\u4e86\u300cAVL \u6811\u300d\u3002\u8bba\u6587\u4e2d\u8be6\u7ec6\u63cf\u8ff0\u4e86\u4e00\u7cfb\u5217\u64cd\u4f5c\uff0c\u786e\u4fdd\u5728\u6301\u7eed\u6dfb\u52a0\u548c\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4e0d\u4f1a\u9000\u5316\uff0c\u4ece\u800c\u4f7f\u5f97\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4fdd\u6301\u5728 \\(O(\\log n)\\) \u7ea7\u522b\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u9700\u8981\u9891\u7e41\u8fdb\u884c\u589e\u5220\u67e5\u6539\u64cd\u4f5c\u7684\u573a\u666f\u4e2d\uff0cAVL \u6811\u80fd\u59cb\u7ec8\u4fdd\u6301\u9ad8\u6548\u7684\u6570\u636e\u64cd\u4f5c\u6027\u80fd\uff0c\u5177\u6709\u5f88\u597d\u7684\u5e94\u7528\u4ef7\u503c\u3002</p>"},{"location":"chapter_tree/avl_tree/#751-avl","title":"7.5.1 \u00a0 AVL \u6811\u5e38\u89c1\u672f\u8bed","text":"<p>AVL \u6811\u65e2\u662f\u4e8c\u53c9\u641c\u7d22\u6811\u4e5f\u662f\u5e73\u8861\u4e8c\u53c9\u6811\uff0c\u540c\u65f6\u6ee1\u8db3\u8fd9\u4e24\u7c7b\u4e8c\u53c9\u6811\u7684\u6240\u6709\u6027\u8d28\uff0c\u56e0\u6b64\u4e5f\u88ab\u79f0\u4e3a\u300c\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811 balanced binary search tree\u300d\u3002</p>"},{"location":"chapter_tree/avl_tree/#1","title":"1. \u00a0 \u8282\u70b9\u9ad8\u5ea6","text":"<p>\u7531\u4e8e AVL \u6811\u7684\u76f8\u5173\u64cd\u4f5c\u9700\u8981\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e3a\u8282\u70b9\u7c7b\u6dfb\u52a0 <code>height</code> \u53d8\u91cf\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class TreeNode:\n    \"\"\"AVL \u6811\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                 # \u8282\u70b9\u503c\n        self.height: int = 0                # \u8282\u70b9\u9ad8\u5ea6\n        self.left: TreeNode | None = None   # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        self.right: TreeNode | None = None  # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nstruct TreeNode {\n    int val{};          // \u8282\u70b9\u503c\n    int height = 0;     // \u8282\u70b9\u9ad8\u5ea6\n    TreeNode *left{};   // \u5de6\u5b50\u8282\u70b9\n    TreeNode *right{};  // \u53f3\u5b50\u8282\u70b9\n    TreeNode() = default;\n    explicit TreeNode(int x) : val(x){}\n};\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    public int val;        // \u8282\u70b9\u503c\n    public int height;     // \u8282\u70b9\u9ad8\u5ea6\n    public TreeNode left;  // \u5de6\u5b50\u8282\u70b9\n    public TreeNode right; // \u53f3\u5b50\u8282\u70b9\n    public TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u8282\u70b9\u503c\n    public int height;      // \u8282\u70b9\u9ad8\u5ea6\n    public TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n    Val    int       // \u8282\u70b9\u503c\n    Height int       // \u8282\u70b9\u9ad8\u5ea6\n    Left   *TreeNode // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    Right  *TreeNode // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    var val: Int // \u8282\u70b9\u503c\n    var height: Int // \u8282\u70b9\u9ad8\u5ea6\n    var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\n    var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\n\n    init(x: Int) {\n        val = x\n        height = 0\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val; // \u8282\u70b9\u503c\n    height; //\u8282\u70b9\u9ad8\u5ea6\n    left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val, left, right, height) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val: number;            // \u8282\u70b9\u503c\n    height: number;         // \u8282\u70b9\u9ad8\u5ea6\n    left: TreeNode | null;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right: TreeNode | null; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val;\n        this.height = height === undefined ? 0 : height; \n        this.left = left === undefined ? null : left; \n        this.right = right === undefined ? null : right; \n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n  int val;         // \u8282\u70b9\u503c\n  int height;      // \u8282\u70b9\u9ad8\u5ea6\n  TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\n  TreeNode? right; // \u53f3\u5b50\u8282\u70b9\n  TreeNode(this.val, [this.height = 0, this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    val: i32,                               // \u8282\u70b9\u503c\n    height: i32,                            // \u8282\u70b9\u9ad8\u5ea6\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            height: 0,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* AVL \u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nTreeNode struct TreeNode {\n    int val;\n    int height;\n    struct TreeNode *left;\n    struct TreeNode *right;\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u201c\u8282\u70b9\u9ad8\u5ea6\u201d\u662f\u6307\u4ece\u8be5\u8282\u70b9\u5230\u5176\u6700\u8fdc\u53f6\u8282\u70b9\u7684\u8ddd\u79bb\uff0c\u5373\u6240\u7ecf\u8fc7\u7684\u201c\u8fb9\u201d\u7684\u6570\u91cf\u3002\u9700\u8981\u7279\u522b\u6ce8\u610f\u7684\u662f\uff0c\u53f6\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(0\\) \uff0c\u800c\u7a7a\u8282\u70b9\u7684\u9ad8\u5ea6\u4e3a \\(-1\\) \u3002\u6211\u4eec\u5c06\u521b\u5efa\u4e24\u4e2a\u5de5\u5177\u51fd\u6570\uff0c\u5206\u522b\u7528\u4e8e\u83b7\u53d6\u548c\u66f4\u65b0\u8282\u70b9\u7684\u9ad8\u5ea6\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def height(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node is not None:\n        return node.height\n    return -1\n\ndef update_height(self, node: TreeNode | None):\n    \"\"\"\u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\"\"\"\n    # \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = max([self.height(node.left), self.height(node.right)]) + 1\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == nullptr ? -1 : node-&gt;height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node-&gt;height = max(height(node-&gt;left), height(node-&gt;right)) + 1;\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint Height(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid UpdateHeight(TreeNode node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height = Math.Max(Height(node.left), Height(node.right)) + 1;\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) height(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if node != nil {\n        return node.Height\n    }\n    return -1\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc (t *aVLTree) updateHeight(node *TreeNode) {\n    lh := t.height(node.Left)\n    rh := t.height(node.Right)\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if lh &gt; rh {\n        node.Height = lh + 1\n    } else {\n        node.Height = rh + 1\n    }\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfunc height(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    node == nil ? -1 : node!.height\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfunc updateHeight(node: TreeNode?) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node?.height = max(height(node: node?.left), height(node: node?.right)) + 1\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\n#updateHeight(node) {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nheight(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return node === null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nupdateHeight(node: TreeNode): void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.height =\n        Math.max(this.height(node.left), this.height(node.right)) + 1;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n  return node == null ? -1 : node.height;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode? node) {\n  // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n  node!.height = max(height(node.left), height(node.right)) + 1;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nfn height(node: OptionTreeNodeRc) -&gt; i32 {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    match node {\n        Some(node) =&gt; node.borrow().height,\n        None =&gt; -1,\n    }\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nfn update_height(node: OptionTreeNodeRc) {\n    if let Some(node) = node {\n        let left = node.borrow().left.clone();\n        let right = node.borrow().right.clone();\n        // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n        node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6 */\nint height(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    if (node != NULL) {\n        return node-&gt;height;\n    }\n    return -1;\n}\n\n/* \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6 */\nvoid updateHeight(TreeNode *node) {\n    int lh = height(node-&gt;left);\n    int rh = height(node-&gt;right);\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    if (lh &gt; rh) {\n        node-&gt;height = lh + 1;\n    } else {\n        node-&gt;height = rh + 1;\n    }\n}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u8282\u70b9\u9ad8\u5ea6\nfn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    _ = self;\n    // \u7a7a\u8282\u70b9\u9ad8\u5ea6\u4e3a -1 \uff0c\u53f6\u8282\u70b9\u9ad8\u5ea6\u4e3a 0\n    return if (node == null) -1 else node.?.height;\n}\n\n// \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\nfn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {\n    // \u8282\u70b9\u9ad8\u5ea6\u7b49\u4e8e\u6700\u9ad8\u5b50\u6811\u9ad8\u5ea6 + 1\n    node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2","title":"2. \u00a0 \u8282\u70b9\u5e73\u8861\u56e0\u5b50","text":"<p>\u8282\u70b9\u7684\u300c\u5e73\u8861\u56e0\u5b50 balance factor\u300d\u5b9a\u4e49\u4e3a\u8282\u70b9\u5de6\u5b50\u6811\u7684\u9ad8\u5ea6\u51cf\u53bb\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\uff0c\u540c\u65f6\u89c4\u5b9a\u7a7a\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4e3a \\(0\\) \u3002\u6211\u4eec\u540c\u6837\u5c06\u83b7\u53d6\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u7684\u529f\u80fd\u5c01\u88c5\u6210\u51fd\u6570\uff0c\u65b9\u4fbf\u540e\u7eed\u4f7f\u7528\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def balance_factor(self, node: TreeNode | None) -&gt; int:\n    \"\"\"\u83b7\u53d6\u5e73\u8861\u56e0\u5b50\"\"\"\n    # \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node is None:\n        return 0\n    # \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.left) - self.height(node.right)\n</code></pre> avl_tree.cpp<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == nullptr)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.java<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null)\n        return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.cs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint BalanceFactor(TreeNode? node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return Height(node.left) - Height(node.right);\n}\n</code></pre> avl_tree.go<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc (t *aVLTree) balanceFactor(node *TreeNode) int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if node == nil {\n        return 0\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return t.height(node.Left) - t.height(node.Right)\n}\n</code></pre> avl_tree.swift<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfunc balanceFactor(node: TreeNode?) -&gt; Int {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    guard let node = node else { return 0 }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node: node.left) - height(node: node.right)\n}\n</code></pre> avl_tree.js<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.ts<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nbalanceFactor(node: TreeNode): number {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node === null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return this.height(node.left) - this.height(node.right);\n}\n</code></pre> avl_tree.dart<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode? node) {\n  // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n  if (node == null) return 0;\n  // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n  return height(node.left) - height(node.right);\n}\n</code></pre> avl_tree.rs<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nfn balance_factor(node: OptionTreeNodeRc) -&gt; i32 {\n    match node {\n        // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n        None =&gt; 0,\n        // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n        Some(node) =&gt; {\n            Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())\n        }\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u83b7\u53d6\u5e73\u8861\u56e0\u5b50 */\nint balanceFactor(TreeNode *node) {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == NULL) {\n        return 0;\n    }\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return height(node-&gt;left) - height(node-&gt;right);\n}\n</code></pre> avl_tree.zig<pre><code>// \u83b7\u53d6\u5e73\u8861\u56e0\u5b50\nfn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {\n    // \u7a7a\u8282\u70b9\u5e73\u8861\u56e0\u5b50\u4e3a 0\n    if (node == null) return 0;\n    // \u8282\u70b9\u5e73\u8861\u56e0\u5b50 = \u5de6\u5b50\u6811\u9ad8\u5ea6 - \u53f3\u5b50\u6811\u9ad8\u5ea6\n    return self.height(node.?.left) - self.height(node.?.right);\n}\n</code></pre> <p>Note</p> <p>\u8bbe\u5e73\u8861\u56e0\u5b50\u4e3a \\(f\\) \uff0c\u5219\u4e00\u68f5 AVL \u6811\u7684\u4efb\u610f\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7686\u6ee1\u8db3 \\(-1 \\le f \\le 1\\) \u3002</p>"},{"location":"chapter_tree/avl_tree/#752-avl","title":"7.5.2 \u00a0 AVL \u6811\u65cb\u8f6c","text":"<p>AVL \u6811\u7684\u7279\u70b9\u5728\u4e8e\u201c\u65cb\u8f6c\u201d\u64cd\u4f5c\uff0c\u5b83\u80fd\u591f\u5728\u4e0d\u5f71\u54cd\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u7684\u524d\u63d0\u4e0b\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u65cb\u8f6c\u64cd\u4f5c\u65e2\u80fd\u4fdd\u6301\u201c\u4e8c\u53c9\u641c\u7d22\u6811\u201d\u7684\u6027\u8d28\uff0c\u4e5f\u80fd\u4f7f\u6811\u91cd\u65b0\u53d8\u4e3a\u201c\u5e73\u8861\u4e8c\u53c9\u6811\u201d\u3002</p> <p>\u6211\u4eec\u5c06\u5e73\u8861\u56e0\u5b50\u7edd\u5bf9\u503c \\(&gt; 1\\) \u7684\u8282\u70b9\u79f0\u4e3a\u201c\u5931\u8861\u8282\u70b9\u201d\u3002\u6839\u636e\u8282\u70b9\u5931\u8861\u60c5\u51b5\u7684\u4e0d\u540c\uff0c\u65cb\u8f6c\u64cd\u4f5c\u5206\u4e3a\u56db\u79cd\uff1a\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3002\u4e0b\u9762\u8be6\u7ec6\u4ecb\u7ecd\u8fd9\u4e9b\u65cb\u8f6c\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/avl_tree/#1_1","title":"1. \u00a0 \u53f3\u65cb","text":"<p>\u5982\u56fe 7-26 \u6240\u793a\uff0c\u8282\u70b9\u4e0b\u65b9\u4e3a\u5e73\u8861\u56e0\u5b50\u3002\u4ece\u5e95\u81f3\u9876\u770b\uff0c\u4e8c\u53c9\u6811\u4e2d\u9996\u4e2a\u5931\u8861\u8282\u70b9\u662f\u201c\u8282\u70b9 3\u201d\u3002\u6211\u4eec\u5173\u6ce8\u4ee5\u8be5\u5931\u8861\u8282\u70b9\u4e3a\u6839\u8282\u70b9\u7684\u5b50\u6811\uff0c\u5c06\u8be5\u8282\u70b9\u8bb0\u4e3a <code>node</code> \uff0c\u5176\u5de6\u5b50\u8282\u70b9\u8bb0\u4e3a <code>child</code> \uff0c\u6267\u884c\u201c\u53f3\u65cb\u201d\u64cd\u4f5c\u3002\u5b8c\u6210\u53f3\u65cb\u540e\uff0c\u5b50\u6811\u6062\u590d\u5e73\u8861\uff0c\u5e76\u4e14\u4ecd\u7136\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u3002</p> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-26 \u00a0 \u53f3\u65cb\u64cd\u4f5c\u6b65\u9aa4 </p> <p>\u5982\u56fe 7-27 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u53f3\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grandChild</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u53f3\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grandChild</code> \u4f5c\u4e3a <code>node</code> \u7684\u5de6\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-27 \u00a0 \u6709 grandChild \u7684\u53f3\u65cb\u64cd\u4f5c </p> <p>\u201c\u5411\u53f3\u65cb\u8f6c\u201d\u662f\u4e00\u79cd\u5f62\u8c61\u5316\u7684\u8bf4\u6cd5\uff0c\u5b9e\u9645\u4e0a\u9700\u8981\u901a\u8fc7\u4fee\u6539\u8282\u70b9\u6307\u9488\u6765\u5b9e\u73b0\uff0c\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def right_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u53f3\u65cb\u64cd\u4f5c\"\"\"\n    child = node.left\n    grand_child = child.right\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node\n    node.left = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;left;\n    TreeNode *grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode rightRotate(TreeNode node) {\n    TreeNode child = node.left;\n    TreeNode grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? RightRotate(TreeNode? node) {\n    TreeNode? child = node?.left;\n    TreeNode? grandChild = child?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {\n    child := node.Left\n    grandChild := child.Right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.Right = node\n    node.Left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfunc rightRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.left\n    let grandChild = child?.right\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child?.right = node\n    node?.left = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\n#rightRotate(node) {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nrightRotate(node: TreeNode): TreeNode {\n    const child = node.left;\n    const grandChild = child.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.right = node;\n    node.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode? rightRotate(TreeNode? node) {\n  TreeNode? child = node!.left;\n  TreeNode? grandChild = child!.right;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n  child.right = node;\n  node.left = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nfn right_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().left.clone().unwrap();\n            let grand_child = child.borrow().right.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n            child.borrow_mut().right = Some(node.clone());\n            node.borrow_mut().left = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u53f3\u65cb\u64cd\u4f5c */\nTreeNode *rightRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;left;\n    grandChild = child-&gt;right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child-&gt;right = node;\n    node-&gt;left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.zig<pre><code>// \u53f3\u65cb\u64cd\u4f5c\nfn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.left;\n    var grandChild = child.?.right;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u53f3\u65cb\u8f6c\n    child.?.right = node;\n    node.?.left = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_1","title":"2. \u00a0 \u5de6\u65cb","text":"<p>\u76f8\u5e94\u5730\uff0c\u5982\u679c\u8003\u8651\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u201c\u955c\u50cf\u201d\uff0c\u5219\u9700\u8981\u6267\u884c\u56fe 7-28 \u6240\u793a\u7684\u201c\u5de6\u65cb\u201d\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-28 \u00a0 \u5de6\u65cb\u64cd\u4f5c </p> <p>\u540c\u7406\uff0c\u5982\u56fe 7-29 \u6240\u793a\uff0c\u5f53\u8282\u70b9 <code>child</code> \u6709\u5de6\u5b50\u8282\u70b9\uff08\u8bb0\u4e3a <code>grandChild</code> \uff09\u65f6\uff0c\u9700\u8981\u5728\u5de6\u65cb\u4e2d\u6dfb\u52a0\u4e00\u6b65\uff1a\u5c06 <code>grandChild</code> \u4f5c\u4e3a <code>node</code> \u7684\u53f3\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-29 \u00a0 \u6709 grandChild \u7684\u5de6\u65cb\u64cd\u4f5c </p> <p>\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u53f3\u65cb\u548c\u5de6\u65cb\u64cd\u4f5c\u5728\u903b\u8f91\u4e0a\u662f\u955c\u50cf\u5bf9\u79f0\u7684\uff0c\u5b83\u4eec\u5206\u522b\u89e3\u51b3\u7684\u4e24\u79cd\u5931\u8861\u60c5\u51b5\u4e5f\u662f\u5bf9\u79f0\u7684\u3002\u57fa\u4e8e\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u53ea\u9700\u5c06\u53f3\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\u4e2d\u7684\u6240\u6709\u7684 <code>left</code> \u66ff\u6362\u4e3a <code>right</code> \uff0c\u5c06\u6240\u6709\u7684 <code>right</code> \u66ff\u6362\u4e3a <code>left</code> \uff0c\u5373\u53ef\u5f97\u5230\u5de6\u65cb\u7684\u5b9e\u73b0\u4ee3\u7801\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def left_rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u5de6\u65cb\u64cd\u4f5c\"\"\"\n    child = node.right\n    grand_child = child.left\n    # \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node\n    node.right = grand_child\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    self.update_height(child)\n    # \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n</code></pre> avl_tree.cpp<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child = node-&gt;right;\n    TreeNode *grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode leftRotate(TreeNode node) {\n    TreeNode child = node.right;\n    TreeNode grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? LeftRotate(TreeNode? node) {\n    TreeNode? child = node?.right;\n    TreeNode? grandChild = child?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    UpdateHeight(node);\n    UpdateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {\n    child := node.Right\n    grandChild := child.Left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.Left = node\n    node.Right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    t.updateHeight(child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfunc leftRotate(node: TreeNode?) -&gt; TreeNode? {\n    let child = node?.right\n    let grandChild = child?.left\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child?.left = node\n    node?.right = grandChild\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node: node)\n    updateHeight(node: child)\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child\n}\n</code></pre> avl_tree.js<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\n#leftRotate(node) {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.#updateHeight(node);\n    this.#updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nleftRotate(node: TreeNode): TreeNode {\n    const child = node.right;\n    const grandChild = child.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.left = node;\n    node.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    this.updateHeight(node);\n    this.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode? leftRotate(TreeNode? node) {\n  TreeNode? child = node!.right;\n  TreeNode? grandChild = child!.left;\n  // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n  child.left = node;\n  node.right = grandChild;\n  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  updateHeight(node);\n  updateHeight(child);\n  // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return child;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nfn left_rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(node) =&gt; {\n            let child = node.borrow().right.clone().unwrap();\n            let grand_child = child.borrow().left.clone();\n            // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n            child.borrow_mut().left = Some(node.clone());\n            node.borrow_mut().right = grand_child;\n            // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            Self::update_height(Some(node));\n            Self::update_height(Some(child.clone()));\n            // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(child)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5de6\u65cb\u64cd\u4f5c */\nTreeNode *leftRotate(TreeNode *node) {\n    TreeNode *child, *grandChild;\n    child = node-&gt;right;\n    grandChild = child-&gt;left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child-&gt;left = node;\n    node-&gt;right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre> avl_tree.zig<pre><code>// \u5de6\u65cb\u64cd\u4f5c\nfn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    var child = node.?.right;\n    var grandChild = child.?.left;\n    // \u4ee5 child \u4e3a\u539f\u70b9\uff0c\u5c06 node \u5411\u5de6\u65cb\u8f6c\n    child.?.left = node;\n    node.?.right = grandChild;\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.updateHeight(node);\n    self.updateHeight(child);\n    // \u8fd4\u56de\u65cb\u8f6c\u540e\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return child;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3","title":"3. \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb","text":"<p>\u5bf9\u4e8e\u56fe 7-30 \u4e2d\u7684\u5931\u8861\u8282\u70b9 3 \uff0c\u4ec5\u4f7f\u7528\u5de6\u65cb\u6216\u53f3\u65cb\u90fd\u65e0\u6cd5\u4f7f\u5b50\u6811\u6062\u590d\u5e73\u8861\u3002\u6b64\u65f6\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-30 \u00a0 \u5148\u5de6\u65cb\u540e\u53f3\u65cb </p>"},{"location":"chapter_tree/avl_tree/#4","title":"4. \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb","text":"<p>\u5982\u56fe 7-31 \u6240\u793a\uff0c\u5bf9\u4e8e\u4e0a\u8ff0\u5931\u8861\u4e8c\u53c9\u6811\u7684\u955c\u50cf\u60c5\u51b5\uff0c\u9700\u8981\u5148\u5bf9 <code>child</code> \u6267\u884c\u201c\u53f3\u65cb\u201d\uff0c\u518d\u5bf9 <code>node</code> \u6267\u884c\u201c\u5de6\u65cb\u201d\u3002</p> <p></p> <p> \u56fe 7-31 \u00a0 \u5148\u53f3\u65cb\u540e\u5de6\u65cb </p>"},{"location":"chapter_tree/avl_tree/#5","title":"5. \u00a0 \u65cb\u8f6c\u7684\u9009\u62e9","text":"<p>\u56fe 7-32 \u5c55\u793a\u7684\u56db\u79cd\u5931\u8861\u60c5\u51b5\u4e0e\u4e0a\u8ff0\u6848\u4f8b\u9010\u4e2a\u5bf9\u5e94\uff0c\u5206\u522b\u9700\u8981\u91c7\u7528\u53f3\u65cb\u3001\u5148\u5de6\u65cb\u540e\u53f3\u65cb\u3001\u5148\u53f3\u65cb\u540e\u5de6\u65cb\u3001\u5de6\u65cb\u7684\u64cd\u4f5c\u3002</p> <p></p> <p> \u56fe 7-32 \u00a0 AVL \u6811\u7684\u56db\u79cd\u65cb\u8f6c\u60c5\u51b5 </p> <p>\u5982\u4e0b\u8868\u6240\u793a\uff0c\u6211\u4eec\u901a\u8fc7\u5224\u65ad\u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u4ee5\u53ca\u8f83\u9ad8\u4e00\u4fa7\u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50\u7684\u6b63\u8d1f\u53f7\uff0c\u6765\u786e\u5b9a\u5931\u8861\u8282\u70b9\u5c5e\u4e8e\u56fe 7-32 \u4e2d\u7684\u54ea\u79cd\u60c5\u51b5\u3002</p> <p> \u8868 7-3 \u00a0 \u56db\u79cd\u65cb\u8f6c\u60c5\u51b5\u7684\u9009\u62e9\u6761\u4ef6 </p> \u5931\u8861\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5b50\u8282\u70b9\u7684\u5e73\u8861\u56e0\u5b50 \u5e94\u91c7\u7528\u7684\u65cb\u8f6c\u65b9\u6cd5 \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(\\geq 0\\) \u53f3\u65cb \\(&gt; 1\\) \uff08\u5de6\u504f\u6811\uff09 \\(&lt;0\\) \u5148\u5de6\u65cb\u540e\u53f3\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(\\leq 0\\) \u5de6\u65cb \\(&lt; -1\\) \uff08\u53f3\u504f\u6811\uff09 \\(&gt;0\\) \u5148\u53f3\u65cb\u540e\u5de6\u65cb <p>\u4e3a\u4e86\u4fbf\u4e8e\u4f7f\u7528\uff0c\u6211\u4eec\u5c06\u65cb\u8f6c\u64cd\u4f5c\u5c01\u88c5\u6210\u4e00\u4e2a\u51fd\u6570\u3002\u6709\u4e86\u8fd9\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5c31\u80fd\u5bf9\u5404\u79cd\u5931\u8861\u60c5\u51b5\u8fdb\u884c\u65cb\u8f6c\uff0c\u4f7f\u5931\u8861\u8282\u70b9\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def rotate(self, node: TreeNode | None) -&gt; TreeNode | None:\n    \"\"\"\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\"\"\"\n    # \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    balance_factor = self.balance_factor(node)\n    # \u5de6\u504f\u6811\n    if balance_factor &gt; 1:\n        if self.balance_factor(node.left) &gt;= 0:\n            # \u53f3\u65cb\n            return self.right_rotate(node)\n        else:\n            # \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = self.left_rotate(node.left)\n            return self.right_rotate(node)\n    # \u53f3\u504f\u6811\n    elif balance_factor &lt; -1:\n        if self.balance_factor(node.right) &lt;= 0:\n            # \u5de6\u65cb\n            return self.left_rotate(node)\n        else:\n            # \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = self.right_rotate(node.right)\n            return self.left_rotate(node)\n    # \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n</code></pre> avl_tree.cpp<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int _balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (_balanceFactor &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (_balanceFactor &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode rotate(TreeNode node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactor = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = leftRotate(node.left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = rightRotate(node.right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? Rotate(TreeNode? node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int balanceFactorInt = BalanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactorInt &gt; 1) {\n        if (BalanceFactor(node?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return RightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node!.left = LeftRotate(node!.left);\n            return RightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactorInt &lt; -1) {\n        if (BalanceFactor(node?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return LeftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node!.right = RightRotate(node!.right);\n            return LeftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc (t *aVLTree) rotate(node *TreeNode) *TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    // Go \u63a8\u8350\u77ed\u53d8\u91cf\uff0c\u8fd9\u91cc bf \u6307\u4ee3 t.balanceFactor\n    bf := t.balanceFactor(node)\n    // \u5de6\u504f\u6811\n    if bf &gt; 1 {\n        if t.balanceFactor(node.Left) &gt;= 0 {\n            // \u53f3\u65cb\n            return t.rightRotate(node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.Left = t.leftRotate(node.Left)\n            return t.rightRotate(node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if bf &lt; -1 {\n        if t.balanceFactor(node.Right) &lt;= 0 {\n            // \u5de6\u65cb\n            return t.leftRotate(node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.Right = t.rightRotate(node.Right)\n            return t.leftRotate(node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfunc rotate(node: TreeNode?) -&gt; TreeNode? {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balanceFactor = balanceFactor(node: node)\n    // \u5de6\u504f\u6811\n    if balanceFactor &gt; 1 {\n        if self.balanceFactor(node: node?.left) &gt;= 0 {\n            // \u53f3\u65cb\n            return rightRotate(node: node)\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node?.left = leftRotate(node: node?.left)\n            return rightRotate(node: node)\n        }\n    }\n    // \u53f3\u504f\u6811\n    if balanceFactor &lt; -1 {\n        if self.balanceFactor(node: node?.right) &lt;= 0 {\n            // \u5de6\u65cb\n            return leftRotate(node: node)\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node?.right = rightRotate(node: node?.right)\n            return leftRotate(node: node)\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n#rotate(node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.#rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.#leftRotate(node.left);\n            return this.#rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.#leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.#rightRotate(node.right);\n            return this.#leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nrotate(node: TreeNode): TreeNode {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    const balanceFactor = this.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balanceFactor &gt; 1) {\n        if (this.balanceFactor(node.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return this.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.left = this.leftRotate(node.left);\n            return this.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balanceFactor &lt; -1) {\n        if (this.balanceFactor(node.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return this.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.right = this.rightRotate(node.right);\n            return this.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode? rotate(TreeNode? node) {\n  // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n  int factor = balanceFactor(node);\n  // \u5de6\u504f\u6811\n  if (factor &gt; 1) {\n    if (balanceFactor(node!.left) &gt;= 0) {\n      // \u53f3\u65cb\n      return rightRotate(node);\n    } else {\n      // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n      node.left = leftRotate(node.left);\n      return rightRotate(node);\n    }\n  }\n  // \u53f3\u504f\u6811\n  if (factor &lt; -1) {\n    if (balanceFactor(node!.right) &lt;= 0) {\n      // \u5de6\u65cb\n      return leftRotate(node);\n    } else {\n      // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n      node.right = rightRotate(node.right);\n      return leftRotate(node);\n    }\n  }\n  // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nfn rotate(node: OptionTreeNodeRc) -&gt; OptionTreeNodeRc {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    let balance_factor = Self::balance_factor(node.clone());\n    // \u5de6\u504f\u6811\n    if balance_factor &gt; 1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().left.clone()) &gt;= 0 {\n            // \u53f3\u65cb\n            Self::right_rotate(Some(node))\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            let left = node.borrow().left.clone();\n            node.borrow_mut().left = Self::left_rotate(left);\n            Self::right_rotate(Some(node))\n        }\n    }\n    // \u53f3\u504f\u6811\n    else if balance_factor &lt; -1 {\n        let node = node.unwrap();\n        if Self::balance_factor(node.borrow().right.clone()) &lt;= 0 {\n            // \u5de6\u65cb\n            Self::left_rotate(Some(node))\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            let right = node.borrow().right.clone();\n            node.borrow_mut().right = Self::right_rotate(right);\n            Self::left_rotate(Some(node))\n        }\n    } else {\n        // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n        node\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\nTreeNode *rotate(TreeNode *node) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    int bf = balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (bf &gt; 1) {\n        if (balanceFactor(node-&gt;left) &gt;= 0) {\n            // \u53f3\u65cb\n            return rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node-&gt;left = leftRotate(node-&gt;left);\n            return rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (bf &lt; -1) {\n        if (balanceFactor(node-&gt;right) &lt;= 0) {\n            // \u5de6\u65cb\n            return leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node-&gt;right = rightRotate(node-&gt;right);\n            return leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre> avl_tree.zig<pre><code>// \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\nfn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {\n    // \u83b7\u53d6\u8282\u70b9 node \u7684\u5e73\u8861\u56e0\u5b50\n    var balance_factor = self.balanceFactor(node);\n    // \u5de6\u504f\u6811\n    if (balance_factor &gt; 1) {\n        if (self.balanceFactor(node.?.left) &gt;= 0) {\n            // \u53f3\u65cb\n            return self.rightRotate(node);\n        } else {\n            // \u5148\u5de6\u65cb\u540e\u53f3\u65cb\n            node.?.left = self.leftRotate(node.?.left);\n            return self.rightRotate(node);\n        }\n    }\n    // \u53f3\u504f\u6811\n    if (balance_factor &lt; -1) {\n        if (self.balanceFactor(node.?.right) &lt;= 0) {\n            // \u5de6\u65cb\n            return self.leftRotate(node);\n        } else {\n            // \u5148\u53f3\u65cb\u540e\u5de6\u65cb\n            node.?.right = self.rightRotate(node.?.right);\n            return self.leftRotate(node);\n        }\n    }\n    // \u5e73\u8861\u6811\uff0c\u65e0\u987b\u65cb\u8f6c\uff0c\u76f4\u63a5\u8fd4\u56de\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#753-avl","title":"7.5.3 \u00a0 AVL \u6811\u5e38\u7528\u64cd\u4f5c","text":""},{"location":"chapter_tree/avl_tree/#1_2","title":"1. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u5728\u4e3b\u4f53\u4e0a\u7c7b\u4f3c\u3002\u552f\u4e00\u7684\u533a\u522b\u5728\u4e8e\uff0c\u5728 AVL \u6811\u4e2d\u63d2\u5165\u8282\u70b9\u540e\uff0c\u4ece\u8be5\u8282\u70b9\u5230\u6839\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u53ef\u80fd\u4f1a\u51fa\u73b0\u4e00\u7cfb\u5217\u5931\u8861\u8282\u70b9\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4ece\u8fd9\u4e2a\u8282\u70b9\u5f00\u59cb\uff0c\u81ea\u5e95\u5411\u4e0a\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def insert(self, val):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    self._root = self.insert_helper(self._root, val)\n\ndef insert_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode:\n    \"\"\"\u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return TreeNode(val)\n    # 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\n    if val &lt; node.val:\n        node.left = self.insert_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.insert_helper(node.right, val)\n    else:\n        # \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    else\n        return node;    // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n    root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode insertHelper(TreeNode node, int val) {\n    if (node == null)\n        return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = insertHelper(node.right, val);\n    else\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int val) {\n    root = InsertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? InsertHelper(TreeNode? node, int val) {\n    if (node == null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val)\n        node.left = InsertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = InsertHelper(node.right, val);\n    else\n        return node;     // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (t *aVLTree) insert(val int) {\n    t.root = t.insertHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return NewTreeNode(val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node.Val.(int) {\n        node.Left = t.insertHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.insertHelper(node.Right, val)\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(val: Int) {\n    root = insertHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc insertHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return TreeNode(x: val)\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if val &lt; node!.val {\n        node?.left = insertHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = insertHelper(node: node?.right, val: val)\n    } else {\n        return node // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val) {\n    this.root = this.#insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#insertHelper(node, val) {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) node.left = this.#insertHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#insertHelper(node.right, val);\n    else return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(val: number): void {\n    this.root = this.insertHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\ninsertHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return new TreeNode(val);\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node.val) {\n        node.left = this.insertHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.insertHelper(node.right, val);\n    } else {\n        return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int val) {\n  root = insertHelper(root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? insertHelper(TreeNode? node, int val) {\n  if (node == null) return TreeNode(val);\n  /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n  if (val &lt; node.val)\n    node.left = insertHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = insertHelper(node.right, val);\n  else\n    return node; // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfn insert(&amp;mut self, val: i32) {\n    self.root = Self::insert_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn insert_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n            match {\n                let node_val = node.borrow().val;\n                node_val\n            }\n            .cmp(&amp;val)\n            {\n                Ordering::Greater =&gt; {\n                    let left = node.borrow().left.clone();\n                    node.borrow_mut().left = Self::insert_helper(left, val);\n                }\n                Ordering::Less =&gt; {\n                    let right = node.borrow().right.clone();\n                    node.borrow_mut().right = Self::insert_helper(right, val);\n                }\n                Ordering::Equal =&gt; {\n                    return Some(node); // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n                }\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; Some(TreeNode::new(val)),\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(AVLTree *tree, int val) {\n    tree-&gt;root = insertHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *insertHelper(TreeNode *node, int val) {\n    if (node == NULL) {\n        return newTreeNode(val);\n    }\n    /* 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9 */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = insertHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = insertHelper(node-&gt;right, val);\n    } else {\n        // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n        return node;\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, val: T) !void {\n    self.root = (try self.insertHelper(self.root, val)).?;\n}\n\n// \u9012\u5f52\u63d2\u5165\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) {\n        var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));\n        tmp_node.init(val);\n        return tmp_node;\n    }\n    // 1. \u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff0c\u5e76\u63d2\u5165\u8282\u70b9\n    if (val &lt; node.?.val) {\n        node.?.left = try self.insertHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = try self.insertHelper(node.?.right, val);\n    } else {\n        return node;            // \u91cd\u590d\u8282\u70b9\u4e0d\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\n    }\n    self.updateHeight(node);    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#2_2","title":"2. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u7c7b\u4f3c\u5730\uff0c\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5220\u9664\u8282\u70b9\u65b9\u6cd5\u7684\u57fa\u7840\u4e0a\uff0c\u9700\u8981\u4ece\u5e95\u81f3\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6240\u6709\u5931\u8861\u8282\u70b9\u6062\u590d\u5e73\u8861\u3002\u4ee3\u7801\u5982\u4e0b\u6240\u793a\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig avl_tree.py<pre><code>def remove(self, val: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    self._root = self.remove_helper(self._root, val)\n\ndef remove_helper(self, node: TreeNode | None, val: int) -&gt; TreeNode | None:\n    \"\"\"\u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\"\"\"\n    if node is None:\n        return None\n    # 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\n    if val &lt; node.val:\n        node.left = self.remove_helper(node.left, val)\n    elif val &gt; node.val:\n        node.right = self.remove_helper(node.right, val)\n    else:\n        if node.left is None or node.right is None:\n            child = node.left or node.right\n            # \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child is None:\n                return None\n            # \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else:\n                node = child\n        else:\n            # \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp = node.right\n            while temp.left is not None:\n                temp = temp.left\n            node.right = self.remove_helper(node.right, temp.val)\n            node.val = temp.val\n    # \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    self.update_height(node)\n    # 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    return self.rotate(node)\n</code></pre> avl_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    if (node == nullptr)\n        return nullptr;\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if (val &lt; node-&gt;val)\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    else if (val &gt; node-&gt;val)\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    else {\n        if (node-&gt;left == nullptr || node-&gt;right == nullptr) {\n            TreeNode *child = node-&gt;left != nullptr ? node-&gt;left : node-&gt;right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == nullptr) {\n                delete node;\n                return nullptr;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                delete node;\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != nullptr) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n    root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode removeHelper(TreeNode node, int val) {\n    if (node == null)\n        return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if (val &lt; node.val)\n        node.left = removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = removeHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode child = node.left != null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int val) {\n    root = RemoveHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? RemoveHelper(TreeNode? node, int val) {\n    if (node == null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if (val &lt; node.val)\n        node.left = RemoveHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = RemoveHelper(node.right, val);\n    else {\n        if (node.left == null || node.right == null) {\n            TreeNode? child = node.left ?? node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null)\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else\n                node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode? temp = node.right;\n            while (temp.left != null) {\n                temp = temp.left;\n            }\n            node.right = RemoveHelper(node.right, temp.val!.Value);\n            node.val = temp.val;\n        }\n    }\n    UpdateHeight(node);  // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = Rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (t *aVLTree) remove(val int) {\n    t.root = t.removeHelper(t.root, val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nfunc (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if val &lt; node.Val.(int) {\n        node.Left = t.removeHelper(node.Left, val)\n    } else if val &gt; node.Val.(int) {\n        node.Right = t.removeHelper(node.Right, val)\n    } else {\n        if node.Left == nil || node.Right == nil {\n            child := node.Left\n            if node.Right != nil {\n                child = node.Right\n            }\n            if child == nil {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                return nil\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            temp := node.Right\n            for temp.Left != nil {\n                temp = temp.Left\n            }\n            node.Right = t.removeHelper(node.Right, temp.Val.(int))\n            node.Val = temp.Val\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    t.updateHeight(node)\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = t.rotate(node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(val: Int) {\n    root = removeHelper(node: root, val: val)\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfunc removeHelper(node: TreeNode?, val: Int) -&gt; TreeNode? {\n    var node = node\n    if node == nil {\n        return nil\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if val &lt; node!.val {\n        node?.left = removeHelper(node: node?.left, val: val)\n    } else if val &gt; node!.val {\n        node?.right = removeHelper(node: node?.right, val: val)\n    } else {\n        if node?.left == nil || node?.right == nil {\n            let child = node?.left != nil ? node?.left : node?.right\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if child == nil {\n                return nil\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else {\n                node = child\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node?.right\n            while temp?.left != nil {\n                temp = temp?.left\n            }\n            node?.right = removeHelper(node: node?.right, val: temp!.val)\n            node?.val = temp!.val\n        }\n    }\n    updateHeight(node: node) // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node: node)\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node\n}\n</code></pre> avl_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val) {\n    this.root = this.#removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\n#removeHelper(node, val) {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if (val &lt; node.val) node.left = this.#removeHelper(node.left, val);\n    else if (val &gt; node.val)\n        node.right = this.#removeHelper(node.right, val);\n    else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            else node = child;\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.#removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.#updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.#rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(val: number): void {\n    this.root = this.removeHelper(this.root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nremoveHelper(node: TreeNode, val: number): TreeNode {\n    if (node === null) return null;\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if (val &lt; node.val) {\n        node.left = this.removeHelper(node.left, val);\n    } else if (val &gt; node.val) {\n        node.right = this.removeHelper(node.right, val);\n    } else {\n        if (node.left === null || node.right === null) {\n            const child = node.left !== null ? node.left : node.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child === null) {\n                return null;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            let temp = node.right;\n            while (temp.left !== null) {\n                temp = temp.left;\n            }\n            node.right = this.removeHelper(node.right, temp.val);\n            node.val = temp.val;\n        }\n    }\n    this.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = this.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int val) {\n  root = removeHelper(root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nTreeNode? removeHelper(TreeNode? node, int val) {\n  if (node == null) return null;\n  /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n  if (val &lt; node.val)\n    node.left = removeHelper(node.left, val);\n  else if (val &gt; node.val)\n    node.right = removeHelper(node.right, val);\n  else {\n    if (node.left == null || node.right == null) {\n      TreeNode? child = node.left ?? node.right;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n      if (child == null)\n        return null;\n      // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n      else\n        node = child;\n    } else {\n      // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n      TreeNode? temp = node.right;\n      while (temp!.left != null) {\n        temp = temp.left;\n      }\n      node.right = removeHelper(node.right, temp.val);\n      node.val = temp.val;\n    }\n  }\n  updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n  /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n  node = rotate(node);\n  // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n  return node;\n}\n</code></pre> avl_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfn remove(&amp;self, val: i32) {\n    Self::remove_helper(self.root.clone(), val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09 */\nfn remove_helper(node: OptionTreeNodeRc, val: i32) -&gt; OptionTreeNodeRc {\n    match node {\n        Some(mut node) =&gt; {\n            /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n            if val &lt; node.borrow().val {\n                let left = node.borrow().left.clone();\n                node.borrow_mut().left = Self::remove_helper(left, val);\n            } else if val &gt; node.borrow().val {\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, val);\n            } else if node.borrow().left.is_none() || node.borrow().right.is_none() {\n                let child = if node.borrow().left.is_some() {\n                    node.borrow().left.clone()\n                } else {\n                    node.borrow().right.clone()\n                };\n                match child {\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n                    None =&gt; {\n                        return None;\n                    }\n                    // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                    Some(child) =&gt; node = child,\n                }\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n                let mut temp = node.borrow().right.clone().unwrap();\n                loop {\n                    let temp_left = temp.borrow().left.clone();\n                    if temp_left.is_none() {\n                        break;\n                    }\n                    temp = temp_left.unwrap();\n                }\n                let right = node.borrow().right.clone();\n                node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);\n                node.borrow_mut().val = temp.borrow().val;\n            }\n            Self::update_height(Some(node.clone())); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n            /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n            node = Self::rotate(Some(node)).unwrap();\n            // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n            Some(node)\n        }\n        None =&gt; None,\n    }\n}\n</code></pre> avl_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(AVLTree *tree, int val) {\n    TreeNode *root = removeHelper(tree-&gt;root, val);\n}\n\n/* \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u51fd\u6570\uff09 */\nTreeNode *removeHelper(TreeNode *node, int val) {\n    TreeNode *child, *grandChild;\n    if (node == NULL) {\n        return NULL;\n    }\n    /* 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b */\n    if (val &lt; node-&gt;val) {\n        node-&gt;left = removeHelper(node-&gt;left, val);\n    } else if (val &gt; node-&gt;val) {\n        node-&gt;right = removeHelper(node-&gt;right, val);\n    } else {\n        if (node-&gt;left == NULL || node-&gt;right == NULL) {\n            child = node-&gt;left;\n            if (node-&gt;right != NULL) {\n                child = node-&gt;right;\n            }\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == NULL) {\n                return NULL;\n            } else {\n                // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            TreeNode *temp = node-&gt;right;\n            while (temp-&gt;left != NULL) {\n                temp = temp-&gt;left;\n            }\n            int tempVal = temp-&gt;val;\n            node-&gt;right = removeHelper(node-&gt;right, temp-&gt;val);\n            node-&gt;val = tempVal;\n        }\n    }\n    // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    updateHeight(node);\n    /* 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861 */\n    node = rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre> avl_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, val: T) void {\n   self.root = self.removeHelper(self.root, val).?;\n}\n\n// \u9012\u5f52\u5220\u9664\u8282\u70b9\uff08\u8f85\u52a9\u65b9\u6cd5\uff09\nfn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {\n    var node = node_;\n    if (node == null) return null;\n    // 1. \u67e5\u627e\u8282\u70b9\uff0c\u5e76\u5220\u9664\u4e4b\n    if (val &lt; node.?.val) {\n        node.?.left = self.removeHelper(node.?.left, val);\n    } else if (val &gt; node.?.val) {\n        node.?.right = self.removeHelper(node.?.right, val);\n    } else {\n        if (node.?.left == null or node.?.right == null) {\n            var child = if (node.?.left != null) node.?.left else node.?.right;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 0 \uff0c\u76f4\u63a5\u5220\u9664 node \u5e76\u8fd4\u56de\n            if (child == null) {\n                return null;\n            // \u5b50\u8282\u70b9\u6570\u91cf = 1 \uff0c\u76f4\u63a5\u5220\u9664 node\n            } else {\n                node = child;\n            }\n        } else {\n            // \u5b50\u8282\u70b9\u6570\u91cf = 2 \uff0c\u5219\u5c06\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e2a\u8282\u70b9\u5220\u9664\uff0c\u5e76\u7528\u8be5\u8282\u70b9\u66ff\u6362\u5f53\u524d\u8282\u70b9\n            var temp = node.?.right;\n            while (temp.?.left != null) {\n                temp = temp.?.left;\n            }\n            node.?.right = self.removeHelper(node.?.right, temp.?.val);\n            node.?.val = temp.?.val;\n        }\n    }\n    self.updateHeight(node); // \u66f4\u65b0\u8282\u70b9\u9ad8\u5ea6\n    // 2. \u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u8be5\u5b50\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\n    node = self.rotate(node);\n    // \u8fd4\u56de\u5b50\u6811\u7684\u6839\u8282\u70b9\n    return node;\n}\n</code></pre>"},{"location":"chapter_tree/avl_tree/#3_1","title":"3. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>AVL \u6811\u7684\u8282\u70b9\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u53c9\u641c\u7d22\u6811\u4e00\u81f4\uff0c\u5728\u6b64\u4e0d\u518d\u8d58\u8ff0\u3002</p>"},{"location":"chapter_tree/avl_tree/#754-avl","title":"7.5.4 \u00a0 AVL \u6811\u5178\u578b\u5e94\u7528","text":"<ul> <li>\u7ec4\u7ec7\u548c\u5b58\u50a8\u5927\u578b\u6570\u636e\uff0c\u9002\u7528\u4e8e\u9ad8\u9891\u67e5\u627e\u3001\u4f4e\u9891\u589e\u5220\u7684\u573a\u666f\u3002</li> <li>\u7528\u4e8e\u6784\u5efa\u6570\u636e\u5e93\u4e2d\u7684\u7d22\u5f15\u7cfb\u7edf\u3002</li> <li>\u7ea2\u9ed1\u6811\u5728\u8bb8\u591a\u5e94\u7528\u4e2d\u6bd4 AVL \u6811\u66f4\u53d7\u6b22\u8fce\u3002\u8fd9\u662f\u56e0\u4e3a\u7ea2\u9ed1\u6811\u7684\u5e73\u8861\u6761\u4ef6\u76f8\u5bf9\u5bbd\u677e\uff0c\u5728\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u6240\u9700\u7684\u65cb\u8f6c\u64cd\u4f5c\u76f8\u5bf9\u8f83\u5c11\uff0c\u5176\u8282\u70b9\u589e\u5220\u64cd\u4f5c\u7684\u5e73\u5747\u6548\u7387\u66f4\u9ad8\u3002</li> </ul>"},{"location":"chapter_tree/binary_search_tree/","title":"7.4 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811","text":"<p>\u5982\u56fe 7-16 \u6240\u793a\uff0c\u300c\u4e8c\u53c9\u641c\u7d22\u6811 binary search tree\u300d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u3002</p> <ol> <li>\u5bf9\u4e8e\u6839\u8282\u70b9\uff0c\u5de6\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c \\(&lt;\\) \u6839\u8282\u70b9\u7684\u503c \\(&lt;\\) \u53f3\u5b50\u6811\u4e2d\u6240\u6709\u8282\u70b9\u7684\u503c\u3002</li> <li>\u4efb\u610f\u8282\u70b9\u7684\u5de6\u3001\u53f3\u5b50\u6811\u4e5f\u662f\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5373\u540c\u6837\u6ee1\u8db3\u6761\u4ef6 <code>1.</code> \u3002</li> </ol> <p></p> <p> \u56fe 7-16 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811 </p>"},{"location":"chapter_tree/binary_search_tree/#741","title":"7.4.1 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u64cd\u4f5c","text":"<p>\u6211\u4eec\u5c06\u4e8c\u53c9\u641c\u7d22\u6811\u5c01\u88c5\u4e3a\u4e00\u4e2a\u7c7b <code>BinarySearchTree</code> \uff0c\u5e76\u58f0\u660e\u4e00\u4e2a\u6210\u5458\u53d8\u91cf <code>root</code> \uff0c\u6307\u5411\u6811\u7684\u6839\u8282\u70b9\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#1","title":"1. \u00a0 \u67e5\u627e\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u76ee\u6807\u8282\u70b9\u503c <code>num</code> \uff0c\u53ef\u4ee5\u6839\u636e\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6027\u8d28\u6765\u67e5\u627e\u3002\u5982\u56fe 7-17 \u6240\u793a\uff0c\u6211\u4eec\u58f0\u660e\u4e00\u4e2a\u8282\u70b9 <code>cur</code> \uff0c\u4ece\u4e8c\u53c9\u6811\u7684\u6839\u8282\u70b9 <code>root</code> \u51fa\u53d1\uff0c\u5faa\u73af\u6bd4\u8f83\u8282\u70b9\u503c <code>cur.val</code> \u548c <code>num</code> \u4e4b\u95f4\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <ul> <li>\u82e5 <code>cur.val &lt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u53f3\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.right</code> \u3002</li> <li>\u82e5 <code>cur.val &gt; num</code> \uff0c\u8bf4\u660e\u76ee\u6807\u8282\u70b9\u5728 <code>cur</code> \u7684\u5de6\u5b50\u6811\u4e2d\uff0c\u56e0\u6b64\u6267\u884c <code>cur = cur.left</code> \u3002</li> <li>\u82e5 <code>cur.val = num</code> \uff0c\u8bf4\u660e\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\u5e76\u8fd4\u56de\u8be5\u8282\u70b9\u3002</li> </ul> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-17 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u67e5\u627e\u8282\u70b9\u793a\u4f8b </p> <p>\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u67e5\u627e\u64cd\u4f5c\u4e0e\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5\u7684\u5de5\u4f5c\u539f\u7406\u4e00\u81f4\uff0c\u90fd\u662f\u6bcf\u8f6e\u6392\u9664\u4e00\u534a\u60c5\u51b5\u3002\u5faa\u73af\u6b21\u6570\u6700\u591a\u4e3a\u4e8c\u53c9\u6811\u7684\u9ad8\u5ea6\uff0c\u5f53\u4e8c\u53c9\u6811\u5e73\u8861\u65f6\uff0c\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py<pre><code>def search(self, num: int) -&gt; TreeNode | None:\n    \"\"\"\u67e5\u627e\u8282\u70b9\"\"\"\n    cur = self._root\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur is not None:\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        elif cur.val &gt; num:\n            cur = cur.left\n        # \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else:\n            break\n    return cur\n</code></pre> binary_search_tree.cpp<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(int num) {\n    TreeNode *cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur-&gt;val &gt; num)\n            cur = cur-&gt;left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode search(int num) {\n    TreeNode cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? Search(int num) {\n    TreeNode? cur = root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur =\n            cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num)\n            cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else\n            break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc (bst *binarySearchTree) search(num int) *TreeNode {\n    node := bst.root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for node != nil {\n        if node.Val.(int) &lt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            node = node.Right\n        } else if node.Val.(int) &gt; num {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            node = node.Left\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return node\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nfunc search(num: Int) -&gt; TreeNode? {\n    var cur = root\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if cur!.val &gt; num {\n            cur = cur?.left\n        }\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else {\n            break\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num) {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nsearch(num: number): TreeNode | null {\n    let cur = this.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if (cur.val &gt; num) cur = cur.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else break;\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode? search(int _num) {\n  TreeNode? cur = _root;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else if (cur.val &gt; _num)\n      cur = cur.left;\n    // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    else\n      break;\n  }\n  // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n  return cur;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u67e5\u627e\u8282\u70b9 */\npub fn search(&amp;self, num: i32) -&gt; Option&lt;TreeNodeRc&gt; {\n    let mut cur = self.root.clone();\n\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if node.borrow().val &lt; num {\n            cur = node.borrow().right.clone();\n        }\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else if node.borrow().val &gt; num {\n            cur = node.borrow().left.clone();\n        }\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    cur\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u67e5\u627e\u8282\u70b9 */\nTreeNode *search(BinarySearchTree *bst, int num) {\n    TreeNode *cur = bst-&gt;root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        if (cur-&gt;val &lt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else if (cur-&gt;val &gt; num) {\n            // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        } else {\n            // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre> binary_search_tree.zig<pre><code>// \u67e5\u627e\u8282\u70b9\nfn search(self: *Self, num: T) ?*inc.TreeNode(T) {\n    var cur = self.root;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u76ee\u6807\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else if (cur.?.val &gt; num) {\n            cur = cur.?.left;\n        // \u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        } else {\n            break;\n        }\n    }\n    // \u8fd4\u56de\u76ee\u6807\u8282\u70b9\n    return cur;\n}\n</code></pre>"},{"location":"chapter_tree/binary_search_tree/#2","title":"2. \u00a0 \u63d2\u5165\u8282\u70b9","text":"<p>\u7ed9\u5b9a\u4e00\u4e2a\u5f85\u63d2\u5165\u5143\u7d20 <code>num</code> \uff0c\u4e3a\u4e86\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u63d2\u5165\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-18 \u6240\u793a\u3002</p> <ol> <li>\u67e5\u627e\u63d2\u5165\u4f4d\u7f6e\uff1a\u4e0e\u67e5\u627e\u64cd\u4f5c\u76f8\u4f3c\uff0c\u4ece\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6839\u636e\u5f53\u524d\u8282\u70b9\u503c\u548c <code>num</code> \u7684\u5927\u5c0f\u5173\u7cfb\u5faa\u73af\u5411\u4e0b\u641c\u7d22\uff0c\u76f4\u5230\u8d8a\u8fc7\u53f6\u8282\u70b9\uff08\u904d\u5386\u81f3 \\(\\text{None}\\) \uff09\u65f6\u8df3\u51fa\u5faa\u73af\u3002</li> <li>\u5728\u8be5\u4f4d\u7f6e\u63d2\u5165\u8282\u70b9\uff1a\u521d\u59cb\u5316\u8282\u70b9 <code>num</code> \uff0c\u5c06\u8be5\u8282\u70b9\u7f6e\u4e8e \\(\\text{None}\\) \u7684\u4f4d\u7f6e\u3002</li> </ol> <p></p> <p> \u56fe 7-18 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u63d2\u5165\u8282\u70b9 </p> <p>\u5728\u4ee3\u7801\u5b9e\u73b0\u4e2d\uff0c\u9700\u8981\u6ce8\u610f\u4ee5\u4e0b\u4e24\u70b9\u3002</p> <ul> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u4e0d\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u8282\u70b9\uff0c\u5426\u5219\u5c06\u8fdd\u53cd\u5176\u5b9a\u4e49\u3002\u56e0\u6b64\uff0c\u82e5\u5f85\u63d2\u5165\u8282\u70b9\u5728\u6811\u4e2d\u5df2\u5b58\u5728\uff0c\u5219\u4e0d\u6267\u884c\u63d2\u5165\uff0c\u76f4\u63a5\u8fd4\u56de\u3002</li> <li>\u4e3a\u4e86\u5b9e\u73b0\u63d2\u5165\u8282\u70b9\uff0c\u6211\u4eec\u9700\u8981\u501f\u52a9\u8282\u70b9 <code>pre</code> \u4fdd\u5b58\u4e0a\u4e00\u8f6e\u5faa\u73af\u7684\u8282\u70b9\u3002\u8fd9\u6837\u5728\u904d\u5386\u81f3 \\(\\text{None}\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u83b7\u53d6\u5230\u5176\u7236\u8282\u70b9\uff0c\u4ece\u800c\u5b8c\u6210\u8282\u70b9\u63d2\u5165\u64cd\u4f5c\u3002</li> </ul> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py<pre><code>def insert(self, num: int):\n    \"\"\"\u63d2\u5165\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self._root is None:\n        self._root = TreeNode(num)\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur.val == num:\n            return\n        pre = cur\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u63d2\u5165\u8282\u70b9\n    node = TreeNode(num)\n    if pre.val &lt; num:\n        pre.right = node\n    else:\n        pre.left = node\n</code></pre> binary_search_tree.cpp<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == nullptr) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = new TreeNode(num);\n    if (pre-&gt;val &lt; num)\n        pre-&gt;right = node;\n    else\n        pre-&gt;left = node;\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new TreeNode(num);\n    if (pre.val &lt; num)\n        pre.right = node;\n    else\n        pre.left = node;\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid Insert(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (root == null) {\n        root = new TreeNode(num);\n        return;\n    }\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val == num)\n            return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode node = new(num);\n    if (pre != null) {\n        if (pre.val &lt; num)\n            pre.right = node;\n        else\n            pre.left = node;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc (bst *binarySearchTree) insert(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if cur == nil {\n        bst.root = NewTreeNode(num)\n        return\n    }\n    // \u5f85\u63d2\u5165\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            return\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            cur = cur.Right\n        } else {\n            cur = cur.Left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    node := NewTreeNode(num)\n    if pre.Val.(int) &lt; num {\n        pre.Right = node\n    } else {\n        pre.Left = node\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nfunc insert(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if root == nil {\n        root = TreeNode(x: num)\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if cur!.val == num {\n            return\n        }\n        pre = cur\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let node = TreeNode(x: num)\n    if pre!.val &lt; num {\n        pre?.right = node\n    } else {\n        pre?.left = node\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre.val &lt; num) pre.right = node;\n    else pre.left = node;\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u63d2\u5165\u8282\u70b9 */\ninsert(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (this.root === null) {\n        this.root = new TreeNode(num);\n        return;\n    }\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.val === num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u63d2\u5165\u8282\u70b9\n    const node = new TreeNode(num);\n    if (pre!.val &lt; num) pre!.right = node;\n    else pre!.left = node;\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n  if (_root == null) {\n    _root = TreeNode(_num);\n    return;\n  }\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n    if (cur.val == _num) return;\n    pre = cur;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u63d2\u5165\u8282\u70b9\n  TreeNode? node = TreeNode(_num);\n  if (pre!.val &lt; _num)\n    pre.right = node;\n  else\n    pre.left = node;\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u63d2\u5165\u8282\u70b9 */\npub fn insert(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if self.root.is_none() {\n        self.root = Some(TreeNode::new(num));\n        return;\n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if node.borrow().val == num {\n            return;\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        pre = cur.clone();\n        if node.borrow().val &lt; num {\n            cur = node.borrow().right.clone();\n        }\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = node.borrow().left.clone();\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    let node = TreeNode::new(num);\n    let pre = pre.unwrap();\n    if pre.borrow().val &lt; num {\n        pre.borrow_mut().right = Some(Rc::clone(&amp;node));\n    } else {\n        pre.borrow_mut().left = Some(Rc::clone(&amp;node));\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u63d2\u5165\u8282\u70b9 */\nvoid insert(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (bst-&gt;root == NULL) {\n        bst-&gt;root = newTreeNode(num);\n        return;\n    }\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur-&gt;val == num) {\n            return;\n        }\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    TreeNode *node = newTreeNode(num);\n    if (pre-&gt;val &lt; num) {\n        pre-&gt;right = node;\n    } else {\n        pre-&gt;left = node;\n    }\n}\n</code></pre> binary_search_tree.zig<pre><code>// \u63d2\u5165\u8282\u70b9\nfn insert(self: *Self, num: T) !void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u5219\u521d\u59cb\u5316\u6839\u8282\u70b9\n    if (self.root == null) {\n        self.root = try self.mem_allocator.create(inc.TreeNode(T));\n        return;\n    }\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u91cd\u590d\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n        if (cur.?.val == num) return;\n        pre = cur;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u63d2\u5165\u4f4d\u7f6e\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u63d2\u5165\u8282\u70b9\n    var node = try self.mem_allocator.create(inc.TreeNode(T));\n    node.init(num);\n    if (pre.?.val &lt; num) {\n        pre.?.right = node;\n    } else {\n        pre.?.left = node;\n    }\n}\n</code></pre> <p>\u4e0e\u67e5\u627e\u8282\u70b9\u76f8\u540c\uff0c\u63d2\u5165\u8282\u70b9\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\u3002</p>"},{"location":"chapter_tree/binary_search_tree/#3","title":"3. \u00a0 \u5220\u9664\u8282\u70b9","text":"<p>\u5148\u5728\u4e8c\u53c9\u6811\u4e2d\u67e5\u627e\u5230\u76ee\u6807\u8282\u70b9\uff0c\u518d\u5c06\u5176\u5220\u9664\u3002</p> <p>\u4e0e\u63d2\u5165\u8282\u70b9\u7c7b\u4f3c\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5728\u5220\u9664\u64cd\u4f5c\u5b8c\u6210\u540e\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u201c\u5de6\u5b50\u6811 &lt; \u6839\u8282\u70b9 &lt; \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\u4ecd\u7136\u6ee1\u8db3\u3002</p> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u6839\u636e\u76ee\u6807\u8282\u70b9\u7684\u5b50\u8282\u70b9\u6570\u91cf\uff0c\u5206 0\u30011 \u548c 2 \u4e09\u79cd\u60c5\u51b5\uff0c\u6267\u884c\u5bf9\u5e94\u7684\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>\u5982\u56fe 7-19 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \u65f6\uff0c\u8868\u793a\u8be5\u8282\u70b9\u662f\u53f6\u8282\u70b9\uff0c\u53ef\u4ee5\u76f4\u63a5\u5220\u9664\u3002</p> <p></p> <p> \u56fe 7-19 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 0 \uff09 </p> <p>\u5982\u56fe 7-20 \u6240\u793a\uff0c\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(1\\) \u65f6\uff0c\u5c06\u5f85\u5220\u9664\u8282\u70b9\u66ff\u6362\u4e3a\u5176\u5b50\u8282\u70b9\u5373\u53ef\u3002</p> <p></p> <p> \u56fe 7-20 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 1 \uff09 </p> <p>\u5f53\u5f85\u5220\u9664\u8282\u70b9\u7684\u5ea6\u4e3a \\(2\\) \u65f6\uff0c\u6211\u4eec\u65e0\u6cd5\u76f4\u63a5\u5220\u9664\u5b83\uff0c\u800c\u9700\u8981\u4f7f\u7528\u4e00\u4e2a\u8282\u70b9\u66ff\u6362\u8be5\u8282\u70b9\u3002\u7531\u4e8e\u8981\u4fdd\u6301\u4e8c\u53c9\u641c\u7d22\u6811\u201c\u5de6\u5b50\u6811 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u6811\u201d\u7684\u6027\u8d28\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8282\u70b9\u53ef\u4ee5\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\u6216\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u3002</p> <p>\u5047\u8bbe\u6211\u4eec\u9009\u62e9\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u8282\u70b9\uff08\u4e2d\u5e8f\u904d\u5386\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff09\uff0c\u5219\u5220\u9664\u64cd\u4f5c\u6d41\u7a0b\u5982\u56fe 7-21 \u6240\u793a\u3002</p> <ol> <li>\u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\u5728\u201c\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u201d\u4e2d\u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\uff0c\u8bb0\u4e3a <code>tmp</code> \u3002</li> <li>\u7528 <code>tmp</code> \u7684\u503c\u8986\u76d6\u5f85\u5220\u9664\u8282\u70b9\u7684\u503c\uff0c\u5e76\u5728\u6811\u4e2d\u9012\u5f52\u5220\u9664\u8282\u70b9 <code>tmp</code> \u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt; <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-21 \u00a0 \u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u5220\u9664\u8282\u70b9\uff08\u5ea6\u4e3a 2 \uff09 </p> <p>\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u540c\u6837\u4f7f\u7528 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d\u67e5\u627e\u5f85\u5220\u9664\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\uff0c\u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u540e\u7ee7\u8282\u70b9\u9700\u8981 \\(O(\\log n)\\) \u65f6\u95f4\u3002\u793a\u4f8b\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_search_tree.py<pre><code>def remove(self, num: int):\n    \"\"\"\u5220\u9664\u8282\u70b9\"\"\"\n    # \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self._root is None:\n        return\n    # \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    cur, pre = self._root, None\n    while cur is not None:\n        # \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur.val == num:\n            break\n        pre = cur\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur.val &lt; num:\n            cur = cur.right\n        # \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else:\n            cur = cur.left\n    # \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur is None:\n        return\n\n    # \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur.left is None or cur.right is None:\n        # \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        child = cur.left or cur.right\n        # \u5220\u9664\u8282\u70b9 cur\n        if cur != self._root:\n            if pre.left == cur:\n                pre.left = child\n            else:\n                pre.right = child\n        else:\n            # \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            self._root = child\n    # \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else:\n        # \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp: TreeNode = cur.right\n        while tmp.left is not None:\n            tmp = tmp.left\n        # \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.val)\n        # \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val\n</code></pre> binary_search_tree.cpp<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == nullptr)\n        return;\n    TreeNode *cur = root, *pre = nullptr;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != nullptr) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur-&gt;val &lt; num)\n            cur = cur-&gt;right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur-&gt;left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == nullptr)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur-&gt;left == nullptr || cur-&gt;right == nullptr) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != nullptr ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre-&gt;left == cur)\n                pre-&gt;left = child;\n            else\n                pre-&gt;right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n        // \u91ca\u653e\u5185\u5b58\n        delete cur;\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != nullptr) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.java<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode child = cur.left != null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.cs<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid Remove(int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (root == null)\n        return;\n    TreeNode? cur = root, pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val == num)\n            break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num)\n            cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else\n            cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null)\n        return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left == null || cur.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        TreeNode? child = cur.left ?? cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur != root) {\n            if (pre!.left == cur)\n                pre.left = child;\n            else\n                pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode? tmp = cur.right;\n        while (tmp.left != null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        Remove(tmp.val!.Value);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.go<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc (bst *binarySearchTree) remove(num int) {\n    cur := bst.root\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5f85\u5220\u9664\u8282\u70b9\u4e4b\u524d\u7684\u8282\u70b9\u4f4d\u7f6e\n    var pre *TreeNode = nil\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    for cur != nil {\n        if cur.Val == num {\n            break\n        }\n        pre = cur\n        if cur.Val.(int) &lt; num {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u53f3\u5b50\u6811\u4e2d\n            cur = cur.Right\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728\u5de6\u5b50\u6811\u4e2d\n            cur = cur.Left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u4e3a 0 \u6216 1\n    if cur.Left == nil || cur.Right == nil {\n        var child *TreeNode = nil\n        // \u53d6\u51fa\u5f85\u5220\u9664\u8282\u70b9\u7684\u5b50\u8282\u70b9\n        if cur.Left != nil {\n            child = cur.Left\n        } else {\n            child = cur.Right\n        }\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur != bst.root {\n            if pre.Left == cur {\n                pre.Left = child\n            } else {\n                pre.Right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            bst.root = child\n        }\n        // \u5b50\u8282\u70b9\u6570\u4e3a 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d\u5f85\u5220\u9664\u8282\u70b9 cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        tmp := cur.Right\n        for tmp.Left != nil {\n            tmp = tmp.Left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        bst.remove(tmp.Val.(int))\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.Val = tmp.Val\n    }\n}\n</code></pre> binary_search_tree.swift<pre><code>/* \u5220\u9664\u8282\u70b9 */\nfunc remove(num: Int) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if root == nil {\n        return\n    }\n    var cur = root\n    var pre: TreeNode?\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while cur != nil {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if cur!.val == num {\n            break\n        }\n        pre = cur\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if cur!.val &lt; num {\n            cur = cur?.right\n        }\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = cur?.left\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur == nil {\n        return\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur?.left == nil || cur?.right == nil {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        let child = cur?.left != nil ? cur?.left : cur?.right\n        // \u5220\u9664\u8282\u70b9 cur\n        if cur !== root {\n            if pre?.left === cur {\n                pre?.left = child\n            } else {\n                pre?.right = child\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            root = child\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur?.right\n        while tmp?.left != nil {\n            tmp = tmp?.left\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        remove(num: tmp!.val)\n        // \u7528 tmp \u8986\u76d6 cur\n        cur?.val = tmp!.val\n    }\n}\n</code></pre> binary_search_tree.js<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur = this.root,\n        pre = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child = cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre.left === cur) pre.left = child;\n            else pre.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp = cur.right;\n        while (tmp.left !== null) {\n            tmp = tmp.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp.val;\n    }\n}\n</code></pre> binary_search_tree.ts<pre><code>/* \u5220\u9664\u8282\u70b9 */\nremove(num: number): void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (this.root === null) return;\n    let cur: TreeNode | null = this.root,\n        pre: TreeNode | null = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur !== null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.val === num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.val &lt; num) cur = cur.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else cur = cur.left;\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur === null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.left === null || cur.right === null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        const child: TreeNode | null =\n            cur.left !== null ? cur.left : cur.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (cur !== this.root) {\n            if (pre!.left === cur) pre!.left = child;\n            else pre!.right = child;\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            this.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let tmp: TreeNode | null = cur.right;\n        while (tmp!.left !== null) {\n            tmp = tmp!.left;\n        }\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        this.remove(tmp!.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.val = tmp!.val;\n    }\n}\n</code></pre> binary_search_tree.dart<pre><code>/* \u5220\u9664\u8282\u70b9 */\nvoid remove(int _num) {\n  // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n  if (_root == null) return;\n  TreeNode? cur = _root;\n  TreeNode? pre = null;\n  // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n  while (cur != null) {\n    // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n    if (cur.val == _num) break;\n    pre = cur;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n    if (cur.val &lt; _num)\n      cur = cur.right;\n    // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n    else\n      cur = cur.left;\n  }\n  // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u76f4\u63a5\u8fd4\u56de\n  if (cur == null) return;\n  // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n  if (cur.left == null || cur.right == null) {\n    // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n    TreeNode? child = cur.left ?? cur.right;\n    // \u5220\u9664\u8282\u70b9 cur\n    if (cur != _root) {\n      if (pre!.left == cur)\n        pre.left = child;\n      else\n        pre.right = child;\n    } else {\n      // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n      _root = child;\n    }\n  } else {\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n    TreeNode? tmp = cur.right;\n    while (tmp!.left != null) {\n      tmp = tmp.left;\n    }\n    // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n    remove(tmp.val);\n    // \u7528 tmp \u8986\u76d6 cur\n    cur.val = tmp.val;\n  }\n}\n</code></pre> binary_search_tree.rs<pre><code>/* \u5220\u9664\u8282\u70b9 */\npub fn remove(&amp;mut self, num: i32) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if self.root.is_none() { \n        return; \n    }\n    let mut cur = self.root.clone();\n    let mut pre = None;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while let Some(node) = cur.clone() {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if node.borrow().val == num {\n            break;\n        }\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        pre = cur.clone();\n        if node.borrow().val &lt; num {\n            cur = node.borrow().right.clone();\n        }\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        else {\n            cur = node.borrow().left.clone();\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if cur.is_none() {\n        return;\n    }\n    let cur = cur.unwrap();\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if cur.borrow().left.is_none() || cur.borrow().right.is_none() {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        let child = cur.borrow().left.clone().or_else(|| cur.borrow().right.clone());\n        let pre = pre.unwrap();\n        let left = pre.borrow().left.clone().unwrap();\n        // \u5220\u9664\u8282\u70b9 cur\n        if !Rc::ptr_eq(&amp;cur, self.root.as_ref().unwrap()) {\n            if Rc::ptr_eq(&amp;left, &amp;cur) {\n                pre.borrow_mut().left = child;\n            } else {\n                pre.borrow_mut().right = child;\n            }\n        } else {\n            // \u82e5\u5220\u9664\u8282\u70b9\u4e3a\u6839\u8282\u70b9\uff0c\u5219\u91cd\u65b0\u6307\u5b9a\u6839\u8282\u70b9\n            self.root = child;\n        }\n    }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        let mut tmp = cur.borrow().right.clone();\n        while let Some(node) = tmp.clone() {\n            if node.borrow().left.is_some() {\n                tmp = node.borrow().left.clone();\n            } else {\n                break;\n            }\n        }\n        let tmpval = tmp.unwrap().borrow().val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmpval);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.borrow_mut().val = tmpval;\n    }\n}\n</code></pre> binary_search_tree.c<pre><code>/* \u5220\u9664\u8282\u70b9 */\n// \u7531\u4e8e\u5f15\u5165\u4e86 stdio.h \uff0c\u6b64\u5904\u65e0\u6cd5\u4f7f\u7528 remove \u5173\u952e\u8bcd\nvoid removeItem(BinarySearchTree *bst, int num) {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (bst-&gt;root == NULL)\n        return;\n    TreeNode *cur = bst-&gt;root, *pre = NULL;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != NULL) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur-&gt;val == num)\n            break;\n        pre = cur;\n        if (cur-&gt;val &lt; num) {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u53f3\u5b50\u6811\u4e2d\n            cur = cur-&gt;right;\n        } else {\n            // \u5f85\u5220\u9664\u8282\u70b9\u5728 root \u7684\u5de6\u5b50\u6811\u4e2d\n            cur = cur-&gt;left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == NULL)\n        return;\n    // \u5224\u65ad\u5f85\u5220\u9664\u8282\u70b9\u662f\u5426\u5b58\u5728\u5b50\u8282\u70b9\n    if (cur-&gt;left == NULL || cur-&gt;right == NULL) {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1 */\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = nullptr / \u8be5\u5b50\u8282\u70b9\n        TreeNode *child = cur-&gt;left != NULL ? cur-&gt;left : cur-&gt;right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre-&gt;left == cur) {\n            pre-&gt;left = child;\n        } else {\n            pre-&gt;right = child;\n        }\n    } else {\n        /* \u5b50\u8282\u70b9\u6570\u91cf = 2 */\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        TreeNode *tmp = cur-&gt;right;\n        while (tmp-&gt;left != NULL) {\n            tmp = tmp-&gt;left;\n        }\n        int tmpVal = tmp-&gt;val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        removeItem(bst, tmp-&gt;val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur-&gt;val = tmpVal;\n    }\n}\n</code></pre> binary_search_tree.zig<pre><code>// \u5220\u9664\u8282\u70b9\nfn remove(self: *Self, num: T) void {\n    // \u82e5\u6811\u4e3a\u7a7a\uff0c\u76f4\u63a5\u63d0\u524d\u8fd4\u56de\n    if (self.root == null) return;\n    var cur = self.root;\n    var pre: ?*inc.TreeNode(T) = null;\n    // \u5faa\u73af\u67e5\u627e\uff0c\u8d8a\u8fc7\u53f6\u8282\u70b9\u540e\u8df3\u51fa\n    while (cur != null) {\n        // \u627e\u5230\u5f85\u5220\u9664\u8282\u70b9\uff0c\u8df3\u51fa\u5faa\u73af\n        if (cur.?.val == num) break;\n        pre = cur;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u53f3\u5b50\u6811\u4e2d\n        if (cur.?.val &lt; num) {\n            cur = cur.?.right;\n        // \u5f85\u5220\u9664\u8282\u70b9\u5728 cur \u7684\u5de6\u5b50\u6811\u4e2d\n        } else {\n            cur = cur.?.left;\n        }\n    }\n    // \u82e5\u65e0\u5f85\u5220\u9664\u8282\u70b9\uff0c\u5219\u76f4\u63a5\u8fd4\u56de\n    if (cur == null) return;\n    // \u5b50\u8282\u70b9\u6570\u91cf = 0 or 1\n    if (cur.?.left == null or cur.?.right == null) {\n        // \u5f53\u5b50\u8282\u70b9\u6570\u91cf = 0 / 1 \u65f6\uff0c child = null / \u8be5\u5b50\u8282\u70b9\n        var child = if (cur.?.left != null) cur.?.left else cur.?.right;\n        // \u5220\u9664\u8282\u70b9 cur\n        if (pre.?.left == cur) {\n            pre.?.left = child;\n        } else {\n            pre.?.right = child;\n        }\n    // \u5b50\u8282\u70b9\u6570\u91cf = 2\n    } else {\n        // \u83b7\u53d6\u4e2d\u5e8f\u904d\u5386\u4e2d cur \u7684\u4e0b\u4e00\u4e2a\u8282\u70b9\n        var tmp = cur.?.right;\n        while (tmp.?.left != null) {\n            tmp = tmp.?.left;\n        }\n        var tmp_val = tmp.?.val;\n        // \u9012\u5f52\u5220\u9664\u8282\u70b9 tmp\n        self.remove(tmp.?.val);\n        // \u7528 tmp \u8986\u76d6 cur\n        cur.?.val = tmp_val;\n    }\n}\n</code></pre>"},{"location":"chapter_tree/binary_search_tree/#4","title":"4. \u00a0 \u4e2d\u5e8f\u904d\u5386\u6709\u5e8f","text":"<p>\u5982\u56fe 7-22 \u6240\u793a\uff0c\u4e8c\u53c9\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u9075\u5faa\u201c\u5de6 \\(\\rightarrow\\) \u6839 \\(\\rightarrow\\) \u53f3\u201d\u7684\u904d\u5386\u987a\u5e8f\uff0c\u800c\u4e8c\u53c9\u641c\u7d22\u6811\u6ee1\u8db3\u201c\u5de6\u5b50\u8282\u70b9 \\(&lt;\\) \u6839\u8282\u70b9 \\(&lt;\\) \u53f3\u5b50\u8282\u70b9\u201d\u7684\u5927\u5c0f\u5173\u7cfb\u3002</p> <p>\u8fd9\u610f\u5473\u7740\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u8fdb\u884c\u4e2d\u5e8f\u904d\u5386\u65f6\uff0c\u603b\u662f\u4f1a\u4f18\u5148\u904d\u5386\u4e0b\u4e00\u4e2a\u6700\u5c0f\u8282\u70b9\uff0c\u4ece\u800c\u5f97\u51fa\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\uff1a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217\u662f\u5347\u5e8f\u7684\u3002</p> <p>\u5229\u7528\u4e2d\u5e8f\u904d\u5386\u5347\u5e8f\u7684\u6027\u8d28\uff0c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u83b7\u53d6\u6709\u5e8f\u6570\u636e\u4ec5\u9700 \\(O(n)\\) \u65f6\u95f4\uff0c\u65e0\u987b\u8fdb\u884c\u989d\u5916\u7684\u6392\u5e8f\u64cd\u4f5c\uff0c\u975e\u5e38\u9ad8\u6548\u3002</p> <p></p> <p> \u56fe 7-22 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u4e2d\u5e8f\u904d\u5386\u5e8f\u5217 </p>"},{"location":"chapter_tree/binary_search_tree/#742","title":"7.4.2 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387","text":"<p>\u7ed9\u5b9a\u4e00\u7ec4\u6570\u636e\uff0c\u6211\u4eec\u8003\u8651\u4f7f\u7528\u6570\u7ec4\u6216\u4e8c\u53c9\u641c\u7d22\u6811\u5b58\u50a8\u3002\u89c2\u5bdf\u8868 7-2 \uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u5404\u9879\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\u5bf9\u6570\u9636\uff0c\u5177\u6709\u7a33\u5b9a\u4e14\u9ad8\u6548\u7684\u6027\u80fd\u3002\u53ea\u6709\u5728\u9ad8\u9891\u6dfb\u52a0\u3001\u4f4e\u9891\u67e5\u627e\u5220\u9664\u6570\u636e\u7684\u573a\u666f\u4e0b\uff0c\u6570\u7ec4\u6bd4\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u6548\u7387\u66f4\u9ad8\u3002</p> <p> \u8868 7-2 \u00a0 \u6570\u7ec4\u4e0e\u641c\u7d22\u6811\u7684\u6548\u7387\u5bf9\u6bd4 </p> \u65e0\u5e8f\u6570\u7ec4 \u4e8c\u53c9\u641c\u7d22\u6811 \u67e5\u627e\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) \u63d2\u5165\u5143\u7d20 \\(O(1)\\) \\(O(\\log n)\\) \u5220\u9664\u5143\u7d20 \\(O(n)\\) \\(O(\\log n)\\) <p>\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u201c\u5e73\u8861\u201d\u7684\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u5728 \\(\\log n\\) \u8f6e\u5faa\u73af\u5185\u67e5\u627e\u4efb\u610f\u8282\u70b9\u3002</p> <p>\u7136\u800c\uff0c\u5982\u679c\u6211\u4eec\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\u4e0d\u65ad\u5730\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\uff0c\u53ef\u80fd\u5bfc\u81f4\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u56fe 7-23 \u6240\u793a\u7684\u94fe\u8868\uff0c\u8fd9\u65f6\u5404\u79cd\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e5f\u4f1a\u9000\u5316\u4e3a \\(O(n)\\) \u3002</p> <p></p> <p> \u56fe 7-23 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316 </p>"},{"location":"chapter_tree/binary_search_tree/#743","title":"7.4.3 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u5e38\u89c1\u5e94\u7528","text":"<ul> <li>\u7528\u4f5c\u7cfb\u7edf\u4e2d\u7684\u591a\u7ea7\u7d22\u5f15\uff0c\u5b9e\u73b0\u9ad8\u6548\u7684\u67e5\u627e\u3001\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u3002</li> <li>\u4f5c\u4e3a\u67d0\u4e9b\u641c\u7d22\u7b97\u6cd5\u7684\u5e95\u5c42\u6570\u636e\u7ed3\u6784\u3002</li> <li>\u7528\u4e8e\u5b58\u50a8\u6570\u636e\u6d41\uff0c\u4ee5\u4fdd\u6301\u5176\u6709\u5e8f\u72b6\u6001\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree/","title":"7.1 \u00a0 \u4e8c\u53c9\u6811","text":"<p>\u300c\u4e8c\u53c9\u6811 binary tree\u300d\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4ee3\u8868\u201c\u7956\u5148\u201d\u4e0e\u201c\u540e\u4ee3\u201d\u4e4b\u95f4\u7684\u6d3e\u751f\u5173\u7cfb\uff0c\u4f53\u73b0\u4e86\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u4e8c\u53c9\u6811\u7684\u57fa\u672c\u5355\u5143\u662f\u8282\u70b9\uff0c\u6bcf\u4e2a\u8282\u70b9\u5305\u542b\u503c\u3001\u5de6\u5b50\u8282\u70b9\u5f15\u7528\u548c\u53f3\u5b50\u8282\u70b9\u5f15\u7528\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig <pre><code>class TreeNode:\n    \"\"\"\u4e8c\u53c9\u6811\u8282\u70b9\u7c7b\"\"\"\n    def __init__(self, val: int):\n        self.val: int = val                # \u8282\u70b9\u503c\n        self.left: TreeNode | None = None  # \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        self.right: TreeNode | None = None # \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    int val;          // \u8282\u70b9\u503c\n    TreeNode *left;   // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    TreeNode *right;  // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}\n};\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    int val;         // \u8282\u70b9\u503c\n    TreeNode left;   // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    TreeNode right;  // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n    TreeNode(int x) { val = x; }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode(int? x) {\n    public int? val = x;    // \u8282\u70b9\u503c\n    public TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    public TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntype TreeNode struct {\n    Val   int\n    Left  *TreeNode\n    Right *TreeNode\n}\n/* \u6784\u9020\u65b9\u6cd5 */\nfunc NewTreeNode(v int) *TreeNode {\n    return &amp;TreeNode{\n        Left:  nil, // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n        Right: nil, // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n        Val:   v,   // \u8282\u70b9\u503c\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    var val: Int // \u8282\u70b9\u503c\n    var left: TreeNode? // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    var right: TreeNode? // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n\n    init(x: Int) {\n        val = x\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val; // \u8282\u70b9\u503c\n    left; // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n    constructor(val, left, right) {\n        this.val = val === undefined ? 0 : val;\n        this.left = left === undefined ? null : left;\n        this.right = right === undefined ? null : right;\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n    val: number;\n    left: TreeNode | null;\n    right: TreeNode | null;\n\n    constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {\n        this.val = val === undefined ? 0 : val; // \u8282\u70b9\u503c\n        this.left = left === undefined ? null : left; // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n        this.right = right === undefined ? null : right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7c7b */\nclass TreeNode {\n  int val;         // \u8282\u70b9\u503c\n  TreeNode? left;  // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n  TreeNode? right; // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n  TreeNode(this.val, [this.left, this.right]);\n}\n</code></pre> <pre><code>use std::rc::Rc;\nuse std::cell::RefCell;\n\n/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\nstruct TreeNode {\n    val: i32,                               // \u8282\u70b9\u503c\n    left: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,    // \u5de6\u5b50\u8282\u70b9\u5f15\u7528\n    right: Option&lt;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;,   // \u53f3\u5b50\u8282\u70b9\u5f15\u7528\n}\n\nimpl TreeNode {\n    /* \u6784\u9020\u65b9\u6cd5 */\n    fn new(val: i32) -&gt; Rc&lt;RefCell&lt;Self&gt;&gt; {\n        Rc::new(RefCell::new(Self {\n            val,\n            left: None,\n            right: None\n        }))\n    }\n}\n</code></pre> <pre><code>/* \u4e8c\u53c9\u6811\u8282\u70b9\u7ed3\u6784\u4f53 */\ntypedef struct TreeNode {\n    int val;                // \u8282\u70b9\u503c\n    int height;             // \u8282\u70b9\u9ad8\u5ea6\n    struct TreeNode *left;  // \u5de6\u5b50\u8282\u70b9\u6307\u9488\n    struct TreeNode *right; // \u53f3\u5b50\u8282\u70b9\u6307\u9488\n} TreeNode;\n\n/* \u6784\u9020\u51fd\u6570 */\nTreeNode *newTreeNode(int val) {\n    TreeNode *node;\n\n    node = (TreeNode *)malloc(sizeof(TreeNode));\n    node-&gt;val = val;\n    node-&gt;height = 0;\n    node-&gt;left = NULL;\n    node-&gt;right = NULL;\n    return node;\n}\n</code></pre> <pre><code>\n</code></pre> <p>\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5f15\u7528\uff08\u6307\u9488\uff09\uff0c\u5206\u522b\u6307\u5411\u300c\u5de6\u5b50\u8282\u70b9 left-child node\u300d\u548c\u300c\u53f3\u5b50\u8282\u70b9 right-child node\u300d\uff0c\u8be5\u8282\u70b9\u88ab\u79f0\u4e3a\u8fd9\u4e24\u4e2a\u5b50\u8282\u70b9\u7684\u300c\u7236\u8282\u70b9 parent node\u300d\u3002\u5f53\u7ed9\u5b9a\u4e00\u4e2a\u4e8c\u53c9\u6811\u7684\u8282\u70b9\u65f6\uff0c\u6211\u4eec\u5c06\u8be5\u8282\u70b9\u7684\u5de6\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u300c\u5de6\u5b50\u6811 left subtree\u300d\uff0c\u540c\u7406\u53ef\u5f97\u300c\u53f3\u5b50\u6811 right subtree\u300d\u3002</p> <p>\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u9664\u53f6\u8282\u70b9\u5916\uff0c\u5176\u4ed6\u6240\u6709\u8282\u70b9\u90fd\u5305\u542b\u5b50\u8282\u70b9\u548c\u975e\u7a7a\u5b50\u6811\u3002\u5982\u56fe 7-1 \u6240\u793a\uff0c\u5982\u679c\u5c06\u201c\u8282\u70b9 2\u201d\u89c6\u4e3a\u7236\u8282\u70b9\uff0c\u5219\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u5206\u522b\u662f\u201c\u8282\u70b9 4\u201d\u548c\u201c\u8282\u70b9 5\u201d\uff0c\u5de6\u5b50\u6811\u662f\u201c\u8282\u70b9 4 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\uff0c\u53f3\u5b50\u6811\u662f\u201c\u8282\u70b9 5 \u53ca\u5176\u4ee5\u4e0b\u8282\u70b9\u5f62\u6210\u7684\u6811\u201d\u3002</p> <p></p> <p> \u56fe 7-1 \u00a0 \u7236\u8282\u70b9\u3001\u5b50\u8282\u70b9\u3001\u5b50\u6811 </p>"},{"location":"chapter_tree/binary_tree/#711","title":"7.1.1 \u00a0 \u4e8c\u53c9\u6811\u5e38\u89c1\u672f\u8bed","text":"<p>\u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed\u5982\u56fe 7-2 \u6240\u793a\u3002</p> <ul> <li>\u300c\u6839\u8282\u70b9 root node\u300d\uff1a\u4f4d\u4e8e\u4e8c\u53c9\u6811\u9876\u5c42\u7684\u8282\u70b9\uff0c\u6ca1\u6709\u7236\u8282\u70b9\u3002</li> <li>\u300c\u53f6\u8282\u70b9 leaf node\u300d\uff1a\u6ca1\u6709\u5b50\u8282\u70b9\u7684\u8282\u70b9\uff0c\u5176\u4e24\u4e2a\u6307\u9488\u5747\u6307\u5411 \\(\\text{None}\\) \u3002</li> <li>\u300c\u8fb9 edge\u300d\uff1a\u8fde\u63a5\u4e24\u4e2a\u8282\u70b9\u7684\u7ebf\u6bb5\uff0c\u5373\u8282\u70b9\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</li> <li>\u8282\u70b9\u6240\u5728\u7684\u300c\u5c42 level\u300d\uff1a\u4ece\u9876\u81f3\u5e95\u9012\u589e\uff0c\u6839\u8282\u70b9\u6240\u5728\u5c42\u4e3a 1 \u3002</li> <li>\u8282\u70b9\u7684\u300c\u5ea6 degree\u300d\uff1a\u8282\u70b9\u7684\u5b50\u8282\u70b9\u7684\u6570\u91cf\u3002\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u5ea6\u7684\u53d6\u503c\u8303\u56f4\u662f 0\u30011\u30012 \u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u6700\u8fdc\u53f6\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u300c\u6df1\u5ea6 depth\u300d\uff1a\u4ece\u6839\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> <li>\u8282\u70b9\u7684\u300c\u9ad8\u5ea6 height\u300d\uff1a\u4ece\u8ddd\u79bb\u8be5\u8282\u70b9\u6700\u8fdc\u7684\u53f6\u8282\u70b9\u5230\u8be5\u8282\u70b9\u6240\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u3002</li> </ul> <p></p> <p> \u56fe 7-2 \u00a0 \u4e8c\u53c9\u6811\u7684\u5e38\u7528\u672f\u8bed </p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u201c\u9ad8\u5ea6\u201d\u548c\u201c\u6df1\u5ea6\u201d\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\uff0c\u4f46\u6709\u4e9b\u9898\u76ee\u6216\u6559\u6750\u53ef\u80fd\u4f1a\u5c06\u5176\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8282\u70b9\u7684\u6570\u91cf\u201d\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9ad8\u5ea6\u548c\u6df1\u5ea6\u90fd\u9700\u8981\u52a0 1 \u3002</p>"},{"location":"chapter_tree/binary_tree/#712","title":"7.1.2 \u00a0 \u4e8c\u53c9\u6811\u57fa\u672c\u64cd\u4f5c","text":""},{"location":"chapter_tree/binary_tree/#1","title":"1. \u00a0 \u521d\u59cb\u5316\u4e8c\u53c9\u6811","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u9996\u5148\u521d\u59cb\u5316\u8282\u70b9\uff0c\u7136\u540e\u6784\u5efa\u5f15\u7528\uff08\u6307\u9488\uff09\u3002</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py<pre><code># \u521d\u59cb\u5316\u4e8c\u53c9\u6811\n# \u521d\u59cb\u5316\u8282\u70b9\nn1 = TreeNode(val=1)\nn2 = TreeNode(val=2)\nn3 = TreeNode(val=3)\nn4 = TreeNode(val=4)\nn5 = TreeNode(val=5)\n# \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.cpp<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode* n1 = new TreeNode(1);\nTreeNode* n2 = new TreeNode(2);\nTreeNode* n3 = new TreeNode(3);\nTreeNode* n4 = new TreeNode(4);\nTreeNode* n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.java<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.cs<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new(1);\nTreeNode n2 = new(2);\nTreeNode n3 = new(3);\nTreeNode n4 = new(4);\nTreeNode n5 = new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.go<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nn1 := NewTreeNode(1)\nn2 := NewTreeNode(2)\nn3 := NewTreeNode(3)\nn4 := NewTreeNode(4)\nn5 := NewTreeNode(5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.Left = n2\nn1.Right = n3\nn2.Left = n4\nn2.Right = n5\n</code></pre> binary_tree.swift<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode(x: 1)\nlet n2 = TreeNode(x: 2)\nlet n3 = TreeNode(x: 3)\nlet n4 = TreeNode(x: 4)\nlet n5 = TreeNode(x: 5)\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2\nn1.right = n3\nn2.left = n4\nn2.right = n5\n</code></pre> binary_tree.js<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.ts<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = new TreeNode(1),\n    n2 = new TreeNode(2),\n    n3 = new TreeNode(3),\n    n4 = new TreeNode(4),\n    n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.dart<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode n1 = new TreeNode(1);\nTreeNode n2 = new TreeNode(2);\nTreeNode n3 = new TreeNode(3);\nTreeNode n4 = new TreeNode(4);\nTreeNode n5 = new TreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.left = n2;\nn1.right = n3;\nn2.left = n4;\nn2.right = n5;\n</code></pre> binary_tree.rs<pre><code>// \u521d\u59cb\u5316\u8282\u70b9\nlet n1 = TreeNode::new(1);\nlet n2 = TreeNode::new(2);\nlet n3 = TreeNode::new(3);\nlet n4 = TreeNode::new(4);\nlet n5 = TreeNode::new(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1.borrow_mut().left = Some(n2.clone());\nn1.borrow_mut().right = Some(n3);\nn2.borrow_mut().left = Some(n4);\nn2.borrow_mut().right = Some(n5);\n</code></pre> binary_tree.c<pre><code>/* \u521d\u59cb\u5316\u4e8c\u53c9\u6811 */\n// \u521d\u59cb\u5316\u8282\u70b9\nTreeNode *n1 = newTreeNode(1);\nTreeNode *n2 = newTreeNode(2);\nTreeNode *n3 = newTreeNode(3);\nTreeNode *n4 = newTreeNode(4);\nTreeNode *n5 = newTreeNode(5);\n// \u6784\u5efa\u8282\u70b9\u4e4b\u95f4\u7684\u5f15\u7528\uff08\u6307\u9488\uff09\nn1-&gt;left = n2;\nn1-&gt;right = n3;\nn2-&gt;left = n4;\nn2-&gt;right = n5;\n</code></pre> binary_tree.zig<pre><code>\n</code></pre>"},{"location":"chapter_tree/binary_tree/#2","title":"2. \u00a0 \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9","text":"<p>\u4e0e\u94fe\u8868\u7c7b\u4f3c\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u6307\u9488\u6765\u5b9e\u73b0\u3002\u56fe 7-3 \u7ed9\u51fa\u4e86\u4e00\u4e2a\u793a\u4f8b\u3002</p> <p></p> <p> \u56fe 7-3 \u00a0 \u5728\u4e8c\u53c9\u6811\u4e2d\u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 </p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree.py<pre><code># \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9\np = TreeNode(0)\n# \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = p\np.left = n2\n# \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.cpp<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode* P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.java<pre><code>TreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.cs<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.go<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\np := NewTreeNode(0)\nn1.Left = p\np.Left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.Left = n2\n</code></pre> binary_tree.swift<pre><code>let P = TreeNode(x: 0)\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P\nP.left = n2\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2\n</code></pre> binary_tree.js<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nlet P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.ts<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nconst P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.dart<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode P = new TreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.left = P;\nP.left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1.left = n2;\n</code></pre> binary_tree.rs<pre><code>let p = TreeNode::new(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1.borrow_mut().left = Some(p.clone());\np.borrow_mut().left = Some(n2.clone());\n// \u5220\u9664\u8282\u70b9 p\nn1.borrow_mut().left = Some(n2);\n</code></pre> binary_tree.c<pre><code>/* \u63d2\u5165\u4e0e\u5220\u9664\u8282\u70b9 */\nTreeNode *P = newTreeNode(0);\n// \u5728 n1 -&gt; n2 \u4e2d\u95f4\u63d2\u5165\u8282\u70b9 P\nn1-&gt;left = P;\nP-&gt;left = n2;\n// \u5220\u9664\u8282\u70b9 P\nn1-&gt;left = n2;\n</code></pre> binary_tree.zig<pre><code>\n</code></pre> <p>Note</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u63d2\u5165\u8282\u70b9\u53ef\u80fd\u4f1a\u6539\u53d8\u4e8c\u53c9\u6811\u7684\u539f\u6709\u903b\u8f91\u7ed3\u6784\uff0c\u800c\u5220\u9664\u8282\u70b9\u901a\u5e38\u610f\u5473\u7740\u5220\u9664\u8be5\u8282\u70b9\u53ca\u5176\u6240\u6709\u5b50\u6811\u3002\u56e0\u6b64\uff0c\u5728\u4e8c\u53c9\u6811\u4e2d\uff0c\u63d2\u5165\u4e0e\u5220\u9664\u901a\u5e38\u662f\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\u7684\uff0c\u4ee5\u5b9e\u73b0\u6709\u5b9e\u9645\u610f\u4e49\u7684\u64cd\u4f5c\u3002</p>"},{"location":"chapter_tree/binary_tree/#713","title":"7.1.3 \u00a0 \u5e38\u89c1\u4e8c\u53c9\u6811\u7c7b\u578b","text":""},{"location":"chapter_tree/binary_tree/#1_1","title":"1. \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-4 \u6240\u793a\uff0c\u300c\u5b8c\u7f8e\u4e8c\u53c9\u6811 perfect binary tree\u300d\u6240\u6709\u5c42\u7684\u8282\u70b9\u90fd\u88ab\u5b8c\u5168\u586b\u6ee1\u3002\u5728\u5b8c\u7f8e\u4e8c\u53c9\u6811\u4e2d\uff0c\u53f6\u8282\u70b9\u7684\u5ea6\u4e3a \\(0\\) \uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u7684\u5ea6\u90fd\u4e3a \\(2\\) \uff1b\u82e5\u6811\u7684\u9ad8\u5ea6\u4e3a \\(h\\) \uff0c\u5219\u8282\u70b9\u603b\u6570\u4e3a \\(2^{h+1} - 1\\) \uff0c\u5448\u73b0\u6807\u51c6\u7684\u6307\u6570\u7ea7\u5173\u7cfb\uff0c\u53cd\u6620\u4e86\u81ea\u7136\u754c\u4e2d\u5e38\u89c1\u7684\u7ec6\u80de\u5206\u88c2\u73b0\u8c61\u3002</p> <p>Tip</p> <p>\u8bf7\u6ce8\u610f\uff0c\u5728\u4e2d\u6587\u793e\u533a\u4e2d\uff0c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u5e38\u88ab\u79f0\u4e3a\u300c\u6ee1\u4e8c\u53c9\u6811\u300d\u3002</p> <p></p> <p> \u56fe 7-4 \u00a0 \u5b8c\u7f8e\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#2_1","title":"2. \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-5 \u6240\u793a\uff0c\u300c\u5b8c\u5168\u4e8c\u53c9\u6811 complete binary tree\u300d\u53ea\u6709\u6700\u5e95\u5c42\u7684\u8282\u70b9\u672a\u88ab\u586b\u6ee1\uff0c\u4e14\u6700\u5e95\u5c42\u8282\u70b9\u5c3d\u91cf\u9760\u5de6\u586b\u5145\u3002</p> <p></p> <p> \u56fe 7-5 \u00a0 \u5b8c\u5168\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#3","title":"3. \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-6 \u6240\u793a\uff0c\u300c\u5b8c\u6ee1\u4e8c\u53c9\u6811 full binary tree\u300d\u9664\u4e86\u53f6\u8282\u70b9\u4e4b\u5916\uff0c\u5176\u4f59\u6240\u6709\u8282\u70b9\u90fd\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\u3002</p> <p></p> <p> \u56fe 7-6 \u00a0 \u5b8c\u6ee1\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#4","title":"4. \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811","text":"<p>\u5982\u56fe 7-7 \u6240\u793a\uff0c\u300c\u5e73\u8861\u4e8c\u53c9\u6811 balanced binary tree\u300d\u4e2d\u4efb\u610f\u8282\u70b9\u7684\u5de6\u5b50\u6811\u548c\u53f3\u5b50\u6811\u7684\u9ad8\u5ea6\u4e4b\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc7 1 \u3002</p> <p></p> <p> \u56fe 7-7 \u00a0 \u5e73\u8861\u4e8c\u53c9\u6811 </p>"},{"location":"chapter_tree/binary_tree/#714","title":"7.1.4 \u00a0 \u4e8c\u53c9\u6811\u7684\u9000\u5316","text":"<p>\u56fe 7-8 \u5c55\u793a\u4e86\u4e8c\u53c9\u6811\u7684\u7406\u60f3\u7ed3\u6784\u4e0e\u9000\u5316\u7ed3\u6784\u3002\u5f53\u4e8c\u53c9\u6811\u7684\u6bcf\u5c42\u8282\u70b9\u90fd\u88ab\u586b\u6ee1\u65f6\uff0c\u8fbe\u5230\u201c\u5b8c\u7f8e\u4e8c\u53c9\u6811\u201d\uff1b\u800c\u5f53\u6240\u6709\u8282\u70b9\u90fd\u504f\u5411\u4e00\u4fa7\u65f6\uff0c\u4e8c\u53c9\u6811\u9000\u5316\u4e3a\u201c\u94fe\u8868\u201d\u3002</p> <ul> <li>\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u7406\u60f3\u60c5\u51b5\uff0c\u53ef\u4ee5\u5145\u5206\u53d1\u6325\u4e8c\u53c9\u6811\u201c\u5206\u6cbb\u201d\u7684\u4f18\u52bf\u3002</li> <li>\u94fe\u8868\u5219\u662f\u53e6\u4e00\u4e2a\u6781\u7aef\uff0c\u5404\u9879\u64cd\u4f5c\u90fd\u53d8\u4e3a\u7ebf\u6027\u64cd\u4f5c\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u9000\u5316\u81f3 \\(O(n)\\) \u3002</li> </ul> <p></p> <p> \u56fe 7-8 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> <p>\u5982\u8868 7-1 \u6240\u793a\uff0c\u5728\u6700\u4f73\u7ed3\u6784\u548c\u6700\u5dee\u7ed3\u6784\u4e0b\uff0c\u4e8c\u53c9\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf\u3001\u8282\u70b9\u603b\u6570\u3001\u9ad8\u5ea6\u7b49\u8fbe\u5230\u6781\u5927\u503c\u6216\u6781\u5c0f\u503c\u3002</p> <p> \u8868 7-1 \u00a0 \u4e8c\u53c9\u6811\u7684\u6700\u4f73\u7ed3\u6784\u4e0e\u6700\u5dee\u7ed3\u6784 </p> \u5b8c\u7f8e\u4e8c\u53c9\u6811 \u94fe\u8868 \u7b2c \\(i\\) \u5c42\u7684\u8282\u70b9\u6570\u91cf \\(2^{i-1}\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u53f6\u8282\u70b9\u6570\u91cf \\(2^h\\) \\(1\\) \u9ad8\u5ea6\u4e3a \\(h\\) \u7684\u6811\u7684\u8282\u70b9\u603b\u6570 \\(2^{h+1} - 1\\) \\(h + 1\\) \u8282\u70b9\u603b\u6570\u4e3a \\(n\\) \u7684\u6811\u7684\u9ad8\u5ea6 \\(\\log_2 (n+1) - 1\\) \\(n - 1\\)"},{"location":"chapter_tree/binary_tree_traversal/","title":"7.2 \u00a0 \u4e8c\u53c9\u6811\u904d\u5386","text":"<p>\u4ece\u7269\u7406\u7ed3\u6784\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u6811\u662f\u4e00\u79cd\u57fa\u4e8e\u94fe\u8868\u7684\u6570\u636e\u7ed3\u6784\uff0c\u56e0\u6b64\u5176\u904d\u5386\u65b9\u5f0f\u662f\u901a\u8fc7\u6307\u9488\u9010\u4e2a\u8bbf\u95ee\u8282\u70b9\u3002\u7136\u800c\uff0c\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u8fd9\u4f7f\u5f97\u904d\u5386\u6811\u6bd4\u904d\u5386\u94fe\u8868\u66f4\u52a0\u590d\u6742\uff0c\u9700\u8981\u501f\u52a9\u641c\u7d22\u7b97\u6cd5\u6765\u5b9e\u73b0\u3002</p> <p>\u4e8c\u53c9\u6811\u5e38\u89c1\u7684\u904d\u5386\u65b9\u5f0f\u5305\u62ec\u5c42\u5e8f\u904d\u5386\u3001\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u7b49\u3002</p>"},{"location":"chapter_tree/binary_tree_traversal/#721","title":"7.2.1 \u00a0 \u5c42\u5e8f\u904d\u5386","text":"<p>\u5982\u56fe 7-9 \u6240\u793a\uff0c\u300c\u5c42\u5e8f\u904d\u5386 level-order traversal\u300d\u4ece\u9876\u90e8\u5230\u5e95\u90e8\u9010\u5c42\u904d\u5386\u4e8c\u53c9\u6811\uff0c\u5e76\u5728\u6bcf\u4e00\u5c42\u6309\u7167\u4ece\u5de6\u5230\u53f3\u7684\u987a\u5e8f\u8bbf\u95ee\u8282\u70b9\u3002</p> <p>\u5c42\u5e8f\u904d\u5386\u672c\u8d28\u4e0a\u5c5e\u4e8e\u300c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386 breadth-first traversal, BFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\u3002</p> <p></p> <p> \u56fe 7-9 \u00a0 \u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u901a\u5e38\u501f\u52a9\u201c\u961f\u5217\u201d\u6765\u5b9e\u73b0\u3002\u961f\u5217\u9075\u5faa\u201c\u5148\u8fdb\u5148\u51fa\u201d\u7684\u89c4\u5219\uff0c\u800c\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5219\u9075\u5faa\u201c\u9010\u5c42\u63a8\u8fdb\u201d\u7684\u89c4\u5219\uff0c\u4e24\u8005\u80cc\u540e\u7684\u601d\u60f3\u662f\u4e00\u81f4\u7684\u3002\u5b9e\u73b0\u4ee3\u7801\u5982\u4e0b\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_bfs.py<pre><code>def level_order(root: TreeNode | None) -&gt; list[int]:\n    \"\"\"\u5c42\u5e8f\u904d\u5386\"\"\"\n    # \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue: deque[TreeNode] = deque()\n    queue.append(root)\n    # \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    res = []\n    while queue:\n        node: TreeNode = queue.popleft()  # \u961f\u5217\u51fa\u961f\n        res.append(node.val)  # \u4fdd\u5b58\u8282\u70b9\u503c\n        if node.left is not None:\n            queue.append(node.left)  # \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if node.right is not None:\n            queue.append(node.right)  # \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    return res\n</code></pre> binary_tree_bfs.cpp<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nvector&lt;int&gt; levelOrder(TreeNode *root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue&lt;TreeNode *&gt; queue;\n    queue.push(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    vector&lt;int&gt; vec;\n    while (!queue.empty()) {\n        TreeNode *node = queue.front();\n        queue.pop();              // \u961f\u5217\u51fa\u961f\n        vec.push_back(node-&gt;val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node-&gt;left != nullptr)\n            queue.push(node-&gt;left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node-&gt;right != nullptr)\n            queue.push(node-&gt;right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return vec;\n}\n</code></pre> binary_tree_bfs.java<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;Integer&gt; levelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new LinkedList&lt;&gt;();\n    queue.add(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;Integer&gt; list = new ArrayList&lt;&gt;();\n    while (!queue.isEmpty()) {\n        TreeNode node = queue.poll(); // \u961f\u5217\u51fa\u961f\n        list.add(node.val);           // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.offer(node.left);   // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.offer(node.right);  // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.cs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; LevelOrder(TreeNode root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    Queue&lt;TreeNode&gt; queue = new();\n    queue.Enqueue(root);\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    List&lt;int&gt; list = [];\n    while (queue.Count != 0) {\n        TreeNode node = queue.Dequeue(); // \u961f\u5217\u51fa\u961f\n        list.Add(node.val!.Value);       // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null)\n            queue.Enqueue(node.left);    // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right != null)\n            queue.Enqueue(node.right);   // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.go<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root *TreeNode) []any {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    queue := list.New()\n    queue.PushBack(root)\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5207\u7247\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    nums := make([]any, 0)\n    for queue.Len() &gt; 0 {\n        // \u961f\u5217\u51fa\u961f\n        node := queue.Remove(queue.Front()).(*TreeNode)\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        nums = append(nums, node.Val)\n        if node.Left != nil {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Left)\n        }\n        if node.Right != nil {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue.PushBack(node.Right)\n        }\n    }\n    return nums\n}\n</code></pre> binary_tree_bfs.swift<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunc levelOrder(root: TreeNode) -&gt; [Int] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    var queue: [TreeNode] = [root]\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list: [Int] = []\n    while !queue.isEmpty {\n        let node = queue.removeFirst() // \u961f\u5217\u51fa\u961f\n        list.append(node.val) // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let left = node.left {\n            queue.append(left) // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let right = node.right {\n            queue.append(right) // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list\n}\n</code></pre> binary_tree_bfs.js<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list = [];\n    while (queue.length) {\n        let node = queue.shift(); // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        if (node.right) queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.ts<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfunction levelOrder(root: TreeNode | null): number[] {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const queue = [root];\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    const list: number[] = [];\n    while (queue.length) {\n        let node = queue.shift() as TreeNode; // \u961f\u5217\u51fa\u961f\n        list.push(node.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left) {\n            queue.push(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right) {\n            queue.push(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }\n    }\n    return list;\n}\n</code></pre> binary_tree_bfs.dart<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nList&lt;int&gt; levelOrder(TreeNode? root) {\n  // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n  Queue&lt;TreeNode?&gt; queue = Queue();\n  queue.add(root);\n  // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n  List&lt;int&gt; res = [];\n  while (queue.isNotEmpty) {\n    TreeNode? node = queue.removeFirst(); // \u961f\u5217\u51fa\u961f\n    res.add(node!.val); // \u4fdd\u5b58\u8282\u70b9\u503c\n    if (node.left != null) queue.add(node.left); // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n    if (node.right != null) queue.add(node.right); // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n  }\n  return res;\n}\n</code></pre> binary_tree_bfs.rs<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nfn level_order(root: &amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    let mut que = VecDeque::new();\n    que.push_back(Rc::clone(&amp;root));\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    let mut vec = Vec::new();\n\n    while let Some(node) = que.pop_front() {                 // \u961f\u5217\u51fa\u961f\n        vec.push(node.borrow().val);                         // \u4fdd\u5b58\u8282\u70b9\u503c\n        if let Some(left) = node.borrow().left.as_ref() {\n            que.push_back(Rc::clone(left));                  // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if let Some(right) = node.borrow().right.as_ref() {\n            que.push_back(Rc::clone(right));                 // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        };\n    }\n    vec\n}\n</code></pre> binary_tree_bfs.c<pre><code>/* \u5c42\u5e8f\u904d\u5386 */\nint *levelOrder(TreeNode *root, int *size) {\n    /* \u8f85\u52a9\u961f\u5217 */\n    int front, rear;\n    int index, *arr;\n    TreeNode *node;\n    TreeNode **queue;\n\n    /* \u8f85\u52a9\u961f\u5217 */\n    queue = (TreeNode **)malloc(sizeof(TreeNode *) * MAX_SIZE);\n    // \u961f\u5217\u6307\u9488\n    front = 0, rear = 0;\n    // \u52a0\u5165\u6839\u8282\u70b9\n    queue[rear++] = root;\n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    /* \u8f85\u52a9\u6570\u7ec4 */\n    arr = (int *)malloc(sizeof(int) * MAX_SIZE);\n    // \u6570\u7ec4\u6307\u9488\n    index = 0;\n    while (front &lt; rear) {\n        // \u961f\u5217\u51fa\u961f\n        node = queue[front++];\n        // \u4fdd\u5b58\u8282\u70b9\u503c\n        arr[index++] = node-&gt;val;\n        if (node-&gt;left != NULL) {\n            // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;left;\n        }\n        if (node-&gt;right != NULL) {\n            // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n            queue[rear++] = node-&gt;right;\n        }\n    }\n    // \u66f4\u65b0\u6570\u7ec4\u957f\u5ea6\u7684\u503c\n    *size = index;\n    arr = realloc(arr, sizeof(int) * (*size));\n\n    // \u91ca\u653e\u8f85\u52a9\u6570\u7ec4\u7a7a\u95f4\n    free(queue);\n    return arr;\n}\n</code></pre> binary_tree_bfs.zig<pre><code>// \u5c42\u5e8f\u904d\u5386\nfn levelOrder(comptime T: type, mem_allocator: std.mem.Allocator, root: *inc.TreeNode(T)) !std.ArrayList(T) {\n    // \u521d\u59cb\u5316\u961f\u5217\uff0c\u52a0\u5165\u6839\u8282\u70b9\n    const L = std.TailQueue(*inc.TreeNode(T));\n    var queue = L{};\n    var root_node = try mem_allocator.create(L.Node);\n    root_node.data = root;\n    queue.append(root_node); \n    // \u521d\u59cb\u5316\u4e00\u4e2a\u5217\u8868\uff0c\u7528\u4e8e\u4fdd\u5b58\u904d\u5386\u5e8f\u5217\n    var list = std.ArrayList(T).init(std.heap.page_allocator);\n    while (queue.len &gt; 0) {\n        var queue_node = queue.popFirst().?;    // \u961f\u5217\u51fa\u961f\n        var node = queue_node.data;\n        try list.append(node.val);              // \u4fdd\u5b58\u8282\u70b9\u503c\n        if (node.left != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.left.?;\n            queue.append(tmp_node);             // \u5de6\u5b50\u8282\u70b9\u5165\u961f\n        }\n        if (node.right != null) {\n            var tmp_node = try mem_allocator.create(L.Node);\n            tmp_node.data = node.right.?;\n            queue.append(tmp_node);             // \u53f3\u5b50\u8282\u70b9\u5165\u961f\n        }        \n    }\n    return list;\n}\n</code></pre>"},{"location":"chapter_tree/binary_tree_traversal/#2","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\uff0c\u5176\u4e2d \\(n\\) \u4e3a\u8282\u70b9\u6570\u91cf\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6ee1\u4e8c\u53c9\u6811\u65f6\uff0c\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u6700\u591a\u540c\u65f6\u5b58\u5728 \\((n + 1) / 2\\) \u4e2a\u8282\u70b9\uff0c\u5360\u7528 \\(O(n)\\) \u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/binary_tree_traversal/#722","title":"7.2.2 \u00a0 \u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386","text":"<p>\u76f8\u5e94\u5730\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u548c\u540e\u5e8f\u904d\u5386\u90fd\u5c5e\u4e8e\u300c\u6df1\u5ea6\u4f18\u5148\u904d\u5386 depth-first traversal, DFS\u300d\uff0c\u5b83\u4f53\u73b0\u4e86\u4e00\u79cd\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\u3002</p> <p>\u56fe 7-10 \u5c55\u793a\u4e86\u5bf9\u4e8c\u53c9\u6811\u8fdb\u884c\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u7684\u5de5\u4f5c\u539f\u7406\u3002\u6df1\u5ea6\u4f18\u5148\u904d\u5386\u5c31\u50cf\u662f\u7ed5\u7740\u6574\u68f5\u4e8c\u53c9\u6811\u7684\u5916\u56f4\u201c\u8d70\u201d\u4e00\u5708\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u90fd\u4f1a\u9047\u5230\u4e09\u4e2a\u4f4d\u7f6e\uff0c\u5206\u522b\u5bf9\u5e94\u524d\u5e8f\u904d\u5386\u3001\u4e2d\u5e8f\u904d\u5386\u548c\u540e\u5e8f\u904d\u5386\u3002</p> <p></p> <p> \u56fe 7-10 \u00a0 \u4e8c\u53c9\u641c\u7d22\u6811\u7684\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386 </p>"},{"location":"chapter_tree/binary_tree_traversal/#1_1","title":"1. \u00a0 \u4ee3\u7801\u5b9e\u73b0","text":"<p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u901a\u5e38\u57fa\u4e8e\u9012\u5f52\u5b9e\u73b0\uff1a</p> PythonC++JavaC#GoSwiftJSTSDartRustCZig binary_tree_dfs.py<pre><code>def pre_order(root: TreeNode | None):\n    \"\"\"\u524d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    res.append(root.val)\n    pre_order(root=root.left)\n    pre_order(root=root.right)\n\ndef in_order(root: TreeNode | None):\n    \"\"\"\u4e2d\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    in_order(root=root.left)\n    res.append(root.val)\n    in_order(root=root.right)\n\ndef post_order(root: TreeNode | None):\n    \"\"\"\u540e\u5e8f\u904d\u5386\"\"\"\n    if root is None:\n        return\n    # \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    post_order(root=root.left)\n    post_order(root=root.right)\n    res.append(root.val)\n</code></pre> binary_tree_dfs.cpp<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    vec.push_back(root-&gt;val);\n    preOrder(root-&gt;left);\n    preOrder(root-&gt;right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left);\n    vec.push_back(root-&gt;val);\n    inOrder(root-&gt;right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root) {\n    if (root == nullptr)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left);\n    postOrder(root-&gt;right);\n    vec.push_back(root-&gt;val);\n}\n</code></pre> binary_tree_dfs.java<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.add(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.add(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode root) {\n    if (root == null)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.add(root.val);\n}\n</code></pre> binary_tree_dfs.cs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid PreOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.Add(root.val!.Value);\n    PreOrder(root.left);\n    PreOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid InOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    InOrder(root.left);\n    list.Add(root.val!.Value);\n    InOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid PostOrder(TreeNode? root) {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    PostOrder(root.left);\n    PostOrder(root.right);\n    list.Add(root.val!.Value);\n}\n</code></pre> binary_tree_dfs.go<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    nums = append(nums, node.Val)\n    preOrder(node.Left)\n    preOrder(node.Right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(node.Left)\n    nums = append(nums, node.Val)\n    inOrder(node.Right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(node *TreeNode) {\n    if node == nil {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(node.Left)\n    postOrder(node.Right)\n    nums = append(nums, node.Val)\n}\n</code></pre> binary_tree_dfs.swift<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunc preOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.append(root.val)\n    preOrder(root: root.left)\n    preOrder(root: root.right)\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunc inOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root: root.left)\n    list.append(root.val)\n    inOrder(root: root.right)\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunc postOrder(root: TreeNode?) {\n    guard let root = root else {\n        return\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root: root.left)\n    postOrder(root: root.right)\n    list.append(root.val)\n}\n</code></pre> binary_tree_dfs.js<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root) {\n    if (root === null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.ts<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfunction preOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    list.push(root.val);\n    preOrder(root.left);\n    preOrder(root.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfunction inOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root.left);\n    list.push(root.val);\n    inOrder(root.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfunction postOrder(root: TreeNode | null): void {\n    if (root === null) {\n        return;\n    }\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root.left);\n    postOrder(root.right);\n    list.push(root.val);\n}\n</code></pre> binary_tree_dfs.dart<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n  list.add(node.val);\n  preOrder(node.left);\n  preOrder(node.right);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n  inOrder(node.left);\n  list.add(node.val);\n  inOrder(node.right);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode? node) {\n  if (node == null) return;\n  // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n  postOrder(node.left);\n  postOrder(node.right);\n  list.add(node.val);\n}\n</code></pre> binary_tree_dfs.rs<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nfn pre_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n        result.push(node.borrow().val);\n        result.append(&amp;mut pre_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut pre_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nfn in_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n        result.append(&amp;mut in_order(node.borrow().left.as_ref()));\n        result.push(node.borrow().val);\n        result.append(&amp;mut in_order(node.borrow().right.as_ref()));\n    }\n    result\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nfn post_order(root: Option&lt;&amp;Rc&lt;RefCell&lt;TreeNode&gt;&gt;&gt;) -&gt; Vec&lt;i32&gt; {\n    let mut result = vec![];\n\n    if let Some(node) = root {\n        // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n        result.append(&amp;mut post_order(node.borrow().left.as_ref()));\n        result.append(&amp;mut post_order(node.borrow().right.as_ref()));\n        result.push(node.borrow().val);\n    }\n    result\n}\n</code></pre> binary_tree_dfs.c<pre><code>/* \u524d\u5e8f\u904d\u5386 */\nvoid preOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    arr[(*size)++] = root-&gt;val;\n    preOrder(root-&gt;left, size);\n    preOrder(root-&gt;right, size);\n}\n\n/* \u4e2d\u5e8f\u904d\u5386 */\nvoid inOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    inOrder(root-&gt;left, size);\n    arr[(*size)++] = root-&gt;val;\n    inOrder(root-&gt;right, size);\n}\n\n/* \u540e\u5e8f\u904d\u5386 */\nvoid postOrder(TreeNode *root, int *size) {\n    if (root == NULL)\n        return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    postOrder(root-&gt;left, size);\n    postOrder(root-&gt;right, size);\n    arr[(*size)++] = root-&gt;val;\n}\n</code></pre> binary_tree_dfs.zig<pre><code>// \u524d\u5e8f\u904d\u5386\nfn preOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u6839\u8282\u70b9 -&gt; \u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811\n    try list.append(root.?.val);\n    try preOrder(T, root.?.left);\n    try preOrder(T, root.?.right);\n}\n\n// \u4e2d\u5e8f\u904d\u5386\nfn inOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u6839\u8282\u70b9 -&gt; \u53f3\u5b50\u6811\n    try inOrder(T, root.?.left);\n    try list.append(root.?.val);\n    try inOrder(T, root.?.right);\n}\n\n// \u540e\u5e8f\u904d\u5386\nfn postOrder(comptime T: type, root: ?*inc.TreeNode(T)) !void {\n    if (root == null) return;\n    // \u8bbf\u95ee\u4f18\u5148\u7ea7\uff1a\u5de6\u5b50\u6811 -&gt; \u53f3\u5b50\u6811 -&gt; \u6839\u8282\u70b9\n    try postOrder(T, root.?.left);\n    try postOrder(T, root.?.right);\n    try list.append(root.?.val);\n}\n</code></pre> <p>Tip</p> <p>\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u4e5f\u53ef\u4ee5\u57fa\u4e8e\u8fed\u4ee3\u5b9e\u73b0\uff0c\u6709\u5174\u8da3\u7684\u8bfb\u8005\u53ef\u4ee5\u81ea\u884c\u7814\u7a76\u3002</p> <p>\u56fe 7-11 \u5c55\u793a\u4e86\u524d\u5e8f\u904d\u5386\u4e8c\u53c9\u6811\u7684\u9012\u5f52\u8fc7\u7a0b\uff0c\u5176\u53ef\u5206\u4e3a\u201c\u9012\u201d\u548c\u201c\u5f52\u201d\u4e24\u4e2a\u9006\u5411\u7684\u90e8\u5206\u3002</p> <ol> <li>\u201c\u9012\u201d\u8868\u793a\u5f00\u542f\u65b0\u65b9\u6cd5\uff0c\u7a0b\u5e8f\u5728\u6b64\u8fc7\u7a0b\u4e2d\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u8282\u70b9\u3002</li> <li>\u201c\u5f52\u201d\u8868\u793a\u51fd\u6570\u8fd4\u56de\uff0c\u4ee3\u8868\u5f53\u524d\u8282\u70b9\u5df2\u7ecf\u8bbf\u95ee\u5b8c\u6bd5\u3002</li> </ol> &lt;1&gt;&lt;2&gt;&lt;3&gt;&lt;4&gt;&lt;5&gt;&lt;6&gt;&lt;7&gt;&lt;8&gt;&lt;9&gt;&lt;10&gt;&lt;11&gt; <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p></p> <p> \u56fe 7-11 \u00a0 \u524d\u5e8f\u904d\u5386\u7684\u9012\u5f52\u8fc7\u7a0b </p>"},{"location":"chapter_tree/binary_tree_traversal/#2_1","title":"2. \u00a0 \u590d\u6742\u5ea6\u5206\u6790","text":"<ul> <li>\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u6240\u6709\u8282\u70b9\u88ab\u8bbf\u95ee\u4e00\u6b21\uff0c\u4f7f\u7528 \\(O(n)\\) \u65f6\u95f4\u3002</li> <li>\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\) \uff1a\u5728\u6700\u5dee\u60c5\u51b5\u4e0b\uff0c\u5373\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u9012\u5f52\u6df1\u5ea6\u8fbe\u5230 \\(n\\) \uff0c\u7cfb\u7edf\u5360\u7528 \\(O(n)\\) \u6808\u5e27\u7a7a\u95f4\u3002</li> </ul>"},{"location":"chapter_tree/summary/","title":"7.6 \u00a0 \u5c0f\u7ed3","text":""},{"location":"chapter_tree/summary/#1","title":"1. \u00a0 \u91cd\u70b9\u56de\u987e","text":"<ul> <li>\u4e8c\u53c9\u6811\u662f\u4e00\u79cd\u975e\u7ebf\u6027\u6570\u636e\u7ed3\u6784\uff0c\u4f53\u73b0\u201c\u4e00\u5206\u4e3a\u4e8c\u201d\u7684\u5206\u6cbb\u903b\u8f91\u3002\u6bcf\u4e2a\u4e8c\u53c9\u6811\u8282\u70b9\u5305\u542b\u4e00\u4e2a\u503c\u4ee5\u53ca\u4e24\u4e2a\u6307\u9488\uff0c\u5206\u522b\u6307\u5411\u5176\u5de6\u5b50\u8282\u70b9\u548c\u53f3\u5b50\u8282\u70b9\u3002</li> <li>\u5bf9\u4e8e\u4e8c\u53c9\u6811\u4e2d\u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u5176\u5de6\uff08\u53f3\uff09\u5b50\u8282\u70b9\u53ca\u5176\u4ee5\u4e0b\u5f62\u6210\u7684\u6811\u88ab\u79f0\u4e3a\u8be5\u8282\u70b9\u7684\u5de6\uff08\u53f3\uff09\u5b50\u6811\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u76f8\u5173\u672f\u8bed\u5305\u62ec\u6839\u8282\u70b9\u3001\u53f6\u8282\u70b9\u3001\u5c42\u3001\u5ea6\u3001\u8fb9\u3001\u9ad8\u5ea6\u548c\u6df1\u5ea6\u7b49\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u521d\u59cb\u5316\u3001\u8282\u70b9\u63d2\u5165\u548c\u8282\u70b9\u5220\u9664\u64cd\u4f5c\u4e0e\u94fe\u8868\u64cd\u4f5c\u65b9\u6cd5\u7c7b\u4f3c\u3002</li> <li>\u5e38\u89c1\u7684\u4e8c\u53c9\u6811\u7c7b\u578b\u6709\u5b8c\u7f8e\u4e8c\u53c9\u6811\u3001\u5b8c\u5168\u4e8c\u53c9\u6811\u3001\u5b8c\u6ee1\u4e8c\u53c9\u6811\u548c\u5e73\u8861\u4e8c\u53c9\u6811\u3002\u5b8c\u7f8e\u4e8c\u53c9\u6811\u662f\u6700\u7406\u60f3\u7684\u72b6\u6001\uff0c\u800c\u94fe\u8868\u662f\u9000\u5316\u540e\u7684\u6700\u5dee\u72b6\u6001\u3002</li> <li>\u4e8c\u53c9\u6811\u53ef\u4ee5\u7528\u6570\u7ec4\u8868\u793a\uff0c\u65b9\u6cd5\u662f\u5c06\u8282\u70b9\u503c\u548c\u7a7a\u4f4d\u6309\u5c42\u5e8f\u904d\u5386\u987a\u5e8f\u6392\u5217\uff0c\u5e76\u6839\u636e\u7236\u8282\u70b9\u4e0e\u5b50\u8282\u70b9\u4e4b\u95f4\u7684\u7d22\u5f15\u6620\u5c04\u5173\u7cfb\u6765\u5b9e\u73b0\u6307\u9488\u3002</li> <li>\u4e8c\u53c9\u6811\u7684\u5c42\u5e8f\u904d\u5386\u662f\u4e00\u79cd\u5e7f\u5ea6\u4f18\u5148\u641c\u7d22\u65b9\u6cd5\uff0c\u5b83\u4f53\u73b0\u4e86\u201c\u4e00\u5708\u4e00\u5708\u5411\u5916\u6269\u5c55\u201d\u7684\u9010\u5c42\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u901a\u8fc7\u961f\u5217\u6765\u5b9e\u73b0\u3002</li> <li>\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u7686\u5c5e\u4e8e\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\uff0c\u5b83\u4eec\u4f53\u73b0\u4e86\u201c\u5148\u8d70\u5230\u5c3d\u5934\uff0c\u518d\u56de\u6eaf\u7ee7\u7eed\u201d\u7684\u904d\u5386\u65b9\u5f0f\uff0c\u901a\u5e38\u4f7f\u7528\u9012\u5f52\u6765\u5b9e\u73b0\u3002</li> <li>\u4e8c\u53c9\u641c\u7d22\u6811\u662f\u4e00\u79cd\u9ad8\u6548\u7684\u5143\u7d20\u67e5\u627e\u6570\u636e\u7ed3\u6784\uff0c\u5176\u67e5\u627e\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5747\u4e3a \\(O(\\log n)\\) \u3002\u5f53\u4e8c\u53c9\u641c\u7d22\u6811\u9000\u5316\u4e3a\u94fe\u8868\u65f6\uff0c\u5404\u9879\u65f6\u95f4\u590d\u6742\u5ea6\u4f1a\u52a3\u5316\u81f3 \\(O(n)\\) \u3002</li> <li>AVL \u6811\uff0c\u4e5f\u79f0\u5e73\u8861\u4e8c\u53c9\u641c\u7d22\u6811\uff0c\u5b83\u901a\u8fc7\u65cb\u8f6c\u64cd\u4f5c\u786e\u4fdd\u5728\u4e0d\u65ad\u63d2\u5165\u548c\u5220\u9664\u8282\u70b9\u540e\u6811\u4ecd\u7136\u4fdd\u6301\u5e73\u8861\u3002</li> <li>AVL \u6811\u7684\u65cb\u8f6c\u64cd\u4f5c\u5305\u62ec\u53f3\u65cb\u3001\u5de6\u65cb\u3001\u5148\u53f3\u65cb\u518d\u5de6\u65cb\u3001\u5148\u5de6\u65cb\u518d\u53f3\u65cb\u3002\u5728\u63d2\u5165\u6216\u5220\u9664\u8282\u70b9\u540e\uff0cAVL \u6811\u4f1a\u4ece\u5e95\u5411\u9876\u6267\u884c\u65cb\u8f6c\u64cd\u4f5c\uff0c\u4f7f\u6811\u91cd\u65b0\u6062\u590d\u5e73\u8861\u3002</li> </ul>"},{"location":"chapter_tree/summary/#2-q-a","title":"2. \u00a0 Q &amp; A","text":"<p>\u5bf9\u4e8e\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u53c9\u6811\uff0c\u6811\u7684\u9ad8\u5ea6\u548c\u6839\u8282\u70b9\u7684\u6df1\u5ea6\u90fd\u662f \\(0\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u56e0\u4e3a\u9ad8\u5ea6\u548c\u6df1\u5ea6\u901a\u5e38\u5b9a\u4e49\u4e3a\u201c\u7ecf\u8fc7\u7684\u8fb9\u7684\u6570\u91cf\u201d\u3002</p> <p>\u4e8c\u53c9\u6811\u4e2d\u7684\u63d2\u5165\u4e0e\u5220\u9664\u4e00\u822c\u7531\u4e00\u5957\u64cd\u4f5c\u914d\u5408\u5b8c\u6210\uff0c\u8fd9\u91cc\u7684\u201c\u4e00\u5957\u64cd\u4f5c\u201d\u6307\u4ec0\u4e48\u5462\uff1f\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d44\u6e90\u7684\u5b50\u8282\u70b9\u7684\u8d44\u6e90\u91ca\u653e\u5417\uff1f</p> <p>\u62ff\u4e8c\u53c9\u641c\u7d22\u6811\u6765\u4e3e\u4f8b\uff0c\u5220\u9664\u8282\u70b9\u64cd\u4f5c\u8981\u5206\u4e09\u79cd\u60c5\u51b5\u5904\u7406\uff0c\u5176\u4e2d\u6bcf\u79cd\u60c5\u51b5\u90fd\u9700\u8981\u8fdb\u884c\u591a\u4e2a\u6b65\u9aa4\u7684\u8282\u70b9\u64cd\u4f5c\u3002</p> <p>\u4e3a\u4ec0\u4e48 DFS \u904d\u5386\u4e8c\u53c9\u6811\u6709\u524d\u3001\u4e2d\u3001\u540e\u4e09\u79cd\u987a\u5e8f\uff0c\u5206\u522b\u6709\u4ec0\u4e48\u7528\u5462\uff1f</p> <p>\u4e0e\u987a\u5e8f\u548c\u9006\u5e8f\u904d\u5386\u6570\u7ec4\u7c7b\u4f3c\uff0c\u524d\u5e8f\u3001\u4e2d\u5e8f\u3001\u540e\u5e8f\u904d\u5386\u662f\u4e09\u79cd\u4e8c\u53c9\u6811\u904d\u5386\u65b9\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5b83\u4eec\u5f97\u5230\u4e00\u4e2a\u7279\u5b9a\u987a\u5e8f\u7684\u904d\u5386\u7ed3\u679c\u3002\u4f8b\u5982\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4e2d\uff0c\u7531\u4e8e\u8282\u70b9\u5927\u5c0f\u6ee1\u8db3 <code>\u5de6\u5b50\u8282\u70b9\u503c &lt; \u6839\u8282\u70b9\u503c &lt; \u53f3\u5b50\u8282\u70b9\u503c</code> \uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u8981\u6309\u7167 <code>\u5de6 $\\rightarrow$ \u6839 $\\rightarrow$ \u53f3</code> \u7684\u4f18\u5148\u7ea7\u904d\u5386\u6811\uff0c\u5c31\u53ef\u4ee5\u83b7\u5f97\u6709\u5e8f\u7684\u8282\u70b9\u5e8f\u5217\u3002</p> <p>\u53f3\u65cb\u64cd\u4f5c\u662f\u5904\u7406\u5931\u8861\u8282\u70b9 <code>node</code>\u3001<code>child</code>\u3001<code>grand_child</code> \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u90a3 <code>node</code> \u7684\u7236\u8282\u70b9\u548c <code>node</code> \u539f\u6765\u7684\u8fde\u63a5\u4e0d\u9700\u8981\u7ef4\u62a4\u5417\uff1f\u53f3\u65cb\u64cd\u4f5c\u540e\u5c82\u4e0d\u662f\u65ad\u6389\u4e86\uff1f</p> <p>\u6211\u4eec\u9700\u8981\u4ece\u9012\u5f52\u7684\u89c6\u89d2\u6765\u770b\u8fd9\u4e2a\u95ee\u9898\u3002\u53f3\u65cb\u64cd\u4f5c <code>right_rotate(root)</code> \u4f20\u5165\u7684\u662f\u5b50\u6811\u7684\u6839\u8282\u70b9\uff0c\u6700\u7ec8 <code>return child</code> \u8fd4\u56de\u65cb\u8f6c\u4e4b\u540e\u7684\u5b50\u6811\u7684\u6839\u8282\u70b9\u3002\u5b50\u6811\u7684\u6839\u8282\u70b9\u548c\u5176\u7236\u8282\u70b9\u7684\u8fde\u63a5\u662f\u5728\u8be5\u51fd\u6570\u8fd4\u56de\u540e\u5b8c\u6210\u7684\uff0c\u4e0d\u5c5e\u4e8e\u53f3\u65cb\u64cd\u4f5c\u7684\u7ef4\u62a4\u8303\u56f4\u3002</p> <p>\u5728 C++ \u4e2d\uff0c\u51fd\u6570\u88ab\u5212\u5206\u5230 <code>private</code> \u548c <code>public</code> \u4e2d\uff0c\u8fd9\u65b9\u9762\u6709\u4ec0\u4e48\u8003\u91cf\u5417\uff1f\u4e3a\u4ec0\u4e48\u8981\u5c06 <code>height()</code> \u51fd\u6570\u548c <code>updateHeight()</code> \u51fd\u6570\u5206\u522b\u653e\u5728 <code>public</code> \u548c <code>private</code> \u4e2d\u5462\uff1f</p> <p>\u4e3b\u8981\u770b\u65b9\u6cd5\u7684\u4f7f\u7528\u8303\u56f4\uff0c\u5982\u679c\u65b9\u6cd5\u53ea\u5728\u7c7b\u5185\u90e8\u4f7f\u7528\uff0c\u90a3\u4e48\u5c31\u8bbe\u8ba1\u4e3a <code>private</code> \u3002\u4f8b\u5982\uff0c\u7528\u6237\u5355\u72ec\u8c03\u7528 <code>updateHeight()</code> \u662f\u6ca1\u6709\u610f\u4e49\u7684\uff0c\u5b83\u53ea\u662f\u63d2\u5165\u3001\u5220\u9664\u64cd\u4f5c\u4e2d\u7684\u4e00\u6b65\u3002\u800c <code>height()</code> \u662f\u8bbf\u95ee\u8282\u70b9\u9ad8\u5ea6\uff0c\u7c7b\u4f3c\u4e8e <code>vector.size()</code> \uff0c\u56e0\u6b64\u8bbe\u7f6e\u6210 <code>public</code> \u4ee5\u4fbf\u4f7f\u7528\u3002</p> <p>\u5982\u4f55\u4ece\u4e00\u7ec4\u8f93\u5165\u6570\u636e\u6784\u5efa\u4e00\u68f5\u4e8c\u53c9\u641c\u7d22\u6811\uff1f\u6839\u8282\u70b9\u7684\u9009\u62e9\u662f\u4e0d\u662f\u5f88\u91cd\u8981\uff1f</p> <p>\u662f\u7684\uff0c\u6784\u5efa\u6811\u7684\u65b9\u6cd5\u5df2\u5728\u4e8c\u53c9\u641c\u7d22\u6811\u4ee3\u7801\u4e2d\u7684 <code>build_tree()</code> \u65b9\u6cd5\u4e2d\u7ed9\u51fa\u3002\u81f3\u4e8e\u6839\u8282\u70b9\u7684\u9009\u62e9\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5c06\u8f93\u5165\u6570\u636e\u6392\u5e8f\uff0c\u7136\u540e\u5c06\u4e2d\u70b9\u5143\u7d20\u4f5c\u4e3a\u6839\u8282\u70b9\uff0c\u518d\u9012\u5f52\u5730\u6784\u5efa\u5de6\u53f3\u5b50\u6811\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u6700\u5927\u7a0b\u5ea6\u4fdd\u8bc1\u6811\u7684\u5e73\u8861\u6027\u3002</p> <p>\u5728 Java \u4e2d\uff0c\u5b57\u7b26\u4e32\u5bf9\u6bd4\u662f\u5426\u4e00\u5b9a\u8981\u7528 <code>equals()</code> \u65b9\u6cd5\uff1f</p> <p>\u5728 Java \u4e2d\uff0c\u5bf9\u4e8e\u57fa\u672c\u6570\u636e\u7c7b\u578b\uff0c<code>==</code> \u7528\u4e8e\u5bf9\u6bd4\u4e24\u4e2a\u53d8\u91cf\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002\u5bf9\u4e8e\u5f15\u7528\u7c7b\u578b\uff0c\u4e24\u79cd\u7b26\u53f7\u7684\u5de5\u4f5c\u539f\u7406\u662f\u4e0d\u540c\u7684\u3002</p> <ul> <li><code>==</code> \uff1a\u7528\u6765\u6bd4\u8f83\u4e24\u4e2a\u53d8\u91cf\u662f\u5426\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u5373\u5b83\u4eec\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u662f\u5426\u76f8\u540c\u3002</li> <li><code>equals()</code>\uff1a\u7528\u6765\u5bf9\u6bd4\u4e24\u4e2a\u5bf9\u8c61\u7684\u503c\u662f\u5426\u76f8\u7b49\u3002</li> </ul> <p>\u56e0\u6b64\uff0c\u5982\u679c\u8981\u5bf9\u6bd4\u503c\uff0c\u6211\u4eec\u5e94\u8be5\u4f7f\u7528 <code>equals()</code> \u3002\u7136\u800c\uff0c\u901a\u8fc7 <code>String a = \"hi\"; String b = \"hi\";</code> \u521d\u59cb\u5316\u7684\u5b57\u7b26\u4e32\u90fd\u5b58\u50a8\u5728\u5b57\u7b26\u4e32\u5e38\u91cf\u6c60\u4e2d\uff0c\u5b83\u4eec\u6307\u5411\u540c\u4e00\u4e2a\u5bf9\u8c61\uff0c\u56e0\u6b64\u4e5f\u53ef\u4ee5\u7528 <code>a == b</code> \u6765\u6bd4\u8f83\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u5185\u5bb9\u3002</p> <p>\u5e7f\u5ea6\u4f18\u5148\u904d\u5386\u5230\u6700\u5e95\u5c42\u4e4b\u524d\uff0c\u961f\u5217\u4e2d\u7684\u8282\u70b9\u6570\u91cf\u662f \\(2^h\\) \u5417\uff1f</p> <p>\u662f\u7684\uff0c\u4f8b\u5982\u9ad8\u5ea6 \\(h = 2\\) \u7684\u6ee1\u4e8c\u53c9\u6811\uff0c\u5176\u8282\u70b9\u603b\u6570 \\(n = 7\\) \uff0c\u5219\u5e95\u5c42\u8282\u70b9\u6570\u91cf \\(4 = 2^h = (n + 1) / 2\\) \u3002</p>"}]}
\ No newline at end of file
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