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<p><img alt="算法使用的相关空间" src="../space_complexity.assets/space_types.png" /></p>
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<p align="center"> Fig. 算法使用的相关空间 </p>
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<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="cm">/* 类 */</span>
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<li><strong>以最差输入数据为准</strong>。当 <span class="arithmatex">\(n < 10\)</span> 时，空间复杂度为 <span class="arithmatex">\(O(1)\)</span> ；但当 <span class="arithmatex">\(n > 10\)</span> 时，初始化的数组 <code>nums</code> 占用 <span class="arithmatex">\(O(n)\)</span> 空间；因此最差空间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。</li>
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<li><strong>以算法运行过程中的峰值内存为准</strong>。例如，程序在执行最后一行之前，占用 <span class="arithmatex">\(O(1)\)</span> 空间；当初始化数组 <code>nums</code> 时，程序占用 <span class="arithmatex">\(O(n)\)</span> 空间；因此最差空间复杂度为 <span class="arithmatex">\(O(n)\)</span> 。</li>
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</ul>
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<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
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<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
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<p><strong>在递归函数中，需要注意统计栈帧空间</strong>。例如，函数 <code>loop()</code> 在循环中调用了 <span class="arithmatex">\(n\)</span> 次 <code>function()</code> ，每轮中的 <code>function()</code> 都返回并释放了栈帧空间，因此空间复杂度仍为 <span class="arithmatex">\(O(1)\)</span> 。而递归函数 <code>recur()</code> 在运行过程中会同时存在 <span class="arithmatex">\(n\)</span> 个未返回的 <code>recur()</code> ，从而占用 <span class="arithmatex">\(O(n)\)</span> 的栈帧空间。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
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<div class="highlight"><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="kt">int</span><span class="w"> </span><span class="nf">function</span><span class="p">()</span><span class="w"> </span><span class="p">{</span>
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@@ -4081,7 +4081,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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<h3 id="o1">常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#o1" title="Permanent link">¶</a></h3>
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<p>常数阶常见于数量与输入数据大小 <span class="arithmatex">\(n\)</span> 无关的常量、变量、对象。</p>
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<p>需要注意的是，在循环中初始化变量或调用函数而占用的内存，在进入下一循环后就会被释放，即不会累积占用空间，空间复杂度仍为 <span class="arithmatex">\(O(1)\)</span> 。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JavaScript</label><label for="__tabbed_4_6">TypeScript</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label><label for="__tabbed_4_11">Dart</label><label for="__tabbed_4_12">Rust</label></div>
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<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="cm">/* 常数阶 */</span>
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@@ -4322,7 +4322,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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<h3 id="on">线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#on" title="Permanent link">¶</a></h3>
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<p>线性阶常见于元素数量与 <span class="arithmatex">\(n\)</span> 成正比的数组、链表、栈、队列等。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JavaScript</label><label for="__tabbed_5_6">TypeScript</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
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<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* 线性阶 */</span>
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@@ -4565,7 +4565,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
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<p>以下递归函数会同时存在 <span class="arithmatex">\(n\)</span> 个未返回的 <code>algorithm()</code> 函数，使用 <span class="arithmatex">\(O(n)\)</span> 大小的栈帧空间。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
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|
||||
<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* 线性阶（递归实现） */</span>
|
||||
@@ -4689,7 +4689,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
||||
|
||||
<h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">¶</a></h3>
|
||||
<p>平方阶常见于矩阵和图，元素数量与 <span class="arithmatex">\(n\)</span> 成平方关系。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JavaScript</label><label for="__tabbed_7_6">TypeScript</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JS</label><label for="__tabbed_7_6">TS</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
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||||
<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* 平方阶 */</span>
|
||||
@@ -4883,7 +4883,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
||||
</div>
|
||||
</div>
|
||||
<p>在以下递归函数中，同时存在 <span class="arithmatex">\(n\)</span> 个未返回的 <code>algorithm()</code> ，并且每个函数中都初始化了一个数组，长度分别为 <span class="arithmatex">\(n, n-1, n-2, ..., 2, 1\)</span> ，平均长度为 <span class="arithmatex">\(\frac{n}{2}\)</span> ，因此总体占用 <span class="arithmatex">\(O(n^2)\)</span> 空间。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="8:12"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Java</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Python</label><label for="__tabbed_8_4">Go</label><label for="__tabbed_8_5">JavaScript</label><label for="__tabbed_8_6">TypeScript</label><label for="__tabbed_8_7">C</label><label for="__tabbed_8_8">C#</label><label for="__tabbed_8_9">Swift</label><label for="__tabbed_8_10">Zig</label><label for="__tabbed_8_11">Dart</label><label for="__tabbed_8_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="8:12"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Java</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Python</label><label for="__tabbed_8_4">Go</label><label for="__tabbed_8_5">JS</label><label for="__tabbed_8_6">TS</label><label for="__tabbed_8_7">C</label><label for="__tabbed_8_8">C#</label><label for="__tabbed_8_9">Swift</label><label for="__tabbed_8_10">Zig</label><label for="__tabbed_8_11">Dart</label><label for="__tabbed_8_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* 平方阶（递归实现） */</span>
|
||||
@@ -5025,7 +5025,7 @@ O(1) < O(\log n) < O(n) < O(n^2) < O(2^n) \newline
|
||||
|
||||
<h3 id="o2n">指数阶 <span class="arithmatex">\(O(2^n)\)</span><a class="headerlink" href="#o2n" title="Permanent link">¶</a></h3>
|
||||
<p>指数阶常见于二叉树。高度为 <span class="arithmatex">\(n\)</span> 的「满二叉树」的节点数量为 <span class="arithmatex">\(2^n - 1\)</span> ，占用 <span class="arithmatex">\(O(2^n)\)</span> 空间。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="9:12"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JavaScript</label><label for="__tabbed_9_6">TypeScript</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label><label for="__tabbed_9_9">Swift</label><label for="__tabbed_9_10">Zig</label><label for="__tabbed_9_11">Dart</label><label for="__tabbed_9_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="9:12"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JS</label><label for="__tabbed_9_6">TS</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label><label for="__tabbed_9_9">Swift</label><label for="__tabbed_9_10">Zig</label><label for="__tabbed_9_11">Dart</label><label for="__tabbed_9_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">space_complexity.java</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* 指数阶（建立满二叉树） */</span>
|
||||
|
||||
@@ -3591,7 +3591,7 @@
|
||||
<div class="arithmatex">\[
|
||||
1 + 1 + 10 + (1 + 5) \times n = 6n + 12
|
||||
\]</div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JavaScript</label><label for="__tabbed_1_6">TypeScript</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="1:12"><input checked="checked" id="__tabbed_1_1" name="__tabbed_1" type="radio" /><input id="__tabbed_1_2" name="__tabbed_1" type="radio" /><input id="__tabbed_1_3" name="__tabbed_1" type="radio" /><input id="__tabbed_1_4" name="__tabbed_1" type="radio" /><input id="__tabbed_1_5" name="__tabbed_1" type="radio" /><input id="__tabbed_1_6" name="__tabbed_1" type="radio" /><input id="__tabbed_1_7" name="__tabbed_1" type="radio" /><input id="__tabbed_1_8" name="__tabbed_1" type="radio" /><input id="__tabbed_1_9" name="__tabbed_1" type="radio" /><input id="__tabbed_1_10" name="__tabbed_1" type="radio" /><input id="__tabbed_1_11" name="__tabbed_1" type="radio" /><input id="__tabbed_1_12" name="__tabbed_1" type="radio" /><div class="tabbed-labels"><label for="__tabbed_1_1">Java</label><label for="__tabbed_1_2">C++</label><label for="__tabbed_1_3">Python</label><label for="__tabbed_1_4">Go</label><label for="__tabbed_1_5">JS</label><label for="__tabbed_1_6">TS</label><label for="__tabbed_1_7">C</label><label for="__tabbed_1_8">C#</label><label for="__tabbed_1_9">Swift</label><label for="__tabbed_1_10">Zig</label><label for="__tabbed_1_11">Dart</label><label for="__tabbed_1_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
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|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1">// 在某运行平台下</span>
|
||||
@@ -3740,7 +3740,7 @@
|
||||
<li>算法 <code>B</code> 中的打印操作需要循环 <span class="arithmatex">\(n\)</span> 次，算法运行时间随着 <span class="arithmatex">\(n\)</span> 增大呈线性增长。此算法的时间复杂度被称为「线性阶」。</li>
|
||||
<li>算法 <code>C</code> 中的打印操作需要循环 <span class="arithmatex">\(1000000\)</span> 次，但运行时间仍与输入数据大小 <span class="arithmatex">\(n\)</span> 无关。因此 <code>C</code> 的时间复杂度和 <code>A</code> 相同，仍为「常数阶」。</li>
|
||||
</ul>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JavaScript</label><label for="__tabbed_2_6">TypeScript</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="2:12"><input checked="checked" id="__tabbed_2_1" name="__tabbed_2" type="radio" /><input id="__tabbed_2_2" name="__tabbed_2" type="radio" /><input id="__tabbed_2_3" name="__tabbed_2" type="radio" /><input id="__tabbed_2_4" name="__tabbed_2" type="radio" /><input id="__tabbed_2_5" name="__tabbed_2" type="radio" /><input id="__tabbed_2_6" name="__tabbed_2" type="radio" /><input id="__tabbed_2_7" name="__tabbed_2" type="radio" /><input id="__tabbed_2_8" name="__tabbed_2" type="radio" /><input id="__tabbed_2_9" name="__tabbed_2" type="radio" /><input id="__tabbed_2_10" name="__tabbed_2" type="radio" /><input id="__tabbed_2_11" name="__tabbed_2" type="radio" /><input id="__tabbed_2_12" name="__tabbed_2" type="radio" /><div class="tabbed-labels"><label for="__tabbed_2_1">Java</label><label for="__tabbed_2_2">C++</label><label for="__tabbed_2_3">Python</label><label for="__tabbed_2_4">Go</label><label for="__tabbed_2_5">JS</label><label for="__tabbed_2_6">TS</label><label for="__tabbed_2_7">C</label><label for="__tabbed_2_8">C#</label><label for="__tabbed_2_9">Swift</label><label for="__tabbed_2_10">Zig</label><label for="__tabbed_2_11">Dart</label><label for="__tabbed_2_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a><span class="c1">// 算法 A 时间复杂度：常数阶</span>
|
||||
@@ -3951,7 +3951,7 @@
|
||||
<div class="arithmatex">\[
|
||||
T(n) = 3 + 2n
|
||||
\]</div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JavaScript</label><label for="__tabbed_3_6">TypeScript</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="3:12"><input checked="checked" id="__tabbed_3_1" name="__tabbed_3" type="radio" /><input id="__tabbed_3_2" name="__tabbed_3" type="radio" /><input id="__tabbed_3_3" name="__tabbed_3" type="radio" /><input id="__tabbed_3_4" name="__tabbed_3" type="radio" /><input id="__tabbed_3_5" name="__tabbed_3" type="radio" /><input id="__tabbed_3_6" name="__tabbed_3" type="radio" /><input id="__tabbed_3_7" name="__tabbed_3" type="radio" /><input id="__tabbed_3_8" name="__tabbed_3" type="radio" /><input id="__tabbed_3_9" name="__tabbed_3" type="radio" /><input id="__tabbed_3_10" name="__tabbed_3" type="radio" /><input id="__tabbed_3_11" name="__tabbed_3" type="radio" /><input id="__tabbed_3_12" name="__tabbed_3" type="radio" /><div class="tabbed-labels"><label for="__tabbed_3_1">Java</label><label for="__tabbed_3_2">C++</label><label for="__tabbed_3_3">Python</label><label for="__tabbed_3_4">Go</label><label for="__tabbed_3_5">JS</label><label for="__tabbed_3_6">TS</label><label for="__tabbed_3_7">C</label><label for="__tabbed_3_8">C#</label><label for="__tabbed_3_9">Swift</label><label for="__tabbed_3_10">Zig</label><label for="__tabbed_3_11">Dart</label><label for="__tabbed_3_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
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||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-24-1" name="__codelineno-24-1" href="#__codelineno-24-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
@@ -4118,7 +4118,7 @@ T(n) & = n^2 + n & \text{偷懒统计 (o.O)}
|
||||
\end{aligned}
|
||||
\]</div>
|
||||
<p>最终，两者都能推出相同的时间复杂度结果，即 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JavaScript</label><label for="__tabbed_4_6">TypeScript</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label><label for="__tabbed_4_11">Dart</label><label for="__tabbed_4_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="4:12"><input checked="checked" id="__tabbed_4_1" name="__tabbed_4" type="radio" /><input id="__tabbed_4_2" name="__tabbed_4" type="radio" /><input id="__tabbed_4_3" name="__tabbed_4" type="radio" /><input id="__tabbed_4_4" name="__tabbed_4" type="radio" /><input id="__tabbed_4_5" name="__tabbed_4" type="radio" /><input id="__tabbed_4_6" name="__tabbed_4" type="radio" /><input id="__tabbed_4_7" name="__tabbed_4" type="radio" /><input id="__tabbed_4_8" name="__tabbed_4" type="radio" /><input id="__tabbed_4_9" name="__tabbed_4" type="radio" /><input id="__tabbed_4_10" name="__tabbed_4" type="radio" /><input id="__tabbed_4_11" name="__tabbed_4" type="radio" /><input id="__tabbed_4_12" name="__tabbed_4" type="radio" /><div class="tabbed-labels"><label for="__tabbed_4_1">Java</label><label for="__tabbed_4_2">C++</label><label for="__tabbed_4_3">Python</label><label for="__tabbed_4_4">Go</label><label for="__tabbed_4_5">JS</label><label for="__tabbed_4_6">TS</label><label for="__tabbed_4_7">C</label><label for="__tabbed_4_8">C#</label><label for="__tabbed_4_9">Swift</label><label for="__tabbed_4_10">Zig</label><label for="__tabbed_4_11">Dart</label><label for="__tabbed_4_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
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||||
<div class="highlight"><pre><span></span><code><a id="__codelineno-36-1" name="__codelineno-36-1" href="#__codelineno-36-1"></a><span class="kt">void</span><span class="w"> </span><span class="nf">algorithm</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
|
||||
@@ -4349,7 +4349,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
<h3 id="o1">常数阶 <span class="arithmatex">\(O(1)\)</span><a class="headerlink" href="#o1" title="Permanent link">¶</a></h3>
|
||||
<p>常数阶的操作数量与输入数据大小 <span class="arithmatex">\(n\)</span> 无关，即不随着 <span class="arithmatex">\(n\)</span> 的变化而变化。</p>
|
||||
<p>对于以下算法，尽管操作数量 <code>size</code> 可能很大，但由于其与数据大小 <span class="arithmatex">\(n\)</span> 无关，因此时间复杂度仍为 <span class="arithmatex">\(O(1)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JavaScript</label><label for="__tabbed_5_6">TypeScript</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="5:12"><input checked="checked" id="__tabbed_5_1" name="__tabbed_5" type="radio" /><input id="__tabbed_5_2" name="__tabbed_5" type="radio" /><input id="__tabbed_5_3" name="__tabbed_5" type="radio" /><input id="__tabbed_5_4" name="__tabbed_5" type="radio" /><input id="__tabbed_5_5" name="__tabbed_5" type="radio" /><input id="__tabbed_5_6" name="__tabbed_5" type="radio" /><input id="__tabbed_5_7" name="__tabbed_5" type="radio" /><input id="__tabbed_5_8" name="__tabbed_5" type="radio" /><input id="__tabbed_5_9" name="__tabbed_5" type="radio" /><input id="__tabbed_5_10" name="__tabbed_5" type="radio" /><input id="__tabbed_5_11" name="__tabbed_5" type="radio" /><input id="__tabbed_5_12" name="__tabbed_5" type="radio" /><div class="tabbed-labels"><label for="__tabbed_5_1">Java</label><label for="__tabbed_5_2">C++</label><label for="__tabbed_5_3">Python</label><label for="__tabbed_5_4">Go</label><label for="__tabbed_5_5">JS</label><label for="__tabbed_5_6">TS</label><label for="__tabbed_5_7">C</label><label for="__tabbed_5_8">C#</label><label for="__tabbed_5_9">Swift</label><label for="__tabbed_5_10">Zig</label><label for="__tabbed_5_11">Dart</label><label for="__tabbed_5_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-48-1" name="__codelineno-48-1" href="#__codelineno-48-1"></a><span class="cm">/* 常数阶 */</span>
|
||||
@@ -4494,7 +4494,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
</div>
|
||||
<h3 id="on">线性阶 <span class="arithmatex">\(O(n)\)</span><a class="headerlink" href="#on" title="Permanent link">¶</a></h3>
|
||||
<p>线性阶的操作数量相对于输入数据大小以线性级别增长。线性阶通常出现在单层循环中。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JavaScript</label><label for="__tabbed_6_6">TypeScript</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="6:12"><input checked="checked" id="__tabbed_6_1" name="__tabbed_6" type="radio" /><input id="__tabbed_6_2" name="__tabbed_6" type="radio" /><input id="__tabbed_6_3" name="__tabbed_6" type="radio" /><input id="__tabbed_6_4" name="__tabbed_6" type="radio" /><input id="__tabbed_6_5" name="__tabbed_6" type="radio" /><input id="__tabbed_6_6" name="__tabbed_6" type="radio" /><input id="__tabbed_6_7" name="__tabbed_6" type="radio" /><input id="__tabbed_6_8" name="__tabbed_6" type="radio" /><input id="__tabbed_6_9" name="__tabbed_6" type="radio" /><input id="__tabbed_6_10" name="__tabbed_6" type="radio" /><input id="__tabbed_6_11" name="__tabbed_6" type="radio" /><input id="__tabbed_6_12" name="__tabbed_6" type="radio" /><div class="tabbed-labels"><label for="__tabbed_6_1">Java</label><label for="__tabbed_6_2">C++</label><label for="__tabbed_6_3">Python</label><label for="__tabbed_6_4">Go</label><label for="__tabbed_6_5">JS</label><label for="__tabbed_6_6">TS</label><label for="__tabbed_6_7">C</label><label for="__tabbed_6_8">C#</label><label for="__tabbed_6_9">Swift</label><label for="__tabbed_6_10">Zig</label><label for="__tabbed_6_11">Dart</label><label for="__tabbed_6_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-60-1" name="__codelineno-60-1" href="#__codelineno-60-1"></a><span class="cm">/* 线性阶 */</span>
|
||||
@@ -4627,7 +4627,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
<p class="admonition-title">如何确定输入数据大小 <span class="arithmatex">\(n\)</span> ？</p>
|
||||
<p><strong>数据大小 <span class="arithmatex">\(n\)</span> 需根据输入数据的类型来具体确定</strong>。例如，在上述示例中，我们直接将 <span class="arithmatex">\(n\)</span> 视为输入数据大小；在下面遍历数组的示例中，数据大小 <span class="arithmatex">\(n\)</span> 为数组的长度。</p>
|
||||
</div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JavaScript</label><label for="__tabbed_7_6">TypeScript</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="7:12"><input checked="checked" id="__tabbed_7_1" name="__tabbed_7" type="radio" /><input id="__tabbed_7_2" name="__tabbed_7" type="radio" /><input id="__tabbed_7_3" name="__tabbed_7" type="radio" /><input id="__tabbed_7_4" name="__tabbed_7" type="radio" /><input id="__tabbed_7_5" name="__tabbed_7" type="radio" /><input id="__tabbed_7_6" name="__tabbed_7" type="radio" /><input id="__tabbed_7_7" name="__tabbed_7" type="radio" /><input id="__tabbed_7_8" name="__tabbed_7" type="radio" /><input id="__tabbed_7_9" name="__tabbed_7" type="radio" /><input id="__tabbed_7_10" name="__tabbed_7" type="radio" /><input id="__tabbed_7_11" name="__tabbed_7" type="radio" /><input id="__tabbed_7_12" name="__tabbed_7" type="radio" /><div class="tabbed-labels"><label for="__tabbed_7_1">Java</label><label for="__tabbed_7_2">C++</label><label for="__tabbed_7_3">Python</label><label for="__tabbed_7_4">Go</label><label for="__tabbed_7_5">JS</label><label for="__tabbed_7_6">TS</label><label for="__tabbed_7_7">C</label><label for="__tabbed_7_8">C#</label><label for="__tabbed_7_9">Swift</label><label for="__tabbed_7_10">Zig</label><label for="__tabbed_7_11">Dart</label><label for="__tabbed_7_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-72-1" name="__codelineno-72-1" href="#__codelineno-72-1"></a><span class="cm">/* 线性阶（遍历数组） */</span>
|
||||
@@ -4775,7 +4775,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
</div>
|
||||
<h3 id="on2">平方阶 <span class="arithmatex">\(O(n^2)\)</span><a class="headerlink" href="#on2" title="Permanent link">¶</a></h3>
|
||||
<p>平方阶的操作数量相对于输入数据大小以平方级别增长。平方阶通常出现在嵌套循环中，外层循环和内层循环都为 <span class="arithmatex">\(O(n)\)</span> ，因此总体为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="8:12"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Java</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Python</label><label for="__tabbed_8_4">Go</label><label for="__tabbed_8_5">JavaScript</label><label for="__tabbed_8_6">TypeScript</label><label for="__tabbed_8_7">C</label><label for="__tabbed_8_8">C#</label><label for="__tabbed_8_9">Swift</label><label for="__tabbed_8_10">Zig</label><label for="__tabbed_8_11">Dart</label><label for="__tabbed_8_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="8:12"><input checked="checked" id="__tabbed_8_1" name="__tabbed_8" type="radio" /><input id="__tabbed_8_2" name="__tabbed_8" type="radio" /><input id="__tabbed_8_3" name="__tabbed_8" type="radio" /><input id="__tabbed_8_4" name="__tabbed_8" type="radio" /><input id="__tabbed_8_5" name="__tabbed_8" type="radio" /><input id="__tabbed_8_6" name="__tabbed_8" type="radio" /><input id="__tabbed_8_7" name="__tabbed_8" type="radio" /><input id="__tabbed_8_8" name="__tabbed_8" type="radio" /><input id="__tabbed_8_9" name="__tabbed_8" type="radio" /><input id="__tabbed_8_10" name="__tabbed_8" type="radio" /><input id="__tabbed_8_11" name="__tabbed_8" type="radio" /><input id="__tabbed_8_12" name="__tabbed_8" type="radio" /><div class="tabbed-labels"><label for="__tabbed_8_1">Java</label><label for="__tabbed_8_2">C++</label><label for="__tabbed_8_3">Python</label><label for="__tabbed_8_4">Go</label><label for="__tabbed_8_5">JS</label><label for="__tabbed_8_6">TS</label><label for="__tabbed_8_7">C</label><label for="__tabbed_8_8">C#</label><label for="__tabbed_8_9">Swift</label><label for="__tabbed_8_10">Zig</label><label for="__tabbed_8_11">Dart</label><label for="__tabbed_8_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-84-1" name="__codelineno-84-1" href="#__codelineno-84-1"></a><span class="cm">/* 平方阶 */</span>
|
||||
@@ -4953,7 +4953,7 @@ O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!
|
||||
<div class="arithmatex">\[
|
||||
O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
\]</div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="9:12"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JavaScript</label><label for="__tabbed_9_6">TypeScript</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label><label for="__tabbed_9_9">Swift</label><label for="__tabbed_9_10">Zig</label><label for="__tabbed_9_11">Dart</label><label for="__tabbed_9_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="9:12"><input checked="checked" id="__tabbed_9_1" name="__tabbed_9" type="radio" /><input id="__tabbed_9_2" name="__tabbed_9" type="radio" /><input id="__tabbed_9_3" name="__tabbed_9" type="radio" /><input id="__tabbed_9_4" name="__tabbed_9" type="radio" /><input id="__tabbed_9_5" name="__tabbed_9" type="radio" /><input id="__tabbed_9_6" name="__tabbed_9" type="radio" /><input id="__tabbed_9_7" name="__tabbed_9" type="radio" /><input id="__tabbed_9_8" name="__tabbed_9" type="radio" /><input id="__tabbed_9_9" name="__tabbed_9" type="radio" /><input id="__tabbed_9_10" name="__tabbed_9" type="radio" /><input id="__tabbed_9_11" name="__tabbed_9" type="radio" /><input id="__tabbed_9_12" name="__tabbed_9" type="radio" /><div class="tabbed-labels"><label for="__tabbed_9_1">Java</label><label for="__tabbed_9_2">C++</label><label for="__tabbed_9_3">Python</label><label for="__tabbed_9_4">Go</label><label for="__tabbed_9_5">JS</label><label for="__tabbed_9_6">TS</label><label for="__tabbed_9_7">C</label><label for="__tabbed_9_8">C#</label><label for="__tabbed_9_9">Swift</label><label for="__tabbed_9_10">Zig</label><label for="__tabbed_9_11">Dart</label><label for="__tabbed_9_12">Rust</label></div>
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<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-96-1" name="__codelineno-96-1" href="#__codelineno-96-1"></a><span class="cm">/* 平方阶（冒泡排序） */</span>
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@@ -5211,7 +5211,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
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||||
<p>生物学的“细胞分裂”是指数阶增长的典型例子：初始状态为 <span class="arithmatex">\(1\)</span> 个细胞，分裂一轮后变为 <span class="arithmatex">\(2\)</span> 个，分裂两轮后变为 <span class="arithmatex">\(4\)</span> 个，以此类推，分裂 <span class="arithmatex">\(n\)</span> 轮后有 <span class="arithmatex">\(2^n\)</span> 个细胞。</p>
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</div>
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||||
<p>指数阶增长非常迅速，在实际应用中通常是不可接受的。若一个问题使用「暴力枚举」求解的时间复杂度为 <span class="arithmatex">\(O(2^n)\)</span> ，那么通常需要使用「动态规划」或「贪心算法」等方法来解决。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="10:12"><input checked="checked" id="__tabbed_10_1" name="__tabbed_10" type="radio" /><input id="__tabbed_10_2" name="__tabbed_10" type="radio" /><input id="__tabbed_10_3" name="__tabbed_10" type="radio" /><input id="__tabbed_10_4" name="__tabbed_10" type="radio" /><input id="__tabbed_10_5" name="__tabbed_10" type="radio" /><input id="__tabbed_10_6" name="__tabbed_10" type="radio" /><input id="__tabbed_10_7" name="__tabbed_10" type="radio" /><input id="__tabbed_10_8" name="__tabbed_10" type="radio" /><input id="__tabbed_10_9" name="__tabbed_10" type="radio" /><input id="__tabbed_10_10" name="__tabbed_10" type="radio" /><input id="__tabbed_10_11" name="__tabbed_10" type="radio" /><input id="__tabbed_10_12" name="__tabbed_10" type="radio" /><div class="tabbed-labels"><label for="__tabbed_10_1">Java</label><label for="__tabbed_10_2">C++</label><label for="__tabbed_10_3">Python</label><label for="__tabbed_10_4">Go</label><label for="__tabbed_10_5">JavaScript</label><label for="__tabbed_10_6">TypeScript</label><label for="__tabbed_10_7">C</label><label for="__tabbed_10_8">C#</label><label for="__tabbed_10_9">Swift</label><label for="__tabbed_10_10">Zig</label><label for="__tabbed_10_11">Dart</label><label for="__tabbed_10_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="10:12"><input checked="checked" id="__tabbed_10_1" name="__tabbed_10" type="radio" /><input id="__tabbed_10_2" name="__tabbed_10" type="radio" /><input id="__tabbed_10_3" name="__tabbed_10" type="radio" /><input id="__tabbed_10_4" name="__tabbed_10" type="radio" /><input id="__tabbed_10_5" name="__tabbed_10" type="radio" /><input id="__tabbed_10_6" name="__tabbed_10" type="radio" /><input id="__tabbed_10_7" name="__tabbed_10" type="radio" /><input id="__tabbed_10_8" name="__tabbed_10" type="radio" /><input id="__tabbed_10_9" name="__tabbed_10" type="radio" /><input id="__tabbed_10_10" name="__tabbed_10" type="radio" /><input id="__tabbed_10_11" name="__tabbed_10" type="radio" /><input id="__tabbed_10_12" name="__tabbed_10" type="radio" /><div class="tabbed-labels"><label for="__tabbed_10_1">Java</label><label for="__tabbed_10_2">C++</label><label for="__tabbed_10_3">Python</label><label for="__tabbed_10_4">Go</label><label for="__tabbed_10_5">JS</label><label for="__tabbed_10_6">TS</label><label for="__tabbed_10_7">C</label><label for="__tabbed_10_8">C#</label><label for="__tabbed_10_9">Swift</label><label for="__tabbed_10_10">Zig</label><label for="__tabbed_10_11">Dart</label><label for="__tabbed_10_12">Rust</label></div>
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<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-108-1" name="__codelineno-108-1" href="#__codelineno-108-1"></a><span class="cm">/* 指数阶（循环实现） */</span>
|
||||
@@ -5417,7 +5417,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
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||||
<p align="center"> Fig. 指数阶的时间复杂度 </p>
|
||||
|
||||
<p>在实际算法中，指数阶常出现于递归函数。例如以下代码，不断地一分为二，经过 <span class="arithmatex">\(n\)</span> 次分裂后停止。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="11:12"><input checked="checked" id="__tabbed_11_1" name="__tabbed_11" type="radio" /><input id="__tabbed_11_2" name="__tabbed_11" type="radio" /><input id="__tabbed_11_3" name="__tabbed_11" type="radio" /><input id="__tabbed_11_4" name="__tabbed_11" type="radio" /><input id="__tabbed_11_5" name="__tabbed_11" type="radio" /><input id="__tabbed_11_6" name="__tabbed_11" type="radio" /><input id="__tabbed_11_7" name="__tabbed_11" type="radio" /><input id="__tabbed_11_8" name="__tabbed_11" type="radio" /><input id="__tabbed_11_9" name="__tabbed_11" type="radio" /><input id="__tabbed_11_10" name="__tabbed_11" type="radio" /><input id="__tabbed_11_11" name="__tabbed_11" type="radio" /><input id="__tabbed_11_12" name="__tabbed_11" type="radio" /><div class="tabbed-labels"><label for="__tabbed_11_1">Java</label><label for="__tabbed_11_2">C++</label><label for="__tabbed_11_3">Python</label><label for="__tabbed_11_4">Go</label><label for="__tabbed_11_5">JavaScript</label><label for="__tabbed_11_6">TypeScript</label><label for="__tabbed_11_7">C</label><label for="__tabbed_11_8">C#</label><label for="__tabbed_11_9">Swift</label><label for="__tabbed_11_10">Zig</label><label for="__tabbed_11_11">Dart</label><label for="__tabbed_11_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="11:12"><input checked="checked" id="__tabbed_11_1" name="__tabbed_11" type="radio" /><input id="__tabbed_11_2" name="__tabbed_11" type="radio" /><input id="__tabbed_11_3" name="__tabbed_11" type="radio" /><input id="__tabbed_11_4" name="__tabbed_11" type="radio" /><input id="__tabbed_11_5" name="__tabbed_11" type="radio" /><input id="__tabbed_11_6" name="__tabbed_11" type="radio" /><input id="__tabbed_11_7" name="__tabbed_11" type="radio" /><input id="__tabbed_11_8" name="__tabbed_11" type="radio" /><input id="__tabbed_11_9" name="__tabbed_11" type="radio" /><input id="__tabbed_11_10" name="__tabbed_11" type="radio" /><input id="__tabbed_11_11" name="__tabbed_11" type="radio" /><input id="__tabbed_11_12" name="__tabbed_11" type="radio" /><div class="tabbed-labels"><label for="__tabbed_11_1">Java</label><label for="__tabbed_11_2">C++</label><label for="__tabbed_11_3">Python</label><label for="__tabbed_11_4">Go</label><label for="__tabbed_11_5">JS</label><label for="__tabbed_11_6">TS</label><label for="__tabbed_11_7">C</label><label for="__tabbed_11_8">C#</label><label for="__tabbed_11_9">Swift</label><label for="__tabbed_11_10">Zig</label><label for="__tabbed_11_11">Dart</label><label for="__tabbed_11_12">Rust</label></div>
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<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-120-1" name="__codelineno-120-1" href="#__codelineno-120-1"></a><span class="cm">/* 指数阶（递归实现） */</span>
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||||
@@ -5530,7 +5530,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<p>与指数阶相反，对数阶反映了“每轮缩减到一半的情况”。对数阶仅次于常数阶，时间增长缓慢，是理想的时间复杂度。</p>
|
||||
<p>对数阶常出现于「二分查找」和「分治算法」中，体现了“一分为多”和“化繁为简”的算法思想。</p>
|
||||
<p>设输入数据大小为 <span class="arithmatex">\(n\)</span> ，由于每轮缩减到一半，因此循环次数是 <span class="arithmatex">\(\log_2 n\)</span> ，即 <span class="arithmatex">\(2^n\)</span> 的反函数。</p>
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||||
<div class="tabbed-set tabbed-alternate" data-tabs="12:12"><input checked="checked" id="__tabbed_12_1" name="__tabbed_12" type="radio" /><input id="__tabbed_12_2" name="__tabbed_12" type="radio" /><input id="__tabbed_12_3" name="__tabbed_12" type="radio" /><input id="__tabbed_12_4" name="__tabbed_12" type="radio" /><input id="__tabbed_12_5" name="__tabbed_12" type="radio" /><input id="__tabbed_12_6" name="__tabbed_12" type="radio" /><input id="__tabbed_12_7" name="__tabbed_12" type="radio" /><input id="__tabbed_12_8" name="__tabbed_12" type="radio" /><input id="__tabbed_12_9" name="__tabbed_12" type="radio" /><input id="__tabbed_12_10" name="__tabbed_12" type="radio" /><input id="__tabbed_12_11" name="__tabbed_12" type="radio" /><input id="__tabbed_12_12" name="__tabbed_12" type="radio" /><div class="tabbed-labels"><label for="__tabbed_12_1">Java</label><label for="__tabbed_12_2">C++</label><label for="__tabbed_12_3">Python</label><label for="__tabbed_12_4">Go</label><label for="__tabbed_12_5">JavaScript</label><label for="__tabbed_12_6">TypeScript</label><label for="__tabbed_12_7">C</label><label for="__tabbed_12_8">C#</label><label for="__tabbed_12_9">Swift</label><label for="__tabbed_12_10">Zig</label><label for="__tabbed_12_11">Dart</label><label for="__tabbed_12_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="12:12"><input checked="checked" id="__tabbed_12_1" name="__tabbed_12" type="radio" /><input id="__tabbed_12_2" name="__tabbed_12" type="radio" /><input id="__tabbed_12_3" name="__tabbed_12" type="radio" /><input id="__tabbed_12_4" name="__tabbed_12" type="radio" /><input id="__tabbed_12_5" name="__tabbed_12" type="radio" /><input id="__tabbed_12_6" name="__tabbed_12" type="radio" /><input id="__tabbed_12_7" name="__tabbed_12" type="radio" /><input id="__tabbed_12_8" name="__tabbed_12" type="radio" /><input id="__tabbed_12_9" name="__tabbed_12" type="radio" /><input id="__tabbed_12_10" name="__tabbed_12" type="radio" /><input id="__tabbed_12_11" name="__tabbed_12" type="radio" /><input id="__tabbed_12_12" name="__tabbed_12" type="radio" /><div class="tabbed-labels"><label for="__tabbed_12_1">Java</label><label for="__tabbed_12_2">C++</label><label for="__tabbed_12_3">Python</label><label for="__tabbed_12_4">Go</label><label for="__tabbed_12_5">JS</label><label for="__tabbed_12_6">TS</label><label for="__tabbed_12_7">C</label><label for="__tabbed_12_8">C#</label><label for="__tabbed_12_9">Swift</label><label for="__tabbed_12_10">Zig</label><label for="__tabbed_12_11">Dart</label><label for="__tabbed_12_12">Rust</label></div>
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<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-132-1" name="__codelineno-132-1" href="#__codelineno-132-1"></a><span class="cm">/* 对数阶（循环实现） */</span>
|
||||
@@ -5683,7 +5683,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
|
||||
<p align="center"> Fig. 对数阶的时间复杂度 </p>
|
||||
|
||||
<p>与指数阶类似，对数阶也常出现于递归函数。以下代码形成了一个高度为 <span class="arithmatex">\(\log_2 n\)</span> 的递归树。</p>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="13:12"><input checked="checked" id="__tabbed_13_1" name="__tabbed_13" type="radio" /><input id="__tabbed_13_2" name="__tabbed_13" type="radio" /><input id="__tabbed_13_3" name="__tabbed_13" type="radio" /><input id="__tabbed_13_4" name="__tabbed_13" type="radio" /><input id="__tabbed_13_5" name="__tabbed_13" type="radio" /><input id="__tabbed_13_6" name="__tabbed_13" type="radio" /><input id="__tabbed_13_7" name="__tabbed_13" type="radio" /><input id="__tabbed_13_8" name="__tabbed_13" type="radio" /><input id="__tabbed_13_9" name="__tabbed_13" type="radio" /><input id="__tabbed_13_10" name="__tabbed_13" type="radio" /><input id="__tabbed_13_11" name="__tabbed_13" type="radio" /><input id="__tabbed_13_12" name="__tabbed_13" type="radio" /><div class="tabbed-labels"><label for="__tabbed_13_1">Java</label><label for="__tabbed_13_2">C++</label><label for="__tabbed_13_3">Python</label><label for="__tabbed_13_4">Go</label><label for="__tabbed_13_5">JavaScript</label><label for="__tabbed_13_6">TypeScript</label><label for="__tabbed_13_7">C</label><label for="__tabbed_13_8">C#</label><label for="__tabbed_13_9">Swift</label><label for="__tabbed_13_10">Zig</label><label for="__tabbed_13_11">Dart</label><label for="__tabbed_13_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="13:12"><input checked="checked" id="__tabbed_13_1" name="__tabbed_13" type="radio" /><input id="__tabbed_13_2" name="__tabbed_13" type="radio" /><input id="__tabbed_13_3" name="__tabbed_13" type="radio" /><input id="__tabbed_13_4" name="__tabbed_13" type="radio" /><input id="__tabbed_13_5" name="__tabbed_13" type="radio" /><input id="__tabbed_13_6" name="__tabbed_13" type="radio" /><input id="__tabbed_13_7" name="__tabbed_13" type="radio" /><input id="__tabbed_13_8" name="__tabbed_13" type="radio" /><input id="__tabbed_13_9" name="__tabbed_13" type="radio" /><input id="__tabbed_13_10" name="__tabbed_13" type="radio" /><input id="__tabbed_13_11" name="__tabbed_13" type="radio" /><input id="__tabbed_13_12" name="__tabbed_13" type="radio" /><div class="tabbed-labels"><label for="__tabbed_13_1">Java</label><label for="__tabbed_13_2">C++</label><label for="__tabbed_13_3">Python</label><label for="__tabbed_13_4">Go</label><label for="__tabbed_13_5">JS</label><label for="__tabbed_13_6">TS</label><label for="__tabbed_13_7">C</label><label for="__tabbed_13_8">C#</label><label for="__tabbed_13_9">Swift</label><label for="__tabbed_13_10">Zig</label><label for="__tabbed_13_11">Dart</label><label for="__tabbed_13_12">Rust</label></div>
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<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-144-1" name="__codelineno-144-1" href="#__codelineno-144-1"></a><span class="cm">/* 对数阶（递归实现） */</span>
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@@ -5795,7 +5795,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
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||||
<h3 id="on-log-n">线性对数阶 <span class="arithmatex">\(O(n \log n)\)</span><a class="headerlink" href="#on-log-n" title="Permanent link">¶</a></h3>
|
||||
<p>线性对数阶常出现于嵌套循环中，两层循环的时间复杂度分别为 <span class="arithmatex">\(O(\log n)\)</span> 和 <span class="arithmatex">\(O(n)\)</span> 。</p>
|
||||
<p>主流排序算法的时间复杂度通常为 <span class="arithmatex">\(O(n \log n)\)</span> ，例如快速排序、归并排序、堆排序等。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="14:12"><input checked="checked" id="__tabbed_14_1" name="__tabbed_14" type="radio" /><input id="__tabbed_14_2" name="__tabbed_14" type="radio" /><input id="__tabbed_14_3" name="__tabbed_14" type="radio" /><input id="__tabbed_14_4" name="__tabbed_14" type="radio" /><input id="__tabbed_14_5" name="__tabbed_14" type="radio" /><input id="__tabbed_14_6" name="__tabbed_14" type="radio" /><input id="__tabbed_14_7" name="__tabbed_14" type="radio" /><input id="__tabbed_14_8" name="__tabbed_14" type="radio" /><input id="__tabbed_14_9" name="__tabbed_14" type="radio" /><input id="__tabbed_14_10" name="__tabbed_14" type="radio" /><input id="__tabbed_14_11" name="__tabbed_14" type="radio" /><input id="__tabbed_14_12" name="__tabbed_14" type="radio" /><div class="tabbed-labels"><label for="__tabbed_14_1">Java</label><label for="__tabbed_14_2">C++</label><label for="__tabbed_14_3">Python</label><label for="__tabbed_14_4">Go</label><label for="__tabbed_14_5">JavaScript</label><label for="__tabbed_14_6">TypeScript</label><label for="__tabbed_14_7">C</label><label for="__tabbed_14_8">C#</label><label for="__tabbed_14_9">Swift</label><label for="__tabbed_14_10">Zig</label><label for="__tabbed_14_11">Dart</label><label for="__tabbed_14_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="14:12"><input checked="checked" id="__tabbed_14_1" name="__tabbed_14" type="radio" /><input id="__tabbed_14_2" name="__tabbed_14" type="radio" /><input id="__tabbed_14_3" name="__tabbed_14" type="radio" /><input id="__tabbed_14_4" name="__tabbed_14" type="radio" /><input id="__tabbed_14_5" name="__tabbed_14" type="radio" /><input id="__tabbed_14_6" name="__tabbed_14" type="radio" /><input id="__tabbed_14_7" name="__tabbed_14" type="radio" /><input id="__tabbed_14_8" name="__tabbed_14" type="radio" /><input id="__tabbed_14_9" name="__tabbed_14" type="radio" /><input id="__tabbed_14_10" name="__tabbed_14" type="radio" /><input id="__tabbed_14_11" name="__tabbed_14" type="radio" /><input id="__tabbed_14_12" name="__tabbed_14" type="radio" /><div class="tabbed-labels"><label for="__tabbed_14_1">Java</label><label for="__tabbed_14_2">C++</label><label for="__tabbed_14_3">Python</label><label for="__tabbed_14_4">Go</label><label for="__tabbed_14_5">JS</label><label for="__tabbed_14_6">TS</label><label for="__tabbed_14_7">C</label><label for="__tabbed_14_8">C#</label><label for="__tabbed_14_9">Swift</label><label for="__tabbed_14_10">Zig</label><label for="__tabbed_14_11">Dart</label><label for="__tabbed_14_12">Rust</label></div>
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<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-156-1" name="__codelineno-156-1" href="#__codelineno-156-1"></a><span class="cm">/* 线性对数阶 */</span>
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@@ -5966,7 +5966,7 @@ O((n - 1) \frac{n}{2}) = O(n^2)
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n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
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\]</div>
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<p>阶乘通常使用递归实现。例如以下代码，第一层分裂出 <span class="arithmatex">\(n\)</span> 个，第二层分裂出 <span class="arithmatex">\(n - 1\)</span> 个，以此类推，直至第 <span class="arithmatex">\(n\)</span> 层时终止分裂。</p>
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<div class="tabbed-set tabbed-alternate" data-tabs="15:12"><input checked="checked" id="__tabbed_15_1" name="__tabbed_15" type="radio" /><input id="__tabbed_15_2" name="__tabbed_15" type="radio" /><input id="__tabbed_15_3" name="__tabbed_15" type="radio" /><input id="__tabbed_15_4" name="__tabbed_15" type="radio" /><input id="__tabbed_15_5" name="__tabbed_15" type="radio" /><input id="__tabbed_15_6" name="__tabbed_15" type="radio" /><input id="__tabbed_15_7" name="__tabbed_15" type="radio" /><input id="__tabbed_15_8" name="__tabbed_15" type="radio" /><input id="__tabbed_15_9" name="__tabbed_15" type="radio" /><input id="__tabbed_15_10" name="__tabbed_15" type="radio" /><input id="__tabbed_15_11" name="__tabbed_15" type="radio" /><input id="__tabbed_15_12" name="__tabbed_15" type="radio" /><div class="tabbed-labels"><label for="__tabbed_15_1">Java</label><label for="__tabbed_15_2">C++</label><label for="__tabbed_15_3">Python</label><label for="__tabbed_15_4">Go</label><label for="__tabbed_15_5">JavaScript</label><label for="__tabbed_15_6">TypeScript</label><label for="__tabbed_15_7">C</label><label for="__tabbed_15_8">C#</label><label for="__tabbed_15_9">Swift</label><label for="__tabbed_15_10">Zig</label><label for="__tabbed_15_11">Dart</label><label for="__tabbed_15_12">Rust</label></div>
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<div class="tabbed-set tabbed-alternate" data-tabs="15:12"><input checked="checked" id="__tabbed_15_1" name="__tabbed_15" type="radio" /><input id="__tabbed_15_2" name="__tabbed_15" type="radio" /><input id="__tabbed_15_3" name="__tabbed_15" type="radio" /><input id="__tabbed_15_4" name="__tabbed_15" type="radio" /><input id="__tabbed_15_5" name="__tabbed_15" type="radio" /><input id="__tabbed_15_6" name="__tabbed_15" type="radio" /><input id="__tabbed_15_7" name="__tabbed_15" type="radio" /><input id="__tabbed_15_8" name="__tabbed_15" type="radio" /><input id="__tabbed_15_9" name="__tabbed_15" type="radio" /><input id="__tabbed_15_10" name="__tabbed_15" type="radio" /><input id="__tabbed_15_11" name="__tabbed_15" type="radio" /><input id="__tabbed_15_12" name="__tabbed_15" type="radio" /><div class="tabbed-labels"><label for="__tabbed_15_1">Java</label><label for="__tabbed_15_2">C++</label><label for="__tabbed_15_3">Python</label><label for="__tabbed_15_4">Go</label><label for="__tabbed_15_5">JS</label><label for="__tabbed_15_6">TS</label><label for="__tabbed_15_7">C</label><label for="__tabbed_15_8">C#</label><label for="__tabbed_15_9">Swift</label><label for="__tabbed_15_10">Zig</label><label for="__tabbed_15_11">Dart</label><label for="__tabbed_15_12">Rust</label></div>
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<div class="tabbed-content">
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<div class="tabbed-block">
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<div class="highlight"><span class="filename">time_complexity.java</span><pre><span></span><code><a id="__codelineno-168-1" name="__codelineno-168-1" href="#__codelineno-168-1"></a><span class="cm">/* 阶乘阶（递归实现） */</span>
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@@ -6144,7 +6144,7 @@ n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1
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<li>当 <code>nums = [1, ?, ?, ...]</code> ，即当首个数字为 <span class="arithmatex">\(1\)</span> 时，无论数组多长都不需要继续遍历，此时达到 <strong>最佳时间复杂度 <span class="arithmatex">\(\Omega(1)\)</span></strong> 。</li>
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||||
</ul>
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||||
<p>“函数渐近上界”使用大 <span class="arithmatex">\(O\)</span> 记号表示，代表「最差时间复杂度」。相应地，“函数渐近下界”用 <span class="arithmatex">\(\Omega\)</span> 记号来表示，代表「最佳时间复杂度」。</p>
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||||
<div class="tabbed-set tabbed-alternate" data-tabs="16:12"><input checked="checked" id="__tabbed_16_1" name="__tabbed_16" type="radio" /><input id="__tabbed_16_2" name="__tabbed_16" type="radio" /><input id="__tabbed_16_3" name="__tabbed_16" type="radio" /><input id="__tabbed_16_4" name="__tabbed_16" type="radio" /><input id="__tabbed_16_5" name="__tabbed_16" type="radio" /><input id="__tabbed_16_6" name="__tabbed_16" type="radio" /><input id="__tabbed_16_7" name="__tabbed_16" type="radio" /><input id="__tabbed_16_8" name="__tabbed_16" type="radio" /><input id="__tabbed_16_9" name="__tabbed_16" type="radio" /><input id="__tabbed_16_10" name="__tabbed_16" type="radio" /><input id="__tabbed_16_11" name="__tabbed_16" type="radio" /><input id="__tabbed_16_12" name="__tabbed_16" type="radio" /><div class="tabbed-labels"><label for="__tabbed_16_1">Java</label><label for="__tabbed_16_2">C++</label><label for="__tabbed_16_3">Python</label><label for="__tabbed_16_4">Go</label><label for="__tabbed_16_5">JavaScript</label><label for="__tabbed_16_6">TypeScript</label><label for="__tabbed_16_7">C</label><label for="__tabbed_16_8">C#</label><label for="__tabbed_16_9">Swift</label><label for="__tabbed_16_10">Zig</label><label for="__tabbed_16_11">Dart</label><label for="__tabbed_16_12">Rust</label></div>
|
||||
<div class="tabbed-set tabbed-alternate" data-tabs="16:12"><input checked="checked" id="__tabbed_16_1" name="__tabbed_16" type="radio" /><input id="__tabbed_16_2" name="__tabbed_16" type="radio" /><input id="__tabbed_16_3" name="__tabbed_16" type="radio" /><input id="__tabbed_16_4" name="__tabbed_16" type="radio" /><input id="__tabbed_16_5" name="__tabbed_16" type="radio" /><input id="__tabbed_16_6" name="__tabbed_16" type="radio" /><input id="__tabbed_16_7" name="__tabbed_16" type="radio" /><input id="__tabbed_16_8" name="__tabbed_16" type="radio" /><input id="__tabbed_16_9" name="__tabbed_16" type="radio" /><input id="__tabbed_16_10" name="__tabbed_16" type="radio" /><input id="__tabbed_16_11" name="__tabbed_16" type="radio" /><input id="__tabbed_16_12" name="__tabbed_16" type="radio" /><div class="tabbed-labels"><label for="__tabbed_16_1">Java</label><label for="__tabbed_16_2">C++</label><label for="__tabbed_16_3">Python</label><label for="__tabbed_16_4">Go</label><label for="__tabbed_16_5">JS</label><label for="__tabbed_16_6">TS</label><label for="__tabbed_16_7">C</label><label for="__tabbed_16_8">C#</label><label for="__tabbed_16_9">Swift</label><label for="__tabbed_16_10">Zig</label><label for="__tabbed_16_11">Dart</label><label for="__tabbed_16_12">Rust</label></div>
|
||||
<div class="tabbed-content">
|
||||
<div class="tabbed-block">
|
||||
<div class="highlight"><span class="filename">worst_best_time_complexity.java</span><pre><span></span><code><a id="__codelineno-180-1" name="__codelineno-180-1" href="#__codelineno-180-1"></a><span class="cm">/* 生成一个数组，元素为 { 1, 2, ..., n }，顺序被打乱 */</span>
|
||||
|
||||
Reference in New Issue
Block a user