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<title>2.3. 空间复杂度 - Hello 算法</title>
<title>2.3.   空间复杂度 - Hello 算法</title>
@@ -111,7 +111,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
2.3. 空间复杂度
2.3. &nbsp; 空间复杂度
</span>
</div>
@@ -268,14 +268,14 @@
<label class="md-nav__link" for="__nav_1" id="__nav_1_label" tabindex="0">
0. 写在前面
0. &nbsp; &nbsp; 写在前面
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_1">
<span class="md-nav__icon md-icon"></span>
0. 写在前面
0. &nbsp; &nbsp; 写在前面
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -286,7 +286,7 @@
<li class="md-nav__item">
<a href="../../chapter_preface/about_the_book/" class="md-nav__link">
0.1. 关于本书
0.1. &nbsp; 关于本书
</a>
</li>
@@ -300,7 +300,7 @@
<li class="md-nav__item">
<a href="../../chapter_preface/suggestions/" class="md-nav__link">
0.2. 如何使用本书
0.2. &nbsp; 如何使用本书
</a>
</li>
@@ -314,7 +314,7 @@
<li class="md-nav__item">
<a href="../../chapter_preface/installation/" class="md-nav__link">
0.3. 编程环境安装
0.3. &nbsp; 编程环境安装
</a>
</li>
@@ -328,7 +328,7 @@
<li class="md-nav__item">
<a href="../../chapter_preface/contribution/" class="md-nav__link">
0.4. 一起参与创作
0.4. &nbsp; 一起参与创作
</a>
</li>
@@ -367,14 +367,14 @@
<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
1. 引言
1. &nbsp; &nbsp; 引言
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_2">
<span class="md-nav__icon md-icon"></span>
1. 引言
1. &nbsp; &nbsp; 引言
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -385,7 +385,7 @@
<li class="md-nav__item">
<a href="../../chapter_introduction/algorithms_are_everywhere/" class="md-nav__link">
1.1. 算法无处不在
1.1. &nbsp; 算法无处不在
</a>
</li>
@@ -399,7 +399,7 @@
<li class="md-nav__item">
<a href="../../chapter_introduction/what_is_dsa/" class="md-nav__link">
1.2. 算法是什么
1.2. &nbsp; 算法是什么
</a>
</li>
@@ -446,14 +446,14 @@
<label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
2. 计算复杂度
2. &nbsp; &nbsp; 计算复杂度
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
<label class="md-nav__title" for="__nav_3">
<span class="md-nav__icon md-icon"></span>
2. 计算复杂度
2. &nbsp; &nbsp; 计算复杂度
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -464,7 +464,7 @@
<li class="md-nav__item">
<a href="../performance_evaluation/" class="md-nav__link">
2.1. 算法效率评估
2.1. &nbsp; 算法效率评估
</a>
</li>
@@ -478,7 +478,7 @@
<li class="md-nav__item">
<a href="../time_complexity/" class="md-nav__link">
2.2. 时间复杂度
2.2. &nbsp; 时间复杂度
</a>
</li>
@@ -501,12 +501,12 @@
<label class="md-nav__link md-nav__link--active" for="__toc">
2.3. 空间复杂度
2.3. &nbsp; 空间复杂度
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
2.3. 空间复杂度
2.3. &nbsp; 空间复杂度
</a>
@@ -526,24 +526,24 @@
<li class="md-nav__item">
<a href="#231" class="md-nav__link">
2.3.1. 算法相关空间
2.3.1. &nbsp; 算法相关空间
</a>
</li>
<li class="md-nav__item">
<a href="#232" class="md-nav__link">
2.3.2. 推算方法
2.3.2. &nbsp; 推算方法
</a>
</li>
<li class="md-nav__item">
<a href="#233" class="md-nav__link">
2.3.3. 常见类型
2.3.3. &nbsp; 常见类型
</a>
<nav class="md-nav" aria-label="2.3.3. 常见类型">
<nav class="md-nav" aria-label="2.3.3. &nbsp; 常见类型">
<ul class="md-nav__list">
<li class="md-nav__item">
@@ -602,7 +602,7 @@
<li class="md-nav__item">
<a href="../space_time_tradeoff/" class="md-nav__link">
2.4. 权衡时间与空间
2.4. &nbsp; 权衡时间与空间
</a>
</li>
@@ -616,7 +616,7 @@
<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
2.5. 小结
2.5. &nbsp; 小结
</a>
</li>
@@ -657,14 +657,14 @@
<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
3. 数据结构简介
3. &nbsp; &nbsp; 数据结构简介
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_4">
<span class="md-nav__icon md-icon"></span>
3. 数据结构简介
3. &nbsp; &nbsp; 数据结构简介
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -675,7 +675,7 @@
<li class="md-nav__item">
<a href="../../chapter_data_structure/data_and_memory/" class="md-nav__link">
3.1. 数据与内存
3.1. &nbsp; 数据与内存
</a>
</li>
@@ -689,7 +689,7 @@
<li class="md-nav__item">
<a href="../../chapter_data_structure/classification_of_data_structure/" class="md-nav__link">
3.2. 数据结构分类
3.2. &nbsp; 数据结构分类
</a>
</li>
@@ -703,7 +703,7 @@
<li class="md-nav__item">
<a href="../../chapter_data_structure/summary/" class="md-nav__link">
3.3. 小结
3.3. &nbsp; 小结
</a>
</li>
@@ -746,14 +746,14 @@
<label class="md-nav__link" for="__nav_5" id="__nav_5_label" tabindex="0">
4. 数组与链表
4. &nbsp; &nbsp; 数组与链表
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_5_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_5">
<span class="md-nav__icon md-icon"></span>
4. 数组与链表
4. &nbsp; &nbsp; 数组与链表
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -764,7 +764,7 @@
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/array/" class="md-nav__link">
4.1. 数组（Array）
4.1. &nbsp; 数组（Array）
</a>
</li>
@@ -778,7 +778,7 @@
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/linked_list/" class="md-nav__link">
4.2. 链表（LinkedList）
4.2. &nbsp; 链表（LinkedList）
</a>
</li>
@@ -792,7 +792,7 @@
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/list/" class="md-nav__link">
4.3. 列表（List）
4.3. &nbsp; 列表（List）
</a>
</li>
@@ -806,7 +806,7 @@
<li class="md-nav__item">
<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
4.4. 小结
4.4. &nbsp; 小结
</a>
</li>
@@ -849,14 +849,14 @@
<label class="md-nav__link" for="__nav_6" id="__nav_6_label" tabindex="0">
5. 栈与队列
5. &nbsp; &nbsp; 栈与队列
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_6">
<span class="md-nav__icon md-icon"></span>
5. 栈与队列
5. &nbsp; &nbsp; 栈与队列
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -867,7 +867,7 @@
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
5.1. 栈（Stack）
5.1. &nbsp; 栈（Stack）
</a>
</li>
@@ -881,7 +881,7 @@
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
5.2. 队列（Queue）
5.2. &nbsp; 队列（Queue）
</a>
</li>
@@ -895,7 +895,7 @@
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
5.3. 双向队列（Deque）
5.3. &nbsp; 双向队列（Deque）
</a>
</li>
@@ -909,7 +909,7 @@
<li class="md-nav__item">
<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
5.4. 小结
5.4. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_7" id="__nav_7_label" tabindex="0">
6. 散列表
6. &nbsp; &nbsp; 散列表
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_7">
<span class="md-nav__icon md-icon"></span>
6. 散列表
6. &nbsp; &nbsp; 散列表
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -968,7 +968,7 @@
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
6.1. 哈希表（HashMap）
6.1. &nbsp; 哈希表（HashMap）
</a>
</li>
@@ -982,7 +982,7 @@
<li class="md-nav__item">
<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
6.2. 哈希冲突处理
6.2. &nbsp; 哈希冲突处理
</a>
</li>
@@ -996,7 +996,7 @@
<li class="md-nav__item">
<a href="../../chapter_hashing/summary/" class="md-nav__link">
6.3. 小结
6.3. &nbsp; 小结
</a>
</li>
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<label class="md-nav__link" for="__nav_8" id="__nav_8_label" tabindex="0">
7. 二叉树
7. &nbsp; &nbsp; 二叉树
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_8">
<span class="md-nav__icon md-icon"></span>
7. 二叉树
7. &nbsp; &nbsp; 二叉树
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1059,7 +1059,7 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
7.1. 二叉树（Binary Tree）
7.1. &nbsp; 二叉树（Binary Tree）
</a>
</li>
@@ -1073,7 +1073,7 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
7.2. 二叉树遍历
7.2. &nbsp; 二叉树遍历
</a>
</li>
@@ -1087,7 +1087,7 @@
<li class="md-nav__item">
<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
7.3. 二叉搜索树
7.3. &nbsp; 二叉搜索树
</a>
</li>
@@ -1101,7 +1101,7 @@
<li class="md-nav__item">
<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
7.4. AVL 树 *
7.4. &nbsp; AVL 树 *
</a>
</li>
@@ -1115,7 +1115,7 @@
<li class="md-nav__item">
<a href="../../chapter_tree/summary/" class="md-nav__link">
7.5. 小结
7.5. &nbsp; 小结
</a>
</li>
@@ -1152,14 +1152,14 @@
<label class="md-nav__link" for="__nav_9" id="__nav_9_label" tabindex="0">
8. 堆
8. &nbsp; &nbsp; 堆
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_9">
<span class="md-nav__icon md-icon"></span>
8. 堆
8. &nbsp; &nbsp; 堆
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1170,7 +1170,7 @@
<li class="md-nav__item">
<a href="../../chapter_heap/heap/" class="md-nav__link">
8.1. 堆（Heap）
8.1. &nbsp; 堆（Heap）
</a>
</li>
@@ -1211,14 +1211,14 @@
<label class="md-nav__link" for="__nav_10" id="__nav_10_label" tabindex="0">
9. 图
9. &nbsp; &nbsp; 图
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_10">
<span class="md-nav__icon md-icon"></span>
9. 图
9. &nbsp; &nbsp; 图
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1229,7 +1229,7 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph/" class="md-nav__link">
9.1. 图（Graph）
9.1. &nbsp; 图（Graph）
</a>
</li>
@@ -1243,7 +1243,7 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
9.2. 图基础操作
9.2. &nbsp; 图基础操作
</a>
</li>
@@ -1257,7 +1257,7 @@
<li class="md-nav__item">
<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
9.3. 图的遍历
9.3. &nbsp; 图的遍历
</a>
</li>
@@ -1300,14 +1300,14 @@
<label class="md-nav__link" for="__nav_11" id="__nav_11_label" tabindex="0">
10. 查找算法
10. &nbsp; &nbsp; 查找算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_11">
<span class="md-nav__icon md-icon"></span>
10. 查找算法
10. &nbsp; &nbsp; 查找算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1318,7 +1318,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/linear_search/" class="md-nav__link">
10.1. 线性查找
10.1. &nbsp; 线性查找
</a>
</li>
@@ -1332,7 +1332,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/binary_search/" class="md-nav__link">
10.2. 二分查找
10.2. &nbsp; 二分查找
</a>
</li>
@@ -1346,7 +1346,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/hashing_search/" class="md-nav__link">
10.3. 哈希查找
10.3. &nbsp; 哈希查找
</a>
</li>
@@ -1360,7 +1360,7 @@
<li class="md-nav__item">
<a href="../../chapter_searching/summary/" class="md-nav__link">
10.4. 小结
10.4. &nbsp; 小结
</a>
</li>
@@ -1407,14 +1407,14 @@
<label class="md-nav__link" for="__nav_12" id="__nav_12_label" tabindex="0">
11. 排序算法
11. &nbsp; &nbsp; 排序算法
<span class="md-nav__icon md-icon"></span>
</label>
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="false">
<label class="md-nav__title" for="__nav_12">
<span class="md-nav__icon md-icon"></span>
11. 排序算法
11. &nbsp; &nbsp; 排序算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1425,7 +1425,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/intro_to_sort/" class="md-nav__link">
11.1. 排序简介
11.1. &nbsp; 排序简介
</a>
</li>
@@ -1439,7 +1439,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/bubble_sort/" class="md-nav__link">
11.2. 冒泡排序
11.2. &nbsp; 冒泡排序
</a>
</li>
@@ -1453,7 +1453,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/insertion_sort/" class="md-nav__link">
11.3. 插入排序
11.3. &nbsp; 插入排序
</a>
</li>
@@ -1467,7 +1467,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/quick_sort/" class="md-nav__link">
11.4. 快速排序
11.4. &nbsp; 快速排序
</a>
</li>
@@ -1481,7 +1481,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/merge_sort/" class="md-nav__link">
11.5. 归并排序
11.5. &nbsp; 归并排序
</a>
</li>
@@ -1495,7 +1495,7 @@
<li class="md-nav__item">
<a href="../../chapter_sorting/summary/" class="md-nav__link">
11.6. 小结
11.6. &nbsp; 小结
</a>
</li>
@@ -1583,24 +1583,24 @@
<li class="md-nav__item">
<a href="#231" class="md-nav__link">
2.3.1. 算法相关空间
2.3.1. &nbsp; 算法相关空间
</a>
</li>
<li class="md-nav__item">
<a href="#232" class="md-nav__link">
2.3.2. 推算方法
2.3.2. &nbsp; 推算方法
</a>
</li>
<li class="md-nav__item">
<a href="#233" class="md-nav__link">
2.3.3. 常见类型
2.3.3. &nbsp; 常见类型
</a>
<nav class="md-nav" aria-label="2.3.3. 常见类型">
<nav class="md-nav" aria-label="2.3.3. &nbsp; 常见类型">
<ul class="md-nav__list">
<li class="md-nav__item">
@@ -1667,9 +1667,9 @@
<h1 id="23">2.3. 空间复杂度<a class="headerlink" href="#23" title="Permanent link">&para;</a></h1>
<h1 id="23">2.3. &nbsp; 空间复杂度<a class="headerlink" href="#23" title="Permanent link">&para;</a></h1>
<p>「空间复杂度 Space Complexity」统计 <strong>算法使用内存空间随着数据量变大时的增长趋势</strong>。这个概念与时间复杂度很类似。</p>
<h2 id="231">2.3.1. 算法相关空间<a class="headerlink" href="#231" title="Permanent link">&para;</a></h2>
<h2 id="231">2.3.1. &nbsp; 算法相关空间<a class="headerlink" href="#231" title="Permanent link">&para;</a></h2>
<p>算法运行中，使用的内存空间主要有以下几种：</p>
<ul>
<li>「输入空间」用于存储算法的输入数据；</li>
@@ -1897,7 +1897,7 @@
</div>
</div>
</div>
<h2 id="232">2.3.2. 推算方法<a class="headerlink" href="#232" title="Permanent link">&para;</a></h2>
<h2 id="232">2.3.2. &nbsp; 推算方法<a class="headerlink" href="#232" title="Permanent link">&para;</a></h2>
<p>空间复杂度的推算方法和时间复杂度总体类似，只是从统计“计算操作数量”变为统计“使用空间大小”。与时间复杂度不同的是，<strong>我们一般只关注「最差空间复杂度」</strong>。这是因为内存空间是一个硬性要求，我们必须保证在所有输入数据下都有足够的内存空间预留。</p>
<p><strong>最差空间复杂度中的“最差”有两层含义</strong>，分别为输入数据的最差分布、算法运行中的最差时间点。</p>
<ul>
@@ -2164,7 +2164,7 @@
</div>
</div>
</div>
<h2 id="233">2.3.3. 常见类型<a class="headerlink" href="#233" title="Permanent link">&para;</a></h2>
<h2 id="233">2.3.3. &nbsp; 常见类型<a class="headerlink" href="#233" title="Permanent link">&para;</a></h2>
<p>设输入数据大小为 <span class="arithmatex">\(n\)</span> ，常见的空间复杂度类型有（从低到高排列）</p>
<div class="arithmatex">\[
\begin{aligned}
@@ -3083,7 +3083,7 @@ O(1) &lt; O(\log n) &lt; O(n) &lt; O(n^2) &lt; O(2^n) \newline
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