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<title>2.4. 权衡时间与空间 - Hello 算法</title>
<title>2.4.   权衡时间与空间 - Hello 算法</title>
@@ -111,7 +111,7 @@
<div class="md-header__topic" data-md-component="header-topic">
<span class="md-ellipsis">
2.4. 权衡时间与空间
2.4. &nbsp; 权衡时间与空间
</span>
</div>
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<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>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_1_label" aria-expanded="false">
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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>
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@@ -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>
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<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>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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1. 引言
1. &nbsp; &nbsp; 引言
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@@ -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>
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@@ -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
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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>
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<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>
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<li class="md-nav__item">
<a href="../space_complexity/" class="md-nav__link">
2.3. 空间复杂度
2.3. &nbsp; 空间复杂度
</a>
</li>
@@ -515,12 +515,12 @@
<label class="md-nav__link md-nav__link--active" for="__toc">
2.4. 权衡时间与空间
2.4. &nbsp; 权衡时间与空间
<span class="md-nav__icon md-icon"></span>
</label>
<a href="./" class="md-nav__link md-nav__link--active">
2.4. 权衡时间与空间
2.4. &nbsp; 权衡时间与空间
</a>
@@ -540,10 +540,10 @@
<li class="md-nav__item">
<a href="#241" class="md-nav__link">
2.4.1. 示例题目 *
2.4.1. &nbsp; 示例题目 *
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<nav class="md-nav" aria-label="2.4.1. 示例题目 *">
<nav class="md-nav" aria-label="2.4.1. &nbsp; 示例题目 *">
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<li class="md-nav__item">
<a href="../summary/" class="md-nav__link">
2.5. 小结
2.5. &nbsp; 小结
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<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
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3. 数据结构简介
3. &nbsp; &nbsp; 数据结构简介
</label>
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<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>
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<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>
@@ -668,7 +668,7 @@
<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>
@@ -711,14 +711,14 @@
<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>
@@ -729,7 +729,7 @@
<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>
@@ -743,7 +743,7 @@
<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>
@@ -757,7 +757,7 @@
<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>
@@ -771,7 +771,7 @@
<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>
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<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>
@@ -832,7 +832,7 @@
<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>
@@ -846,7 +846,7 @@
<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>
@@ -860,7 +860,7 @@
<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>
@@ -874,7 +874,7 @@
<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>
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<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>
@@ -947,7 +947,7 @@
<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>
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<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>
@@ -1024,7 +1024,7 @@
<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>
@@ -1038,7 +1038,7 @@
<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>
@@ -1052,7 +1052,7 @@
<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>
@@ -1066,7 +1066,7 @@
<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>
@@ -1080,7 +1080,7 @@
<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>
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<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>
@@ -1135,7 +1135,7 @@
<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>
@@ -1176,14 +1176,14 @@
<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">
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9. 图
9. &nbsp; &nbsp; 图
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1194,7 +1194,7 @@
<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>
@@ -1208,7 +1208,7 @@
<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>
@@ -1222,7 +1222,7 @@
<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>
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<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">
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10. 查找算法
10. &nbsp; &nbsp; 查找算法
</label>
<ul class="md-nav__list" data-md-scrollfix>
@@ -1283,7 +1283,7 @@
<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>
@@ -1297,7 +1297,7 @@
<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>
@@ -1311,7 +1311,7 @@
<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>
@@ -1325,7 +1325,7 @@
<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>
@@ -1372,14 +1372,14 @@
<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">
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<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>
@@ -1390,7 +1390,7 @@
<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>
@@ -1404,7 +1404,7 @@
<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>
@@ -1418,7 +1418,7 @@
<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>
@@ -1432,7 +1432,7 @@
<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>
@@ -1446,7 +1446,7 @@
<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>
@@ -1460,7 +1460,7 @@
<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>
@@ -1548,10 +1548,10 @@
<li class="md-nav__item">
<a href="#241" class="md-nav__link">
2.4.1. 示例题目 *
2.4.1. &nbsp; 示例题目 *
</a>
<nav class="md-nav" aria-label="2.4.1. 示例题目 *">
<nav class="md-nav" aria-label="2.4.1. &nbsp; 示例题目 *">
<ul class="md-nav__list">
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@@ -1597,11 +1597,11 @@
<h1 id="24">2.4. 权衡时间与空间<a class="headerlink" href="#24" title="Permanent link">&para;</a></h1>
<h1 id="24">2.4. &nbsp; 权衡时间与空间<a class="headerlink" href="#24" title="Permanent link">&para;</a></h1>
<p>理想情况下，我们希望算法的时间复杂度和空间复杂度都能够达到最优，而实际上，同时优化时间复杂度和空间复杂度是非常困难的。</p>
<p><strong>降低时间复杂度，往往是以提升空间复杂度为代价的，反之亦然</strong>。我们把牺牲内存空间来提升算法运行速度的思路称为「以空间换时间」；反之，称之为「以时间换空间」。选择哪种思路取决于我们更看重哪个方面。</p>
<p>大多数情况下，时间都是比空间更宝贵的，只要空间复杂度不要太离谱、能接受就行，<strong>因此以空间换时间最为常用</strong>。</p>
<h2 id="241">2.4.1. 示例题目 *<a class="headerlink" href="#241" title="Permanent link">&para;</a></h2>
<h2 id="241">2.4.1. &nbsp; 示例题目 *<a class="headerlink" href="#241" title="Permanent link">&para;</a></h2>
<p>以 LeetCode 全站第一题 <a href="https://leetcode.cn/problems/two-sum/">两数之和</a> 为例。</p>
<div class="admonition question">
<p class="admonition-title">两数之和</p>
@@ -2002,7 +2002,7 @@
<nav class="md-footer__inner md-grid" aria-label="页脚" >
<a href="../space_complexity/" class="md-footer__link md-footer__link--prev" aria-label="上一页: 2.3. 空间复杂度" rel="prev">
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2.5. 小结
2.5. &nbsp; 小结
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