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<h1 id="162">16.2 &nbsp; 一起参与创作<a class="headerlink" href="#162" title="Permanent link">&para;</a></h1>
<p>由于笔者能力有限，书中难免存在一些遗漏和错误，请您谅解。如果您发现了笔误、链接失效、内容缺失、文字歧义、解释不清晰或行文结构不合理等问题，请协助我们进行修正，以给读者提供更优质的学习资源。</p>

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<h1 id="16">第 16 章 &nbsp; 附录<a class="headerlink" href="#16" title="Permanent link">&para;</a></h1>
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<h1 id="161">16.1 &nbsp; 编程环境安装<a class="headerlink" href="#161" title="Permanent link">&para;</a></h1>
<h2 id="1611-ide">16.1.1 &nbsp; 安装 IDE<a class="headerlink" href="#1611-ide" title="Permanent link">&para;</a></h2>

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<h1 id="163">16.3 &nbsp; 术语表<a class="headerlink" href="#163" title="Permanent link">&para;</a></h1>
<p>表 16-1 列出了书中出现的重要术语。建议你同时记住它们的中英文叫法，以便阅读英文文献。</p>

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<h1 id="41">4.1 &nbsp; 数组<a class="headerlink" href="#41" title="Permanent link">&para;</a></h1>
<p>「数组 array」是一种线性数据结构，其将相同类型的元素存储在连续的内存空间中。我们将元素在数组中的位置称为该元素的「索引 index」。图 4-1 展示了数组的主要术语和概念。</p>

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<p>内存空间是所有程序的公共资源，在一个复杂的系统运行环境下，空闲的内存空间可能散落在内存各处。我们知道，存储数组的内存空间必须是连续的，而当数组非常大时，内存可能无法提供如此大的连续空间。此时链表的灵活性优势就体现出来了。</p>

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<p>「列表 list」是一个抽象的数据结构概念，它表示元素的有序集合，支持元素访问、修改、添加、删除和遍历等操作，无须使用者考虑容量限制的问题。列表可以基于链表或数组实现。</p>

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<p>在本章的前两节中，我们探讨了数组和链表这两种基础且重要的数据结构，它们分别代表了“连续存储”和“分散存储”这两种不同的物理结构。</p>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<p>「回溯算法 backtracking algorithm」是一种通过穷举来解决问题的方法，它的核心思想是从一个初始状态出发，暴力搜索所有可能的解决方案，当遇到正确的解则将其记录，直到找到解或者尝试了所有可能的选择都无法找到解为止。</p>

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<h1 id="13">第 13 章 &nbsp; 回溯<a class="headerlink" href="#13" title="Permanent link">&para;</a></h1>
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<p>全排列问题是回溯算法的一个典型应用。它的定义是在给定一个集合（如一个数组或字符串）的情况下，找出其中元素的所有可能的排列。</p>

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<h2 id="1331">13.3.1 &nbsp; 无重复元素的情况<a class="headerlink" href="#1331" title="Permanent link">&para;</a></h2>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<h1 id="2">第 2 章 &nbsp; 复杂度分析<a class="headerlink" href="#2" title="Permanent link">&para;</a></h1>
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<p>在算法中，重复执行某个任务是很常见的，其与复杂度分析息息相关。因此，在展开介绍时间复杂度和空间复杂度之前，我们先来了解如何在程序中实现重复执行任务，即两种基本的程序控制结构：迭代、递归。</p>

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<p>「空间复杂度 space complexity」用于衡量算法占用内存空间随着数据量变大时的增长趋势。这个概念与时间复杂度非常类似，只需将“运行时间”替换为“占用内存空间”。</p>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<p>运行时间可以直观且准确地反映算法的效率。如果我们想准确预估一段代码的运行时间，应该如何操作呢？</p>

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<p>在计算机中，所有数据都是以二进制数的形式存储的，字符 <code>char</code> 也不例外。为了表示字符，我们需要建立一套“字符集”，规定每个字符和二进制数之间的一一对应关系。有了字符集之后，计算机就可以通过查表完成二进制数到字符的转换。</p>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<p>我们已经学过，搜索算法分为两大类。</p>

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<p>「分治 divide and conquer」，全称分而治之，是一种非常重要且常见的算法策略。分治通常基于递归实现，包括“分”和“治”两个步骤。</p>

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<p>在归并排序和构建二叉树中，我们都是将原问题分解为两个规模为原问题一半的子问题。然而对于汉诺塔问题，我们采用不同的分解策略。</p>

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<ul>

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<p>在上一节中，我们学习了动态规划是如何通过子问题分解来求解原问题的。实际上，子问题分解是一种通用的算法思路，在分治、动态规划、回溯中的侧重点不同。</p>

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<p>上两节介绍了动态规划问题的主要特征，接下来我们一起探究两个更加实用的问题。</p>

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<p>编辑距离，也称 Levenshtein 距离，指两个字符串之间互相转换的最少修改次数，通常用于在信息检索和自然语言处理中度量两个序列的相似度。</p>

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<h1 id="14">第 14 章 &nbsp; 动态规划<a class="headerlink" href="#14" title="Permanent link">&para;</a></h1>
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<h1 id="141">14.1 &nbsp; 初探动态规划<a class="headerlink" href="#141" title="Permanent link">&para;</a></h1>
<p>「动态规划 dynamic programming」是一个重要的算法范式，它将一个问题分解为一系列更小的子问题，并通过存储子问题的解来避免重复计算，从而大幅提升时间效率。</p>

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<p>背包问题是一个非常好的动态规划入门题目，是动态规划中最常见的问题形式。其具有很多变种，例如 0-1 背包问题、完全背包问题、多重背包问题等。</p>

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<h1 id="147">14.7 &nbsp; 小结<a class="headerlink" href="#147" title="Permanent link">&para;</a></h1>
<ul>

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<p>在本节中，我们先求解另一个常见的背包问题：完全背包，再了解它的一种特例：零钱兑换。</p>

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<p>「图 graph」是一种非线性数据结构，由「顶点 vertex」和「边 edge」组成。我们可以将图 <span class="arithmatex">\(G\)</span> 抽象地表示为一组顶点 <span class="arithmatex">\(V\)</span> 和一组边 <span class="arithmatex">\(E\)</span> 的集合。以下示例展示了一个包含 5 个顶点和 7 条边的图。</p>

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<p>图的基础操作可分为对“边”的操作和对“顶点”的操作。在“邻接矩阵”和“邻接表”两种表示方法下，实现方式有所不同。</p>

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<p>树代表的是“一对多”的关系，而图则具有更高的自由度，可以表示任意的“多对多”关系。因此，我们可以把树看作图的一种特例。显然，<strong>树的遍历操作也是图的遍历操作的一种特例</strong>。</p>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<p>「贪心算法 greedy algorithm」是一种常见的解决优化问题的算法，其基本思想是在问题的每个决策阶段，都选择当前看起来最优的选择，即贪心地做出局部最优的决策，以期获得全局最优解。贪心算法简洁且高效，在许多实际问题中有着广泛的应用。</p>

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<h1 id="63">6.3 &nbsp; 哈希算法<a class="headerlink" href="#63" title="Permanent link">&para;</a></h1>
<p>前两节介绍了哈希表的工作原理和哈希冲突的处理方法。然而无论是开放寻址还是链式地址，<strong>它们只能保证哈希表可以在发生冲突时正常工作，而无法减少哈希冲突的发生</strong>。</p>

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<p>上一节提到，<strong>通常情况下哈希函数的输入空间远大于输出空间</strong>，因此理论上哈希冲突是不可避免的。比如，输入空间为全体整数，输出空间为数组容量大小，则必然有多个整数映射至同一桶索引。</p>

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<h1 id="61">6.1 &nbsp; 哈希表<a class="headerlink" href="#61" title="Permanent link">&para;</a></h1>
<p>「哈希表 hash table」，又称「散列表」，其通过建立键 <code>key</code> 与值 <code>value</code> 之间的映射，实现高效的元素查询。具体而言，我们向哈希表输入一个键 <code>key</code> ，则可以在 <span class="arithmatex">\(O(1)\)</span> 时间内获取对应的值 <code>value</code> 。</p>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<p>在某些情况下，我们希望使用一个列表的所有元素来构建一个堆，这个过程被称为“建堆操作”。</p>

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<p>「堆 heap」是一种满足特定条件的完全二叉树，主要可分为两种类型，如图 8-1 所示。</p>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<p>当我们听到“算法”这个词时，很自然地会想到数学。然而实际上，许多算法并不涉及复杂数学，而是更多地依赖基本逻辑，这些逻辑在我们的日常生活中处处可见。</p>

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<h2 id="121">1.2.1 &nbsp; 算法定义<a class="headerlink" href="#121" title="Permanent link">&para;</a></h2>

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<p>本项目旨在创建一本开源、免费、新手友好的数据结构与算法入门教程。</p>

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<p>[1] Thomas H. Cormen, et al. Introduction to Algorithms (3<sup>rd</sup> Edition).</p>

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<p>「二分查找 binary search」是一种基于分治策略的高效搜索算法。它利用数据的有序性，每轮缩小一半搜索范围，直至找到目标元素或搜索区间为空为止。</p>

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<h2 id="1031">10.3.1 &nbsp; 查找左边界<a class="headerlink" href="#1031" title="Permanent link">&para;</a></h2>

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<p>二分查找不仅可用于搜索目标元素，还可用于解决许多变种问题，比如搜索目标元素的插入位置。</p>

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<h1 id="10">第 10 章 &nbsp; 搜索<a class="headerlink" href="#10" title="Permanent link">&para;</a></h1>
<div class="center-table">

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<p>在算法题中，<strong>我们常通过将线性查找替换为哈希查找来降低算法的时间复杂度</strong>。我们借助一个算法题来加深理解。</p>

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<p>「搜索算法 searching algorithm」用于在数据结构（例如数组、链表、树或图）中搜索一个或一组满足特定条件的元素。</p>

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<ul>

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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><!--! Font Awesome Free 6.4.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2023 Fonticons, Inc.--><path d="M441 58.9 453.1 71c9.4 9.4 9.4 24.6 0 33.9L424 134.1 377.9 88 407 58.9c9.4-9.4 24.6-9.4 33.9 0zM209.8 256.2 344 121.9l46.1 46.1-134.3 134.2c-2.9 2.9-6.5 5-10.4 6.1L186.9 325l16.7-58.5c1.1-3.9 3.2-7.5 6.1-10.4zM373.1 25 175.8 222.2c-8.7 8.7-15 19.4-18.3 31.1l-28.6 100c-2.4 8.4-.1 17.4 6.1 23.6s15.2 8.5 23.6 6.1l100-28.6c11.8-3.4 22.5-9.7 31.1-18.3L487 138.9c28.1-28.1 28.1-73.7 0-101.8L474.9 25c-28.1-28.1-73.7-28.1-101.8 0zM88 64c-48.6 0-88 39.4-88 88v272c0 48.6 39.4 88 88 88h272c48.6 0 88-39.4 88-88V312c0-13.3-10.7-24-24-24s-24 10.7-24 24v112c0 22.1-17.9 40-40 40H88c-22.1 0-40-17.9-40-40V152c0-22.1 17.9-40 40-40h112c13.3 0 24-10.7 24-24s-10.7-24-24-24H88z"/></svg>
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<p>「冒泡排序 bubble sort」通过连续地比较与交换相邻元素实现排序。这个过程就像气泡从底部升到顶部一样，因此得名冒泡排序。</p>

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<p>前述几种排序算法都属于“基于比较的排序算法”，它们通过比较元素间的大小来实现排序。此类排序算法的时间复杂度无法超越 <span class="arithmatex">\(O(n \log n)\)</span> 。接下来，我们将探讨几种“非比较排序算法”，它们的时间复杂度可以达到线性阶。</p>

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<p>「计数排序 counting sort」通过统计元素数量来实现排序，通常应用于整数数组。</p>

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<p>「插入排序 insertion sort」是一种简单的排序算法，它的工作原理与手动整理一副牌的过程非常相似。</p>

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<p>「归并排序 merge sort」是一种基于分治策略的排序算法，包含图 11-10 所示的“划分”和“合并”阶段。</p>

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<p>「快速排序 quick sort」是一种基于分治策略的排序算法，运行高效，应用广泛。</p>

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<p>上一节介绍了计数排序，它适用于数据量 <span class="arithmatex">\(n\)</span> 较大但数据范围 <span class="arithmatex">\(m\)</span> 较小的情况。假设我们需要对 <span class="arithmatex">\(n = 10^6\)</span> 个学号进行排序，而学号是一个 <span class="arithmatex">\(8\)</span> 位数字，这意味着数据范围 <span class="arithmatex">\(m = 10^8\)</span> 非常大，使用计数排序需要分配大量内存空间，而基数排序可以避免这种情况。</p>

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<p>「选择排序 selection sort」的工作原理非常简单：开启一个循环，每轮从未排序区间选择最小的元素，将其放到已排序区间的末尾。</p>

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<p>「排序算法 sorting algorithm」用于对一组数据按照特定顺序进行排列。排序算法有着广泛的应用，因为有序数据通常能够被更高效地查找、分析和处理。</p>

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<p>在队列中，我们仅能删除头部元素或在尾部添加元素。如图 5-7 所示，「双向队列 double-ended queue」提供了更高的灵活性，允许在头部和尾部执行元素的添加或删除操作。</p>

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<p>「队列 queue」是一种遵循先入先出规则的线性数据结构。顾名思义，队列模拟了排队现象，即新来的人不断加入队列尾部，而位于队列头部的人逐个离开。</p>

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<p>「栈 stack」是一种遵循先入后出的逻辑的线性数据结构。</p>

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<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>

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<h1 id="73">7.3 &nbsp; 二叉树数组表示<a class="headerlink" href="#73" title="Permanent link">&para;</a></h1>
<p>在链表表示下，二叉树的存储单元为节点 <code>TreeNode</code> ，节点之间通过指针相连接。上一节介绍了链表表示下的二叉树的各项基本操作。</p>

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<h1 id="75-avl">7.5 &nbsp; AVL 树 *<a class="headerlink" href="#75-avl" title="Permanent link">&para;</a></h1>
<p>在“二叉搜索树”章节中，我们提到，在多次插入和删除操作后，二叉搜索树可能退化为链表。在这种情况下，所有操作的时间复杂度将从 <span class="arithmatex">\(O(\log n)\)</span> 恶化为 <span class="arithmatex">\(O(n)\)</span> 。</p>

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<h1 id="74">7.4 &nbsp; 二叉搜索树<a class="headerlink" href="#74" title="Permanent link">&para;</a></h1>
<p>如图 7-16 所示，「二叉搜索树 binary search tree」满足以下条件。</p>

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<h1 id="71">7.1 &nbsp; 二叉树<a class="headerlink" href="#71" title="Permanent link">&para;</a></h1>
<p>「二叉树 binary tree」是一种非线性数据结构，代表“祖先”与“后代”之间的派生关系，体现了“一分为二”的分治逻辑。与链表类似，二叉树的基本单元是节点，每个节点包含值、左子节点引用和右子节点引用。</p>

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