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<li>数组支持随机访问、内存空间占用小；但插入与删除元素效率低，且初始化后长度不可变。</li>
<li>链表可通过更改指针实现高效的结点插入与删除，并且可以灵活地修改长度；但结点访问效率低、占用内存多。常见的链表类型有单向链表、循环链表、双向链表。</li>
<li>列表又称动态数组，是基于数组实现的一种数据结构，其保存了数组的优势，且可以灵活改变长度。列表的出现大大提升了数组的实用性，但副作用是会造成部分内存空间浪费。</li>
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<h2 id="441-vs">4.4.1. &nbsp; 数组 VS 链表<a class="headerlink" href="#441-vs" title="Permanent link">&para;</a></h2>
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<p>「缓存局部性（Cache locality）」涉及到了计算机操作系统，在本书不做展开介绍，建议有兴趣的同学 Google / Baidu 一下。</p>
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<p class="admonition-title">缓存局部性的简单解释</p>
<p>在计算机中，数据读写速度排序是“硬盘 &lt; 内存 &lt; CPU 缓存”。当我们访问数组元素时，计算机不仅会加载它，还会缓存其周围的其它数据，从而借助高速缓存来提升后续操作的执行速度。链表则不然，计算机只能挨个地缓存各个结点，这样的多次“搬运”降低了整体效率。</p>
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<h2 id="912">9.1.2. &nbsp; 图常用术语<a class="headerlink" href="#912" title="Permanent link">&para;</a></h2>
<ul>
<li>「邻接 Adjacency」：当两顶点之间有边相连时，称此两顶点“邻接”。</li>
<li>「路径 Path」：从顶点 A 到顶点 B 走过的边构成的序列，被称为从 A 到 B 的“路径”。</li>
<li>「邻接 Adjacency」：当两顶点之间有边相连时，称此两顶点“邻接”。例如，上图中顶点 1 的邻接顶点为顶点 2, 3, 5 。</li>
<li>「路径 Path」：从顶点 A 到顶点 B 走过的边构成的序列，被称为从 A 到 B 的“路径”。例如，上图中 1, 5, 2, 4 是顶点 1 到顶点 4 的一个路径。</li>
<li>「度 Degree」表示一个顶点具有多少条边。对于有向图，「入度 In-Degree」表示有多少条边指向该顶点，「出度 Out-Degree」表示有多少条边从该顶点指出。</li>
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<h2 id="913">9.1.3. &nbsp; 图的表示<a class="headerlink" href="#913" title="Permanent link">&para;</a></h2>
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<p>使用邻接矩阵表示图时，我们可以直接通过访问矩阵元素来获取边，因此增删查操作的效率很高，时间复杂度均为 <span class="arithmatex">\(O(1)\)</span> 。然而，矩阵的空间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> ，内存占用较大。</p>
<h3 id="_2">邻接表<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
<p>「邻接表 Adjacency List」使用 <span class="arithmatex">\(n\)</span> 个链表来表示图，链表结点表示顶点。第 <span class="arithmatex">\(i\)</span> 条链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了所有与该顶点相连的顶点。</p>
<p>「邻接表 Adjacency List」使用 <span class="arithmatex">\(n\)</span> 个链表来表示图，链表结点表示顶点。第 <span class="arithmatex">\(i\)</span> 条链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（即与该顶点相连的顶点）。</p>
<p><img alt="图的邻接表表示" src="../graph.assets/adjacency_list.png" /></p>
<p align="center"> Fig. 图的邻接表表示 </p>
<p>邻接表仅存储存在的边，而边的总数往往远小于 <span class="arithmatex">\(n^2\)</span> ，因此更加节省空间。但是，因为在邻接表中需要通过遍历链表来查找边，所以其时间效率不如邻接矩阵。</p>
<p>观察上图发现，<strong>邻接表结构与哈希表「链地址法」非常相似，因此我们也可以用类似方法来优化效率</strong>。比如，当链表较长时，可以把链表转化为「AVL 树」，从而将时间效率从 <span class="arithmatex">\(O(n)\)</span> 优化至 <span class="arithmatex">\(O(\log n)\)</span> ，还可以通过中序遍历获取有序序列；还可以将链表转化为 HashSet（即哈希表），将时间复杂度降低至 <span class="arithmatex">\(O(1)\)</span> 。</p>
<p>观察上图发现，<strong>邻接表结构与哈希表「链地址法」非常相似，因此我们也可以用类似方法来优化效率</strong>。比如，当链表较长时，可以把链表转化为 AVL 树或红黑树，从而将时间效率从 <span class="arithmatex">\(O(n)\)</span> 优化至 <span class="arithmatex">\(O(\log n)\)</span> ，还可以通过中序遍历获取有序序列；还可以将链表转化为哈希表，将时间复杂度降低至 <span class="arithmatex">\(O(1)\)</span> 。</p>
<h2 id="914">9.1.4. &nbsp; 图常见应用<a class="headerlink" href="#914" title="Permanent link">&para;</a></h2>
<p>现实中的许多系统都可以使用图来建模，对应的待求解问题也可以被约化为图计算问题。</p>
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<h1 id="62">6.2. &nbsp; 哈希冲突<a class="headerlink" href="#62" title="Permanent link">&para;</a></h1>
<p>理想情况下，哈希函数应该为每个输入产生唯一的输出，使得 key 和 value 一一对应。而实际上，往往存在向哈希函数输入不同的 key 而产生相同输出的情况，这种情况被称为「哈希冲突 Hash Collision」。哈希冲突会导致查询结果错误，从而严重影响哈希表的可用性。</p>
<p>那么，为什么会出现哈希冲突呢？本质上看，<strong>由于哈希函数的输入空间往往远大于输出空间</strong>，因此不可避免地会出现多个输入产生相同输出的情况，即为哈希冲突。比如，输入空间是全体整数，输出空间是一个固定大小的桶（数组）的索引范围，那么必定会有多个整数同时映射到一个桶索引。</p>
<p>为了缓解哈希冲突，一方面，我们可以通过「哈希表扩容」来减小冲突概率。极端情况下，当输入空间和输出空间大小相等时，哈希表就等价于数组了，可谓“大力出奇迹”。</p>
<p>另一方面，<strong>考虑通过优化数据结构以缓解哈希冲突</strong>，常见的方法有「链式地址」和「开放寻址」。</p>
<p>为了缓解哈希冲突，一方面，<strong>我们可以通过哈希表扩容来减小冲突概率</strong>。极端情况下，当输入空间和输出空间大小相等时，哈希表就等价于数组了，可谓“大力出奇迹”。</p>
<p>另一方面，<strong>考虑通过优化哈希表的表示方式以缓解哈希冲突</strong>，常见的方法有「链式地址」和「开放寻址」。</p>
<h2 id="621">6.2.1. &nbsp; 哈希表扩容<a class="headerlink" href="#621" title="Permanent link">&para;</a></h2>
<p>「负载因子 Load Factor」定义为 <strong>哈希表中元素数量除以桶槽数量（即数组大小）</strong>，代表哈希冲突的严重程度。</p>
<p><strong>负载因子常用作哈希表扩容的触发条件</strong>。比如在 Java 中，当负载因子 <span class="arithmatex">\(&gt; 0.75\)</span> 时则触发扩容，将 HashMap 大小扩充至原先的 <span class="arithmatex">\(2\)</span> 倍。</p>
@ -1666,7 +1714,7 @@
<li><strong>占用空间变大</strong>，因为链表或二叉树包含结点指针，相比于数组更加耗费内存空间；</li>
<li><strong>查询效率降低</strong>，因为需要线性遍历链表来查找对应元素；</li>
</ul>
<p>为了缓解时间效率问题，<strong>可以把「链表」转化为「AVL 树」或「红黑树」</strong>，将查询操作的时间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
<p>为了提升操作效率，<strong>可以把「链表」转化为「AVL 树」或「红黑树」</strong>，将查询操作的时间复杂度优化至 <span class="arithmatex">\(O(\log n)\)</span> 。</p>
<h2 id="623">6.2.3. &nbsp; 开放寻址<a class="headerlink" href="#623" title="Permanent link">&para;</a></h2>
<p>「开放寻址」不引入额外数据结构，而是通过“多次探测”来解决哈希冲突。根据探测方法的不同，主要分为 <strong>线性探测、平方探测、多次哈希</strong>。</p>
<h3 id="_1">线性探测<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>

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@ -1528,6 +1576,17 @@
<h1 id="63">6.3. &nbsp; 小结<a class="headerlink" href="#63" title="Permanent link">&para;</a></h1>
<ul>
<li>向哈希表中输入一个键 key ，查询到值 value 的时间复杂度为 <span class="arithmatex">\(O(1)\)</span> ，非常高效。</li>
<li>哈希表的常用操作包括查询、添加与删除键值对、遍历键值对等。</li>
<li>哈希函数将 key 映射到桶（数组）索引，从而访问到对应的值 value 。</li>
<li>两个不同的 key 经过哈希函数可能得到相同的桶索引，进而发生哈希冲突，导致查询错误。</li>
<li>缓解哈希冲突的途径有两种：哈希表扩容、优化哈希表的表示方式。</li>
<li>负载因子定义为哈希表中元素数量除以桶槽数量，体现哈希冲突的严重程度，常用作哈希表扩容的触发条件。与数组扩容的原理类似，哈希表扩容操作开销也很大。</li>
<li>链式地址考虑将单个元素转化成一个链表，将所有冲突元素都存储在一个链表中，从而解决哈希冲突。而为了提升查询效率，可以把链表转化为 AVL 树或红黑树，</li>
<li>开放寻址通过多次探测来解决哈希冲突。线性探测使用固定步长，缺点是不能删除元素且容易产生聚集。多次哈希使用多个哈希函数进行探测，相对线性探测不容易产生聚集，代价是多个哈希函数增加了计算量。</li>
<li>在工业界中，Java 的 HashMap 采用链式地址、Python 的 Dict 采用开放寻址。</li>
</ul>

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@ -1633,7 +1681,7 @@
<h3 id="_1">借助入堆方法实现<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<p>最直接地，考虑借助「元素入堆」方法，先建立一个空堆，<strong>再将列表元素依次入堆即可</strong>。</p>
<h3 id="_2">基于堆化操作实现<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
<p>然而，<strong>存在一种更加高效的建堆方法</strong>。设结点数量为 <span class="arithmatex">\(n\)</span> ，我们先将列表所有元素原封不动添加进堆，<strong>然后迭代地对各个结点执行「从顶至底堆化」</strong>。当然，<strong>无需对叶结点执行堆化</strong>，因为其没有子结点。</p>
<p>然而，<strong>存在一种更加高效的建堆方法</strong>。设元素数量为 <span class="arithmatex">\(n\)</span> ，我们先将列表所有元素原封不动添加进堆，<strong>然后迭代地对各个结点执行「从顶至底堆化」</strong>。当然，<strong>无需对叶结点执行堆化</strong>，因为其没有子结点。</p>
<div class="tabbed-set tabbed-alternate" data-tabs="1:10"><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" /><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></div>
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@ -1767,7 +1815,7 @@
<div class="arithmatex">\[
T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1
\]</div>
<p><img alt="完美二叉树的各层结点数量" src="../heap.assets/heapify_operations_count.png" /></p>
<p><img alt="完美二叉树的各层结点数量" src="../build_heap.assets/heapify_operations_count.png" /></p>
<p align="center"> Fig. 完美二叉树的各层结点数量 </p>
<p>化简上式需要借助中学的数列知识，先对 <span class="arithmatex">\(T(h)\)</span> 乘以 <span class="arithmatex">\(2\)</span> ，易得</p>
@ -1883,13 +1931,13 @@ T(h) &amp; = 2 \frac{1 - 2^h}{1 - 2} - h \newline
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<h1 id="116">11.6. &nbsp; 小结<a class="headerlink" href="#116" title="Permanent link">&para;</a></h1>
<ul>
<li>冒泡排序通过交换相邻元素来实现排序。通过增加标志位实现提前返回，我们可将冒泡排序的最佳时间复杂度优化至 <span class="arithmatex">\(O(N)\)</span> 。</li>
<li>插入排序每轮将待排序区间内元素插入至已排序区间的正确位置，从而实现排序。插入排序的时间复杂度虽为 <span class="arithmatex">\(O(N^2)\)</span> ，但因为总体操作少而很受欢迎，一般用于小数据量的排序工作。</li>
<li>快速排序基于哨兵划分操作实现排序。在哨兵划分中，有可能每次都选取到最差的基准数，从而导致时间复杂度劣化至 <span class="arithmatex">\(O(N^2)\)</span> ，通过引入中位数基准数或随机基准数可大大降低劣化概率。尾递归方法可以有效减小递归深度，将空间复杂度优化至 <span class="arithmatex">\(O(\log N)\)</span> 。</li>
<li>归并排序包含划分和合并两个阶段，是分而治之的标准体现。对于归并排序，排序数组需要借助辅助数组，空间复杂度为 <span class="arithmatex">\(O(N)\)</span> ；而排序链表的空间复杂度可以被优化至 <span class="arithmatex">\(O(1)\)</span> 。</li>
<li>下图总结对比了各个排序算法的运行效率与特性。其中，桶排序中 <span class="arithmatex">\(k\)</span> 为桶的数量；基数排序仅适用于正整数、字符串、特定格式的浮点数，<span class="arithmatex">\(k\)</span> 为最大数字的位数。</li>
</ul>
<p><img alt="排序算法对比" src="../summary.assets/sorting_algorithms_comparison.png" /></p>
<p align="center"> Fig. 排序算法对比 </p>
<ul>
<li>总体来看，我们追求运行快、稳定、原地、正向自适应性的排序。显然，如同其它数据结构与算法一样，同时满足这些条件的排序算法并不存在，我们需要根据问题特点来选择排序算法。</li>
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<h3 id="_1">二叉树<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
<ul>
<li>二叉树是一种非线性数据结构，代表着“一分为二”的分治逻辑。二叉树的结点包含「值」和两个「指针」，分别指向左子结点和右子结点。</li>
<li>选定二叉树中某结点，将其左（右）子结点以下形成的树称为左（右）子树。</li>
<li>二叉树的术语较多，包括根结点、叶结点、层、度、边、高度、深度等。</li>
<li>二叉树的初始化、结点插入、结点删除操作与链表的操作方法类似。</li>
<li>常见的二叉树类型包括完美二叉树、完全二叉树、完满二叉树、平衡二叉树。完美二叉树是理想状态，链表则是退化后的最差状态。</li>
<li>
<p>二叉树可以使用数组表示，具体做法是将结点值和空位按照层序遍历的顺序排列，并基于父结点和子结点之间的索引映射公式实现指针。</p>
</li>
<li>
<p>二叉树层序遍历是一种广度优先搜索，体现着“一圈一圈向外”的层进式遍历方式，通常借助队列来实现。</p>
</li>
<li>二叉树可以使用数组表示，具体做法是将结点值和空位按照层序遍历的顺序排列，并基于父结点和子结点之间的索引映射公式实现指针。</li>
</ul>
<h3 id="_2">二叉树遍历<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
<ul>
<li>二叉树层序遍历是一种广度优先搜索，体现着“一圈一圈向外”的层进式遍历方式，通常借助队列来实现。</li>
<li>前序、中序、后序遍历是深度优先搜索，体现着“走到头、再回头继续”的回溯遍历方式，通常使用递归实现。</li>
</ul>
<h3 id="_3">二叉搜索树<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h3>
<ul>
<li>二叉搜索树是一种高效的元素查找数据结构，查找、插入、删除操作的时间复杂度皆为 <span class="arithmatex">\(O(\log n)\)</span> 。二叉搜索树退化为链表后，各项时间复杂度劣化至 <span class="arithmatex">\(O(n)\)</span> ，因此如何避免退化是非常重要的课题。</li>
<li>AVL 树又称平衡二叉搜索树，其通过旋转操作，使得在不断插入与删除结点后，仍然可以保持二叉树的平衡（不退化）。</li>
<li>AVL 树的旋转操作分为右旋、左旋、先右旋后左旋、先左旋后右旋。在插入或删除结点后，AVL 树会从底至顶地执行旋转操作，使树恢复平衡。</li>

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