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第 6 章 &nbsp; 哈希表
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第 8 章 &nbsp; 堆
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8.1 &nbsp; 堆
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第 9 章 &nbsp; 图
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9.1 &nbsp; 图
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9.1 &nbsp; 图
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9.1.1 &nbsp; 图的常见类型与术语
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9.1.2 &nbsp; 图的表示
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1. &nbsp; 邻接矩阵
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2. &nbsp; 邻接表
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9.2 &nbsp; 图基础操作
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9.3 &nbsp; 图的遍历
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第 10 章 &nbsp; 搜索
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第 10 章 &nbsp; 搜索
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10.1 &nbsp; 二分查找
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10.2 &nbsp; 二分查找插入点
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10.3 &nbsp; 二分查找边界
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10.4 &nbsp; 哈希优化策略
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10.5 &nbsp; 重识搜索算法
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第 11 章 &nbsp; 排序
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11.1 &nbsp; 排序算法
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11.2 &nbsp; 选择排序
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11.3 &nbsp; 冒泡排序
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11.4 &nbsp; 插入排序
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11.5 &nbsp; 快速排序
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11.6 &nbsp; 归并排序
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11.7 &nbsp; 堆排序
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11.8 &nbsp; 桶排序
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11.9 &nbsp; 计数排序
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第 12 章 &nbsp; 分治
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12.1 &nbsp; 分治算法
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12.2 &nbsp; 分治搜索策略
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12.3 &nbsp; 构建树问题
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12.4 &nbsp; 汉诺塔问题
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12.5 &nbsp; 小结
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第 13 章 &nbsp; 回溯
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13.2 &nbsp; 全排列问题
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13.3 &nbsp; 子集和问题
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13.4 &nbsp; N 皇后问题
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第 14 章 &nbsp; 动态规划
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14.1 &nbsp; 初探动态规划
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14.2 &nbsp; DP 问题特性
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14.3 &nbsp; DP 解题思路
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<h1 id="91">9.1 &nbsp; 图<a class="headerlink" href="#91" title="Permanent link">&para;</a></h1>
<p><u>图（graph）</u>是一种非线性数据结构，由<u>顶点（vertex）</u>和<u>边（edge）</u>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>
<div class="arithmatex">\[
\begin{aligned}
V &amp; = \{ 1, 2, 3, 4, 5 \} \newline
E &amp; = \{ (1,2), (1,3), (1,5), (2,3), (2,4), (2,5), (4,5) \} \newline
G &amp; = \{ V, E \} \newline
\end{aligned}
\]</div>
<p>如果将顶点看作节点，将边看作连接各个节点的引用（指针），我们就可以将图看作一种从链表拓展而来的数据结构。如图 9-1 所示，<strong>相较于线性关系（链表）和分治关系（树），网络关系（图）的自由度更高</strong>，因而更为复杂。</p>
<p><a class="glightbox" href="../graph.assets/linkedlist_tree_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="链表、树、图之间的关系" class="animation-figure" src="../graph.assets/linkedlist_tree_graph.png" /></a></p>
<p align="center"> 图 9-1 &nbsp; 链表、树、图之间的关系 </p>
<h2 id="911">9.1.1 &nbsp; 图的常见类型与术语<a class="headerlink" href="#911" title="Permanent link">&para;</a></h2>
<p>根据边是否具有方向，可分为<u>无向图（undirected graph）</u>和<u>有向图（directed graph）</u>，如图 9-2 所示。</p>
<ul>
<li>在无向图中，边表示两顶点之间的“双向”连接关系，例如微信或 QQ 中的“好友关系”。</li>
<li>在有向图中，边具有方向性，即 <span class="arithmatex">\(A \rightarrow B\)</span> 和 <span class="arithmatex">\(A \leftarrow B\)</span> 两个方向的边是相互独立的，例如微博或抖音上的“关注”与“被关注”关系。</li>
</ul>
<p><a class="glightbox" href="../graph.assets/directed_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有向图与无向图" class="animation-figure" src="../graph.assets/directed_graph.png" /></a></p>
<p align="center"> 图 9-2 &nbsp; 有向图与无向图 </p>
<p>根据所有顶点是否连通，可分为<u>连通图（connected graph）</u>和<u>非连通图（disconnected graph）</u>，如图 9-3 所示。</p>
<ul>
<li>对于连通图，从某个顶点出发，可以到达其余任意顶点。</li>
<li>对于非连通图，从某个顶点出发，至少有一个顶点无法到达。</li>
</ul>
<p><a class="glightbox" href="../graph.assets/connected_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="连通图与非连通图" class="animation-figure" src="../graph.assets/connected_graph.png" /></a></p>
<p align="center"> 图 9-3 &nbsp; 连通图与非连通图 </p>
<p>我们还可以为边添加“权重”变量，从而得到如图 9-4 所示的<u>有权图（weighted graph）</u>。例如在《王者荣耀》等手游中，系统会根据共同游戏时间来计算玩家之间的“亲密度”，这种亲密度网络就可以用有权图来表示。</p>
<p><a class="glightbox" href="../graph.assets/weighted_graph.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="有权图与无权图" class="animation-figure" src="../graph.assets/weighted_graph.png" /></a></p>
<p align="center"> 图 9-4 &nbsp; 有权图与无权图 </p>
<p>图数据结构包含以下常用术语。</p>
<ul>
<li><u>邻接（adjacency）</u>：当两顶点之间存在边相连时，称这两顶点“邻接”。在图 9-4 中，顶点 1 的邻接顶点为顶点 2、3、5。</li>
<li><u>路径（path）</u>：从顶点 A 到顶点 B 经过的边构成的序列被称为从 A 到 B 的“路径”。在图 9-4 中，边序列 1-5-2-4 是顶点 1 到顶点 4 的一条路径。</li>
<li><u>度（degree）</u>：一个顶点拥有的边数。对于有向图，<u>入度（in-degree）</u>表示有多少条边指向该顶点，<u>出度（out-degree）</u>表示有多少条边从该顶点指出。</li>
</ul>
<h2 id="912">9.1.2 &nbsp; 图的表示<a class="headerlink" href="#912" title="Permanent link">&para;</a></h2>
<p>图的常用表示方式包括“邻接矩阵”和“邻接表”。以下使用无向图进行举例。</p>
<h3 id="1">1. &nbsp; 邻接矩阵<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<p>设图的顶点数量为 <span class="arithmatex">\(n\)</span> ，<u>邻接矩阵（adjacency matrix）</u>使用一个 <span class="arithmatex">\(n \times n\)</span> 大小的矩阵来表示图，每一行（列）代表一个顶点，矩阵元素代表边，用 <span class="arithmatex">\(1\)</span> 或 <span class="arithmatex">\(0\)</span> 表示两个顶点之间是否存在边。</p>
<p>如图 9-5 所示，设邻接矩阵为 <span class="arithmatex">\(M\)</span>、顶点列表为 <span class="arithmatex">\(V\)</span> ，那么矩阵元素 <span class="arithmatex">\(M[i, j] = 1\)</span> 表示顶点 <span class="arithmatex">\(V[i]\)</span> 到顶点 <span class="arithmatex">\(V[j]\)</span> 之间存在边，反之 <span class="arithmatex">\(M[i, j] = 0\)</span> 表示两顶点之间无边。</p>
<p><a class="glightbox" href="../graph.assets/adjacency_matrix.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的邻接矩阵表示" class="animation-figure" src="../graph.assets/adjacency_matrix.png" /></a></p>
<p align="center"> 图 9-5 &nbsp; 图的邻接矩阵表示 </p>
<p>邻接矩阵具有以下特性。</p>
<ul>
<li>在简单图中，顶点不能与自身相连，此时邻接矩阵主对角线元素没有意义。</li>
<li>对于无向图，两个方向的边等价，此时邻接矩阵关于主对角线对称。</li>
<li>将邻接矩阵的元素从 <span class="arithmatex">\(1\)</span> 和 <span class="arithmatex">\(0\)</span> 替换为权重，则可表示有权图。</li>
</ul>
<p>使用邻接矩阵表示图时，我们可以直接访问矩阵元素以获取边，因此增删查改操作的效率很高，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> ，内存占用较多。</p>
<h3 id="2">2. &nbsp; 邻接表<a class="headerlink" href="#2" title="Permanent link">&para;</a></h3>
<p><u>邻接表（adjacency list）</u>使用 <span class="arithmatex">\(n\)</span> 个链表来表示图，链表节点表示顶点。第 <span class="arithmatex">\(i\)</span> 个链表对应顶点 <span class="arithmatex">\(i\)</span> ，其中存储了该顶点的所有邻接顶点（与该顶点相连的顶点）。图 9-6 展示了一个使用邻接表存储的图的示例。</p>
<p><a class="glightbox" href="../graph.assets/adjacency_list.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="图的邻接表表示" class="animation-figure" src="../graph.assets/adjacency_list.png" /></a></p>
<p align="center"> 图 9-6 &nbsp; 图的邻接表表示 </p>
<p>邻接表仅存储实际存在的边，而边的总数通常远小于 <span class="arithmatex">\(n^2\)</span> ，因此它更加节省空间。然而，在邻接表中需要通过遍历链表来查找边，因此其时间效率不如邻接矩阵。</p>
<p>观察图 9-6 ，<strong>邻接表结构与哈希表中的“链式地址”非常相似，因此我们也可以采用类似的方法来优化效率</strong>。比如当链表较长时，可以将链表转化为 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> ；还可以把链表转换为哈希表，从而将时间复杂度降至 <span class="arithmatex">\(O(1)\)</span> 。</p>
<h2 id="913">9.1.3 &nbsp; 图的常见应用<a class="headerlink" href="#913" title="Permanent link">&para;</a></h2>
<p>如表 9-1 所示，许多现实系统可以用图来建模，相应的问题也可以约化为图计算问题。</p>
<p align="center"> 表 9-1 &nbsp; 现实生活中常见的图 </p>
<div class="center-table">
<table>
<thead>
<tr>
<th></th>
<th>顶点</th>
<th>边</th>
<th>图计算问题</th>
</tr>
</thead>
<tbody>
<tr>
<td>社交网络</td>
<td>用户</td>
<td>好友关系</td>
<td>潜在好友推荐</td>
</tr>
<tr>
<td>地铁线路</td>
<td>站点</td>
<td>站点间的连通性</td>
<td>最短路线推荐</td>
</tr>
<tr>
<td>太阳系</td>
<td>星体</td>
<td>星体间的万有引力作用</td>
<td>行星轨道计算</td>
</tr>
</tbody>
</table>
</div>
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