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第 6 章 &nbsp; 哈希表
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第 8 章 &nbsp; 堆
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第 8 章 &nbsp; 堆
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8.1 &nbsp; 堆
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8.2 &nbsp; 建堆操作
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8.3 &nbsp; Top-k 问题
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第 9 章 &nbsp; 图
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第 9 章 &nbsp; 图
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9.1 &nbsp; 图
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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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9.4 &nbsp; 小结
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1. &nbsp; 重点回顾
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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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10.6 &nbsp; 小结
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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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11.10 &nbsp; 基数排序
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11.11 &nbsp; 小结
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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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13.1 &nbsp; 回溯算法
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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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14.4 &nbsp; 0-1 背包问题
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14.6 &nbsp; 编辑距离问题
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<h1 id="94">9.4 &nbsp; 小结<a class="headerlink" href="#94" title="Permanent link">&para;</a></h1>
<h3 id="1">1. &nbsp; 重点回顾<a class="headerlink" href="#1" title="Permanent link">&para;</a></h3>
<ul>
<li>图由顶点和边组成，可以表示为一组顶点和一组边构成的集合。</li>
<li>相较于线性关系（链表）和分治关系（树），网络关系（图）具有更高的自由度，因而更为复杂。</li>
<li>有向图的边具有方向性，连通图中的任意顶点均可达，有权图的每条边都包含权重变量。</li>
<li>邻接矩阵利用矩阵来表示图，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> 表示两个顶点之间有边或无边。邻接矩阵在增删查改操作上效率很高，但空间占用较多。</li>
<li>邻接表使用多个链表来表示图，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。邻接表相对于邻接矩阵更加节省空间，但由于需要遍历链表来查找边，因此时间效率较低。</li>
<li>当邻接表中的链表过长时，可以将其转换为红黑树或哈希表，从而提升查询效率。</li>
<li>从算法思想的角度分析，邻接矩阵体现了“以空间换时间”，邻接表体现了“以时间换空间”。</li>
<li>图可用于建模各类现实系统，如社交网络、地铁线路等。</li>
<li>树是图的一种特例，树的遍历也是图的遍历的一种特例。</li>
<li>图的广度优先遍历是一种由近及远、层层扩张的搜索方式，通常借助队列实现。</li>
<li>图的深度优先遍历是一种优先走到底、无路可走时再回溯的搜索方式，常基于递归来实现。</li>
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<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>：路径的定义是顶点序列还是边序列？</p>
<p>维基百科上不同语言版本的定义不一致：英文版是“路径是一个边序列”，而中文版是“路径是一个顶点序列”。以下是英文版原文：In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p>
<p>在本文中，路径被视为一个边序列，而不是一个顶点序列。这是因为两个顶点之间可能存在多条边连接，此时每条边都对应一条路径。</p>
<p><strong>Q</strong>：非连通图中是否会有无法遍历到的点？</p>
<p>在非连通图中，从某个顶点出发，至少有一个顶点无法到达。遍历非连通图需要设置多个起点，以遍历到图的所有连通分量。</p>
<p><strong>Q</strong>：在邻接表中，“与该顶点相连的所有顶点”的顶点顺序是否有要求？</p>
<p>可以是任意顺序。但在实际应用中，可能需要按照指定规则来排序，比如按照顶点添加的次序，或者按照顶点值大小的顺序等，这样有助于快速查找“带有某种极值”的顶点。</p>
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