月更 技术博客 CXX优先 Zola 2026-08-11T00:00:00+00:00 https://suchaharcan.github.io/tags/graph/atom.xml 2026-08-11T00:00:00+00:00 2026-08-11T00:00:00+00:00 Suchaharcan https://suchaharcan.github.io/01Write_blog_AlgorithmDesign/408-graph/ <h1 id="kao-yan-408-tu-he-xin-zhi-shi-dian">考研 408 · 图核心知识点</h1> <blockquote> <p>图是 408 大题最爱：<strong>存储 → 遍历 → 生成树 → 最短路 → 拓扑/关键路径</strong>。</p> </blockquote> <hr /> <h2 id="1-tu-de-cun-chu">1. 图的存储</h2> <h3 id="lin-jie-ju-zhen">邻接矩阵</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span> 1 2 3 4</span></span> <span class="giallo-l"><span>1 [ 0 1 1 0 ]</span></span> <span class="giallo-l"><span>2 [ 1 0 1 1 ]</span></span> <span class="giallo-l"><span>3 [ 1 1 0 1 ]</span></span> <span class="giallo-l"><span>4 [ 0 1 1 0 ]</span></span></code></pre> <p>无向图对称；有向图按出边填写。<strong>判断两点是否相邻 O(1)，求度方便；但空间 O(n²)</strong>。</p> <h3 id="lin-jie-biao">邻接表</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>1 → 2 → 3</span></span> <span class="giallo-l"><span>2 → 1 → 3 → 4</span></span> <span class="giallo-l"><span>3 → 1 → 2 → 4</span></span> <span class="giallo-l"><span>4 → 2 → 3</span></span></code></pre> <p><strong>空间 O(n+e)，适合稀疏图</strong>；求某点出度方便，判断相邻需扫描。</p> <table><thead><tr><th>存储</th><th>空间</th><th>判断相邻</th><th>求所有邻点</th></tr></thead><tbody> <tr><td>邻接矩阵</td><td>O(n²)</td><td>O(1)</td><td>O(n)</td></tr> <tr><td>邻接表</td><td>O(n+e)</td><td>O(degree)</td><td>O(degree)</td></tr> </tbody></table> <hr /> <h2 id="2-bian-li">2. 遍历</h2> <h3 id="dfs-shen-du-you-xian-yong-zhan-di-gui">DFS（深度优先，用栈/递归）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>访问顶点 → 递归访问第一个未访问邻点 → 回溯</span></span></code></pre><h3 id="bfs-yan-du-you-xian-yong-dui-lie">BFS（广度优先，用队列）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>访问顶点 → 邻点全部入队 → 逐个出队重复</span></span></code></pre> <p><strong>复杂度</strong>：邻接表 O(n+e)，邻接矩阵 O(n²)。</p> <p>BFS 还能求<strong>无权图最短路径（边数）</strong>、判断连通分量数。</p> <hr /> <h2 id="3-zui-xiao-sheng-cheng-shu-mst">3. 最小生成树 MST</h2> <h3 id="prim-jia-dian">Prim（加点）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>从一个顶点出发，每次选择&quot;连接已选集合与未选集合的最短边&quot;加入集合，直到覆盖全部顶点。</span></span> <span class="giallo-l"><span>复杂度：邻接矩阵 O(n²)；适合稠密图。</span></span></code></pre><h3 id="kruskal-jia-bian">Kruskal（加边）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>边按权排序，从小到大选边，用并查集避免成环，选 n−1 条为止。</span></span> <span class="giallo-l"><span>复杂度：O(e·log e)；适合稀疏图。</span></span></code></pre><h3 id="shi-li">示例</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span> 1 --2-- 2</span></span> <span class="giallo-l"><span> | \ |</span></span> <span class="giallo-l"><span> 2 4 3</span></span> <span class="giallo-l"><span> | \ |</span></span> <span class="giallo-l"><span> 3 --1-- 4</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span>按 Kruskal：先取 3-4(1)、1-2(2)、1-3(2)（跳过 2-4(3)） ⇒ MST 总权 5</span></span></code></pre> <hr /> <h2 id="4-zui-duan-lu-jing">4. 最短路径</h2> <h3 id="dijkstra-dan-yuan-fei-fu-quan">Dijkstra（单源，非负权）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>维护 dist[]，每次取未确定顶点中 dist 最小者，松弛其邻边。</span></span> <span class="giallo-l"><span>复杂度：邻接矩阵 O(n²)；堆优化 O((n+e)log n)。</span></span> <span class="giallo-l"><span>不能处理负权边。</span></span></code></pre><h3 id="floyd-duo-yuan">Floyd（多源）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>三重循环：d[i][j] = min(d[i][j], d[i][k] + d[k][j])</span></span> <span class="giallo-l"><span>复杂度 O(n³)；可处理负权（无负环）。</span></span></code></pre><h3 id="shi-li-dijkstra-cong-1-chu-fa">示例（Dijkstra 从 1 出发）</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span> 2</span></span> <span class="giallo-l"><span> 1 ───── 3</span></span> <span class="giallo-l"><span> | 1 1 |</span></span> <span class="giallo-l"><span> 2 ───── 4</span></span> <span class="giallo-l"><span> 3</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span>dist: 1→1=0, 1→2=1, 1→3=2, 1→4=3（经 3）</span></span></code></pre> <hr /> <h2 id="5-tuo-bu-pai-xu-aov-wang">5. 拓扑排序（AOV 网）</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>无环有向图，顶点表示活动。</span></span> <span class="giallo-l"><span>反复：选入度为 0 的顶点输出，删除其出边，直到全部输出。</span></span> <span class="giallo-l"><span>若剩余顶点仍有入度 ⇒ 存在环。</span></span> <span class="giallo-l"><span>复杂度：邻接表 O(n+e)。</span></span></code></pre><h3 id="shi-li-1">示例</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>课程依赖：A→C, B→C, C→D</span></span> <span class="giallo-l"><span>拓扑序之一：A B C D（不唯一）</span></span></code></pre> <hr /> <h2 id="6-guan-jian-lu-jing-aoe-wang">6. 关键路径（AOE 网）</h2> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>顶点 = 事件，边 = 活动（带权=耗时）</span></span> <span class="giallo-l"><span>ve(最早发生) 正向取 max；vl(最迟发生) 逆向取 min</span></span> <span class="giallo-l"><span>活动最早 e = ve(弧尾)；活动最迟 l = vl(弧头) − 权</span></span> <span class="giallo-l"><span>e == l 的活动为关键活动，串联成关键路径</span></span> <span class="giallo-l"><span>关键路径长度 = 工程最短工期</span></span></code></pre><h3 id="shi-li-2">示例</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span> v1 ─2─▶ v2 ─3─▶ v4</span></span> <span class="giallo-l"><span> \ ▲</span></span> <span class="giallo-l"><span> 4 ────────▶ 3</span></span> <span class="giallo-l"><span>ve: v1=0, v2=2, v4=5, v3=4</span></span> <span class="giallo-l"><span>vl: v1=0, v2=2, v4=5, v3=2（逆向）</span></span> <span class="giallo-l"><span>关键活动：v1→v2(2), v2→v4(3)，关键路径长 5</span></span></code></pre> <hr /> <h2 id="7-chang-kao-ti-xing">7. 常考题型</h2> <ol> <li>画邻接矩阵/邻接表；</li> <li>给图写 DFS/BFS 序列；</li> <li>Prim/Kruskal 过程与总权；</li> <li>Dijkstra/Floyd 逐步表；</li> <li>拓扑排序与环检测；</li> <li>求 ve/vl/e/l 与关键路径；</li> <li>判断有向图强连通、无向图连通分量。</li> </ol> <hr /> <h2 id="8-yi-cuo-dian">8. 易错点</h2> <ul> <li>Dijkstra <strong>不适用负权</strong>，Floyd 可；</li> <li>MST 不含权值相等的唯一性讨论（可能不唯一）；</li> <li>拓扑排序不唯一；</li> <li>关键路径可能<strong>不止一条</strong>；</li> <li>BFS 求最短路只对<strong>无权</strong>图。</li> </ul> <blockquote> <p>上一篇：<a href="https://suchaharcan.github.io/01Write_blog_AlgorithmDesign/408-tree/">树</a> ｜ 下一篇：<a href="https://suchaharcan.github.io/01Write_blog_AlgorithmDesign/408-sort/">排序</a></p> </blockquote>