{"id":103244,"date":"2026-09-10T06:17:03","date_gmt":"2026-09-09T22:17:03","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/103244.html"},"modified":"2026-09-10T06:17:03","modified_gmt":"2026-09-09T22:17:03","slug":"%e6%b4%9b%e8%b0%b7-b3609%ef%bc%9a%e5%9b%be%e8%ae%ba%e4%b8%8e%e4%bb%a3%e6%95%b0%e7%bb%93%e6%9e%84-701-%e5%bc%ba%e8%bf%9e%e9%80%9a%e5%88%86%e9%87%8f-%e2%86%90-kosaraju-%e7%ae%97%e6%b3%95","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/103244.html","title":{"rendered":"\u6d1b\u8c37 B3609\uff1a[\u56fe\u8bba\u4e0e\u4ee3\u6570\u7ed3\u6784 701] \u5f3a\u8fde\u901a\u5206\u91cf \u2190 Kosaraju \u7b97\u6cd5"},"content":{"rendered":"

\u3010\u9898\u76ee\u6765\u6e90\u3011<\/span> https:\/\/www.luogu.com.cn\/problem\/B3609 \u3010\u9898\u76ee\u63cf\u8ff0\u3011 \u7ed9\u5b9a\u4e00\u5f20 n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe，\u6c42\u51fa\u5176\u6240\u6709\u7684\u5f3a\u8fde\u901a\u5206\u91cf\u3002 \u6ce8\u610f，\u672c\u9898\u53ef\u80fd\u5b58\u5728<\/span>\u91cd\u8fb9<\/span>\u548c<\/span>\u81ea\u73af<\/span>\u3002 \u3010\u8f93\u5165\u683c\u5f0f\u3011 \u7b2c\u4e00\u884c\u4e24\u4e2a\u6b63\u6574\u6570 n，m，\u8868\u793a\u56fe\u7684\u70b9\u6570\u548c\u8fb9\u6570\u3002 \u63a5\u4e0b\u6765 m \u884c，\u6bcf\u884c\u4e24\u4e2a\u6b63\u6574\u6570 u \u548c v \u8868\u793a\u4e00\u6761\u8fb9\u3002 \u3010\u8f93\u51fa\u683c\u5f0f\u3011 \u7b2c\u4e00\u884c\u4e00\u4e2a\u6574\u6570\u8868\u793a\u8fd9\u5f20\u56fe\u7684\u5f3a\u8fde\u901a\u5206\u91cf\u6570\u76ee\u3002 \u63a5\u4e0b\u6765\u6bcf\u884c\u8f93\u51fa\u4e00\u4e2a\u5f3a\u8fde\u901a\u5206\u91cf\u3002\u7b2c\u4e00\u884c\u8f93\u51fa 1 \u53f7\u70b9\u6240\u5728\u5f3a\u8fde\u901a\u5206\u91cf，\u7b2c\u4e8c\u884c\u8f93\u51fa 2 \u53f7\u70b9\u6240\u5728\u5f3a\u8fde\u901a\u5206\u91cf，\u82e5\u5df2\u88ab\u8f93\u51fa，\u5219\u6539\u4e3a\u8f93\u51fa 3 \u53f7\u70b9\u6240\u5728\u5f3a\u8fde\u901a\u5206\u91cf，\u4ee5\u6b64\u7c7b\u63a8\u3002\u6bcf\u4e2a\u5f3a\u8fde\u901a\u5206\u91cf\u6309\u8282\u70b9\u7f16\u53f7\u5927\u5c0f\u8f93\u51fa\u3002 \u3010\u8f93\u5165\u6837\u4f8b\u3011 6 8 1 2 1 5 2 6 5 6 6 1 5 3 6 4 3 4 \u3010\u8f93\u51fa\u6837\u4f8b\u3011 3 1 2 5 6 3 4<\/span> \u3010\u6570\u636e\u8303\u56f4\u3011 \u5bf9\u4e8e\u6240\u6709\u6570\u636e，1\u2264n\u226410000，1\u2264m\u2264100000\u3002 \u3010\u7b97\u6cd5\u5206\u6790\u3011 Kosaraju \u7b97\u6cd5\u662f\u4e00\u79cd\u7528\u4e8e\u67e5\u627e\u6709\u5411\u56fe\u4e2d\u5f3a\u8fde\u901a\u5206\u91cf（Strongly Connected Components, SCC）\u7684\u7ecf\u5178\u7b97\u6cd5\u3002\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u901a\u8fc7\u4e24\u6b21\u6df1\u5ea6\u4f18\u5148\u641c\u7d22（DFS）\u6765\u5b9e\u73b0，\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(V+E)，\u5176\u4e2d V \u662f\u9876\u70b9\u6570，E \u662f\u8fb9\u6570\u3002 Kosaraju \u7b97\u6cd5\u6a21\u677f\u9898\u4ee3\u7801\u8be6\u89c1：<\/span>https:\/\/blog.csdn.net\/hnjzsyjyj\/article\/details\/164698263 \u25cf \u7b97\u6cd5\u6b65\u9aa4 （\u4e00）\u7b2c\u4e00\u6b21 DFS（\u6b63\u5411\u56fe） \u5bf9\u539f\u59cb\u6709\u5411\u56fe\u8fdb\u884c DFS \u904d\u5386，\u8bb0\u5f55\u6bcf\u4e2a\u9876\u70b9\u7684\u5b8c\u6210\u65f6\u95f4（\u5373\u9000\u51fa\u9012\u5f52\u7684\u987a\u5e8f）\u3002 \u5c06\u9876\u70b9\u6309\u5b8c\u6210\u65f6\u95f4\u7684\u9006\u5e8f\u538b\u5165\u4e00\u4e2a\u6808\u4e2d（\u5b8c\u6210\u65f6\u95f4\u8d8a\u665a\u7684\u9876\u70b9\u8d8a\u65e9\u5165\u6808）\u3002 （\u4e8c）\u8f6c\u7f6e\u56fe \u6784\u5efa\u539f\u56fe\u7684\u8f6c\u7f6e\u56fe（\u5c06\u6240\u6709\u8fb9\u7684\u65b9\u5411\u53cd\u8f6c）\u3002 （\u4e09）\u7b2c\u4e8c\u6b21 DFS（\u53cd\u5411\u56fe） \u4ece\u6808\u9876\u4f9d\u6b21\u5f39\u51fa\u9876\u70b9，\u5bf9\u8f6c\u7f6e\u56fe\u8fdb\u884c DFS\u3002 \u6bcf\u6b21 DFS \u8bbf\u95ee\u5230\u7684\u6240\u6709\u9876\u70b9\u6784\u6210\u4e00\u4e2a\u5f3a\u8fde\u901a\u5206\u91cf\u3002 \u25cf \u4e3a\u4ec0\u4e48\u6709\u6548？ \u7b2c\u4e00\u6b21 DFS \u786e\u5b9a\u4e86\u9876\u70b9\u7684\u62d3\u6251\u987a\u5e8f（\u57fa\u4e8e\u5b8c\u6210\u65f6\u95f4）\u3002 \u8f6c\u7f6e\u56fe\u5c06\u539f\u56fe\u4e2d\u7684\u5f3a\u8fde\u901a\u5206\u91cf\u5185\u90e8\u7684\u73af\u4fdd\u6301\u4e0d\u53d8，\u4f46\u6539\u53d8\u4e86\u4e0d\u540c\u5206\u91cf\u4e4b\u95f4\u7684\u8fde\u63a5\u65b9\u5411\u3002 \u7b2c\u4e8c\u6b21 DFS \u5728\u8f6c\u7f6e\u56fe\u4e0a\u6309\u9006\u5e8f\u8bbf\u95ee，\u53ef\u4ee5\u786e\u4fdd\u6bcf\u6b21\u53ea\u63a2\u7d22\u540c\u4e00\u4e2a\u5f3a\u8fde\u901a\u5206\u91cf\u5185\u7684\u9876\u70b9，\u4e0d\u4f1a\u8de8\u5230\u5176\u4ed6\u5206\u91cf\u3002 \u3010\u7b97\u6cd5\u4ee3\u7801\u3011<\/span><\/p>\n

#include<bits\/stdc++.h>
\nusing namespace std;<\/p>\n

const int N=1e5+5;
\nvector<int> G[N],G2[N];
\nvector<int> group[N];
\nint st[N];
\nint post[N],post_cnt;
\nint scc[N],scc_cnt;<\/p>\n

void dfs1(int u) {
\n st[u]=1;
\n for(int v:G[u]) {
\n if(!st[v]) dfs1(v);
\n }
\n post[++post_cnt]=u;
\n}<\/p>\n

void dfs2(int u) {
\n st[u]=1;
\n scc[u]=scc_cnt;
\n group[scc_cnt].push_back(u);
\n for(int v:G2[u]) {
\n if(!st[v]) dfs2(v);
\n }
\n}<\/p>\n

void kosaraju(int n) {
\n memset(st,0,sizeof st);
\n post_cnt=0;
\n for(int i=1; i<=n; i++) {
\n if(!st[i]) dfs1(i);
\n }<\/p>\n

memset(st,0,sizeof st);
\n scc_cnt=0;
\n for(int i=post_cnt; i>=1; i–) {
\n int u=post[i];
\n if(!st[u]) {
\n ++scc_cnt;
\n dfs2(u);
\n }
\n }
\n}<\/p>\n

int main() {
\n ios::sync_with_stdio(0);
\n cin.tie(0);
\n int n,m;
\n cin>>n>>m;
\n for(int i=1; i<=m; i++) {
\n int x,y;
\n cin>>x>>y;
\n G[x].push_back(y);
\n G2[y].push_back(x);
\n }<\/p>\n

kosaraju(n);<\/p>\n

vector<vector<int>> all_scc;
\n for(int id=1; id<=scc_cnt; id++) {
\n sort(group[id].begin(), group[id].end());
\n all_scc.push_back(group[id]);
\n }<\/p>\n

int total=all_scc.size();
\n for(int i=0; i<total; i++) {
\n for(int j=i+1; j<total; j++) {
\n int a=all_scc[i][0];
\n int b=all_scc[j][0];
\n if(a>b) swap(all_scc[i],all_scc[j]);
\n }
\n }<\/p>\n

cout<<all_scc.size()<<"\\\\n";
\n for(int i=0; i<all_scc.size(); i++) {
\n for(int j=0; j<all_scc[i].size(); j++) {
\n cout<<all_scc[i][j]<<" ";
\n }
\n cout<<"\\\\n";
\n }<\/p>\n

return 0;
\n}<\/p>\n

\/*
\nin:
\n6 8
\n1 2
\n1 5
\n2 6
\n5 6
\n6 1
\n5 3
\n6 4
\n3 4<\/p>\n

out:
\n3
\n1 2 5 6
\n3
\n4
\n*\/ <\/p>\n

\u3010\u53c2\u8003\u6587\u732e\u3011<\/span> https:\/\/blog.csdn.net\/hnjzsyjyj\/article\/details\/164698263 https:\/\/blog.csdn.net\/hnjzsyjyj\/article\/details\/164631569 \u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"

\u3010\u9898\u76ee\u6765\u6e90\u3011 https:\/\/www.luogu.com.cn\/problem\/B3609 \u3010\u9898\u76ee\u63cf\u8ff0\u3011 \u7ed9\u5b9a\u4e00\u5f20 n \u4e2a\u70b9 m \u6761\u8fb9\u7684\u6709\u5411\u56fe，\u6c42\u51fa\u5176\u6240\u6709\u7684\u5f3a\u8fde\u901a\u5206\u91cf\u3002 \u6ce8\u610f，\u672c\u9898\u53ef\u80fd\u5b58\u5728\u91cd\u8fb9\u548c\u81ea\u73af\u3002 \u3010\u8f93\u5165\u683c\u5f0f\u3011 \u7b2c\u4e00\u884c\u4e24\u4e2a\u6b63\u6574\u6570 n，m，\u8868\u793a\u56fe\u7684\u70b9\u6570\u548c\u8fb9\u6570\u3002 \u63a5\u4e0b\u6765 m \u884c，\u6bcf\u884c\u4e24\u4e2a\u6b63\u6574\u6570 u \u548c v \u8868\u793a\u4e00\u6761\u8fb9\u3002 \u3010\u8f93\u51fa\u683c\u5f0f\u3011 \u7b2c\u4e00\u884c\u4e00\u4e2a\u6574\u6570\u8868\u793a\u8fd9\u5f20\u56fe\u7684\u5f3a\u8fde\u901a\u5206\u91cf\u6570\u76ee\u3002 \u63a5\u4e0b\u6765\u6bcf\u884c\u8f93\u51fa\u4e00\u4e2a\u5f3a\u8fde\u901a<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[11766],"topic":[],"class_list":["post-103244","post","type-post","status-publish","format-standard","hentry","category-server","tag-kosaraju"],"yoast_head":"\n\u6d1b\u8c37 B3609\uff1a[\u56fe\u8bba\u4e0e\u4ee3\u6570\u7ed3\u6784 701] \u5f3a\u8fde\u901a\u5206\u91cf \u2190 Kosaraju \u7b97\u6cd5 - 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