![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260130213246-697d237eee621.png\")
<\/p>\n\u5355\u6e90\u6700\u77ed\u8def\u95ee\u9898\u662f\u6307\u56fa\u5b9a\u4e00\u4e2a\u8d77\u70b9，\u6c42\u5b83\u5230\u5176\u4ed6\u4efb\u610f\u4e00\u70b9\u7684\u6700\u77ed\u8def\u95ee\u9898\u3002<\/p>\n
\u9996\u5148\u8981\u6ce8\u610f\u7684\u662f，\u8fd9\u4e2a\u7b97\u6cd5\u4ec5\u4ec5\u7528\u4e8e\u56fe\u4e2d\u6ca1\u6709\u8d1f\u8fb9\u7684\u60c5\u51b5\u3002<\/p>\n
Dijkstra\u672c\u8d28\u903b\u8f91\u5176\u5b9e\u662f\u52a8\u6001\u89c4\u5212，\u6211\u4eec\u8981\u6c42\u4efb\u610f\u70b9i\u5230\u8d77\u70b9（\u5047\u8bbe\u4e3a1）\u7684\u8ddd\u79bb，\u6211\u4eec\u4e0d\u59a8\u7528\u4e00\u4e2a\u6570\u7ec4dist\u8bb0\u5f55\u8fd9\u4e00\u8ddd\u79bb，\u5373dist[i]\u4e3a1->i\u7684\u6700\u77ed\u8def\u8ddd\u79bb，\u6240\u4ee5\u73b0\u5728\u6211\u4eec\u9700\u8981\u8003\u8651\u7684\u95ee\u9898\u5c31\u662f\u5982\u4f55\u66f4\u65b0\u5e76\u6c42\u5f97dist，\u8fd9\u4e00\u90e8\u5206\u5c31\u662f\u7b97\u6cd5\u7684\u91cd\u70b9\u3002<\/p>\n
\u6211\u4eec\u5c06\u56fe\u7684\u70b9\u5206\u4e3a\u4e24\u4e2a\u96c6\u5408，\u96c6\u5408A\u4e3a\u6700\u77ed\u8def\u5f84\u5df2\u7ecf\u786e\u5b9a\u7684\u70b9，\u96c6\u5408B\u4e3a\u6700\u77ed\u8def\u5f84\u5c1a\u672a\u786e\u5b9a\u7684\u70b9\u3002<\/p>\n
\u6bcf\u4e00\u6b21\u6211\u4eec\u4eceB\u4e2d\u9009\u53d6dist\u6700\u5c0f\u7684\u70b9（\u8bbe\u4e3ax），\u5c06\u4ed6\u8f6c\u79fb\u5230A\u4e2d，\u540c\u65f6\u6211\u4eec\u5229\u7528x\u7684dist\u6765\u66f4\u65b0x\u7684\u90bb\u5c45（\u8bbe\u4e3ap）\u7684dist，\u5373<\/p>\n
dist[p]=min(dist[p],dist[x]+e(x,p))\/\/e(x,p)\u4e3a\u8fb9xp\u7684\u6743\u503c<\/p>\n
\u901a\u8fc7\u53cd\u590d\u8fdb\u884cB->A\u64cd\u4f5c，\u6700\u7ec8\u5f97\u5230\u5168\u90e8\u7684dist，\u6ca1\u770b\u61c2\u6ca1\u5173\u7cfb，\u6211\u4eec\u7528\u56fe\u6765\u6f14\u793a\u4e00\u904d\u3002<\/p>\n
<\/p>\n
\u8fd9\u662f\u56fe\u7684\u521d\u59cb\u72b6\u6001，dist\u7b80\u5199\u4e3ad，\u6839\u636e\u5b9a\u4e49\u6211\u4eec\u53ef\u5f97\u8d77\u70b9\u7684d=0，\u800c\u5176\u4ed6\u7684dist\u6211\u4eec\u5168\u90e8\u8bbe\u4e3a\u4e00\u4e2a\u6781\u5927\u7684\u6570，\u65b9\u4fbf\u540e\u7eed\u66f4\u65b0，\u6b64\u65f6\u6211\u4eec\u9009\u53d61\u5e76\u5165A，\u540c\u65f6\u66f4\u65b0\u90bb\u5c45\u7684d\u503c<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260130213247-697d237f2f215.png\")
<\/p>\n
\u7ea2\u8272\u4e3a\u8fd9\u4e00\u6b21\u53d1\u751f\u4e86\u66f4\u65b0\u7684d，\u9ec4\u8272\u4e3a\u5df2\u7ecf\u56fa\u5b9a\u7684d，\u4e5f\u5c31\u662f\u96c6\u5408A\u4e2d\u7684\u5143\u7d20\u3002\u63a5\u4e0b\u6765\u6211\u4eec\u5728B\u4e2d\u9009\u62e9min\u8fdb\u884c\u66f4\u65b0，\u663e\u7136\u8fd9\u6b21\u6211\u4eec\u5e94\u8be5\u9009\u62e92，\u6211\u4eec\u4f1a\u8fdb\u884c\u5982\u4e0b\u66f4\u65b0<\/p>\n
d[3]:5->5(d[2]+4>5)<\/p>\n
d[4]:\u221e->d[2]+4=6<\/p>\n
d[5]:\u221e->d[2]+10=12<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260130213247-697d237f716ea.png\")
<\/p>\n
\u9009\u62e9\u6700\u5c0f\u7684d[3]<\/p>\n
d[4]:6->6(d[3]+2>6)<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260130213247-697d237fb556a.png\")
<\/p>\n
\u4e0d\u65ad\u91cd\u590d\u8fd9\u4e00\u64cd\u4f5c，\u6700\u7ec8\u4f7f\u5f97\u5168\u90e8d\u53d8\u6210\u9ec4\u8272，\u5373\u5168\u90e8\u5e76\u5165\u96c6\u5408A<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/01\/20260130213247-697d237fd0612.png\")
<\/p>\n
const int INF=0x3f3f3f3f;
\nconst int N=1e3+7;\/\/\u4e0a\u9650
\nstruct edge{
\n int to,w;
\n edge(int x,int y){to=x;w=y;}
\n};
\nvector<vector<edge>> g(N,vector<edge>());
\n\/\/\u90bb\u63a5\u8868\u5b58\u50a8\u56fe，to\u4e3a\u8fb9\u7684\u7ec8\u70b9\u7aef，w\u4e3a\u6743\u503c
\nbool vis[N];
\nint dist[N];
\nvoid dijkstra(int n,vector<vector<edge>>& g){
\n \/\/n\u4e3a\u56fe\u7684\u70b9\u6570,N>n
\n for(int i=1;i<=n;i++){
\n dist[i]=INF;
\n vis[i]=false;
\n }
\n dist[1]=0;
\n \/\/\u521d\u59cb\u5316
\n while(true){
\n int v=-1;
\n for(int i=1;i<=n;i++){
\n if(vis[i]) continue;
\n if(v==-1||dist[i]<dist[v]) v=i;
\n }
\n if(v==-1) break;
\n \/\/\u5982\u679c\u4e3a-1\u4ee3\u8868\u5168\u90e8\u90fd\u88ab\u6807\u8bb0，\u5373B\u96c6\u5408\u4e3a\u7a7a
\n vis[v]=true;
\n for(auto p:g[v])\/\/\u66f4\u65b0\u64cd\u4f5c
\n dist[p.to]=min(dist[p.to],dist[v]+p.w);
\n }
\n} <\/p>\n
\u6ce8\u610f\u6b64\u5904\u8d77\u70b9\u8bbe\u4e3a1，\u5b9e\u9645\u64cd\u4f5c\u65f6\u6211\u4eec\u53ef\u4ee5\u6309\u7167\u8981\u6c42\u4fee\u6539\u8d77\u70b9\u3002<\/p>\n
\u6700\u7ec8\u5b9e\u73b0\u590d\u6742\u5ea6\u4e3aO(V\u00b2)<\/p>\n
\u53ef\u4ee5\u53d1\u73b0，\u590d\u6742\u5ea6\u6765\u6e90\u4e3b\u8981\u662f\u6700\u5c0fdist\u7684\u9009\u53d6\u4ee5\u53ca\u90bb\u5c45\u7684\u66f4\u65b0，\u90bb\u5c45\u66f4\u65b0\u6211\u4eec\u65e0\u6cd5\u7ee7\u7eed\u4f18\u5316，\u4f46\u662f\u6700\u5c0fdist\u7684\u9009\u53d6\u6211\u4eec\u53ef\u4ee5\u7528\u6700\u5c0f\u5806\u8fdb\u884c\u4f18\u5316\u3002<\/p>\n
\u6700\u5c0f\u5806\u6211\u4eec\u9009\u7528STL\u4e2d\u4f18\u5148\u961f\u5217\u3002<\/p>\n
typedef pair<int,int> P;
\nconst int INF=0x3f3f3f3f;
\nconst int N=1e3+7;\/\/\u4e0a\u9650
\nstruct edge{
\n int to,w;
\n edge(int x,int y){to=x;w=y;}
\n};
\nvector<vector<edge>> g(N,vector<edge>());
\n\/\/\u90bb\u63a5\u8868\u5b58\u50a8\u56fe，to\u4e3a\u8fb9\u7684\u7ec8\u70b9\u7aef，w\u4e3a\u6743\u503c
\nbool vis[N];
\nint dist[N];
\nvoid dijkstra(int n,vector<vector<edge>>& g){
\n \/\/n\u4e3a\u56fe\u7684\u70b9\u6570,N>n
\n priority_queue< P,vector<P>,greater<P> > q;
\n \/\/first\u662f\u6700\u77ed\u8ddd\u79bb，sceond\u662f\u70b9\u7684\u7f16\u53f7
\n for(int i=1;i<=n;i++)
\n dist[i]=INF;
\n dist[1]=0;
\n q.push(P(0,1));
\n \/\/\u521d\u59cb\u5316
\n while(!q.empty()){
\n P x=q.top();q.pop();
\n int v=x.second;
\n if(dist[v]<x.first) continue;
\n\/\/\u5982\u679cx\u7684\u8ddd\u79bb\u5927\u4e8edist[v]\u90a3\u5c31\u4e0d\u80fd\u7528\u5b83\u66f4\u65b0\u90bb\u5c45
\n for(auto p:g[v]){
\n if(dist[p.to]>dist[v]+p.w){
\n dist[p.to]=dist[v]+p.w;
\n q.push(P(dist[p.to],p.to));
\n }
\n }
\n }
\n} <\/p>\n
\u5176\u5b9e\u53ef\u4ee5\u53d1\u73b0，\u6211\u4eec\u5728\u8fd9\u91cc\u5220\u9664\u4e86vis\u6570\u7ec4，\u5b9e\u9645\u4e0a，\u5165\u961f\u64cd\u4f5c\u5b9e\u73b0\u4e86\u8fd9\u4e00\u4f5c\u7528，\u5728\u66f4\u65b0\u8fc7\u7a0b\u4e2d\u6211\u4eec\u5c06\u53d1\u751f\u66f4\u65b0\u7684\u5143\u7d20\u5165\u961f，\u4e5f\u5c31\u662f\u4e0a\u6587\u56fe\u4e2d\u7ea2\u8272\u5143\u7d20，\u800c\u5df2\u7ecf\u56fa\u5b9a\u7684\u9ec4\u8272\u5143\u7d20\u4e0d\u4f1a\u53d1\u751f\u66f4\u65b0\u5219\u88ab\u5254\u9664，\u501f\u6b64\u5b9e\u73b0\u4e86vis\u6570\u7ec4\u7684\u529f\u80fd\u3002<\/p>\n
\u6700\u7ec8\u5b9e\u73b0O(ElogV)\u3002<\/p>\n
\u76f8\u6bd4\u4e8e\u8fea\u514b\u65af\u62c9\u7279\u7b97\u6cd5，\u8fd9\u4e00\u7b97\u6cd5\u53ef\u4ee5\u5904\u7406\u8d1f\u8fb9\u7684\u60c5\u51b5\u3002<\/p>\n
\u8fd9\u4e2a\u539f\u7406\u7279\u522b\u7b80\u5355，\u6211\u4eec\u7ef4\u62a4\u4e00\u4e2a\u6570\u7ec4d，d[i]\u4ee3\u8868\u4e86\u4ece\u8d77\u70b9s\u5230\u70b9i\u7684\u6700\u77ed\u8def\u5f84，\u90a3\u4e48\u6211\u4eec\u5f97\u5230<\/p>\n
d[i]=min{d[j]+e(j,i)}\/\/e(j,i)\u4e3aj\u5230i\u7684\u6743\u503c<\/p>\n
\u5982\u679c\u662fDAG（\u6709\u5411\u65e0\u73af\u56fe）\u6211\u4eec\u5c31\u53ef\u4ee5\u6309\u7167\u62d3\u6251\u5e8f\u904d\u5386\u3002\u4f46\u5982\u679c\u6709\u73af，\u5982\u679c\u4e0d\u662f\u8d1f\u73af，\u8fd9\u4e5f\u5e76\u4e0d\u5f71\u54cd\u7b97\u6cd5\u8fd0\u4f5c，\u5982\u679c\u6709\u8d1f\u73af\u8fd9\u5c31\u662f\u6211\u4eec\u9700\u8981\u7740\u91cd\u8003\u8651\u7684\u90e8\u5206\u4e86\u3002<\/p>\n
\u8d1f\u73af：\u73af\u4e0a\u6240\u6709\u7684\u8fb9\u6743\u548c\u4e3a\u8d1f\u6570\u3002<\/p>\n
\u5982\u679c\u6709\u8d1f\u73af，\u6211\u4eec\u5c06\u4e00\u6761\u6700\u77ed\u8def\u5f84\u7b80\u5316\u6210\u4e09\u4e2a\u70b9<\/p>\n
\u8d77\u70b9——>\u8d1f\u73af——>\u7ec8\u70b9<\/p>\n
\u7531\u4e8e\u8d1f\u73af\u7684\u6743\u503c\u4e3a\u8d1f，\u6240\u4ee5\u8d77\u70b9\u5230\u7ec8\u70b9\u7684\u8ddd\u79bb\u53ef\u4ee5\u65e0\u9650\u5c0f，\u56e0\u4e3a\u6bcf\u8d70\u4e00\u6b21\u8d1f\u73af\u8ddd\u79bb\u90fd\u4f1a\u7f29\u77ed（\u6709\u70b9\u795e\u5947\u4e0d\u662f\u5417），\u5b9e\u73b0\u8d77\u6765\u5c31\u662f<\/p>\n
\u8d77\u70b9——>\u8d1f\u73af\u73af\u73af\u73af\u73af\u73af……..——>\u7ec8\u70b9<\/p>\n
\u6240\u4ee5\u6211\u4eec\u9700\u8981\u68c0\u6d4b\u662f\u5426\u5b58\u5728\u8d1f\u73af，\u6709\u8d1f\u73af\u5c31\u65e0\u89e3（\u6700\u77ed\u8def\u53ef\u4ee5\u65e0\u7a77\u5c0f）\u3002<\/p>\n
\u5bf9\u4e8e\u4efb\u610f\u6709V\u4e2a\u9876\u70b9\u7684\u56fe，\u8d77\u70b9\u5230\u7ec8\u70b9\u7684\u6700\u77ed\u8def\u6700\u591a\u7ecf\u8fc7（v-1)\u6761\u8fb9，\u56e0\u4e3a\u6700\u77ed\u8def\u4e0d\u4f1a\u540c\u65f6\u7ecf\u8fc7\u540c\u4e00\u9876\u70b9\u4e24\u6b21，\u9664\u4e86\u4e00\u79cd\u60c5\u51b5，\u5c31\u662f\u8d1f\u73af\u3002\u56e0\u4e3a\u5bf9\u4e8e\u8d1f\u73af\u6765\u8bf4，\u7ecf\u8fc7\u540c\u4e00\u9876\u70b9\u7684\u4ee3\u4ef7\u662f\u8d1f\u7684\u3002\u6211\u4eec\u5c31\u53ef\u4ee5\u501f\u52a9\u8fd9\u4e00\u7279\u6027\u6765\u68c0\u6d4b\u8d1f\u73af\u3002<\/p>\n
struct edge{
\n int from,to,w;
\n};
\nvector<edge> g;\/\/\u6211\u4eec\u9009\u62e9\u8bb0\u5f55\u8fb9
\nint V,E,s;\/\/\u9876\u70b9\u6570，\u8fb9\u6570，\u8d77\u70b9\u6570
\nint d[MAXV];
\nbool check(){
\n for(int i=1;i<=V;i++) d[i]=0;\/\/\u521d\u59cb\u5316\u4e3a0，\u4e3a\u4e86\u5e94\u5bf9\u8d1f\u8fb9
\n for(int i=1;i<=V;i++){
\n for(int j=1;j<=E;j++){
\n edge e=g[j];
\n if(d[e.to]>d[e.from]+e.w){
\n d[e.to]=d[e.from]+e.w;
\n if(i==V) return false;
\n\/\/\u5982\u679c\u7b2cV\u6b21\u90fd\u66f4\u65b0\u4e86，\u5c31\u4ee3\u8868\u6709\u8d1f\u73af，\u56e0\u4e3a\u6b63\u5e38\u56fe\u6700\u591a\u66f4\u65b0V-1\u6b21
\n }
\n }
\n }
\n return true;
\n}
\nvoid Bellman(){
\n for(int i=1;i<=V;i++) d[i]=INF;
\n d[s]=0;
\n while(true){
\n bool update=false;
\n for(int i=1;i<=E;i++){
\n edge e=g[i];
\n if(d[e.from]!=INF&&d[e.to]>d[e.from]+e.w){
\n d[e.to]=d[e.from]+e.w;
\n update=true;
\n }
\n }
\n if(!update) break;
\n }
\n} <\/p>\n
\u7531\u4e8ewhile\u6700\u591a\u5faa\u73af（V-1）\u6b21\u6240\u4ee5\u5b9e\u73b0\u4e3aO(V\u00b2)<\/p>\n
\u5bf9\u4e8e\u4efb\u610f\u4e24\u70b9\u7684\u95ee\u9898，\u6211\u4eec\u7528Floyd\u7b97\u6cd5\u6765\u89e3\u51b3，\u672c\u8d28\u662f\u52a8\u6001\u89c4\u5212\u3002<\/p>\n
\u6211\u4eec\u5b9a\u4e49\u72b6\u6001\u00a0 f[k][i][j]\u00a0\u00a0,\u8868\u793a\u4e3a\u5728\u53ea\u4f7f\u7528\u7f16\u53f71-k\u7684\u9876\u70b9\u5f53\u4f5c\u4e2d\u95f4\u8282\u70b9\u7684\u60c5\u51b5\u4e0b，i\u5230j\u7684\u6700\u77ed\u8def\u3002<\/p>\n
\u800cf[0][i][j]\u5219\u4ee3\u8868e(i,j)，\u6ca1\u6709\u5c31\u8bbe\u4e3a\u65e0\u7a77\u5927\u3002<\/p>\n
\u6240\u4ee5\u6211\u4eec\u5c31\u53ef\u4ee5\u628a\u53ea\u75281-k\u9876\u70b9\u7684\u95ee\u9898\u7ea6\u675f\u5230\u53ea\u75281-k-1\u9876\u70b9\u7684\u95ee\u9898\u4e2d<\/p>\n
\u5bf9\u4e8e\u9876\u70b9k，\u6211\u4eec\u6709\u4e24\u79cd\u60c5\u51b5，\u7ecf\u8fc7k，\u4e0d\u7ecf\u8fc7k，\u4e8e\u662f\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u72b6\u6001\u8f6c\u79fb\u65b9\u7a0b<\/p>\n
f[k][i][j] = min( f[k-1][i][j] , f[k-1][i][k] + f[k-1][k][j] )<\/p>\n
\u5b9e\u9645\u64cd\u4f5c\u4e2d，\u7531\u4e8e\u7b2c\u4e00\u7ef4\u4e0d\u4f1a\u5f71\u54cd\u7b97\u6cd5\u8fdb\u884c，\u6211\u4eec\u53ef\u4ee5\u53bb\u6389\u7b2c\u4e00\u4f4d\u51cf\u5c0f\u7a7a\u95f4\u5f00\u9500\u3002<\/p>\n
\u800cFloyd\u5bf9\u4e8e\u8d1f\u73af\u5904\u7406\u6bd4\u8f83\u7b80\u5355，\u53ea\u9700\u8981\u68c0\u6d4be(i,j)\u662f\u5426\u5c0f\u4e8e0\u5373\u53ef，\u8fd9\u91cc\u4e0d\u518d\u5c55\u793a\u3002<\/p>\n
struct edge{
\n int from,to,w;
\n};
\nint n,E;\/\/\u70b9\u6570，\u8fb9\u6570
\nint f[N][N];
\nvoid init(){
\n for(int i=1;i<=n;i++){
\n for(int j=1;j<=n;j++){
\n f[N][N]=INF;
\n }
\n }
\n for(int i=1;i<=E;i++){
\n int u,v;cin>>u>>v;
\n cin>>f[u][v];
\n }
\n}
\nvoid Floyd(){
\n for(int k=1;k<=n;k++){
\n for(int i=1;i<=n;i++){
\n for(int j=1;j<=n;j++){
\n f[i][j]=min(f[i][j],f[i][k]+f[k][j]);
\n }
\n }
\n }
\n} <\/p>\n
\u6700\u7ec8\u5b9e\u73b0O(n\u00b3)<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p>\n<\/p><\/p>\n","protected":false},"excerpt":{"rendered":"
\u5355\u6e90\u6700\u77ed\u8def\u95ee\u9898
\n\u5355\u6e90\u6700\u77ed\u8def\u95ee\u9898\u662f\u6307\u56fa\u5b9a\u4e00\u4e2a\u8d77\u70b9，\u6c42\u5b83\u5230\u5176\u4ed6\u4efb\u610f\u4e00\u70b9\u7684\u6700\u77ed\u8def\u95ee\u9898\u3002
\nDijkstra\u7b97\u6cd5（\u8fea\u514b\u65af\u62c9\u7279\u7b97\u6cd5）
\n\u9996\u5148\u8981\u6ce8\u610f\u7684\u662f，\u8fd9\u4e2a\u7b97\u6cd5\u4ec5\u4ec5\u7528\u4e8e\u56fe\u4e2d\u6ca1\u6709\u8d1f\u8fb9\u7684\u60c5\u51b5\u3002
\n\u539f\u7406
\nDijkstra\u672c\u8d28\u903b\u8f91\u5176\u5b9e\u662f\u52a8\u6001\u89c4\u5212，\u6211\u4eec\u8981\u6c42\u4efb\u610f\u70b9i\u5230\u8d77\u70b9（\u5047\u8bbe\u4e3a1）\u7684\u8ddd\u79bb，\u6211\u4eec\u4e0d\u59a8\u7528\u4e00\u4e2a\u6570\u7ec4dist\u8bb0\u5f55\u8fd9\u4e00\u8ddd\u79bb，\u5373dist[i]\u4e3a1->i\u7684\u6700\u77ed\u8def\u8ddd\u79bb&<\/p>\n","protected":false},"author":2,"featured_media":69105,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[1984,427],"topic":[],"class_list":["post-69110","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-1984","tag-427"],"yoast_head":"\n