图论算法----网络流
模板：
最大流：
普通增广：
1 #include <iostream> 2 #include <cstdio> 3 #include <cstring> 4 #include <algorithm> 5 #include <vector> 6 #include <queue> 7 #define maxn 1010 8 #define INF 0x3f3f3f3f 9 using namespace std; 10 struct edge{ 11 int to,cap,rev; 12 }; 13 vector<edge>G[maxn]; 14 bool used[maxn]; 15 void addedge(int from,int to,int cap){ 16 G[from].push_back((edge){to,cap,G[to].size()}); 17 G[to].push_back((edge){from,0,G[from].size()-1}); 18 } 19 int dfs(int v,int t,int f){ 20 if (v==t)return f; 21 used[v]=true; 22 for (int i=0;i<G[v].size();++i){ 23 edge &e = G[v][i]; 24 if(!used[e.to]&&e.cap>0){ 25 int d = dfs(e.to,t,min(f,e.cap)); 26 if (d>0){ 27 e.cap-=d; 28 G[e.to][e.rev].cap +=d; 29 return d; 30 } 31 } 32 } 33 return 0; 34 } 35 int max_flow(int s,int t){ 36 int flow = 0; 37 while(1){ 38 memset(used,0,sizeof(used)); 39 int f = dfs(s,t,INF); 40 if(f==0)return flow; 41 flow+=f; 42 } 43 } 44 int main (){ 45 46 47 }
Dinic：
1 #include<stdio.h> 2 #include<stdlib.h> 3 #include<string.h> 4 #include<math.h> 5 #include<iostream> 6 #include<string> 7 #include<algorithm> 8 #include<vector> 9 #include<queue> 10 #include<stack> 11 #include<map> 12 #define maxn 200001 13 #define INF 0x3f3f3f3f 14 using namespace std; 15 struct edge{ 16 int to,cap,rev; 17 }; 18 vector<edge>G[maxn]; 19 void addedge(int from,int to,int cap){ 20 G[from].push_back((edge){to,cap,G[to].size()}); 21 G[to].push_back((edge){from,0,G[from].size()-1}); 22 } 23 int level[maxn];//点的深度 24 int iter[maxn];//当前遍历到的边的下标 25 void bfs(int s){ 26 memset(level,-1,sizeof(level)); 27 queue<int>q; 28 level[s]=0; 29 q.push(s); 30 while(!q.empty()){ 31 int v = q.front(); 32 q.pop(); 33 for (int i=0;i<G[v].size();++i){ 34 edge &e = G[v][i]; 35 if (e.cap>0&&level[e.to]<0){ 36 level[e.to]=level[v]+1; 37 q.push(e.to); 38 } 39 } 40 } 41 } 42 int dfs(int v,int t,int f){ 43 if (v==t)return f; 44 for (int &i =iter[v];i<G[v].size();++i){ 45 edge &e = G[v][i]; 46 if (e.cap>0&&level[v]<level[e.to]){ 47 int d = dfs(e.to,t,min(f,e.cap)); 48 if(d>0){ 49 e.cap-=d; 50 G[e.to][e.rev].cap += d; 51 return d; 52 } 53 } 54 } 55 return 0; 56 } 57 int max_flow(int s,int t){ 58 int flow = 0; 59 while(1){ 60 bfs(s); 61 if(level[t]<0)return flow; 62 memset(iter,0,sizeof(iter)); 63 int f ; 64 while((f=dfs(s,t,INF))>0){ 65 flow+=f; 66 } 67 } 68 } 69 int N,M; 70 int A[maxn],B[maxn]; 71 int a[maxn],b[maxn],w[maxn]; 72 void solve(){ 73 int s = N,t= s+1; 74 for(int i=0;i<N;++i){ 75 addedge(i,t,A[i]); 76 addedge(s,i,B[i]); 77 } 78 for(int i=0;i<M;++i){ 79 addedge(a[i]-1,b[i]-1,w[i]); 80 addedge(b[i]-1,a[i]-1,w[i]); 81 } 82 int ans = max_flow(s,t); 83 printf("%d\n",ans); 84 } 85 int main (){ 86 while(scanf("%d%d",&N,&M)!=EOF){ 87 for(int i=0;i<N;++i)scanf("%d%d",&A[i],&B[i]); 88 for(int i=0;i<M;++i)scanf("%d%d%d",&a[i],&b[i],&w[i]); 89 solve(); 90 } 91 }
SAP：
1 #include<stdio.h> 2 #include<stdlib.h> 3 #include<string.h> 4 #include<math.h> 5 #include<iostream> 6 #include<string> 7 #include<algorithm> 8 #include<vector> 9 #include<queue> 10 #include<stack> 11 #include<map> 12 #define maxn 200001 13 #define maxm 1800000 14 #define INF 0x3f3f3f3f 15 using namespace std; 16 const int inf=(1<<29); 17 struct EDGE{ 18 int v,next; 19 int cap; 20 }ee[maxm]; 21 int head[maxn],gap[maxn]; //dep[maxn]; 22 int n,m,src,des,siz;//src=start,des=end; 23 void init(){ 24 siz=0; 25 memset(head,-1,sizeof head); 26 } 27 void addedge(int u,int v,int cap){ 28 ee[siz].v=v,ee[siz].cap=cap; 29 ee[siz].next=head[u]; 30 head[u]=siz++; 31 32 ee[siz].v=u,ee[siz].cap=0; 33 ee[siz].next=head[v]; 34 head[v]=siz++; 35 } 36 int dis[maxn],pre[maxn]; 37 int cur[maxn],aug[maxn]; 38 39 int SAP(int s, int e, int n) 40 { 41 int max_flow = 0, v, u = s; 42 int id, mindis; 43 aug[s] = inf; 44 pre[s] = -1; 45 memset(dis, 0, sizeof(dis)); 46 memset(gap, 0, sizeof(gap)); 47 gap[0] = n; 48 for (int i = 0; i <= n; ++i){//初始化当前弧为第一条弧 49 cur[i] = head[i]; 50 } 51 52 while (dis[s] < n) 53 { 54 bool flag = false; 55 if (u == e) 56 { 57 max_flow += aug[e]; 58 for (v = pre[e]; v != -1; v = pre[v])//路径回溯更新残留网络 59 { 60 id = cur[v]; 61 ee[id].cap -= aug[e]; 62 ee[id^1].cap += aug[e]; 63 aug[v] -= aug[e]; //修改可增广量，以后会用到 64 if (ee[id].cap == 0) u = v; //不回退到源点，仅回退到容量为0的弧的弧尾 65 } 66 } 67 for (id = cur[u]; id != -1; id = ee[id].next) 68 { // 从当前弧开始查找允许弧 69 v = ee[id].v; 70 if (ee[id].cap > 0 && dis[u] == dis[v] + 1) //找到允许弧 71 { 72 flag = true; 73 pre[v] = u; 74 cur[u] = id; 75 aug[v] = min(aug[u], ee[id].cap); 76 u = v; 77 break; 78 } 79 } 80 if (flag == false) 81 { 82 if (--gap[dis[u]] == 0) break; /*gap优化层次树出现断层则结束算法*/ 83 mindis = n; 84 cur[u] = head[u]; 85 for (id = head[u]; id != -1; id = ee[id].next) 86 { 87 v = ee[id].v; 88 if (ee[id].cap > 0 && dis[v] < mindis) 89 { 90 mindis = dis[v]; 91 cur[u] = id; //修改标号的同时修改当前弧 92 } 93 } 94 dis[u] = mindis + 1; 95 gap[dis[u]]++; 96 if (u != s) u = pre[u]; //回溯继续寻找允许弧 97 } 98 } 99 return max_flow; 100 } 101 int N,M; 102 int A[maxn],B[maxn]; 103 int a[maxn],b[maxn],w[maxn]; 104 void solve(){ 105 int s = N,t= s+1; 106 for(int i=0;i<N;++i){ 107 addedge(i,t,A[i]); 108 addedge(s,i,B[i]); 109 } 110 for(int i=0;i<M;++i){ 111 addedge(a[i]-1,b[i]-1,w[i]); 112 addedge(b[i]-1,a[i]-1,w[i]); 113 } 114 int ans = SAP(s,t,t+1); 115 printf("%d\n",ans); 116 } 117 int main (){ 118 while(scanf("%d%d",&N,&M)!=EOF){ 119 init(); 120 for(int i=0;i<N;++i)scanf("%d%d",&A[i],&B[i]); 121 for(int i=0;i<M;++i)scanf("%d%d%d",&a[i],&b[i],&w[i]); 122 solve(); 123 } 124 }
最小费用流：
bellman-ford：
1 #include <iostream> 2 #include <cstdio> 3 #include <cstring> 4 #include <algorithm> 5 #include <vector> 6 #include <queue> 7 #define maxn 1010 8 #define maxm 1010 9 #define INF 0x3f3f3f3f 10 using namespace std; 11 struct edge{int to,cap,cost,rev;}; 12 int V; //顶点数 13 vector<edge> G[maxn];//邻接表 14 int dist[maxn];//最短距离 15 int prevv[maxn],preve[maxn];//最短路中前驱节点以及对应的边 16 void addedge(int from ,int to, int cap, int cost){ 17 G[from].push_back((edge){to,cap,cost,G[to].size()}); 18 G[to].push_back((edge){from,0,-cost,G[from].size()-1}); 19 } 20 21 //求从s到t的流量为f的最小费用流, 22 //如果无法增广返回-1 23 //复杂度：O(F|V||E|) 24 int min_cost_flow(int s, int t,int f){ 25 int res = 0; 26 while(f>0){ 27 //利用bellman-ford求解最短路 28 fill (dist,dist+V,INF); 29 dist[s]=0; 30 bool update = true; 31 while(update){ 32 update = false; 33 for(int v = 0; v<V; v++){ 34 if(dist[v]==INF)continue; 35 for (int i=0;i<G[v].size();++i){ 36 edge &e =G[v][i]; 37 if(e.cap>0&&dist[e.to]>dist[v]+e.cost){ 38 dist[e.to]=dist[v]+e.cost; 39 prevv[e.to]=v; 40 preve[e.to]=i; 41 update = true; 42 } 43 } 44 } 45 } 46 if (dist[t]==INF){ 47 //无法再增广 48 return -1; 49 } 50 51 //沿着最短路尽量增广 52 int d = f; 53 for(int v = t; v!=s;v = prevv[v]){ 54 d = min(d,G[prevv[v]][preve[v]].cap); 55 } 56 f-=d; 57 res+=d*dist[t]; 58 for(int v = t;v!=s;v = prevv[v]){ 59 edge &e = G[prevv[v]][preve[v]]; 60 e.cap -=d; 61 G[v][e.rev].cap+=d; 62 } 63 } 64 return res; 65 } 66 int main (){ 67 int n,m; 68 while(scanf("%d%d",&n,&m)!=EOF){ 69 V=n+1; 70 for(int i=0;i<m;++i){ 71 int a,b,c; 72 scanf("%d%d%d",&a,&b,&c); 73 addedge(a,b,1,c); 74 addedge(b,a,1,c); 75 } 76 int ans = min_cost_flow(1,n,2); 77 printf("%d\n",ans); 78 } 79 }
Dijkstra：
1 #include <iostream> 2 #include <cstdio> 3 #include <cstring> 4 #include <algorithm> 5 #include <vector> 6 #include <queue> 7 #define maxn 1010 8 #define maxm 1010 9 #define INF 0x3f3f3f3f 10 using namespace std; 11 typedef pair<int,int>P;//first 为最短距离，second为顶点编号 12 struct edge{int to,cap,cost,rev; }; 13 int V; //顶点数 14 vector<edge> G[maxn];//邻接表 15 int dist[maxn];//最短距离 16 int h[maxn];//顶点的势 17 int prevv[maxn],preve[maxn];//最短路中前驱节点以及对应的边 18 void addedge(int from ,int to, int cap, int cost){ 19 G[from].push_back((edge){to,cap,cost,G[to].size()}); 20 G[to].push_back((edge){from,0,-cost,G[from].size()-1}); 21 } 22 23 //求从s到t的流量为f的最小费用流, 24 //如果无法增广返回-1 25 //复杂度：O(F|V||E|) 26 int min_cost_flow(int s, int t,int f){ 27 int res = 0; 28 fill(h,h+V,0); 29 while(f>0){ 30 //利用Dijkstra更新h 31 priority_queue<P, vector<P>,greater<P> >q; 32 fill (dist,dist+V,INF); 33 dist[s]=0; 34 q.push(P(0,s)); 35 while(!q.empty()){ 36 P p = q.top(); 37 q.pop(); 38 int v = p.second; 39 if (dist[v]<p.first)continue; 40 for (int i=0;i<G[v].size();++i){ 41 edge &e = G[v][i]; 42 if (e.cap>0&&dist[e.to]>dist[v]+e.cost+h[v]-h[e.to]){ 43 dist[e.to]=dist[v]+e.cost+h[v]-h[e.to]; 44 prevv[e.to]=v; 45 preve[e.to]=i; 46 q.push(P(dist[e.to],e.to)); 47 } 48 } 49 } 50 if (dist[t]==INF){ 51 //无法再增广 52 return -1; 53 } 54 for (int v=0;v<V;++v){ 55 h[v]+=dist[v]; 56 } 57 58 //沿着最短路尽量增广 59 int d = f; 60 for(int v = t; v!=s;v = prevv[v]){ 61 d = min(d,G[prevv[v]][preve[v]].cap); 62 } 63 f-=d; 64 res+=d*h[t]; 65 for(int v = t;v!=s;v = prevv[v]){ 66 edge &e = G[prevv[v]][preve[v]]; 67 e.cap -=d; 68 G[v][e.rev].cap+=d; 69 } 70 } 71 return res; 72 } 73 74 int main (){ 75 int n,m; 76 while(scanf("%d%d",&n,&m)!=EOF){ 77 V=n+1; 78 for(int i=0;i<m;++i){ 79 int a,b,c; 80 scanf("%d%d%d",&a,&b,&c); 81 addedge(a,b,1,c); 82 addedge(b,a,1,c); 83 } 84 int ans = min_cost_flow(1,n,2); 85 printf("%d\n",ans); 86 } 87 }
