Poj--1273（最大流）

2014-12-20 00:56:43

思路：最大流启蒙题（启蒙好晚orz....）

　　研究了一下Fork-Fulkerson、EK和Dinic，前两者差别就在与用DFS还是BFS来找增广路，Dinic的话采用了分层图的概念，优化了增广路多次查找过程。

　　总结下：（1）基于流的边反对称性、非源/汇点出入守恒、增广路定理（同理于二分匹配）三大原理（2）通过构造“反向边”的概念使得找增广路算法可以“反悔”

　　　　　　（3）通过不断增广来增加流量，最终达到最大流。

分别用EK、FF、Dinic实现了下。

EK：

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>
#include <vector>
#include <map>
#include <set>
#include <stack>
#include <queue>
#include <iostream>
#include <algorithm>
using namespace std;
#define lp (p << 1)
#define rp (p << 1|1)
#define getmid(l,r) (l + (r - l) / 2)
#define MP(a,b) make_pair(a,b)
typedef long long ll;
typedef unsigned long long ull;
const int INF = 1 << 30;
const int maxn = 210;

int pre[maxn],c[maxn][maxn],n,m;
bool vis[maxn];
queue<int> Q;

bool Bfs(){
    while(!Q.empty()) Q.pop();
    memset(vis,false,sizeof(vis));
    memset(pre,-1,sizeof(pre));
    vis[1] = true;
    Q.push(1);
    while(!Q.empty()){
        int x = Q.front(); Q.pop();
        for(int i = 1; i <= m; ++i) if(!vis[i] && c[x][i] > 0){
            pre[i] = x;
            vis[i] = true;
            if(i == m) return true;
            Q.push(i);
        }
    }
    return false;
}

int EK(){
    int max_flow = 0,plus;
    while(Bfs()){
        plus = INF;
        for(int i = m; i != 1; i = pre[i]) plus = min(plus,c[pre[i]][i]);
        for(int i = m; i != 1; i = pre[i]){
            c[pre[i]][i] -= plus;
            c[i][pre[i]] += plus;
        }
        max_flow += plus;
    }
    return max_flow;
}

int main(){
    int v1,v2,v3;
    while(scanf("%d%d",&n,&m) != EOF){
        memset(c,0,sizeof(c));
        for(int i = 1; i <= n; ++i){
            scanf("%d%d%d",&v1,&v2,&v3);
            c[v1][v2] += v3;
        }
        printf("%d\n",EK());
    }
    return 0;
}

FF：

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>
#include <vector>
#include <map>
#include <set>
#include <stack>
#include <queue>
#include <iostream>
#include <algorithm>
using namespace std;
#define lp (p << 1)
#define rp (p << 1|1)
#define getmid(l,r) (l + (r - l) / 2)
#define MP(a,b) make_pair(a,b)
typedef long long ll;
typedef unsigned long long ull;
const int INF = 1 << 30;
const int maxn = 210;

int c[maxn][maxn],n,m;
bool vis[maxn];

int Dfs(int p,int minc){
    if(p == m) return minc;
    vis[p] = true;
    for(int i = 1; i <= m; ++i) if(!vis[i] && c[p][i] > 0){
        int d = Dfs(i,min(minc,c[p][i]));
        if(d > 0){ //一旦发现就该返回，因为就找一条增广路
            c[p][i] -= d;
            c[i][p] += d;
            return d;
        }
    }
    return 0;
}

int FF(){
    int plus,max_flow = 0;
    while(1){
        memset(vis,false,sizeof(vis));
        plus = Dfs(1,INF);
        if(plus == 0) break;
        max_flow += plus;
    }
    return max_flow;
}

int main(){
    int t1,t2,t3;
    while(scanf("%d%d",&n,&m) != EOF){
        memset(c,0,sizeof(c));
        for(int i = 1; i <= n; ++i){
            scanf("%d%d%d",&t1,&t2,&t3);
            c[t1][t2] += t3;
        }
        printf("%d\n",FF());
    }
    return 0;
}

Dinic：

#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <cmath>
#include <vector>
#include <map>
#include <set>
#include <stack>
#include <queue>
#include <iostream>
#include <algorithm>
using namespace std;
#define lp (p << 1)
#define rp (p << 1|1)
#define getmid(l,r) (l + (r - l) / 2)
#define MP(a,b) make_pair(a,b)
typedef long long ll;
typedef unsigned long long ull;
const int INF = 1 << 30;
const int maxn = 210;

int c[maxn][maxn],n,m,lev[maxn];
int Q[maxn],head,rear;

void Bfs(){
    memset(lev,-1,sizeof(lev));
    lev[1] = 0;
    Q[head = rear = 1] = 1;
    while(head <= rear){
        int x = Q[head++];
        for(int i = 1; i <= m; ++i) if(lev[i] < 0 && c[x][i] > 0){
            lev[i] = lev[x] + 1;
            Q[++rear] = i;
        }
    }
}

int Dfs(int p,int minc){
    if(p == m) return minc;
    for(int i = 1; i <= m; ++i) if(lev[i] > lev[p] && c[p][i] > 0){
        int d = Dfs(i,min(c[p][i],minc));
        if(d > 0){
            c[p][i] -= d;
            c[i][p] += d;
            return d;
        }
    }
    return 0;
}
        

int Dinic(){
    int max_flow = 0,plus;
    while(1){
        Bfs();
        if(lev[m] < 0) break;
        while((plus = Dfs(1,INF)) > 0) max_flow += plus;
    }
    return max_flow;
}

int main(){
    int t1,t2,t3;
    while(scanf("%d%d",&n,&m) != EOF){
        memset(c,0,sizeof(c));
        for(int i = 1; i <= n; ++i){
            scanf("%d%d%d",&t1,&t2,&t3);
                c[t1][t2] += t3;
        }
        printf("%d\n",Dinic());
    }
    return 0;
}