ZOJ 2676 Network Wars(01分数规划-二分+最小割)

红太狼 2022-03-28 16:08 189阅读 0赞

题目链接：http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemId=1676

题意：给出一个无向图，每个边有权值w，起点s和终点t。找出一个割集S，使得sigama(w)/|S|最小？

思路：二分答案x，则每条边的权值为w'=w-x，求最大流。最大流大于等于0，则说明x偏小。否则偏大。另外，对于w'为负的边，直接加入答案。最后从源点DFS一次。

#include <iostream>
    #include <cstdio>
    #include <string.h>
    #include <algorithm>
    #include <cmath>
    #include <vector>
    #include <queue>
    #include <set>
    #include <stack>
    #include <string>
    #include <map>
    
    
    #define max(x,y) ((x)>(y)?(x):(y))
    #define min(x,y) ((x)<(y)?(x):(y))
    #define abs(x) ((x)>=0?(x):-(x))
    #define i64 long long
    #define u32 unsigned int
    #define u64 unsigned long long
    #define clr(x,y) memset(x,y,sizeof(x))
    #define CLR(x) x.clear()
    #define ph(x) push(x)
    #define pb(x) push_back(x)
    #define Len(x) x.length()
    #define SZ(x) x.size()
    #define PI acos(-1.0)
    #define sqr(x) ((x)*(x))
    #define MP(x,y) make_pair(x,y)
    #define EPS 1e-5
    
    #define FOR0(i,x) for(i=0;i<x;i++)
    #define FOR1(i,x) for(i=1;i<=x;i++)
    #define FOR(i,a,b) for(i=a;i<=b;i++)
    #define FORL0(i,a) for(i=a;i>=0;i--)
    #define FORL1(i,a) for(i=a;i>=1;i--)
    #define FORL(i,a,b)for(i=a;i>=b;i--)
    using namespace std;
    
    
    void RD(int &x){scanf("%d",&x);}
    void RD(u32 &x){scanf("%u",&x);}
    void RD(double &x){scanf("%lf",&x);}
    void RD(int &x,int &y){scanf("%d%d",&x,&y);}
    void RD(u32 &x,u32 &y){scanf("%u%u",&x,&y);}
    void RD(double &x,double &y){scanf("%lf%lf",&x,&y);}
    void RD(int &x,int &y,int &z){scanf("%d%d%d",&x,&y,&z);}
    void RD(int &x,int &y,int &z,int &t){scanf("%d%d%d%d",&x,&y,&z,&t);}
    void RD(u32 &x,u32 &y,u32 &z){scanf("%u%u%u",&x,&y,&z);}
    void RD(double &x,double &y,double &z){scanf("%lf%lf%lf",&x,&y,&z);}
    void RD(char &x){x=getchar();}
    void RD(char *s){scanf("%s",s);}
    void RD(string &s){cin>>s;}
    
    
    void PR(int x) {printf("%d\n",x);}
    void PR(int x,int y) {printf("%d %d\n",x,y);}
    void PR(i64 x) {printf("%lld\n",x);}
    void PR(u32 x) {printf("%u\n",x);}
    void PR(double x) {printf("%.5lf\n",x);}
    void PR(char x) {printf("%c\n",x);}
    void PR(char *x) {printf("%s\n",x);}
    void PR(string x) {cout<<x<<endl;}
    
    
    const int INF=1000000000;
    const int N=100005;
    
    struct Node
    {
    	int v,next;
    	double cap,flow;
    };
    
    
    Node edges[N];
    int n,m,s,t,head[N],e;
    int num[N],h[N],curedge[N],pre[N];
    
    
    void add(int u,int v,double cap)
    {
        edges[e].v=v;
        edges[e].cap=cap;
        edges[e].flow=0;
        edges[e].next=head[u];
        head[u]=e++;
    }
    
    void Add(int u,int v,double cap)
    {
        add(u,v,cap);
        add(v,u,0);
    }
    
    
    double Maxflow(int s,int t,int n)
    {
    	double ans=0.0,d;
    	int i,k,x,u;
    
    	memset(num,0,sizeof(num));
    	memset(h,0,sizeof(h));
    	for(i=0;i<=n;i++) curedge[i]=head[i];
    	num[n]=n;u=s;
    	while(h[u]<n)
    	{
    		if(u==t)
    		{
    			d=INF+1;
    			for(i=s;i!=t;i=edges[curedge[i]].v) if(d>edges[curedge[i]].cap)
    				k=i,d=edges[curedge[i]].cap;
    			for(i=s;i!=t;i=edges[curedge[i]].v)
    			{
    				x=curedge[i];
    				edges[x].cap-=d;
    				edges[x].flow+=d;
    				edges[x^1].cap+=d;
    				edges[x^1].flow-=d;
    			}
    			ans+=d;u=k;
    		}
    		for(i=curedge[u];i!=-1;i=edges[i].next) if(edges[i].cap>0&&h[u]==h[edges[i].v]+1)
    			break;
    		if(i!=-1)
    		{
    			curedge[u]=i;
    			pre[edges[i].v]=u;
    			u=edges[i].v;
    		}
    		else
    		{
    			if(--num[h[u]]==0) break;
    			curedge[u]=head[u];
    			for(x=n,i=head[u];i!=-1;i=edges[i].next) if(edges[i].cap>0&&h[edges[i].v]<x)
    				x=h[edges[i].v];
    			h[u]=x+1;num[h[u]]++;
    			if(u!=s) u=pre[u];
    		}
    	}
    	return ans;
    }
    
    
    int visit[10005],a[405],b[405];
    double c[405];
    
    int OK(double mid)
    {
        clr(head,-1); e=0;
    	double sum=0.0;
    	int i;
    	s=1,t=n;
    	FOR1(i,m)
    	{
    		if(c[i]<=mid) sum+=c[i]-mid;
    		else
    		{
    			Add(a[i],b[i],c[i]-mid);
    			Add(b[i],a[i],c[i]-mid);
    		}
    	}
    	sum+=Maxflow(s,t,t);
    	return sum>=0;
    }
    
    
    
    void DFS(int u)
    {
    	visit[u]=1;
    	for(int i=head[u];i!=-1;i=edges[i].next)
    	{
    		if(visit[edges[i].v]||edges[i].cap<=0) continue;
    		DFS(edges[i].v);
    	}
    }
    
    int main()
    {
    	while(scanf("%d%d",&n,&m)!=-1)
    	{
    		int i;
    		double low=INF,high=1.0,mid;
    		FOR1(i,m)
    		{
    		    RD(a[i],b[i]); RD(c[i]);
                low=min(low,c[i]);
    			high=max(high,c[i]);
    		}
    		while(high-low>EPS)
    		{
    			mid=(low+high)/2;
    			if(OK(mid)) low=mid;
    			else high=mid;
    		}
    		clr(visit,0); DFS(1);
    		int ans=0;
    		for(i=1;i<=m;i++) if(c[i]<=mid||visit[a[i]]+visit[b[i]]==1)
            {
                ans++;
            }
            PR(ans);
    		int flag=0;
    		for(i=1;i<=m;i++) if(c[i]<=mid||visit[a[i]]+visit[b[i]]==1)
    		{
    			if(flag) putchar(' ');
    			else flag=1;
    			printf("%d",i);
    		}
    		puts("");
    	}
    	return 0;
    }