UVA1658 - Admiral(最小費用最大流+拆點)

題目鏈接： UVA1658

【題意】給出v(<=1000)個點和e(<=10000)條邊的有向圖，求1~v的兩條不相交(除了起點終點外沒有公共點)的路徑，使得權和最小。

【分析】求的是兩條不相交的路徑，這樣就是除了起點和終點能訪問兩次外，其他點都只能訪問一次，如果直接用最小費用流求的話，因爲算法只能對邊限制而不能限制點，這裏每個點只能訪問一次是限制了點，用最小費用流算法的話會讓每個點訪問不止一次，所以需要創建邊，把每個點都拆成兩個，然後中間連一條容量爲1的邊(起點終點爲2)，這樣就限制了每個點的訪問次數，直接求最小費用流即可。（對於求最大流限制了點的話也可以這樣拆點）

【AC CODE】49ms

#include <cstdio>
#include <cstring>
#include <queue>
#include <vector>
using namespace std;
#define rep(i,a,n) for(int i = a; i < n; i++)
#define repe(i,a,n) for(int i = a; i <= n; i++)
#define clc(a,b) memset(a,b,sizeof(a))
#define MAXN 2010
#define INF 0x3f3f3f3f
typedef long long LL;
struct MCMF{
	int n,m;
	struct Edge{
		int from, to, cap, flow, cost;
		Edge(int a, int b, int c, int d, int e){
			from = a, to = b, cap = c, flow = d, cost = e;
		}
	};
	vector<Edge> edge;
	vector<int> g[MAXN];
	bool inq[MAXN];
	int d[MAXN]/*spfa*/, p[MAXN]/*上一條弧*/, a[MAXN]/*可改進量*/;
	void init(int n){
		this->n = n;
		repe(i,1,n) g[i].clear();
		edge.clear();
	}
	void add_edge(int from, int to, int cap, int cost)
	{
		edge.push_back(Edge(from,to,cap,0,cost));
		edge.push_back(Edge(to,from,0,0,-cost));
		m = edge.size();
		g[from].push_back(m-2);
		g[to].push_back(m-1);
	}
	bool spfa(int s, int t, int& flow, LL& cost)
	{
		clc(d,0x3f);
		clc(inq,0);
		d[s] = 0, inq[s] = true, p[s] = 0, a[s] = INF;
		queue<int> q;
		q.push(s);
		while(!q.empty())
		{
			int u = q.front();q.pop();
			inq[u] = false;
			int sz = g[u].size();
			rep(i,0,sz)
			{
				Edge& e = edge[g[u][i]];
				if(e.cap > e.flow && d[e.to] > d[u]+e.cost)
				{
					d[e.to] = d[u]+e.cost;
					p[e.to] = g[u][i];
					a[e.to] = min(a[u], e.cap-e.flow);
					if(!inq[e.to]) q.push(e.to), inq[e.to] = true;
				}
			}
		}
		if(INF == d[t]) return false;
		flow += a[t];
		cost += (LL)d[t]*(LL)a[t];
		for(int u = t; u != s; u = edge[p[u]].from)
		{
			edge[p[u]].flow += a[t];
			edge[p[u]^1].flow -= a[t];
		}
		return true;
	}
	//需要保證初始網絡沒有負圈，返回最大流量，cost纔是最小花費
	int mincostmaxflow(int s, int t, LL & cost)
	{
		int flow = 0;
		cost = 0;
		while(spfa(s,t,flow,cost));
		return flow;
	}
}mcmf;

int main()
{
#ifdef SHY
	freopen("e:\\1.txt","r",stdin);
#endif
	int n,m;
	while(~scanf("%d %d%*c", &n, &m))
	{
		int a,b,c;
		mcmf.init(n<<1);
		rep(i,2,n) mcmf.add_edge(i,i+n,1,0);
		mcmf.add_edge(1,n+1,2,0);
		mcmf.add_edge(n,n<<1,2,0);
		rep(i,0,m)
		{
			scanf("%d %d %d%*c", &a, &b, &c);
			mcmf.add_edge(a+n,b,1,c);
		}
		LL ans;
		mcmf.mincostmaxflow(1,n<<1,ans);
		printf("%lld\n", ans);
	}
	return 0;
}