0 or 1 HDU - 4370(最短路)
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2022-05-03 19:59:02
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Given a n*n matrix C ij (1<=i,j<=n),We want to find a n*n matrix X ij (1<=i,j<=n),which is 0 or 1.
Besides,X ij meets the following conditions:
1.X 12+X 13+...X 1n=1
2.X 1n+X 2n+...X n-1n=1
3.for each i (1<i<n), satisfies ∑X ki (1<=k<=n)=∑X ij (1<=j<=n).
For example, if n=4,we can get the following equality:
X 12+X 13+X 14=1
X 14+X 24+X 34=1
X 12+X 22+X 32+X 42=X 21+X 22+X 23+X 24
X 13+X 23+X 33+X 43=X 31+X 32+X 33+X 34
Now ,we want to know the minimum of ∑C ij*X ij(1<=i,j<=n) you can get.
Hint
题意:表示1号点出度为1,表示n号点入度为1,表示k( 1 < k < n )号点出度等于入度。∑Cij*Xij(1<=i,j<=n),很明显,这是某个路径的花费,而路径的含义可以有以下两种:
一:1号点到n号点的花费
二:1号点经过其它点成环,n号点经过其它点成环,这两个环的花费之和
于是,就变成了一道简单的最短路问题
关于环花费的算法,可以改进spfa算法,初始化dis[start] = INF,且一开始让源点之外的点入队
4
1 2 4 10
2 0 1 1
2 2 0 5
6 3 1 2
Sample Output
3
Input
The input consists of multiple test cases (less than 35 case).
For each test case ,the first line contains one integer n (1
#include<stdio.h>
#include<stdlib.h>
#include<string.h>
#include<stack>
#define INF 0x3f3f3f3f
using namespace std;
int cost[550][550];
int dis[500],vis[500];
int n;
int min(int a,int b)
{
return a < b ? a : b;
}
void spfa(int start)
{
stack<int>Q;
memset(vis,0,sizeof(vis));
for(int i=1;i<=n;i++)
{
dis[i] = cost[start][i];
if(i!=start)
{
Q.push(i);
vis[i] = 1;
}
}
dis[start] = INF;
while(!Q.empty())
{
int x = Q.top();
Q.pop();
vis[x] = 0;
for(int y = 1;y<=n;y++)
{
if(x==y) continue;
if(dis[x]+cost[x][y]<dis[y])
{
dis[y] = dis[x]+cost[x][y];
if(!vis[y])
{
vis[y] = 1;
Q.push(y);
}
}
}
}
}
int main()
{
while(~scanf("%d",&n))
{
for(int i=1;i<=n;i++)
{
for(int j=1;j<=n;j++)
{
scanf("%d",&cost[i][j]);
}
}
int ans,c1,cn;
spfa(1);
ans = dis[n];
c1 = dis[1];//从1到1
spfa(n);
cn = dis[n];//从n到n
ans = min(ans,c1+cn);
printf("%d\n",ans);
}
return 0;
}