POJ3613 Cow Relays [矩阵乘法 floyd类似]

最新推荐文章于 2020-09-09 21:41:47 发布

转载最新推荐文章于 2020-09-09 21:41:47 发布 · 131 阅读

本文介绍了一种解决特定图论问题的方法：寻找通过指定数量的边连接两个点的最短路径。采用矩阵乘法及自定义运算来实现算法，并提供了一个具体的编程实现案例。

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Cow Relays

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 7335		Accepted: 2878

Description

For their physical fitness program, N (2 ≤ N ≤ 1,000,000) cows have decided to run a relay race using the T (2 ≤ T ≤ 100) cow trails throughout the pasture.

Each trail connects two different intersections (1 ≤ I_1i ≤ 1,000; 1 ≤ I_2i ≤ 1,000), each of which is the termination for at least two trails. The cows know the length_i of each trail (1 ≤ length_i ≤ 1,000), the two intersections the trail connects, and they know that no two intersections are directly connected by two different trails. The trails form a structure known mathematically as a graph.

To run the relay, the N cows position themselves at various intersections (some intersections might have more than one cow). They must position themselves properly so that they can hand off the baton cow-by-cow and end up at the proper finishing place.

Write a program to help position the cows. Find the shortest path that connects the starting intersection (S) and the ending intersection (E) and traverses exactly N cow trails.

Input

* Line 1: Four space-separated integers: N, T, S, and E
* Lines 2..T+1: Line i+1 describes trail i with three space-separated integers: length_i , I_1i , and I_2i

Output

* Line 1: A single integer that is the shortest distance from intersection S to intersection E that traverses exactly N cow trails.

Sample Input

Sample Output

Source

USACO 2007 November Gold

b在大很多也可以

求经过b条边的最短路

貌似直接想floyd不太明白了，一遍floyd怎么是经过一条边的最短路呢？

还是从矩阵乘法考虑，构造一个向量表示距离，不停乘邻接矩阵

定义一种新矩阵乘法，不是加起来而是求最小值

初始化时用0x3f

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
using namespace std;
const int N=105;
typedef long long ll;
inline int read(){
    char c=getchar();int x=0,f=1;
    while(c<'0'||c>'9'){if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9'){x=x*10+c-'0';c=getchar();}
    return x*f;
}
int b,n,m,s,t,u,v,w;
int mp[1005];
struct Mat{
    int a[N][N];
    Mat(){memset(a,0x3f,sizeof(a));}
}G,ans;
Mat operator *(Mat A,Mat B){
    Mat C;
    for(int k=1;k<=n;k++)
        for(int i=1;i<=n;i++) if(A.a[i][k])
            for(int j=1;j<=n;j++) if(B.a[k][j])
                C.a[i][j]=min(C.a[i][j],A.a[i][k]+B.a[k][j]);
    return C;
}
int main(){
    //freopen("in.txt","r",stdin);
    b=read();m=read();s=read();t=read();
    if(!mp[s]) mp[s]=++n;s=mp[s];
    if(!mp[t]) mp[t]=++n;t=mp[t];
    for(int i=1;i<=m;i++){
        w=read();u=read();v=read();
        if(!mp[u]) mp[u]=++n;u=mp[u];
        if(!mp[v]) mp[v]=++n;v=mp[v];
        G.a[u][v]=G.a[v][u]=w;
    }
    ans=G;b--;
    for(;b;b>>=1,G=G*G)
        if(b&1) ans=ans*G;
    printf("%d",ans.a[s][t]);
}