洛谷——P2865 [USACO06NOV]路障Roadblocks

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原文链接：http://www.cnblogs.com/z360/p/7469102.html

本文介绍了一道USACO编程题“路障”及其解题思路。问题要求找到两点间第二短路径，允许重复经过同一节点或边。文章提供了一个通过两次SPFA算法求解的C++实现。

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P2865 [USACO06NOV]路障Roadblocks

题目描述

Bessie has moved to a small farm and sometimes enjoys returning to visit one of her best friends. She does not want to get to her old home too quickly, because she likes the scenery along the way. She has decided to take the second-shortest rather than the shortest path. She knows there must be some second-shortest path.

The countryside consists of R (1 ≤ R ≤ 100,000) bidirectional roads, each linking two of the N (1 ≤ N ≤ 5000) intersections, conveniently numbered 1..N. Bessie starts at intersection 1, and her friend (the destination) is at intersection N.

The second-shortest path may share roads with any of the shortest paths, and it may backtrack i.e., use the same road or intersection more than once. The second-shortest path is the shortest path whose length is longer than the shortest path(s) (i.e., if two or more shortest paths exist, the second-shortest path is the one whose length is longer than those but no longer than any other path).

贝茜把家搬到了一个小农场，但她常常回到FJ的农场去拜访她的朋友。贝茜很喜欢路边的风景，不想那么快地结束她的旅途，于是她每次回农场，都会选择第二短的路径，而不象我们所习惯的那样，选择最短路。贝茜所在的乡村有R(1<=R<=100,000)条双向道路，每条路都联结了所有的N(1<=N<=5000)个农场中的某两个。贝茜居住在农场1，她的朋友们居住在农场N（即贝茜每次旅行的目的地）。贝茜选择的第二短的路径中，可以包含任何一条在最短路中出现的道路，并且，一条路可以重复走多次。当然咯，第二短路的长度必须严格大于最短路（可能有多条）的长度，但它的长度必须不大于所有除最短路外的路径的长度。

输入输出格式

输入格式：

Line 1: Two space-separated integers: N and R

Lines 2..R+1: Each line contains three space-separated integers: A, B, and D that describe a road that connects intersections A and B and has length D (1 ≤ D ≤ 5000)

输出格式：

Line 1: The length of the second shortest path between node 1 and node N

输入输出样例

输入样例#1：

输出样例#1：

说明

Two routes: 1 -> 2 -> 4 (length 100+200=300) and 1 -> 2 -> 3 -> 4 (length 100+250+100=450)

次短路裸题

次短路：

先跑两遍spfa，一遍正向，一遍逆向。然后枚举每一条必须加入的边，计算出加入这条边后的路径长度为从前面跑到该边的前一个节点的最短路+从该边的后一个节点到最后的最短路+改边的长度；判断这条边是不是比最短路大，且为比最短路大的中的最小的。

#include<queue>
#include<cstdio>
#include<cstring>
#include<cstdlib>
#include<iostream>
#include<algorithm>
#define N 510000
#define maxn 99999999
using namespace std;
bool vis[N];
int n,m,x,y,z,sum,ans,tot,d1[N],d2[N],head[N];
queue<int>q;
struct Edge
{
    int to,dis,from,next;
}edge[N<<1];
int add(int x,int y,int z)
{
    tot++;
    edge[tot].to=y;
    edge[tot].dis=z;
    edge[tot].next=head[x];
    head[x]=tot;
}
int read()
{
    int x=0,f=1; char ch=getchar();
    while(ch<'0'||ch>'9'){if(ch=='-')f=-1; ch=getchar();}
    while(ch>='0'&&ch<='9'){x=x*10+ch-'0'; ch=getchar();}
    return x*f;
}
int spfa(int s,int *dis)
{
    for(int i=1;i<=n;i++) dis[i]=maxn,vis[i]=false;
    vis[s]=true,dis[s]=0; q.push(s);
    while(!q.empty())
    {
        int x=q.front();q.pop();vis[x]=false;
        for(int i=head[x];i;i=edge[i].next)
        {
            int t=edge[i].to;
            if(dis[t]>dis[x]+edge[i].dis)
            {
                dis[t]=dis[x]+edge[i].dis;
                if(!vis[t]) vis[t]=true,q.push(t);
            }
        }
    }
}
int main()
{
    n=read(),m=read();
    for(int i=1;i<=m;i++)
    {
        x=read(),y=read(),z=read();
        add(x,y,z),add(y,x,z);
     } 
    spfa(1,d1),spfa(n,d2);
    ans=maxn;
    for(int i=1;i<=n;i++)
     for(int j=head[i];j;j=edge[j].next)
     {
         int t=edge[j].to;
         sum=d1[i]+d2[t]+edge[j].dis;
         if(sum>d1[n]&&sum<ans) ans=sum;
     }
    printf("%d",ans);
    return 0;
}