POJ 2449Remmarguts' Date 第K短路

Remmarguts' Date

Time Limit: 4000MS		Memory Limit: 65536K
Total Submissions: 29625		Accepted: 8034

Description

"Good man never makes girls wait or breaks an appointment!" said the mandarin duck father. Softly touching his little ducks' head, he told them a story.

"Prince Remmarguts lives in his kingdom UDF – United Delta of Freedom. One day their neighboring country sent them Princess Uyuw on a diplomatic mission."

"Erenow, the princess sent Remmarguts a letter, informing him that she would come to the hall and hold commercial talks with UDF if and only if the prince go and meet her via the K-th shortest path. (in fact, Uyuw does not want to come at all)"

Being interested in the trade development and such a lovely girl, Prince Remmarguts really became enamored. He needs you - the prime minister's help!

DETAILS: UDF's capital consists of N stations. The hall is numbered S, while the station numbered T denotes prince' current place. M muddy directed sideways connect some of the stations. Remmarguts' path to welcome the princess might include the same station twice or more than twice, even it is the station with number S or T. Different paths with same length will be considered disparate.

Input

The first line contains two integer numbers N and M (1 <= N <= 1000, 0 <= M <= 100000). Stations are numbered from 1 to N. Each of the following M lines contains three integer numbers A, B and T (1 <= A, B <= N, 1 <= T <= 100). It shows that there is a directed sideway from A-th station to B-th station with time T.

The last line consists of three integer numbers S, T and K (1 <= S, T <= N, 1 <= K <= 1000).

Output

A single line consisting of a single integer number: the length (time required) to welcome Princess Uyuw using the K-th shortest path. If K-th shortest path does not exist, you should output "-1" (without quotes) instead.

Sample Input

2 2
1 2 5
2 1 4
1 2 2

Sample Output

14

Source

POJ Monthly,Zeyuan Zhu

 

题目链接：http://poj.org/problem?id=2449

题意：求s到t的第K短路

/*
*算法思想：
*单源点最短路径+高级搜索A*;
*A*算法结合了启发式方法和形式化方法;
*启发式方法通过充分利用图给出的信息来动态地做出决定而使搜索次数大大降低;
*形式化方法不利用图给出的信息,而仅通过数学的形式分析;
*
*算法通过一个估价函数f(h)来估计图中的当前点p到终点的距离,并由此决定它的搜索方向;
*当这条路径失败时,它会尝试其他路径;
*对于A*,估价函数=当前值+当前位置到终点的距离,即f(p)=g(p)+h(p),每次扩展估价函数值最小的一个;
*
*对于K短路算法来说,g(p)为当前从s到p所走的路径的长度;h(p)为点p到t的最短路的长度;
*f(p)的意义为从s按照当前路径走到p后再走到终点t一共至少要走多远;
*
*为了加速计算,h(p)需要在A*搜索之前进行预处理,只要将原图的所有边反向,再从终点t做一次单源点最短路径就能得到每个点的h(p)了;
*
*算法步骤：
*(1)将有向图转置即所有边反向,以原终点t为源点,求解t到所有点的最短距离;
*(2)新建一个优先队列,将源点s加入到队列中;
*(3)从优先级队列中弹出f(p)最小的点p,如果点p就是t,则计算t出队的次数;
*如果当前为t的第k次出队,则当前路径的长度就是s到t的第k短路的长度,算法结束;
*否则遍历与p相连的所有的边,将扩展出的到p的邻接点信息加入到优先级队列;
代码：

#include<iostream>
#include<cstdio>
#include<cmath>
#include<cstring>
#include<algorithm>
#include<map>
#include<queue>
#include<stack>
#include<vector>
#include<set>
using namespace std;
typedef long long ll;
typedef pair<ll,int> P;
const int maxn=2e5+,maxm=2e5+,inf=0x3f3f3f3f,mod=1e9+;
const ll INF=1e17+;
struct edge
{
    int from,to;
    ll w;
};
vector<edge>G[maxn],T[maxn];
priority_queue<P,vector<P>,greater<P> >q;
ll dist[maxn];
void addedge(int u,int v,ll w)
{
    G[u].push_back((edge)
    {
        u,v,w
    });
    T[v].push_back((edge)
    {
        v,u,w
    });
}
void dij(int s)
{
    dist[s]=0LL;
    q.push(P(dist[s],s));
    while(!q.empty())
    {
        P p=q.top();
        q.pop();
        int u=p.second;
        for(int i=; i<T[u].size(); i++)
        {
            edge e=T[u][i];
            if(dist[e.to]>dist[u]+e.w)
            {
                dist[e.to]=dist[u]+e.w;
                q.push(P(dist[e.to],e.to));
            }
        }
    }
}
struct node
{
    int to;
    ///g(p)为当前从s到p所走的路径的长度;dist[p]为点p到t的最短路的长度;
    ll g,f;///f=g+dist,f(p)的意义为从s按照当前路径走到p后再走到终点t一共至少要走多远;
    bool operator<(const node &x ) const
    {
        if(x.f==f) return x.g<g;
        return x.f<f;
    }
};
ll A_star(int s,int t,int k)
{
    if(dist[s]==INF) return -;
    priority_queue<node>Q;
    int cnt=;
    if(s==t) k++;
    ll g=0LL;
    ll f=g+dist[s];
    Q.push((node)
    {
        s, g, f
    });
    while(!Q.empty())
    {
        node x=Q.top();
        Q.pop();
        int u=x.to;
        if(u==t) cnt++;
        if(cnt==k) return x.g;
        for(int i=; i<G[u].size(); i++)
        {
            edge e=G[u][i];
            ll g=x.g+e.w;
            ll f=g+dist[e.to];
            Q.push((node)
            {
                e.to, g, f
            });
        }
    }
    return -;
}
void init(int n)
{
    for(int i=; i<=n+; i++) G[i].clear(),T[i].clear();
}
int main()
{
    int n,m;
    scanf("%d%d",&n,&m);
    for(int i=; i<=m; i++)
    {
        int u,v;
        ll w;
        scanf("%d%d%lld",&u,&v,&w);
        addedge(u,v,w);
    }
    int s,t,k;
    scanf("%d%d%d",&s,&t,&k);
    for(int i=; i<=n; i++) dist[i]=INF;
    dij(t);
    printf("%lld\n",A_star(s,t,k));
    init(n);
    return ;
}

第k短路