Remmarguts' Date

Time Limit: 4000MS		Memory Limit: 65536K
Total Submissions:35025		Accepted: 9467

"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.

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).

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.

2 2

1 2 5

2 1 4

1 2 2

14

思路

或许将其分在最短路不是特别合适，但是暂时就这样吧
此题为第k短路，用到了a*算法
这是一个神奇的算法。用它来求第k小的话，就是用它不断搜索，即使遇到了终点也不停止。
首先我们知道，这个东西本来就是求最短路的一个算法，所以第一次遇到终点，一定是最短路。遇到终点后，如果不停止，那么接下来，在优先队列里面第二靠近终点的肯定会及时补充上。总之，在这个算法里面，
没有vis数组的限制，又将耗费最小的放在前面，所以第二个到达终点的路径就是第二短的了。
或许这样太过于抽象，但是你只需要记住一点，那就是在这个题目里面的A*算法，它的估计值是精确的，因为我们估计值的来源，是对终点反向图的Dijkstra求出来的。正是这个原因，这个算法跑到的每一次终点
，都是准确的第几小。

代码

#include<iostream>

#include<vector>

#include<queue>

#include<cstdio>

#include<algorithm>

using namespace std;

const int inf = 2100000000;

vector<int>u[200024],w[200024];

vector<int>ux[200024],wx[200024];

bool book[200024];

int dis[200024];

int n,m,s,t,k;

struct node

{

    int num;

    int dis;

    bool operator<(const node x)const

    {

        return dis>x.dis;

    }

};

struct aa

{

    int num;

    int g,f;

    bool operator<(const aa x)const

    {

        if(x.f==f){return x.g<g;}

        return x.f<f;

    }

};

void Dijkstra()

{

    node exa;

    fill(dis,dis+n+5,inf);

    priority_queue<node>q;

    q.push(node{t,0});

    dis[t]=0;

    while(!q.empty()){

        exa=q.top();q.pop();

        if(book[exa.num]){continue;}

        book[exa.num]=true;

        int siz=ux[exa.num].size();

        for(int i=0;i<siz;i++){

            if(dis[ux[exa.num][i]]>dis[exa.num]+wx[exa.num][i]){

                dis[ux[exa.num][i]]=dis[exa.num]+wx[exa.num][i];

                q.push(node{ux[exa.num][i],dis[ux[exa.num][i]]});

            }

        }

    }

}

int A_star()

{

    aa exa;

    if(dis[s]==inf){return -1;}

    priority_queue<aa>q;

    int cnt = 0;

    if(s==t){k++;}

    int r,g,f;

    g=0;f=g+dis[s];

    q.push(aa{s,g,f});

    while(!q.empty()){

        exa= q.top();

        q.pop();

        r=exa.num;

        if(r==t){cnt++;}

        if(cnt==k){return exa.g;}

        int siz=u[r].size();

        for(int i=0;i<siz;i++){

            g=exa.g+w[r][i];

            f=g+dis[u[r][i]];

            q.push(aa{u[r][i],g,f});

        }

    }

    return -1;

}

int main()

{

    scanf("%d%d",&n,&m);

    int x,y,z;

    for(int i=1;i<=m;i++){

        scanf("%d%d%d",&x,&y,&z);

        u[x].push_back(y);

        w[x].push_back(z);

        ux[y].push_back(x);

        wx[y].push_back(z);

    }

    scanf("%d%d%d",&s,&t,&k);

    Dijkstra();

    printf("%d\n",A_star());

    return 0;

}