How Long Does It Take

How Long Does It Take

Given the relations of all the activities of a project, you are supposed to find the earliest completion time of the project.

Input Specification:

Each input file contains one test case. Each case starts with a line containing two positive integers N (≤ 100), the number of activity check points (hence it is assumed that the check points are numbered from 0 to N−1), and M, the number of activities. Then M lines follow, each gives the description of an activity. For the  i -th activity, three non-negative numbers are given:  S[i] ,  E[i] , and  L[i] , where  S[i]  is the index of the starting check point,  E[i]  of the ending check point, and  L[i]  the lasting time of the activity. The numbers in a line are separated by a space.

Output Specification:

For each test case, if the scheduling is possible, print in a line its earliest completion time; or simply output "Impossible".

Sample Input 1:

9 12
0 1 6
0 2 4
0 3 5
1 4 1
2 4 1
3 5 2
5 4 0
4 6 9
4 7 7
5 7 4
6 8 2
7 8 4

Sample Output 1:

18

Sample Input 2:

4 5
0 1 1
0 2 2
2 1 3
1 3 4
3 2 5

Sample Output 2:

Impossible

 

解题思路

　　这道题其实就是求完成所有活动的最短时间，也就是找一条从源点到终点的带权路径长度最长的路径，即关键路径，这条路径的权值就是所有活动完成的最早时间。

　　只需要在拓扑排序的基础上稍作修改。求最长距离和Dijkstra算法很像，用一个dist数组来记录到点i的最长距离。每次从队列弹出入度为0的点v，扫描一遍与该点相邻的点，如果发现dist[v]与点v到相邻点这条边的权值的和比原来的dist[i]大，就更新dist[i]为最大的那个值。

　　其中如果发现这个图有环，也就是弹出的顶点个数与图中顶点个数不相同，就说明没有关键路径。

　　AC代码如下：

#include <cstdio>
#include <iostream>
#include <vector>
#include <queue>
#include <algorithm>

struct Edge {
    int to, wt;
};

struct Graph {
    std::vector<std::vector<Edge> > G;
    int verN, edgeN;
};

Graph *createGraph(int n, int m);
void topSort(Graph *graph);

int main() {
    int n, m;
    scanf("%d %d", &n, &m);
    Graph *graph = createGraph(n, m);
    topSort(graph);
    
    return 0;
}

Graph *createGraph(int n, int m) {
    Graph *graph = new Graph;
    graph->verN = n;
    graph->edgeN = m;
    graph->G.resize(graph->verN);
    
    for (int i = 0; i < graph->edgeN; i++) {
        int v, w, wt;
        scanf("%d %d %d", &v, &w, &wt);
        graph->G[v].push_back({w, wt});
    }
    
    return graph;
}

void topSort(Graph *graph) {
    int indeg[graph->verN] = {0}, time[graph->verN], cnt = 0;
    std::fill(time, time + graph->verN, -1);
    for (int v = 0; v < graph->verN; v++) {
        for (std::vector<Edge>::iterator it = graph->G[v].begin(); it != graph->G[v].end(); it++) {
            indeg[it->to]++;
        }
    }
    
    std::queue<int> q;
    for (int v = 0; v < graph->verN; v++) {
        if (indeg[v] == 0) {
            q.push(v);
            time[v] = 0;
        }
    }
    while (!q.empty()) {
        int v = q.front();
        q.pop();
        cnt++;
        for (std::vector<Edge>::iterator it = graph->G[v].begin(); it != graph->G[v].end(); it++) {
            if (--indeg[it->to] == 0) q.push(it->to);
            time[it->to] = std::max(time[v] + it->wt , time[it->to]);
        }
    }
    
    if (cnt == graph->verN) {
        int maxId = 0;
        for (int v = 1; v < graph->verN; v++) {
            if (time[v] > time[maxId]) maxId = v;
        }
        printf("%d", time[maxId]);
    }
    else {
        printf("Impossible");
    }
}