#include <iostream>
#include <cstdlib>
#include <cstdio>
#include <cstring>
#include <queue>
#include <stack>
using namespace std;
const int MAXN = 1010; //顶点个数最大
const int MAXM = 10010; //边数最大
struct Edge
{
int v, w;
int id;
int next;
}edge[MAXM];
int n, m; //顶点个数，边条数
int cnt;
int first[MAXN], topo[MAXN]; //表头指针
int ind[MAXN], outd[MAXN]; //顶点入度，出度
int tot;
int Ee[MAXN], El[MAXN], E[MAXN], L[MAXN];
/*Ee表示事件最早可能发生时间，El表示事件最迟允许发生时间*/
/*E表示活动最早可能发生时间，L表示活动最迟允许发生时间*/
void init()
{
cnt = 0;
tot = 0;
memset(first, -1, sizeof(first));
memset(ind, 0, sizeof(ind));
memset(outd, 0, sizeof(outd));
memset(Ee, 0, sizeof(Ee));
memset(E, 0, sizeof(E));
memset(L, 0, sizeof(L));
}
void read_graph(int u, int v, int w, int id)
{
edge[cnt].v = v, edge[cnt].w = w, edge[cnt].id = id;
edge[cnt].next = first[u], first[u] = cnt++;
}
void toposort() //拓扑排序
{
queue<int> q;
for(int i = 0; i < n; i++) if(!ind[i]) q.push(i);
while(!q.empty())
{
int x = q.front();
q.pop();
topo[++tot] = x;
for(int e = first[x]; e != -1; e = edge[e].next)
{
int v = edge[e].v, w = edge[e].w;
if(--ind[v] == 0) q.push(v);
if(Ee[v] < Ee[x] + w) //求出各个顶点Ee值
{
Ee[v] = Ee[x] + w;
}
}
}
}
void CriticalPath()
{
toposort();
int top = tot;
for(int i = 0; i < n; i++) El[i] = Ee[n-1]; //初始化顶点事件的最迟发生时间
while(top) //逆拓扑排序求顶点El的值
{
int x = topo[top--];
for(int e = first[x]; e != -1; e = edge[e].next)
{
int v = edge[e].v, w = edge[e].w;
if(El[x] > El[v] - w)
{
El[x] = El[v] - w;
}
}
}
for(int u = 0; u < n; u++) //求出E,L关键活动
{
for(int e = first[u]; e != -1; e = edge[e].next)
{
int v = edge[e].v, id = edge[e].id, w = edge[e].w; //id代表活动的标号
E[id] = Ee[u], L[id] = El[v] - w;
if(E[id] == L[id]) //相等一定是关键活动
{
printf("a%d : %d->%d\n", id, u, v);
}
}
}
}
void read_case()
{
init();
for(int i = 1; i <= m; i++)
{
int u, v, w; //出，入，权值
cout<<"a"<<i<<":";
scanf("%d%d%d", &u, &v, &w); //输入出，入，权值
read_graph(u, v, w, i); //read_graph
outd[u]++, ind[v]++;
}
}
int main()
{
cout<<"请输入顶点个数和边的条数，以及每条有向边的顶点及权值（出，入，权值）："<<endl;
while(~scanf("%d%d", &n, &m))
{
read_case();
printf("\n关键路径:\n");
CriticalPath();
}
return 0;
}
