AOE网求关键路径（拓扑排序 + dp | 最长路）

$描述$
$求出所给的AOE-网的关键路径。$

输入
若干行整数，第一行有2个数，分别为顶点数v和弧数a，接下来有a行，每一行有3个数，分别是该条弧所关联的两个顶点编号和弧的权值
输出
若干个空格隔开的顶点构成的序列(用小写字母)
样例输入
9 11
1 2 6
1 3 4
1 4 5
2 5 1
3 5 1
4 6 2
5 7 9
5 8 7
6 8 4
7 9 2
8 9 3
样例输出
v1 v2 v5 v7 v9
思路：
拓扑排序 + dp
这是个有向图，我们可以在拓扑排序的过程中记录最大值，同时记录更新路径。
$AC \ Code:$

#include<iostream>
#include<cstdio>
#include<cstring>
#include<map>
#include<queue>
using namespace std;
const int N = 1e5 + 1000;
#define ll long long
struct Edge
{
    int next;
    int to;
    int dis;
}edge[N*2];
int head[N],tot;
int deg[N];
inline void add(int from,int to,int dis){
    edge[++tot].next = head[from];
    edge[tot].to = to;
    edge[tot].dis = dis;
    head[from] = tot;
    deg[to] ++;
}
int a[10000];//记录拓扑序
int cnt = 0;
int f[10000];//f[i] 代表到 i的最大值
int pre[N];
void topsort(int n){
    queue<int> Q;
    for(int i = 1;i <= n;++i) if(!deg[i]) Q.push(i);
    while(Q.size()){
        int x = Q.front();
        Q.pop();
        a[++cnt] = x;
        for(int i = head[x];~i;i = edge[i].next){
            int y = edge[i].to;
            int dis = edge[i].dis;
            if(f[y] < f[x] + dis) f[y] = f[x] + dis,pre[y] = x;
            if(--deg[y] == 0) Q.push(y);
        }
    }
}
inline void init(){
    memset(head,-1,sizeof head);
    tot = 0;
}
void print(int x){
    if(pre[x] == 0) {
        cout<<'v'<<x<<' ';
        return ;
    } ;
    print(pre[x]);
    cout<<'v'<<x<<' ';
}
int main(){
    int n,m;
    init();
    cin>>n>>m;
    for(int i = 1;i <= m;++i){
        int u,v,dis;
        cin>> u >> v >> dis;
        add(u,v,dis);
    }
    topsort(n);
    int M = 0;
    for(int i = 1;i <= n;++i){
        M = max(M,f[i]);
    }
    print(n);
}

$spfa求最长路$
$AC\ Code:$

#include<iostream>
#include<cstring>
#include<queue>
#include<map>
#include<cmath>
#include<set>
#include<stack>
#include<cstdio>
#include<sstream>
#include<vector>
#include<bitset>
#include<algorithm>

using namespace std;
#define read(x) scanf("%d",&x)
#define Read(x,y) scanf("%d%d",&x,&y)
#define gc(x)  scanf(" %c",&x);
#define mmt(x,y)  memset(x,y,sizeof x)
#define write(x) printf("%d\n",x)
#define pii pair<int,int>
#define INF 0x3f3f3f3f
#define ll long long
const ll mod =   1000000007;
const int N = 100000 + 100;
const int M = 3e6 + 1005;
int d[2000];
int vis[2000];
struct Edge
{
    int next;
    int to;
    int dis;
}edge[N];
int head[2000],tot;
inline void add(int from,int to,int dis)
{
    edge[++tot].next = head[from];
    edge[tot].to = to;
    edge[tot].dis = dis;
    head[from] = tot;
}
int pre[2000];
void spfa(int u)
{
    mmt(d,0xcf);
    mmt(vis,0);
    d[u] = 0;
    vis[u] = 1;
    queue<int> Q;
    Q.push(u);
    while(Q.size()){
        int x = Q.front();
        Q.pop();
        vis[x] = 0;
        for(int i = head[x];~i;i = edge[i].next){
            int y = edge[i].to;
            int dis = edge[i].dis;
            if(d[y] < d[x] + dis){
                d[y] = d[x] + dis;
                pre[y] = x;
                if(!vis[y]){
                    vis[y] = 1;
                    Q.push(y);
                }
            }
        }
    }
}
void print(int x)
{
    if(pre[x] == 0) {cout<<'v'<<x<<' ';return ;}
    print(pre[x]);

    cout<<'v'<<x<<' ';
}
int main()
{
    mmt(head,-1);
    int n,m;
    cin>>n>>m;
    int u,v,dis;
    for(int i = 1;i <= m;++i){
        cin>>u>>v>>dis;
        add(u,v,dis);
    }
    spfa(1);
    print(n);

}