动态规划算法：多阶段决策问题——最短路径，c++代码

动态规划是一种解决多阶段决策问题的算法，其基本思想是将问题分解成一系列子问题，通过求解子问题的最优解来得到原问题的最优解。最短路径问题就是一种典型的多阶段决策问题，它的目标是在给定的有向加权图中找到一条从起点到终点的路径，使得路径上所有边的权值之和最小。

下面是该问题的一种动态规划算法：

#include <iostream>
#include <vector>
using namespace std;

const int MAXN = 1005;
const int INF = 0x3f3f3f3f;
int n, m, s, t;
int dis[MAXN], vis[MAXN], pre[MAXN];
vector<pair<int, int>> g[MAXN];

void dijkstra(int s) {
    for (int i = 1; i <= n; ++i) {
        dis[i] = INF;
        vis[i] = 0;
    }
    dis[s] = 0;
    for (int i = 1; i < n; ++i) {
        int u = -1, minn = INF;
        for (int j = 1; j <= n; ++j) {
            if (!vis[j] && dis[j] < minn) {
                u = j;
                minn = dis[j];
            }
        }
        if (u == -1) break;
        vis[u] = 1;
        for (int j = 0; j < g[u].size(); ++j) {
            int v = g[u][j].first, w = g[u][j].second;
            if (!vis[v] && dis[u] + w < dis[v]) {
                dis[v] = dis[u] + w;
                pre[v] = u;
            }
        }
    }
}

void dfs(int u) {
    if (u == s) {
        cout << s << " ";
        return;
    }
    dfs(pre[u]);
    cout << u << " ";
}

int main() {
    cin >> n >> m >> s >> t;
    for (int i = 1; i <= m; ++i) {
        int u, v, w;
        cin >> u >> v >> w;
        g[u].push_back(make_pair(v, w));
    }
    dijkstra(s);
    if (dis[t] == INF) cout << "No solution!";
    else {
        cout << "The shortest path is: ";
        dfs(t);
        cout << "\nThe length of the shortest path is: " << dis[t];
    }
    return 0;
}