图论-最短路（bellman-fod算法）-优快云博客

本文链接：https://blog.youkuaiyun.com/m0_73672990/article/details/129869691

文章包含多个使用Bellman-Ford算法解决最短路径问题的实例，包括简单的两点间最短路径、考虑路径节点的最短路径及租房成本最小化问题。每个问题都通过Dijkstra或Bellman-Ford算法进行求解，并输出结果。

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1.最短路

2.最短路pro（要求求出最短路径上的点，其余条件和第一问完全相同）

//最短路2（求最短距离及其路径）
#include <bits/stdc++.h>
using namespace std;
int n, m, k;
struct node {
	int x, y, v;
} edges[10001];
int dis[5001], pre[5001], path[5001];

inline void bellmanFod(int s, int f) {
	memset(dis, 127, sizeof(dis));
	memset(pre, 127, sizeof(pre));
	dis[s] = 0;
	while (1) {
		bool ok = 0;
		for (int i = 1; i <= m; i++) {
			int x = edges[i].x, y = edges[i].y, v = edges[i].v;
			if (dis[x] < 1 << 30) {
				if (dis[x] + v < dis[y]) {
					dis[y] = dis[x] + v;
					pre[y] = x;
					ok = 1;
				}
			}
		}
		if (!ok) {
			break;
		}
	}
	if (dis[f] < 1 << 30) {
		cout << dis[f] << '\n';
		int l = 0;
		for (int i = f; i != s; i = pre[i]){
			path[++l] = i;
		}
		path[++l] = s;
		for (int i = l; i; i--) {
			cout << path[i] << " \n"[i == 1];
		}
	} else {
		cout << "-1" << '\n';
	}
}

int main() {
	ios::sync_with_stdio(false);
	cin.tie(0);
	cin >> n >> m >> k;
	for (int i = 1; i <= m; i++) {
		cin >> edges[i].x >> edges[i].y >> edges[i].v;
	}
	for (int i = 1; i <= k; i++) {
		int x, y;
		cin >> x >> y;
		bellmanFod(x, y);
	}
}

3.租房

//租房
#include <bits/stdc++.h>
using namespace std;
int n, m, k, a, b, c, cnt;
struct node {
	int x, y, v;
} edges[20001];
int dis[10001], f[3][10001];

inline void bellmanFod(int s) {
	memset(dis, 127, sizeof(dis));
	dis[s] = 0;
	while (1) {
		bool ok = 0;
		for (int i = 1; i <= cnt; i++) {
			int x = edges[i].x, y = edges[i].y, v = edges[i].v;
			if (dis[x] < 1 << 30) {
				if (dis[x] + v < dis[y]) {
					dis[y] = dis[x] + v;
					ok = 1;
				}
			}
		}
		if (!ok) {
			break;
		}
	}
}

int main() {
	ios::sync_with_stdio(false);
	cin.tie(0);
	cin >> n >> m;
	cnt = 0;
	for (int i = 1; i <= m; i++) {
		int x, y, z;
		cin >> x >> y >> z;
		edges[++cnt].x = x, edges[cnt].y = y, edges[cnt].v = z;
		edges[++cnt].x = y, edges[cnt].y = x, edges[cnt].v = z;
	}
	cin >> a >> b >> c;
	bellmanFod(a);
	memcpy(f[0], dis, sizeof(dis));
	bellmanFod(b);
	memcpy(f[1], dis, sizeof(dis));
	bellmanFod(c);
	memcpy(f[2], dis, sizeof(dis));
	int ans = 1 << 30;
	for (int i = 1; i <= n; i++) {
		ans = min(ans, f[0][i] + f[1][i] + f[2][i]);
	}
	cout << ans << '\n';
}

4.小蜗的旅行

//小蜗的旅行
#include <bits/stdc++.h>
using namespace std;
int n, m, k, cnt;
struct node {
	int x, y, v;
} edges[20001];
int dis[501][11];

inline void bellmanFod(int s, int f) {
	memset(dis, 127, sizeof(dis));
	dis[s][0] = 0;
	for (int i = 1; i <= n; i++) {
		bool ok = 0;
		for (int j = 1; j <= cnt; j++) {
			int x = edges[j].x, y = edges[j].y, v = edges[j].v;
			for (int l = 0; l <= k; l++) {
				if (dis[x][l] < 1 << 30) {
					if (dis[x][l] + v < dis[y][l]) {
						dis[y][l] = dis[x][l] + v;
						ok = 1;
					}
					if (l != k && dis[x][l] + v / 2 < dis[y][l + 1]) {
						dis[y][l + 1] = dis[x][l] + v / 2;
						ok = 1;
					}
				}
			}
		}
		if (!ok) {
			break;
		}
	}
	int ans = 1 << 30;
	for (int i = 0; i <= k; i++) {
		ans = min(ans, dis[n][i]);
	}
	cout << ans << '\n';
}

int main() {
	ios::sync_with_stdio(false);
	cin.tie(0);
	cin >> n >> m >> k;
	cnt = 0;
	for (int i = 1; i <= m; i++) {
		int x, y, z;
		cin >> x >> y >> z;
		edges[++cnt].x = x, edges[cnt].y = y, edges[cnt].v = z;
		edges[++cnt].x = y, edges[cnt].y = x, edges[cnt].v = z;
	}
	bellmanFod(1, n);
}