if (visited[v]==0&&graph[u][v]&&dist[u]!=INT_MAX&&dist[u] + graph[u][v] < dist[v]) { dist[v]=dist[u]+graph[u][v]; // 更新最短距离 parent[v]=u; // 更新父节点 }

这段代码是什么意思？

这段代码是 Dijkstra 算法中的一部分，用于更新从起点到各个节点的最短距离和路径。其中，visited 数组表示节点是否被访问过，graph[u][v] 表示 u 到 v 的距离，dist[u] 表示起点到 u 的最短距离，dist[v] 表示起点到 v 的最短距离，parent[v] 表示节点 v 的父节点。

当节点 v 没有被访问过，且 u 到 v 有边相连，且起点到 u 的距离不是无穷大，且通过 u 可以更新起点到 v 的最短距离时，就更新 dist[v] 和 parent[v] 的值，以便后续计算从起点到各个节点的最短路径。

相关问题

#include <stdio.h> #include <stdlib.h> #define MAXN 1000 #define INF 0x3f3f3f3f int n, m; int graph[MAXN][MAXN]; int path[MAXN]; int dist; void dfs(int u, int depth, int distance, int visited[]) { if (depth == n) { if (graph[u][1] != INF && distance + graph[u][1] < dist) { dist = distance + graph[u][1]; for (int i = 1; i <= n; i++) { printf("%d ", path[i]); } printf("1\n"); } return; } for (int v = 2; v <= n; v++) { if (!visited[v] && graph[u][v] != INF) { visited[v] = 1; path[depth+1] = v; dfs(v, depth+1, distance+graph[u][v], visited); visited[v] = 0; } } } int main() { scanf("%d %d", &n, &m); for (int i = 1; i <= n; i++) { for (int j = 1; j <= n; j++) { graph[i][j] = INF; } } for (int i = 1; i <= m; i++) { int x, y, z; scanf("%d,%d,%d", &x, &y, &z); graph[x][y] = graph[y][x] = z; } for (int k = 1; k <= n; k++) { for (int i = 1; i <= n; i++) { for (int j = 1; j <= n; j++) { if (graph[i][k] != INF && graph[k][j] != INF && graph[i][k] + graph[k][j] < graph[i][j]) { graph[i][j] = graph[i][k] + graph[k][j]; } } } } int visited[MAXN] = {0}; visited[1] = 1; path[1] = 1; dist = INF; dfs(1, 1, 0, visited); printf("%d\n", dist); return 0; }修改改代码，使其只显示最短路径

可以在dfs函数中加入一个变量来记录当前的最短路径长度，每次更新路径时判断是否比当前最短路径更短。同时，在找到最短路径后就直接return，不再继续搜索其他路径。

修改后的代码如下：

#include <stdio.h>
#include <stdlib.h>

#define MAXN 1000
#define INF 0x3f3f3f3f

int n, m;
int graph[MAXN][MAXN];
int path[MAXN];
int dist;
int shortest_dist = INF;

void dfs(int u, int depth, int distance, int visited[]) {
    if (depth == n) {
        if (graph[u][1] != INF && distance + graph[u][1] < shortest_dist) {
            shortest_dist = distance + graph[u][1];
            dist = distance + graph[u][1];
            for (int i = 1; i <= n; i++) {
                path[i] = i;
            }
            path[n+1] = 1;
        }
        return;
    }
    for (int v = 2; v <= n; v++) {
        if (!visited[v] && graph[u][v] != INF) {
            visited[v] = 1;
            path[depth+1] = v;
            dfs(v, depth+1, distance+graph[u][v], visited);
            visited[v] = 0;
        }
    }
}

int main() {
    scanf("%d %d", &n, &m);
    for (int i = 1; i <= n; i++) {
        for (int j = 1; j <= n; j++) {
            graph[i][j] = INF;
        }
    }
    for (int i = 1; i <= m; i++) {
        int x, y, z;
        scanf("%d,%d,%d", &x, &y, &z);
        graph[x][y] = graph[y][x] = z;
    }
    for (int k = 1; k <= n; k++) {
        for (int i = 1; i <= n; i++) {
            for (int j = 1; j <= n; j++) {
                if (graph[i][k] != INF && graph[k][j] != INF && graph[i][k] + graph[k][j] < graph[i][j]) {
                    graph[i][j] = graph[i][k] + graph[k][j];
                }
            }
        }
    }
    int visited[MAXN] = {0};
    visited[1] = 1;
    path[1] = 1;
    dfs(1, 1, 0, visited);
    printf("%d\n", dist);
    for (int i = 1; i <= n+1; i++) {
        printf("%d ", path[i]);
    }
    printf("\n");
    return 0;
}

# Dijkstra算法求最短路径 def dijkstra(graph, start, end): n = len(graph) dist = [float('inf')] * n dist[start] = 0 visited = [False] * n heap = [(0, start)] while heap: d, u = heapq.heappop(heap) if visited[u]: continue visited[u] = True for v in range(n): if graph[u][v] > 0: if dist[u] + graph[u][v] < dist[v]: dist[v] = dist[u] + graph[u][v] heapq.heappush(heap, (dist[v], v)) return dist[end] # Bellman-Ford算法求最短路径 def bellman_ford(graph, start, end): n = len(graph) dist = [float('inf')] * n dist[start] = 0 for i in range(n-1): for u in range(n): for v in range(n): if graph[u][v] > 0: if dist[u] + graph[u][v] < dist[v]: dist[v] = dist[u] + graph[u][v] return dist[end] # 求最短距离和最短路径 print("Dijkstra算法：") print("最短距离：", dijkstra(graph, 0, 9)) print("最短路径：") path = [] n = len(graph) dist = [float('inf')] * n dist[0] = 0 prev = [-1] * n heap = [(0, 0)] while heap: d, u = heapq.heappop(heap) if u == 9: break for v in range(n): if graph[u][v] > 0: if dist[u] + graph[u][v] < dist[v]: dist[v] = dist[u] + graph[u][v] prev[v] = u heapq.heappush(heap, (dist[v], v)) if prev[9] == -1: print("不存在路径") else: v = 9 while v != -1: path.append(v) v = prev[v] path.reverse() print(path) print("Bellman-Ford算法：") print("最短距离：", bellman_ford(graph, 0, 9))代码分析

这段代码实现了Dijkstra算法和Bellman-Ford算法来求解最短路径问题。其中，Dijkstra算法是一种贪心算法，每次取距离起点最短的未被访问节点，更新其邻居节点的距离。Bellman-Ford算法则是一种动态规划算法，通过多次松弛操作来得到最短路径。在这段代码中，graph表示图的邻接矩阵，其中graph[i][j]表示节点i到节点j的距离。start和end表示起点和终点的编号。

在Dijkstra算法中，使用一个堆来存储待访问的节点，并按照距离起点的距离从小到大排序。每次取出堆中距离最小的节点，更新其邻居节点的距离。当堆为空或者终点被访问时，算法结束。最后通过prev数组来回溯路径。

在Bellman-Ford算法中，需要进行n-1次松弛操作，其中n为节点数。每次将所有边进行松弛操作，即如果从起点到u的距离加上u到v的距离小于从起点到v的距离，则更新从起点到v的距离。如果还能进行松弛操作，则说明存在负环路，否则得到的dist数组即为最短距离。