dijstra单源最短路径问题解决描述

Dijkstra算法是解决单源最短路径问题的一种经典算法。该算法以一个顶点作为起点，逐步扩大路径长度，直到覆盖所有顶点为止。具体描述如下：

1. 初始化：将起点s加入集合S，其他顶点加入集合V-S。对于每个顶点v∈V-S，如果存在一条边从s到v，则将dist[v]设为边的权值，否则将dist[v]设为无穷大。

2. 迭代：重复执行以下步骤，直到集合S包含所有顶点。

a. 在V-S中找到dist值最小的顶点u，将其加入集合S。

b. 对于所有v∈V-S，如果存在一条边从u到v，则更新dist[v]的值为dist[u]+边的权值。

3. 输出：dist数组中存储了起点s到每个顶点的最短路径长度。

Dijkstra算法的核心思想是贪心策略，每次将距离起点最近的顶点加入集合S，并更新其他顶点的最短路径长度。算法的时间复杂度取决于如何实现寻找dist值最小的顶点u的操作，可以通过使用堆等数据结构来优化。

相关问题

dijkstra单源最短路径算法python

以下是Dijkstra单源最短路径算法的Python实现：

import sys

class Graph:
    def __init__(self, vertices):
        self.V = vertices
        self.graph = [[0 for column in range(vertices)] for row in range(vertices)]

    def printSolution(self, dist):
        print("Vertex \t Distance from Source")
        for node in range(self.V):
            print(node, "\t\t", dist[node])

    def minDistance(self, dist, sptSet):
        min = sys.maxsize

        for v in range(self.V):
            if dist[v] < min and sptSet[v] == False:
                min = dist[v]
                min_index = v

        return min_index

    def dijkstra(self, src):
        dist = [sys.maxsize] * self.V
        dist[src] = 0
        sptSet = [False] * self.V

        for cout in range(self.V):
            u = self.minDistance(dist, sptSet)
            sptSet[u] = True

            for v in range(self.V):
                if self.graph[u][v] > 0 and sptSet[v] == False and dist[v] > dist[u] + self.graph[u][v]:
                    dist[v] = dist[u] + self.graph[u][v]

        self.printSolution(dist)

# 测试
g = Graph(9)
g.graph = [[0, 4, 0, 0, 0, 0, 0, 8, 0],
           [4, 0, 8, 0, 0, 0, 0, 11, 0],
           [0, 8, 0, 7, 0, 4, 0, 0, 2],
           [0, 0, 7, 0, 9, 14, 0, 0, 0],
           [0, 0, 0, 9, 0, 10, 0, 0, 0],
           [0, 0, 4, 14, 10, 0, 2, 0, 0],
           [0, 0, 0, 0, 0, 2, 0, 1, 6],
           [8, 11, 0, 0, 0, 0, 1, 0, 7],
           [0, 0, 2, 0, 0, 0, 6, 7, 0]]

g.dijkstra(0)

输出结果将会是：

Vertex   Distance from Source
0             0
1             4
2             12
3             19
4             21
5             11
6             9
7             8
8             14

这个实现中，我们创建了一个Graph类，其中包含了单源最短路径算法Dijkstra的实现。在测试中，我们创建了一个有9个节点的图，并使用Dijkstra算法计算从第0个节点到其他节点的最短路径。

dijkstra单源最短路径java

Dijkstra算法是一种用于计算图中单源最短路径的贪心算法。下面是Java实现Dijkstra算法的示例代码：

import java.util.*;

public class DijkstraAlgorithm {
    private static final Graph.Edge[] GRAPH = {
            new Graph.Edge("A", "B", 7),
            new Graph.Edge("A", "D", 5),
            new Graph.Edge("B", "C", 8),
            new Graph.Edge("B", "D", 9),
            new Graph.Edge("B", "E", 7),
            new Graph.Edge("C", "E", 5),
            new Graph.Edge("D", "E", 15),
            new Graph.Edge("D", "F", 6),
            new Graph.Edge("E", "F", 8),
            new Graph.Edge("E", "G", 9),
            new Graph.Edge("F", "G", 11)
    };

    private static final String START = "A";
    private static final String END = "G";

    public static void main(String[] args) {
        Graph graph = new Graph(GRAPH);
        graph.dijkstra(START);
        System.out.println(graph.getPath(END));
    }

    static class Graph {
        private final Map<String, Vertex> graph;

        static class Edge {
            final String v1, v2;
            final int dist;

            Edge(String v1, String v2, int dist) {
                this.v1 = v1;
                this.v2 = v2;
                this.dist = dist;
            }
        }

        static class Vertex implements Comparable<Vertex> {
            final String name;
            int dist = Integer.MAX_VALUE;
            Vertex previous = null;
            final Map<Vertex, Integer> neighbours = new HashMap<>();

            Vertex(String name) {
                this.name = name;
            }

            private void printPath() {
                if (this == this.previous) {
                    System.out.printf("%s", this.name);
                } else if (this.previous == null) {
                    System.out.printf("%s(unreached)", this.name);
                } else {
                    this.previous.printPath();
                    System.out.printf(" -> %s(%d)", this.name, this.dist);
                }
            }

            public int compareTo(Vertex other) {
                return Integer.compare(dist, other.dist);
            }
        }

        Graph(Edge[] edges) {
            graph = new HashMap<>(edges.length);

            for (Edge e : edges) {
                if (!graph.containsKey(e.v1)) graph.put(e.v1, new Vertex(e.v1));
                if (!graph.containsKey(e.v2)) graph.put(e.v2, new Vertex(e.v2));
            }

            for (Edge e : edges) {
                graph.get(e.v1).neighbours.put(graph.get(e.v2), e.dist);
            }
        }

        void dijkstra(String startName) {
            if (!graph.containsKey(startName)) {
                System.err.printf("Graph doesn't contain start vertex \"%s\"\n", startName);
                return;
            }
            final Vertex source = graph.get(startName);
            NavigableSet<Vertex> q = new TreeSet<>();

            for (Vertex v : graph.values()) {
                v.previous = v == source ? source : null;
                v.dist = v == source ? 0 : Integer.MAX_VALUE;
                q.add(v);
            }

            dijkstra(q);
        }

        private void dijkstra(final NavigableSet<Vertex> q) {
            Vertex u, v;
            while (!q.isEmpty()) {
                u = q.pollFirst();
                if (u.dist == Integer.MAX_VALUE) break;

                for (Map.Entry<Vertex, Integer> a : u.neighbours.entrySet()) {
                    v = a.getKey();

                    final int alternateDist = u.dist + a.getValue();
                    if (alternateDist < v.dist) {
                        q.remove(v);
                        v.dist = alternateDist;
                        v.previous = u;
                        q.add(v);
                    }
                }
            }
        }

        List<String> getPath(String endName) {
            if (!graph.containsKey(endName)) {
                System.err.printf("Graph doesn't contain end vertex \"%s\"\n", endName);
                return Collections.emptyList();
            }

            return graph.get(endName).printPath();
        }
    }
}

该示例代码实现了一个简单的图，其中包含了一些边和顶点。在main方法中，我们创建了一个Graph对象，并调用了它的dijkstra方法来计算从起点到终点的最短路径。最后，我们打印出了这条路径。