算法导论——24.3 Dijkstra最短路径算法java实现

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介绍

迪杰斯特拉算法是由荷兰计算机科学家狄克斯特拉于1959 年提出的，因此又叫狄克斯特拉算法。是从一个顶点到其余各顶点的最短路径算法，解决的是有向图中最短路径问题。迪杰斯特拉算法主要特点是以起始点为中心向外层层扩展，直到扩展到终点为止。
Dijkstra算法是典型的算法。Dijkstra算法是很有代表性的算法。Dijkstra一般的表述通常有两种方式，一种用永久和临时标号方式，一种是用OPEN, CLOSE表的方式，这里均采用永久和临时标号的方式。注意该算法要求图中不存在负权边。

适用条件&范围

1.单源最短路径(从源点s到其它所有顶点v);

2.有向图&无向图(无向图可以看作(u,v),(v,u)同属于边集E的有向图);

3.边权非负;

4.差分约束系统;

算法描述

时间复杂度

邻接矩阵 O(n^2)
邻接表 O(n^2)
邻接表+binary heap O((n+m)logn)
邻接表+fibonacci heap O(m+nlogn)

Java代码实现

package algorithms;import util.AlgoGraph;import java.util.Stack;/** * Created with IntelliJ IDEA. * Created by The_Sam on 2017/5/8 22:08 */public class Dijkstra extends ShortestAlgorithms {    private Stack<Integer> Dealing ;    private boolean Dealed[];    public static void main(String[] args) {        ShortestAlgorithms dk = new Dijkstra();        System.out.println(dk.analyse(AlgoGraph.defaultMap4()));    }    boolean analyse(AlgoGraph algoGraph) {        ini(algoGraph);        int edgenum = algoGraph.getEdgenum();        while (!Dealing.empty()) {            int i=Dealing.pop();            Dealed[i]=true;            for (int j = 0; j < edgenum; j++) {//这里以枚举边的方式找以i为u的边，或许还有更好的做法，比如邻接表的形式                int u = algoGraph.edge.get(j).u;                if (i != u) continue;                int v = algoGraph.edge.get(j).v;                int cost = algoGraph.edge.get(j).cost;                if (dis[v] > dis[u] + cost) {                    dis[v] = dis[u] + cost;                    pre[v] = u;                    if(!Dealed[v])Dealing.push(v);                }            }        }        return true;    }    void ini(AlgoGraph algoGraph) {        int nodenum = algoGraph.getNodenum();        //初始化，将dis[s]设为0，其他的都设为无限大        dis = new int[nodenum];//dis[i] 为 i点到s的最短距离        pre = new int[nodenum];//pre[i] 为i搭配s的最短距离的上一个节点        Dealed =new boolean[nodenum];        Dealing=new Stack<Integer>();        int s = algoGraph.getOriginal();//原点        Dealing.push(s);        for (int i = 0; i < nodenum; i++) {            dis[i] = i == s ? 0 : MAXN;        }    }}

AlgoGraph.java 表示图的代码附上package util;import lombok.Getter;import lombok.Setter;import java.util.ArrayList;/** * Created by The_Sam on 2017/5/8. * 图是从0到nodenum-1表示相应节点 */public class AlgoGraph {    @Getter    @Setter    private int original;    @Getter    @Setter    private int nodenum;    private int indegrees[];    //    private int indegrees[];//indegree[i]表示i节点的入度    public ArrayList<AlgoEdge> edge;    public int getEdgenum() {        return edge.size();    }    public AlgoGraph(int original, int nodenum, ArrayList<AlgoEdge> edge) {        this.original = original;        this.nodenum = nodenum;        this.edge = edge;    }    /**     * @Description :返回函数各个节点的入度     * @Author : The_Sam     * @Datetime : 2017/5/8 23:04     * @Params :     * @Return : int[]     */    public int[] getIndegrees() {        indegrees=new int[nodenum];        for (AlgoEdge e :                edge) {            indegrees[e.v]++;        }        return indegrees;    }    /**     * @Description: 返回一个图，为算法导论图24-4，带负权无负权回路有向环图     * @Author: The_Sam     * @Datetime: 2017/5/8 21:22     * @Params: []     * @Return: AlgoGraph     */    static public AlgoGraph defaultMap1() {        ArrayList<AlgoEdge> edge = new ArrayList<AlgoEdge>();        edge.add(new AlgoEdge(0, 1, 6));        edge.add(new AlgoEdge(0, 2, 7));        edge.add(new AlgoEdge(1, 2, 8));        edge.add(new AlgoEdge(1, 3, 5));        edge.add(new AlgoEdge(1, 4, -4));        edge.add(new AlgoEdge(2, 3, -3));        edge.add(new AlgoEdge(2, 4, 9));        edge.add(new AlgoEdge(3, 1, -2));        edge.add(new AlgoEdge(4, 0, 2));        edge.add(new AlgoEdge(4, 3, 7));        int original = 0;        int nodenum = 5;        return new AlgoGraph(original, nodenum, edge);    }    /**     * @Description: 返回一个图，为算法导论图24-4改变版，有负权回路有向环图     * @Author: The_Sam     * @Datetime: 2017/5/8 21:22     * @Params: []     * @Return: AlgoGraph     */    static public AlgoGraph defaultMap2() {        ArrayList<AlgoEdge> edge = new ArrayList<AlgoEdge>();        edge.add(new AlgoEdge(0, 1, 6));        edge.add(new AlgoEdge(0, 2, 7));        edge.add(new AlgoEdge(1, 2, 8));        edge.add(new AlgoEdge(1, 3, 5));        edge.add(new AlgoEdge(1, 4, -4));        edge.add(new AlgoEdge(2, 3, -3));        edge.add(new AlgoEdge(2, 4, 9));        edge.add(new AlgoEdge(3, 1, -2));        edge.add(new AlgoEdge(4, 0, 2));        edge.add(new AlgoEdge(4, 3, 5));//此处改变 edge.add(new AlgoEdge(4, 3, 7));        int original = 0;        int nodenum = 5;        return new AlgoGraph(original, nodenum, edge);    }    /**     * @Description: 返回一个图，为算法导论图24-4改变版，有向无环图     * @Author: The_Sam     * @Datetime: 2017/5/8 21:22     * @Params: []     * @Return: AlgoGraph     */    static public AlgoGraph defaultMap3() {        ArrayList<AlgoEdge> edge = new ArrayList<AlgoEdge>();        edge.add(new AlgoEdge(0, 1, 6));        edge.add(new AlgoEdge(0, 2, 7));        edge.add(new AlgoEdge(1, 2, 8));        edge.add(new AlgoEdge(1, 3, 5));        edge.add(new AlgoEdge(1, 4, -4));        edge.add(new AlgoEdge(2, 3, -3));        edge.add(new AlgoEdge(2, 4, 9));        //edge.add(new AlgoEdge(3, 1, -2));        //edge.add(new AlgoEdge(4, 0, 2));        //edge.add(new AlgoEdge(4, 3, 5));//此处改变 edge.add(new AlgoEdge(4, 3, 7));        int original = 0;        int nodenum = 5;        return new AlgoGraph(original, nodenum, edge);    }    /**     * @Description: 返回一个图，为算法导论图24-4改变版，无负权有向环图     * @Author: The_Sam     * @Datetime: 2017/5/8 21:22     * @Params: []     * @Return: AlgoGraph     */    static public AlgoGraph defaultMap4() {        ArrayList<AlgoEdge> edge = new ArrayList<AlgoEdge>();        edge.add(new AlgoEdge(0, 1, 6));        edge.add(new AlgoEdge(0, 2, 7));        edge.add(new AlgoEdge(1, 2, 8));        edge.add(new AlgoEdge(1, 3, 5));        edge.add(new AlgoEdge(1, 4, 4));        edge.add(new AlgoEdge(2, 3, 3));        edge.add(new AlgoEdge(2, 4, 9));        //edge.add(new AlgoEdge(3, 1, -2));        //edge.add(new AlgoEdge(4, 0, 2));        //edge.add(new AlgoEdge(4, 3, 5));//此处改变 edge.add(new AlgoEdge(4, 3, 7));        int original = 0;        int nodenum = 5;        return new AlgoGraph(original, nodenum, edge);    }}

AlgoEdge.java表示边的代码附上package util;/** * Created by The_Sam on 2017/5/8. */public class AlgoEdge {    public int u,v;    public int cost;    public AlgoEdge(int u, int v, int cost) {        this.u = u;        this.v = v;        this.cost = cost;    }}

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