最短路径：迪杰斯特拉算法

简介

英文名Dijkstra

作用：找到路中指定起点到指定终点的带权最短路径

核心步骤

1）确定起点，终点

2）从未走过的点中选取从起点到权值最小点作为中心点

3）如果满足起点到中心点权值 + 中心点到指定其他点的权值 < 起点到其他点的权值，

即Weight[start] [center] +Weight [center] [other] < Weight [start] [center] ,

简言之，有更短的路径就取更短的路

理论模拟

以A为起点，D为终点，如图所示径, 更新记录更短权值路径

从未走过的点中选取权值最小点，即A作为中心点，标记A走过，更新起点到B、F、G的路径

从未走过的点中选取权值最小点，即B, 并且W:B->C + W:A->C = 12 + 10 < +oo ，更新起点A到C的路径和,

即W: A-> C =W: A-> B -> C =12+10 =22

继续从未走过的点中选取权值最小点G, W: A -> E =+oo > W: A->G ->E =14+8 =22 ,

更新W: A->E 为22

选取F, 由于W:A->F->E=16+2 =18 <22 更新 W: A-> E =18 ,

选取E,由于W:A->E->D=18+4=22<+oo,则更新W: A->D =22

选取C,无可更新点

到达终点D! 最短路径为A->F->E->D ,最短路径和为22

Java代码实现

顶点

java 复制代码

//顶点类
public class Vertex {
    public String Number;  //顶点编号
    public List<Vertex>neighborVertexs;    //邻居顶点
    public Map<Vertex,Integer>weights;     //与邻居节点之间的权值

    public Vertex(String number) {
        this.Number = number;
        this.neighborVertexs=new LinkedList<>();
        this.weights=new HashMap<>();
    }
}

边

java 复制代码

public class Edge {
    public Vertex start;
    public Vertex end;
    public Integer weight;

    public Edge(Vertex start, Vertex end, Integer weight) {
        this.start = start;
        this.end = end;
        this.weight = weight;
    }
}

最短路径返回结果

java 复制代码

public class MinPathResult {
    public String minPathString;    //将最短路径拼接成字符串形式，如 A->B->C
    public List<Vertex>minPathList; //将起点到终点的路径储存在List集合中
    public Integer minPathSum;  //记录起点到终点的最短路径和
    public MinPathResult(List<Vertex> minPathList, String minPathString,Integer minPathSum) {
        this.minPathString = minPathString;
        this.minPathList = minPathList;
        this.minPathSum=minPathSum;
    }

    @Override
    public String toString() {
        return "Result{" +
                "minPathString:'" + minPathString +"  minPathSum:"+minPathSum+
                '}';
    }
}

Dijkstra算法的实现,适用于无向图

java 复制代码

import java.util.*;
//适用于无向图
public class DijkstraOperator {
    private List<Vertex>vertexs;    //全部顶点
    private List<Edge>edges;        //所有边
    private Map<String,Vertex>vertexs_map;  //通过顶点编号找到顶点

    private final static Integer INF=Integer.MAX_VALUE;     //代表无穷大

    public DijkstraOperator(List<Vertex> vertexs, List<Edge> edges) {
        this.vertexs = vertexs;
        this.edges = edges;
        this.vertexs_map=new HashMap<>();
        //构建编号映射顶点
        for(Vertex v:vertexs)
        {
            vertexs_map.put(v.Number,v);
        }

        //填充所有顶点的邻居以及权值
        for(int i=0;i<edges.size();i++)
        {
            //填充起点的邻居，以及起点到终点的权值
            edges.get(i).start.neighborVertexs.add(edges.get(i).end);
            edges.get(i).start.weights.put(edges.get(i).end,edges.get(i).weight);

            //填充终点的邻居，以及终点到起点的权值
            edges.get(i).end.neighborVertexs.add(edges.get(i).start);
            edges.get(i).end.weights.put(edges.get(i).start,edges.get(i).weight);
        }
    }
    //获得从起点到终点之间的路径
    public MinPathResult getMinPath(String start, String end){
        //用哈希表标记某个顶点是否走过
        Map<Vertex,Boolean>visited=new HashMap<>();
        //用哈希表记录顶点的前驱
        Map<Vertex,Vertex>preVertex=new HashMap<>();
        //利用哈希表记录起点到任意一点的最短路径
        Map<Vertex,Integer>minPath=new HashMap<>();

        //初始化三个表
        for(int i=0;i<vertexs.size();i++)
        {
            //初始化每一个点都未走过
            visited.put(vertexs.get(i),false);
            //初始化每个点的前驱都是自己
            preVertex.put(vertexs.get(i),vertexs.get(i));
            //初始化起点到任意两个点之间的最短路径都是无穷大
            minPath.put(vertexs.get(i),INF);
        }

        Vertex startVertex=vertexs_map.get(start);
        Vertex endVertex=vertexs_map.get(end);

        //填充存在的路径
        for(int i=0;i<startVertex.neighborVertexs.size();i++)
        {
            //设置起点与邻居节点之间的权值
            minPath.put(startVertex.neighborVertexs.get(i),startVertex.weights.get(startVertex.neighborVertexs.get(i)));
            //设置点前驱
            preVertex.put(startVertex.neighborVertexs.get(i),startVertex);
        }
        //自己到自己的距离为0
        minPath.put(startVertex,0);

        Vertex curVertex=null;
        //一直寻路，直到找到终点
        while(curVertex!=endVertex)
        {
            Integer minWeight=Integer.MAX_VALUE;
            curVertex=null;
            //能看到的点之间选取距离最小的那个,且这个点并没有走过
            for(Vertex v:minPath.keySet())
            {
                if(!visited.get(v)&&minPath.get(v)<minWeight)
                {
                    //切换中心点
                    curVertex=v;
                    //更新最小权值
                    minWeight=minPath.get(v);
                }
            }

            //如果找不到下一个中心点，说明从起点根本到达不来终点
            if(curVertex==null)
                return null;
            //标记选取点
            visited.put(curVertex,true);

            //更新权值
            for(int i=0;i<curVertex.neighborVertexs.size();i++)
            {
                //邻居节点
                Vertex neighborVertex=curVertex.neighborVertexs.get(i);

                //计算起点到邻居节点之间新的权值
                int newWeight=minPath.get(curVertex)+curVertex.weights.get(neighborVertex);

                //找到能移动的点,如果转折之后距离更短，则记录更短的距离
                if(visited.get(neighborVertex)==false&&newWeight<minPath.get(neighborVertex))
                {
                    //记录更短距离
                    minPath.put(neighborVertex,newWeight);

                    //记录邻居节点的前驱
                    preVertex.put(neighborVertex,curVertex);
                }
            }
        }
        //起点到终点之间的最短路径
        LinkedList<Vertex>targetPath=new LinkedList<>();

        for(Vertex curVer=endVertex;curVer!=startVertex;curVer=preVertex.get(curVer))
        {
            targetPath.addFirst(curVer);
        }
        targetPath.addFirst(startVertex);

        //拼接最短路径
        StringBuffer minPathStringBuffer=new StringBuffer();
        Integer pathSum=0;
        for(int i=0;i< targetPath.size();i++)
        {
            minPathStringBuffer.append(targetPath.get(i).Number);
            if(i!= targetPath.size()-1)
            {
                pathSum=pathSum+targetPath.get(i).weights.get(targetPath.get(i+1));
                minPathStringBuffer.append("->");
            }
        }
        return new MinPathResult(targetPath, minPathStringBuffer.toString(),pathSum);
    }
}

测试函数

java 复制代码

import java.util.*;

public class Main {
    public static void main(String[] args) {
        Scanner scanner=new Scanner(System.in);

        List<Vertex>vertexs=new LinkedList<>();
        List<Edge>edges=new LinkedList<>();

        System.out.println("请输入顶点的数量:");
        Integer vexcnt= scanner.nextInt();
        System.out.println("请输入这些顶点编号:");
        for(int i=0;i<vexcnt;i++)
        {
            vertexs.add(new Vertex(scanner.next()));
        }

        System.out.println("请输入边的数量：");
        Integer edgecnt= scanner.nextInt();
        System.out.println("请输入这些边的端点编号和权值:");
        for(int i=0;i<edgecnt;i++)
        {
            String number1= scanner.next();
            String number2= scanner.next();
            Integer weight= scanner.nextInt();

            Vertex v1=null;
            Vertex v2=null;
            for(int j=0;j<vertexs.size();j++)
            {
                if(vertexs.get(j).Number.equals(number1))
                    v1=vertexs.get(j);
                if(vertexs.get(j).Number.equals(number2))
                    v2=vertexs.get(j);
            }
            edges.add(new Edge(v1,v2,weight));
        }

        //定义迪杰斯特拉操作类
        DijkstraOperator dijkstra=new DijkstraOperator(vertexs,edges);

        //获取任意两点之间的最短路径结果集
        List<MinPathResult>minPathResults=new ArrayList<>();

        for(int i=0;i< vertexs.size();i++)
        {
            for(int j=i+1;j< vertexs.size();j++)
            {
                minPathResults.add(dijkstra.getMinPath(vertexs.get(i).Number,vertexs.get(j).Number));
                System.out.println(minPathResults.get(minPathResults.size()-1));
            }
        }

    }
}

测试输入与输出结果

java 复制代码

//输入部分
请输入顶点的数量:
7
请输入这些顶点编号:
A B C D E F G
请输入边的数量：
12
请输入这些边的端点编号和权值:
A B 12
A F 16
A G 14
B C 10
B F 7
G F 9
G E 8
F C 6
F E 2
C D 3
C E 5
E D 4

//输出部分
Result{minPathString:'A->B  minPathSum:12}
Result{minPathString:'A->B->C  minPathSum:22}
Result{minPathString:'A->F->E->D  minPathSum:22}
Result{minPathString:'A->F->E  minPathSum:18}
Result{minPathString:'A->F  minPathSum:16}
Result{minPathString:'A->G  minPathSum:14}
Result{minPathString:'B->C  minPathSum:10}
Result{minPathString:'B->F->E->D  minPathSum:13}
Result{minPathString:'B->F->E  minPathSum:9}
Result{minPathString:'B->F  minPathSum:7}
Result{minPathString:'B->F->G  minPathSum:16}
Result{minPathString:'C->D  minPathSum:3}
Result{minPathString:'C->E  minPathSum:5}
Result{minPathString:'C->F  minPathSum:6}
Result{minPathString:'C->E->G  minPathSum:13}
Result{minPathString:'D->E  minPathSum:4}
Result{minPathString:'D->E->F  minPathSum:6}
Result{minPathString:'D->E->G  minPathSum:12}
Result{minPathString:'E->F  minPathSum:2}
Result{minPathString:'E->G  minPathSum:8}
Result{minPathString:'F->G  minPathSum:9}

进程已结束，退出代码为 0