从A到B,有多条路线,要找出最短路线,应该用哪种数据结构来存储这些数据。

 

这不是显然的考查图论的相关知识了么,

1.图的两种表示方式:

邻接矩阵:二维数组搞定

邻接表:Map<Vertext,List<Edge>>搞定

其中邻接矩阵适用于稠密图,即图上的任意两点之间均(差不多都)存在一条边。

而A到B之间的路线,显然是稀疏图,果断的选用邻接表。

2.加权有向图最短路径问题,典型的dijkstra最短路径算法。

 

说干就干,翻翻《数据结构与算法》,自己用Java大概实现了一下,具体代码如下:

 

实现思路:

1,定义一个类:有向图类:Graph。

有向图类的子类:节点类:Vertex,边类:Vertex。

节点类:保存节点名称,上一个节点,长度等属性。

边节点:保存每条边的两边的节点,通过边找到对应的另一条节点。

2,该类有两个属性:



1,List<Vertex> vertexList:保存图的顶点集合,便于遍历顶点的时候查找对应集合。
2,Map<Vertex, List<Edge>> ver_edgeList_map:图的每个顶点对应的有向边。



3,为了能够记录最短路径,需要为每个节点定义一个属性:父节点,表示父节点到该点的距离最短。

3,每个节点有多个属性:



String name;  //节点名字
boolean known; //此节点之前是否已知,如果未知的话,则需要初始化距离adjuDist和parent属性
int adjuDist; //保存从开始节点到此节点距离
Vertex parent; //当前从初始节点到此节点的最短路径下,的父节点。



4,从起点节点开始查找。

比较规则:从A节点开始比较,对其指向的B节点进行初始化和比较:

如果B节点未被初始化,先设置该B节点的父节点为A节点,距离为边长加上A节点的adjuDist。

如果已经初始化完了,则重新比较:

如果A节点加边长小于B节点的adjuDist,则证明A节点到B节点的距离最短,设置A节点为B节点父节点,并且长度修改为A节点的adjuDist加上边长。

否则不做操作。

5,等所有的节点初始化完了,从终止节点开始,通过终止节点的父节点找到上一个节点,输出节点的路径。

 

代码如下:



package 笔试题;

import java.util.LinkedList;
import java.util.List;
import java.util.Map;
 
public class Graph{
    
    private List<Vertex> vertexList;   //图的顶点集
    private Map<Vertex, List<Edge>> ver_edgeList_map;  //图的每个顶点对应的有向边
    
    public Graph(List<Vertex> vertexList, Map<Vertex, List<Edge>> ver_edgeList_map) {
        super();
        this.vertexList = vertexList;
        this.ver_edgeList_map = ver_edgeList_map;
    }
 
    public List<Vertex> getVertexList() {
        return vertexList;
    }
 
    public void setVertexList(List<Vertex> vertexList) {
        this.vertexList = vertexList;
    }
 
    
    public Map<Vertex, List<Edge>> getVer_edgeList_map() {
        return ver_edgeList_map;
    }
 
    public void setVer_edgeList_map(Map<Vertex, List<Edge>> ver_edgeList_map) {
        this.ver_edgeList_map = ver_edgeList_map;
    }
 
 
    static class Edge{
        private Vertex startVertex;  //此有向边的起始点
        private Vertex endVertex;  //此有向边的终点
        private int weight;  //此有向边的权值
        
        public Edge(Vertex startVertex, Vertex endVertex, int weight) {
            super();
            this.startVertex = startVertex;
            this.endVertex = endVertex;
            this.weight = weight;
        }
        
        public Edge()
        {}
        
        public Vertex getStartVertex() {
            return startVertex;
        }
        public void setStartVertex(Vertex startVertex) {
            this.startVertex = startVertex;
        }
        public Vertex getEndVertex() {
            return endVertex;
        }
        public void setEndVertex(Vertex endVertex) {
            this.endVertex = endVertex;
        }
        public int getWeight() {
            return weight;
        }
        public void setWeight(int weight) {
            this.weight = weight;
        }
    }
    
     static class Vertex {
        private final static int infinite_dis = Integer.MAX_VALUE;
        
        private String name;  //节点名字
        private boolean known; //此节点之前是否已知
        private int adjuDist; //此节点距离
        private Vertex parent; //当前从初始节点到此节点的最短路径下,的父节点。
        
        public Vertex()
        {
            this.known = false;
            this.adjuDist = infinite_dis;
            this.parent = null;
        }
        
        public Vertex(String name)
        {
            this.known = false;
            this.adjuDist = infinite_dis;
            this.parent = null;
            this.name = name;
        }
        
        public String getName() {
            return name;
        }
        public void setName(String name) {
            this.name = name;
        }
        public boolean isKnown() {
            return known;
        }
        public void setKnown(boolean known) {
            this.known = known;
        }
        public int getAdjuDist() {
            return adjuDist;
        }
        public void setAdjuDist(int adjuDist) {
            this.adjuDist = adjuDist;
        }
        
        public Vertex getParent() {
            return parent;
        }
 
        public void setParent(Vertex parent) {
            this.parent = parent;
        }
        
        /**
         * 重新Object父类的equals方法
         */
        @Override
        public boolean equals(Object obj) {
            if (!(obj instanceof Vertex)) {
                throw new ClassCastException("an object to compare with a Vertext must be Vertex");
            }
            
            if (this.name==null) {
                throw new NullPointerException("name of Vertex to be compared cannot be null");
            }
            
            return this.name.equals(obj);
        }
    }
    
    public void setRoot(Vertex v)
    {
        v.setParent(null);
        v.setAdjuDist(0);
    }
    
    
    /**
     * 
     * @param startIndex dijkstra遍历的起点节点下标
     * @param destIndex dijkstra遍历的终点节点下标
     */
    public void dijkstraTravasal(int startIndex,int destIndex)
    {
        Vertex start = vertexList.get(startIndex);
        Vertex dest = vertexList.get(destIndex);
        String path = "["+dest.getName()+"]";
        
        setRoot(start);
        updateChildren(vertexList.get(startIndex));
        
        int shortest_length = dest.getAdjuDist(); 
        
        while((dest.getParent()!=null)&&(!dest.equals(start)))
        {
            path = "["+dest.getParent().getName()+"] --> "+path;
            dest = dest.getParent();
        }
        
        System.out.println("["+vertexList.get(startIndex).getName() +"] to ["+
                vertexList.get(destIndex).getName()+"] dijkstra shortest path :: "+path);
        System.out.println("shortest length::"+shortest_length);
    }
    
    /**
     * 从初始节点开始递归更新邻接表
     * @param v
     */
    private void updateChildren(Vertex v)
    {
        if (v==null) {
            return;
        }
        
        if (ver_edgeList_map.get(v)==null||ver_edgeList_map.get(v).size()==0) {
            return;
        }
        //用来保存每个可达的节点
        List<Vertex> childrenList = new LinkedList<Graph.Vertex>();
        for(Edge e:ver_edgeList_map.get(v))
        {
            Vertex childVertex = e.getEndVertex();
            
            //如果子节点之前未知,则进行初始化,
            //把当前边的开始点默认为子节点的父节点,长度默认为边长加边的起始节点的长度,并修改该点为已经添加过,表示不用初始化
            if(!childVertex.isKnown())
            {
                childVertex.setKnown(true);
                childVertex.setAdjuDist(v.getAdjuDist()+e.getWeight());
                childVertex.setParent(v);
                childrenList.add(childVertex);
            }
            
            //此时该子节点的父节点和之前到该节点的最小长度已经知道了,则比较该边起始节点到该点的距离是否小于子节点的长度,
            //只有小于的情况下,才更新该点为该子节点父节点,并且更新长度。
            int nowDist = v.getAdjuDist()+e.getWeight();
            if(nowDist>=childVertex.getAdjuDist())
            {
                continue;
            }
            else {
                childVertex.setAdjuDist(nowDist);
                childVertex.setParent(v);
            }
        }
        
        //更新每一个子节点
        for(Vertex vc:childrenList)
        {
            updateChildren(vc);
        }
    }
    
}



测试代码:



package 笔试题;

import java.util.HashMap;
import java.util.LinkedList;
import java.util.List;
import java.util.Map;
 
import 笔试题.Graph.Edge;
import 笔试题.Graph.Vertex;
 
/**
 * 测试用main方法
 * @author wuhui.wwh
 *
 */
public class TestGraph {
    public static void main(String[] args) {
        Vertex v1= new Vertex("v1");
        Vertex v2= new Vertex("v2");
        Vertex v3= new Vertex("v3");
        Vertex v4= new Vertex("v4");
        Vertex v5= new Vertex("v5");
        Vertex v6= new Vertex("v6");
        Vertex v7= new Vertex("v7");
        Vertex v8= new Vertex("v8");
        
        List<Vertex> verList = new LinkedList<Graph.Vertex>();
        verList.add(v1);
        verList.add(v2);
        verList.add(v3);
        verList.add(v4);
        verList.add(v5);
        verList.add(v6);
        verList.add(v7);
        verList.add(v8);
        
        Map<Vertex, List<Edge>> vertex_edgeList_map = new HashMap<Graph.Vertex, List<Edge>>();
        
        List<Edge> v1List = new LinkedList<Graph.Edge>();
        v1List.add(new Edge(v1,v2,6));
        v1List.add(new Edge(v1,v4,1));
        v1List.add(new Edge(v1,v4,1));
        
        List<Edge> v2List = new LinkedList<Graph.Edge>();
        v2List.add(new Edge(v2,v3,43));
        v2List.add(new Edge(v2,v4,11));
        v2List.add(new Edge(v2,v5,6));
        
        List<Edge> v3List = new LinkedList<Graph.Edge>();
        v3List.add(new Edge(v3,v8,8));
        
        List<Edge> v4List = new LinkedList<Graph.Edge>();
        v4List.add(new Edge(v4,v3,15));
        v4List.add(new Edge(v4,v5,12));
        
        List<Edge> v5List = new LinkedList<Graph.Edge>();
        v5List.add(new Edge(v5,v3,38));
        v5List.add(new Edge(v5,v8,13));
        v5List.add(new Edge(v5,v7,24));
        
        List<Edge> v6List = new LinkedList<Graph.Edge>();
        v6List.add(new Edge(v6,v5,1));
        v6List.add(new Edge(v6,v7,12));
        
        List<Edge> v7List = new LinkedList<Graph.Edge>();
        v7List.add(new Edge(v7,v8,20));
        
        vertex_edgeList_map.put(v1, v1List);
        vertex_edgeList_map.put(v2, v2List);
        vertex_edgeList_map.put(v3, v3List);
        vertex_edgeList_map.put(v4, v4List);
        vertex_edgeList_map.put(v5, v5List);
        vertex_edgeList_map.put(v6, v6List);
        vertex_edgeList_map.put(v7, v7List);
        
        
        Graph g = new Graph(verList, vertex_edgeList_map);
        g.dijkstraTravasal(0, 7);
    }
}



运行结果:



[v1] to [v8] dijkstra shortest path :: [v1] --> [v2] --> [v5] --> [v8]
shortest length::25