Kruskal算法求解最小生成树的Java实现

假设一个无向图共有V（V>1）个顶点，E条边，那么它的最小生成树（如果有的话）有V个顶点，V-1条边。Krusal算法在求解最小生成树的过程中就是不断地选出一条权重最小的边，如果将这条边插入到目前的最小生成树（此时尚未最终形成）中不形成环路，则将其插入，否则再选择次小边重复以上过程，直到最后最小生成树中有V-1条边，则最小生成树求解成功，否则，原图中不包含最小生成树。

下面是《算法导论》一书中的一个Krusal算法求解最小生成树过程的一个例子。

以下是具体的Java代码实现

package Kruskal;/** * 边 * @author sdu20 * */public class Edge {private int v1;private int v2;private int weight;/** * 为查找最小边专门所设 * @param weight */public Edge(int weight){this.v1 = -1;this.v2 = -1;this.weight = weight;}public Edge(int v1,int v2,int weight){this.v1 = v1;this.v2 = v2;this.weight = weight;}public int getV1(){return v1;}public int getV2(){return v2;}public int getWeight(){return weight;}public String toString(){String str = "[ "+v1+" , "+v2+" , "+weight+" ]";return str;}public boolean equals(Edge edge){boolean equal = this.v1==edge.getV1() && this.v2==edge.getV2() && this.weight==edge.getWeight()|| this.v1==edge.getV2() && this.v2==edge.getV1() && this.weight==edge.getWeight();return equal;}}

package Kruskal;import java.util.*;public class Graph {private int vNum;private int edgeNum;private LinkedList<Edge>[] edgeLinks;private LinkedList<Edge> T;//入选的边集private LinkedList<Integer>[] kindLists;//用于区分是否形成环public Graph(int vNum){this.vNum = vNum;this.edgeNum = 0;edgeLinks = new LinkedList[vNum];for(int i = 0;i<vNum;i++){edgeLinks[i] = new LinkedList<>();}}public void insertEdge(Edge edge){int v1 = edge.getV1();int v2 = edge.getV2();edgeLinks[v1].add(edge);Edge edge2 = new Edge(v2,v1,edge.getWeight());edgeLinks[v2].add(edge2);edgeNum++;}public void deleteEdge(Edge edge){int v1 = edge.getV1();int v2 = edge.getV2();edgeLinks[v1].remove(edge);for(int i = 0;i<edgeLinks[v2].size();i++){Edge edge2 = edgeLinks[v2].get(i);if(edge2.equals(edge)){edgeLinks[v2].remove(edge2);break;}}edgeNum--;}public void bianli(){System.out.println("共有 "+vNum+" 个顶点， "+edgeNum+" 条边。");for(int i = 0;i<vNum;i++){LinkedList<Edge> list = (LinkedList<Edge>) edgeLinks[i].clone();System.out.print(i+" : [");while(!list.isEmpty()){Edge edge = list.pop();System.out.print(edge.getV2()+"("+edge.getWeight()+")"+"  ");}System.out.println("]");}}/** * Kruskal算法实现 */public void Kruskal(){T = new LinkedList<>();kindLists = new LinkedList[vNum];for(int i = 0;i<vNum;i++){kindLists[i] = new LinkedList<>();int num = i;kindLists[i].add(num);}while(edgeNum>0 && T.size()!=vNum-1){Edge edge = this.getMinEdge();this.deleteEdge(edge);int v1 = edge.getV1();int v2 = edge.getV2();int containsV1 = -1;int containsV2 = -1;for(int i = 0;i<vNum;i++){LinkedList<Integer> list = (LinkedList<Integer>) kindLists[i].clone();if(list.contains(v1)){containsV1 = i;}if(list.contains(v2)){containsV2 = i;}}if(containsV1 != containsV2){T.add(edge);while(!kindLists[containsV2].isEmpty()){kindLists[containsV1].add(kindLists[containsV2].pop());}}}if(T.size() == vNum-1){System.out.println("求最小生成树成功");int sumWeight = 0;LinkedList<Edge> TT = (LinkedList<Edge>) T.clone();while(!TT.isEmpty()){Edge ee = TT.pop();sumWeight += ee.getWeight();System.out.println(ee.toString());}System.out.println("最小生成树权重之和为： "+sumWeight);}else{System.out.println("没有最小生成树");}}public Edge getMinEdge(){Edge minEdge = new Edge(10000);for(int i = 0;i<vNum;i++){LinkedList<Edge> list = (LinkedList<Edge>) edgeLinks[i].clone();while(!list.isEmpty()){Edge edge = list.pop();if(minEdge.getWeight()>edge.getWeight()){minEdge = edge;}}}if(minEdge.getWeight()==10000)return null;return minEdge;}}

package Kruskal;public class Main {public static void main(String[] args) {// TODO Auto-generated method stubbookGraph();//randomGraph();}public static void bookGraph(){Graph graph = new Graph(9);Edge[] edges = new Edge[14];edges[0] = new Edge(0,1,4);edges[1] = new Edge(0,7,8);edges[2] = new Edge(1,2,8);edges[3] = new Edge(1,7,11);edges[4] = new Edge(2,3,7);edges[5] = new Edge(2,5,4);edges[6] = new Edge(2,8,2);edges[7] = new Edge(3,4,9);edges[8] = new Edge(3,5,14);edges[9] = new Edge(4,5,10);edges[10] = new Edge(5,6,2);edges[11] = new Edge(6,7,1);edges[12] = new Edge(6,8,6);edges[13] = new Edge(7,8,7);for(int i = 0;i<14;i++){graph.insertEdge(edges[i]);}graph.bianli();graph.Kruskal();}/** * 100个点，1000条边，权重为1~100的随机数 */public static void randomGraph(){Graph graph = new Graph(100);for(int i = 0;i<1000;){int preV = (int)(Math.random()*100);            int folV = (int)(Math.random()*100);            int weight = (int)(Math.random()*100+1);            if(preV != folV){            Edge edge = new Edge(preV,folV,weight);            try{            graph.insertEdge(edge);            i++;            }catch(Exception e){            continue;            }            }}graph.bianli();graph.Kruskal();}}

运行截图如下

该测试用例即为前面所给的例子。