上一篇既然提到了Krusal演算法，這裡就不得不說Prim演算法了，這兩個演算法都是求解最小生成樹的經典的貪婪演算法。與Krusal演算法不同的是，Prim演算法在求解過程中始終保持臨時結果是一顆聯通的樹。該演算法的虛擬碼如下

//假設網路中至少有一個個頂點
設T為所選邊的集合，初始化T為空
設 TV為已在樹中的頂點的集合，置TV={1}
令E為網路中的邊的集合
while(E不為空，並且T 中的邊數不等於n-1){//這裡n指原圖中頂點個數
令（u,v）為最小代價的邊，其中u屬於TV，v不屬於TV
if（沒有這種邊）
break
E=E-{（u，v）}
在T中加入邊（u，v） 
}
if(|T|==n-1）
T是一顆最小生成樹
else
沒有最小生成樹
 

下圖是有一個Prim 演算法求解最小生成樹的過程的一個例子

以下是用Java程式碼點的實現

package Prim;
/**
 * 邊
 * @author sdu20
 *
 */
public class Edge {

	private int v1;
	private int v2;
	private int weight;
	
	/**
	 * 為查詢最小邊專門所設
	 * @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 Prim;

import java.util.*;

public class Graph {

	private int vNum;
	private int edgeNum;
	private LinkedList<Edge>[] edgeLinks;
	private LinkedList<Integer> TV;	//已在樹中的頂點集
	private LinkedList<Edge> T;	//入選的邊集
	
	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();
		Edge edge2 = new Edge(v2,v1,edge.getWeight());
		edgeLinks[v1].remove(edge);
		edgeLinks[v2].remove(edge2);
		edgeNum--;
	}
	
	public void bianli(){
		System.out.println("共有 "+vNum+" 個頂點， "+edgeNum+" 條邊。");
		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("]");
		}
	}
	
	/**
	 * Prim演算法實現
	 */
	public void Prim(){
		
		TV = new LinkedList<>();
		T = new LinkedList<>();
		TV.add(0);
		
		while(edgeNum>0 && T.size()!=vNum-1){
			Edge edge = getMinEdge(TV);
			if(edge==null)
				break;
			this.deleteEdge(edge);
			T.add(edge);
			TV.add(edge.getV2());
		}
		
		if(T.size()==vNum-1){
			System.out.println("求最小生成樹成功");
			LinkedList<Edge> list = (LinkedList<Edge>) T.clone();
			int sumWeight = 0;
			while(!list.isEmpty()){
				Edge edge = list.pop();
				sumWeight += edge.getWeight();
				System.out.println(edge.toString());
			}
			System.out.println("總的權重為： "+sumWeight);
		}else{
			System.out.println("無最小生成樹");
		}
		
		
	}
	
	public Edge getMinEdge(LinkedList<Integer> t){
		
		Edge minEdge = new Edge(10000);
		LinkedList<Integer> tt = (LinkedList<Integer>) t.clone();
		
		while(!tt.isEmpty()){
			int i = tt.pop();
			LinkedList<Edge> list = (LinkedList<Edge>) edgeLinks[i].clone();
			while(!list.isEmpty()){
				Edge edge = list.pop();
				if(minEdge.getWeight()>edge.getWeight() && !t.contains(edge.getV2())){
					minEdge = edge;
				}
			}
		}		

		if(minEdge.getWeight()==10000)
			return null;
		return minEdge;
	}
	
}
 

package Prim;

public class Main {

	public static void main(String[] args) {
		// TODO Auto-generated method stub

		bookGraph();
		//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.Prim();
	}
	
	/**
	 * 100個點，1000條邊，權重為1~100的隨機數
	 */
	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.Prim();
	}

}

執行截圖如下所示