面向对象方式实现最小生成树算法

最小生成树
     一个有 n 个结点的连通图的生成树是原图的极小连通子图，且包含原图中的所有 n 个结点，并且有保持图连通的最少的边。
    
    最小生成树可参考：http://baike.baidu.com/view/288214.htm

    下面实现最小生成树的Prim算法。
 
    网上包括很多论坛里实现最小生成树的算法多为二维数组、面向过程的方式。最近得闲试着用Java代码实现面向对象的最小生成树算法。这样从实用角度看至少没那么书卷气了。

    准备工作：
   
    点、边、图的实现

点

package com.zas.test.tree;

/**
 * 点的定义 暂时为一个标记类 如果有现实的业务需求 再添加点的属性定义
 * @author zas
 */
public class Point {
	//暂时定义为字符串标记
	String point;
	
	public Point() {
		super();
	}

	public Point(String point) {
		super();
		this.point = point;
	}

	public String getPoint() {
		return point;
	}

	public void setPoint(String point) {
		this.point = point;
	}
	
	/* (non-Javadoc)
	 * @see java.lang.Object#equals(java.lang.Object)
	 * 只有点描述相同的点是相同的
	 */
	@Override
	public boolean equals(Object obj) {
		if(obj instanceof Point){
			if(((Point) obj).point.equals(this.point)){
				return true;
			}
		}
		return false;
	}

	@Override
	public String toString() {
		return "Point [point=" + point + "]";
	}

	/**
	 * @param args
	 */
	public static void main(String[] args) {

	}

}

普通边

package com.zas.test.tree;

/**
 * 边的定义
 * @author zas
 */
public class Edge {
	//起点
	protected Point startPoint;
	//终点
	protected Point endPoint;
	
	public Edge() {
		super();
	}

	public Edge(Point startPoint, Point endPoint) {
		super();
		this.startPoint = startPoint;
		this.endPoint = endPoint;
	}

	public Point getStartPoint() {
		return startPoint;
	}

	public void setStartPoint(Point startPoint) {
		this.startPoint = startPoint;
	}

	public Point getEndPoint() {
		return endPoint;
	}

	public void setEndPoint(Point endPoint) {
		this.endPoint = endPoint;
	}
	
	/* (non-Javadoc)
	 * @see java.lang.Object#equals(java.lang.Object)
	 * 只有起止点相同的边才是同一条边
	 */
	@Override
	public boolean equals(Object obj) {
		if(obj instanceof Edge){
			Edge outEdge = (Edge)obj;
			if(outEdge.startPoint.equals(this.startPoint)){
				if(outEdge.endPoint.equals(this.endPoint)){
					return true;
				}
			}
		}
		return false;
	}
	
	@Override
	public String toString() {
		return "Edge [startPoint=" + startPoint + ", endPoint=" + endPoint
				+ "]";
	}

	/**
	 * @param args
	 */
	public static void main(String[] args) {

	}

}

权重

package com.zas.test.tree;

/**
 * 权重的定义
 * @author zas
 */
public class Weight implements Comparable<Weight>{
	//double类型定义一个权重信息
	double weight = 0.0;

	public Weight() {
		super();
	}
	
	public Weight(double weight) {
		super();
		this.weight = weight;
	}

	@Override
	public int compareTo(Weight o) {
		if(o.weight == this.weight){
			return 0;
		}else if(o.weight > this.weight){
			return -1;
		}else{
			return 1;
		}
	}
	@Override
	public boolean equals(Object obj) {
		if(obj instanceof Weight){
			if(((Weight) obj).weight == this.weight){
				return true;
			}
		}
		return false;
	}
	@Override
	public String toString() {
		return "Weight [weight=" + weight + "]";
	}

	/**
	 * @param args
	 */
	public static void main(String[] args) {
		
	}
}

带权边

package com.zas.test.tree;

public class EdgeWithWeight extends Edge implements Comparable<EdgeWithWeight> {
	//边的权重
	Weight weight;
	
	public EdgeWithWeight() {
		super();
	}

	public EdgeWithWeight(Point startPoint, Point endPoint) {
		super(startPoint, endPoint);
	}

	public EdgeWithWeight(Weight weight) {
		super();
		this.weight = weight;
	}
	
	public EdgeWithWeight(Point startPoint, Point endPoint, Weight weight) {
		super(startPoint, endPoint);
		this.weight = weight;
	}

	@Override
	public int compareTo(EdgeWithWeight o) {
		return o.weight.compareTo(this.weight);
	}
	
	public Weight getWeight() {
		return weight;
	}

	public void setWeight(Weight weight) {
		this.weight = weight;
	}

	@Override
	public boolean equals(Object obj) {
		if(obj instanceof EdgeWithWeight){
			if(((EdgeWithWeight) obj).getStartPoint().equals(this.getStartPoint())){
				if(((EdgeWithWeight) obj).getEndPoint().equals(this.getEndPoint())){
					if(((EdgeWithWeight) obj).getWeight().equals(this.getWeight())){
						return true;
					}
				}
			}
		}
		return false;
	}

	@Override
	public String toString() {
		return "EdgeWithWeight [weight=" + weight + ", startPoint="
				+ startPoint + ", endPoint=" + endPoint + "]";
	}

	/**
	 * @param args
	 */
	public static void main(String[] args) {

	}

}

图

package com.zas.test.tree;

import java.util.ArrayList;
import java.util.List;

/**
 * 图的定义
 * @author zas
 */
public class Graph {
	//图的点列表
	List<Point> pointList;
	//图的边列表
	List<EdgeWithWeight> edgeList;
	
	public Graph() {
		super();
	}

	public Graph(List<Point> pointList, List<EdgeWithWeight> edgeList) {
		super();
		this.pointList = pointList;
		this.edgeList = edgeList;
	}
	
	/**
	 * 添加一个点到图中
	 * @param p
	 */
	public void addPoint(Point p) {
		if(pointList == null){
			pointList = new ArrayList<Point>();
		}
		pointList.add(p);
	}
	
	/**
	 * 从图中删除一个点
	 * @param p
	 */
	public void removePoint(Point p){
		for (Point point : pointList) {
			if(p.equals(point)){
				pointList.remove(point);
				break;
			}
		}
	}
	
	/**
	 * 添加一条边到图中
	 * @param p
	 */
	public void addEdge(EdgeWithWeight e) {
		if(edgeList == null){
			edgeList = new ArrayList<EdgeWithWeight>();
		}
		edgeList.add(e);
	}
	
	/**
	 * 从图中删除一条边
	 * @param p
	 */
	public void removeEdge(EdgeWithWeight e) {
		for (EdgeWithWeight edge : edgeList) {
			if(e.equals(edge)){
				edgeList.remove(edge);
				break;
			}
		}
	}
	
	/**
	 * 点是否存在于图中
	 * @param p
	 * @return
	 */
	public boolean isPointInGraph(Point p) {
		if(null == pointList || pointList.size() < 1){
			return false;
		}
		for (Point point : pointList) {
			if(p.equals(point)){
				return true;
			}
		}
		return false;
	}
	
	/**
	 * 点是否存在于图中
	 * @param p
	 * @return
	 */
	public boolean isEdgeInGraph(EdgeWithWeight e) {
		if(null == edgeList || edgeList.size() < 1){
			return false;
		}
		for (EdgeWithWeight edge : edgeList) {
			if(e.equals(edge)){
				return true;
			}
		}
		return false;
	}

	public List<Point> getPointList() {
		return pointList;
	}

	public void setPointList(List<Point> pointList) {
		this.pointList = pointList;
	}

	public List<EdgeWithWeight> getEdgeList() {
		return edgeList;
	}

	public void setEdgeList(List<EdgeWithWeight> edgeList) {
		this.edgeList = edgeList;
	}
	
	@Override
	public String toString() {
		return "Graph [pointList=" + pointList + ", edgeList=" + edgeList + "]";
	}
	
	@Override
	public Graph clone() {
		Graph g = new Graph();
		for (Point p : pointList) {
			g.addPoint(p);
		}
		for (EdgeWithWeight e : edgeList) {
			g.addEdge(e);
		}
		return g;
	}
	/**
	 * @param args
	 */
	public static void main(String[] args) {

	}

}

    Prim算法

         Prim算法实现的是找出一个有权重连通图中的最小生成树，即：具有最小权重且连接到所有结点的树。

         首先以一个结点作为最小生成树的初始结点，然后以迭代的方式找出与最小生成树中各结点权重最小边，并加入到最小生成树中。加入之后如果产生回路则跳过这条边，选择下一个结点。当所有结点都加入到最小生成树中之后，就找出了连通图中的最小生成树了。

Java实现：

package com.zas.test.tree;

import java.util.ArrayList;
import java.util.List;

/**
 * Prim算法实现的是找出一个有权重连通图中的最小生成树，即：具有最小权重且连接到所有结点的树。
 * @author zas
 */
public class Prim {
	//一个要找最小生成树的图
	Graph graph;
	
	public Prim(Graph graph) {
		super();
		this.graph = graph;
	}

	public Graph getGraph() {
		return graph;
	}

	public void setGraph(Graph graph) {
		this.graph = graph;
	}
	
	/**
	 * 首先以一个结点作为最小生成树的初始结点，然后以迭代的方式找出与最小生成树中各结点权重最小边，
	 * 并加入到最小生成树中。加入之后如果产生回路则跳过这条边，选择下一个结点。
	 * 当所有结点都加入到最小生成树中之后，就找出了连通图中的最小生成树了。
	 * @return
	 */
	public Graph prim() {
		Graph minTree = new Graph();
		for (Point p : graph.getPointList()) {
			minTree.addPoint(p);
			//获得该点的最小权重边
			EdgeWithWeight edge = getMinWeightEdgeForPoit(p, minTree);
			if(null != edge){
				//添加该边到最小生成树
				minTree.addEdge(edge);
				//检测是否产生回路
				if(isGraphHasCircle(minTree)){
					minTree.removeEdge(edge);
				}
			}
		}
		return minTree;
	}
	
	/**
	 * 检测是否产生回路
	       如果存在回路，则必存在一个子图，是一个环路。环路中所有顶点的度>=2。  
		n算法：   
			第一步：删除所有度<=1的顶点及相关的边，并将另外与这些边相关的其它顶点的度减一。  
     		第二步：将度数变为1的顶点排入队列，并从该队列中取出一个顶点重复步骤一。  
     		如果最后还有未删除顶点，则存在环，否则没有环。  
		n算法分析：   
			由于有m条边，n个顶点。
     		i)如果m>=n，则根据图论知识可直接判断存在环路。（证明：如果没有环路，则该图必然是k棵树 k>=1。根据树的性质，边的数目m = n-k。k>=1，所以：m<n）              
     		ii)如果m<n 则按照上面的算法每删除一个度为0的顶点操作一次（最多n次），或每删除一个度为1的顶点（同时删一条边）操作一次（最多m次）。这两种操作的总数不会超过m+n。由于m<n，所以算法复杂度为O(n)。
	 * @param minTree
	 * @return
	 */
	private boolean isGraphHasCircle(Graph minTree) {
		//若图中没有顶点，或者只有一个顶点则没有回路
		if(minTree.getPointList() == null || minTree.getPointList().size() < 2){
			return false;
		}
		if(minTree.getEdgeList().size() > minTree.getPointList().size()){
			return true;
		}
		
		Graph g = minTree.clone();
		
		int pointsLeftCount = g.getPointList().size();
		while(pointsLeftCount > 0){
			//一次遍历如没有删除一个度小于2的点，则结束循环
			boolean endFlag = true;
			Point pointForRemove = null;
			for (Point p : g.getPointList()) {
				//计算顶点的度
				
				if(getCountForPoint(p, g) <= 1){
					//为了规避最后一个顶点被删除是的并发异常 采用标记删除
					pointForRemove = p;
					//删除之后从新遍历顶点
					endFlag = false;
					break;
				}
			}
			if(endFlag){
				break;
			}else{
				g.removePoint(pointForRemove);
				List<EdgeWithWeight> edgeForRemoveList = new ArrayList<EdgeWithWeight>();
				for (EdgeWithWeight e : g.getEdgeList()) {
					if(isPointInEdge(pointForRemove, e)){
						edgeForRemoveList.add(e);
					}
				}
				for (EdgeWithWeight edgeWithWeight : edgeForRemoveList) {
					g.removeEdge(edgeWithWeight);
				}
			}
			pointsLeftCount = g.getPointList().size();
		}
		if(g.getPointList().size() > 0){
			return true;
		}else{
			return false;
		}
	}

	/**
	 * 计算顶点的度
	 * @param p
	 * @return
	 */
	private int getCountForPoint(Point p, Graph g) {
		int count = 0;
		for (EdgeWithWeight e : g.getEdgeList()) {
			if(isPointInEdge(p, e)){
				count = count + 1;
			}
		}
		return count;
	}

	/**
	 * 获取权重最小边
	 * @param p
	 * @param minTree
	 * @return
	 */
	private EdgeWithWeight getMinWeightEdgeForPoit(Point p, Graph minTree) {
		EdgeWithWeight e = null;
		for (EdgeWithWeight edge : graph.getEdgeList()) {
			if(!minTree.isEdgeInGraph(edge)){
				if(isPointInEdge(p, edge)){
					if(e == null){
						e = edge;
					}else{
						if(e.compareTo(edge) == -1){
							e = edge;
						}
					}
				}
			}
		}
		return e;
	}

	/**
	 * 点是否在边中
	 * @param p
	 * @param edge
	 * @return
	 */
	private boolean isPointInEdge(Point p, EdgeWithWeight edge) {
		if(p.equals(edge.getStartPoint()) || p.equals(edge.getEndPoint())){
			return true;
		}
		return false;
	}

	/**
	 * @param args
	 */
	public static void main(String[] args) {
		//构建一个图
		Graph graph = new Graph();
		
		Point a = new Point("A");
		Point b = new Point("B");
		Point c = new Point("C");
		Point d = new Point("D");
		Point e = new Point("E");
		Point f = new Point("F");
		
		graph.addPoint(a);
		graph.addPoint(b);
		graph.addPoint(c);
		graph.addPoint(d);
		graph.addPoint(e);
		graph.addPoint(f);
		
		//所有边权重相同
		graph.addEdge(new EdgeWithWeight(a, b, new Weight()));
		graph.addEdge(new EdgeWithWeight(a, c, new Weight()));
		graph.addEdge(new EdgeWithWeight(a, d, new Weight()));
		graph.addEdge(new EdgeWithWeight(b, c, new Weight()));
		graph.addEdge(new EdgeWithWeight(b, e, new Weight()));
		graph.addEdge(new EdgeWithWeight(c, d, new Weight()));
		graph.addEdge(new EdgeWithWeight(c, e, new Weight()));
		graph.addEdge(new EdgeWithWeight(c, f, new Weight()));
		graph.addEdge(new EdgeWithWeight(d, f, new Weight()));
		graph.addEdge(new EdgeWithWeight(e, f, new Weight()));
		
		Prim prim = new Prim(graph);
		Graph minTree = prim.prim();
		System.out.println(minTree);
		
		
		Graph graphWithWeight = new Graph();
		
		graphWithWeight.addPoint(a);
		graphWithWeight.addPoint(b);
		graphWithWeight.addPoint(c);
		graphWithWeight.addPoint(d);
		graphWithWeight.addPoint(e);
		graphWithWeight.addPoint(f);
		
		graphWithWeight.addEdge(new EdgeWithWeight(a, b, new Weight(6)));
		graphWithWeight.addEdge(new EdgeWithWeight(a, c, new Weight(1)));
		graphWithWeight.addEdge(new EdgeWithWeight(a, d, new Weight(5)));
		graphWithWeight.addEdge(new EdgeWithWeight(b, c, new Weight(5)));
		graphWithWeight.addEdge(new EdgeWithWeight(b, e, new Weight(3)));
		graphWithWeight.addEdge(new EdgeWithWeight(c, d, new Weight(7)));
		graphWithWeight.addEdge(new EdgeWithWeight(c, e, new Weight(5)));
		graphWithWeight.addEdge(new EdgeWithWeight(c, f, new Weight(4)));
		graphWithWeight.addEdge(new EdgeWithWeight(d, f, new Weight(2)));
		graphWithWeight.addEdge(new EdgeWithWeight(e, f, new Weight(6)));
		
		Prim primForWeigtTree = new Prim(graphWithWeight);
		Graph minTreeForWeightTree = primForWeigtTree.prim();
		System.out.println(minTreeForWeightTree);
	}

}

测试结果

边权重一样的图

Graph [
pointList=[Point [point=A], Point [point=B], Point [point=C], Point [point=D], Point [point=E], Point [point=F]], edgeList=[
EdgeWithWeight [weight=Weight [weight=0.0], startPoint=Point [point=A], endPoint=Point [point=B]], 
EdgeWithWeight [weight=Weight [weight=0.0], startPoint=Point [point=B], endPoint=Point [point=C]], 
EdgeWithWeight [weight=Weight [weight=0.0], startPoint=Point [point=A], endPoint=Point [point=D]], 
EdgeWithWeight [weight=Weight [weight=0.0], startPoint=Point [point=B], endPoint=Point [point=E]], 
EdgeWithWeight [weight=Weight [weight=0.0], startPoint=Point [point=C], endPoint=Point [point=F]]]]

边权重不一样的图

Graph [
pointList=[Point [point=A], Point [point=B], Point [point=C], Point [point=D], Point [point=E], Point [point=F]], edgeList=[
EdgeWithWeight [weight=Weight [weight=1.0], startPoint=Point [point=A], endPoint=Point [point=C]], 
EdgeWithWeight [weight=Weight [weight=3.0], startPoint=Point [point=B], endPoint=Point [point=E]], 
EdgeWithWeight [weight=Weight [weight=4.0], startPoint=Point [point=C], endPoint=Point [point=F]], 
EdgeWithWeight [weight=Weight [weight=2.0], startPoint=Point [point=D], endPoint=Point [point=F]], 
EdgeWithWeight [weight=Weight [weight=5.0], startPoint=Point [point=C], endPoint=Point [point=E]]]]

  以上代码还有重构优化的余地，如：计算顶点的度，获取权重最小边可移入图类中；点是否在边中可移入边类中；图是否有环的代码看起来不够一目了然等。

  另外最小生成树算法还有Kruskal算法；Dijkstra（迪杰斯特拉）算法也可以用来实现最小生成树程序。

  以后有空再实现之。



参考资料：
可从下面链接找到我的测试用例所用的图，以及Prim算法的解释
http://squirrelrao.iteye.com/blog/1044867


可从下面链接找到无向图的环查找算法：
http://blog.csdn.net/sunmenggmail/article/details/7324646