【数据结构】【图论】【最小生成树】Kruskal算法

一、Kruskal算法核心

1. Kruskal算法和Prime算法一样也是计算最小生成树的一种算法。考虑问题的出发点: 为使生成树上边的权值之和达到最小，则应使生成树中每一条边的权值尽可能地小。
2. 具体做法: 先构造一个只含 n 个顶点的子图 SG，然后从权值最小的边开始，若它的添加不使SG 中产生回路，则在 SG 上加上这条边，如此重复，直至加上 n-1 条边为止。
3. 算法演示

 

 二、代码实现（Java版）

import java.util.Comparator;
import java.util.HashMap;
import java.util.Iterator;
import java.util.Map;
import java.util.Scanner;
import java.util.Set;
import java.util.TreeSet;

class Edge {
	int node1;
	int node2;
	int edgeValue;
}

class MySort implements Comparator<Edge> {
	public int compare(Edge o1, Edge o2) {
		return o1.edgeValue > o2.edgeValue ? 1 : -1;
	}

}

public class Kruskal {
	static final int maxNodeValue = (1 << 31) - 1;

	public static void main(String args[]) {
		Scanner scanner = new Scanner(System.in);
		Set<Edge> edges = new TreeSet<Edge>(new MySort());
		Map<Integer, Integer> preNode = new HashMap<Integer, Integer>();
		while (scanner.hasNext()) {
			int edgeCounts = scanner.nextInt();
			for (int i = 0; i < edgeCounts; i++) {
				int node1 = scanner.nextInt();
				int node2 = scanner.nextInt();
				int edgeValue = scanner.nextInt();
				Edge oldEdge = findOldEdge(node1, node2, edges);
				if (oldEdge != null) {
					if (oldEdge.edgeValue > edgeValue) {
						Edge edge = new Edge();
						edge.edgeValue = edgeValue;
						edge.node1 = node1;
						edge.node2 = node2;
						edges.add(edge);
					}
				} else {
					Edge edge = new Edge();
					edge.edgeValue = edgeValue;
					edge.node1 = node1;
					edge.node2 = node2;
					edges.add(edge);
				}
			}
			int cost = kruskal(edges, preNode);
			System.out.println(cost);
			edges.clear();
			preNode.clear();
		}
	}

	private static Edge findOldEdge(int node1, int node2, Set<Edge> edges) {
		Iterator<Edge> iterator = edges.iterator();
		while (iterator.hasNext()) {
			Edge edge = iterator.next();
			if ((edge.node1 == node1 && edge.node2 == node2) || (edge.node1 == node2 && edge.node2 == node1)) {
				return edge;
			}
		}
		return null;
	}

	public static int findRootNode(Integer node, Map<Integer, Integer> preNode) {
		while (preNode.get(node) != null) {
			node = preNode.get(node);
		}
		return node;
	}

	public static int kruskal(Set<Edge> edges, Map<Integer, Integer> preNode) {
		Iterator<Edge> it = edges.iterator();
		int cost = 0;
		while (it.hasNext()) {
			Edge edge = it.next();
			int node1 = edge.node1;
			int node2 = edge.node2;
			int edgeValue = edge.edgeValue;
			int node1_parent=findRootNode(node1, preNode) ;
			int node2_parent=findRootNode(node2, preNode);
			if (node1_parent!=node2_parent) {
				preNode.put(node1_parent, node2_parent);
				cost += edgeValue;
			}
		}
		return cost;
	}
}

 三、测试用例

输入
11
1 2 19
1 5 14
1 7 18
2 3 5
2 4 7
2 5 12
3 4 3
4 5 8
4 6 21
5 7 16
6 7 27

输出
67

 拓扑图和算法筛选过程：

 四、ACM

题目链接：http://acm.sdut.edu.cn/sdutoj/problem.php?action=showproblem&problemid=2144
AC代码：

import java.util.Comparator;
import java.util.HashMap;
import java.util.Iterator;
import java.util.Map;
import java.util.Scanner;
import java.util.Set;
import java.util.TreeSet;

class Edge {
	int node1;
	int node2;
	int edgeValue;
}

class MySort implements Comparator<Edge> {
	public int compare(Edge o1, Edge o2) {
		return o1.edgeValue > o2.edgeValue ? 1 : -1;
	}

}

public class Main{
	static final int maxNodeValue = (1 << 31) - 1;

	public static void main(String args[]) {
		Scanner scanner = new Scanner(System.in);
		Set<Edge> edges = new TreeSet<Edge>(new MySort());
		Map<Integer, Integer> preNode = new HashMap<Integer, Integer>();
		while (scanner.hasNext()) {
			int nodeCounts = scanner.nextInt();
			int edgeCounts = scanner.nextInt();
			for (int i = 0; i < edgeCounts; i++) {
				int node1 = scanner.nextInt();
				int node2 = scanner.nextInt();
				int edgeValue = scanner.nextInt();
				Edge oldEdge = findOldEdge(node1, node2, edges);
				if (oldEdge != null) {
					if (oldEdge.edgeValue > edgeValue) {
						Edge edge = new Edge();
						edge.edgeValue = edgeValue;
						edge.node1 = node1;
						edge.node2 = node2;
						edges.add(edge);
					}
				} else {
					Edge edge = new Edge();
					edge.edgeValue = edgeValue;
					edge.node1 = node1;
					edge.node2 = node2;
					edges.add(edge);
				}
			}
			int cost = kruskal(edges, preNode);
			System.out.println(cost);
			edges.clear();
			preNode.clear();
		}
	}

	private static Edge findOldEdge(int node1, int node2, Set<Edge> edges) {
		Iterator<Edge> iterator = edges.iterator();
		while (iterator.hasNext()) {
			Edge edge = iterator.next();
			if ((edge.node1 == node1 && edge.node2 == node2) || (edge.node1 == node2 && edge.node2 == node1)) {
				return edge;
			}
		}
		return null;
	}

	public static int findRootNode(Integer node, Map<Integer, Integer> preNode) {
		while (preNode.get(node) != null) {
			node = preNode.get(node);
		}
		return node;
	}

	public static int kruskal(Set<Edge> edges, Map<Integer, Integer> preNode) {
		Iterator<Edge> it = edges.iterator();
		int cost = 0;
		while (it.hasNext()) {
			Edge edge = it.next();
			int node1 = edge.node1;
			int node2 = edge.node2;
			int edgeValue = edge.edgeValue;
			int node1_parent=findRootNode(node1, preNode) ;
			int node2_parent=findRootNode(node2, preNode);
			if (node1_parent!=node2_parent) {
				preNode.put(node1_parent, node2_parent);
				cost += edgeValue;
			}
		}
		return cost;
	}
}

        AC代码和上面的代码只有一点不同，那就是输入了节点的个数nodeCount，然而这个数在创建生成树的过程中并没有被使用到。