图的最小生成树(MST)之Kruskal算法

克鲁斯卡尔（Kruskal）算法（只与边相关）

        Kruskal算法通过每次向当前最小代价生成树T中加入一条边的方法构造最终的最小生成树T。算法按照边的权值非递减顺序选取边，并将其加入T中。如果所选取的边与T中的边不形成环路，则这条边加入T中。由于图G是联通的，且具有n个顶点（n>0），所以最终恰好选取n-1条边加入T中。

实现代码：

#include<iostream>
#include<cstring>
#include<string>
#include<cstdio>
#include<algorithm>
using namespace std;
#define MAX 50
int parent[MAX], family[MAX];
int vertex_num, edge_num;

typedef struct Edge //Edge info storage
{
	int a;
	int b;
	int weight;
};

bool cmp(const Edge & a, const Edge & b)
{
	return a.weight < b.weight;
}

int find(int x) 
{
	/* if vertex x is single : parent[x] = x family[x] = 1
	if vertex x is attached to another vertex y: 
	parent[x] = y  , family[y] = family[y] + family[x]   */
	for(;parent[x]!=x;x=parent[x])
		;
	return x; 
}

bool join(int x, int y) //join two vertex set
{
	/* vertex set join : if parent[x] = parent[y]
	vertex x and vertex y are in the same set  */
	int root1, root2;
	root1 = find(x);
	root2 = find(y);
	if(root1 == root2) //in the same vertex set
		return false;
	else if(family[root1] >= family[root2])
		{
			parent[root2] = root1;
			family[root1] += family[root2];
		}
		else
		{
			parent[root1] = root2;
			family[root2] += family[root1];
		}
	return true;
}

int main()
{
	int ltotal, sum, flag;
	Edge edge[MAX];

	scanf("%d%d", &vertex_num, &edge_num);
	ltotal = 0, sum = 0, flag = 0;
	for(int i = 1; i <= vertex_num; ++i) //initialize parent[] and family[] 
	{
		parent[i] = i;
		family[i] = 1;
	}
	for(int i = 1; i <= edge_num ; ++i)
	{
		scanf("%d%d%d", &edge[i].a, &edge[i].b, &edge[i].weight);
	}
	sort(edge + 1, edge + 1 + edge_num, cmp); //sort edge by weight of each edge
	for(int i = 1; i <= edge_num; ++i)
	{
		if(join(edge[i].a, edge[i].b))
		{
			ltotal++; // edge num increment by 1
			sum += edge[i].weight; //total weight 
			cout<<edge[i].a<<"->"<<edge[i].b<<endl;
		}
		if(ltotal == vertex_num - 1) //MST if edge num = vertex num - 1
		{
			flag = 1;
			break;
		}
	}
	if(flag) printf("%d\n", sum);
	else printf("data error.\n");
	
	return 0;
}

输入数据：

7 9
1 2 28
1 6 10
2 3 16
2 7 14
3 4 12
4 5 22
4 7 18
5 6 25
5 7 24

输入数据代表的图及Kruskal算法的结果图：

输出数据：

1->6
3->4
2->7
2->3
4->5
5->6
99

REF：

1，http://blog.youkuaiyun.com/niushuai666/article/details/6689285

2，数据结构（C语言版） Ellis Horowitz