最小生成树---普里姆算法（Prim算法)和克鲁斯卡尔算法（Kruskal算法）

普里姆算法（Prim算法)

#include<bits/stdc++.h>
using namespace std;
#define MAXVEX 100
#define INF 65535
typedef char VertexType;
typedef int EdgeType;
typedef struct {
	VertexType vexs[MAXVEX];
	EdgeType arc[MAXVEX][MAXVEX];
	int numVertexes, numEdges;
}MGraph;

void CreateMGraph(MGraph *G) {
	int m, n, w;	//vm-vn的权重w
	scanf("%d %d", &G->numVertexes, &G->numEdges);

	for(int i = 0; i < G->numVertexes; i++) {
		getchar();
		scanf("%c", &G->vexs[i]);
	}

	for(int i = 0; i < G->numVertexes; i++) {
		for(int j = 0; j < G->numVertexes; j++) {
			if(i == j)	G->arc[i][j] = 0;
			else	G->arc[i][j] = INF;
		}
	}
	for(int k = 0; k < G->numEdges; k++) {
		scanf("%d %d %d", &m, &n, &w);
		G->arc[m][n] = w;
		G->arc[n][m] = G->arc[m][n];
	}
}

void MiniSpanTree_Prim(MGraph G) {
	int min, j, k;
	int arjvex[MAXVEX];		//最小边在 U集合中的那个顶点的下标
	int lowcost[MAXVEX];	// 最小边上的权值
	//初始化,从点 V0开始找最小生成树T
	arjvex[0] = 0;		//arjvex[i] = j表示 V-U中集合中的 Vi点的最小边在U集合中的点为 Vj
	lowcost[0] = 0;		//lowcost[i] = 0表示将点Vi纳入集合 U ，lowcost[i] = w表示 V-U中 Vi点到 U的最小权值
	for(int i = 1; i < G.numVertexes; i++) {
		lowcost[i] = G.arc[0][i];
		arjvex[i] = 0;
	}
	//根据最小生成树的定义：从n个顶点中，找出 n - 1条连线，使得各边权值最小
	for(int i = 1; i < G.numVertexes; i++) {
		min = INF, j = 1, k = 0;
		//寻找  V-U到 U的最小权值min
		for(j; j < G.numVertexes; j++) {
			// lowcost[j] != 0保证顶点在 V-U中，用k记录此时的最小权值边在 V-U中顶点的下标
			if(lowcost[j] != 0 && lowcost[j] < min) {
				min = lowcost[j];
				k = j;
			}
		}
	}
	printf("V[%d]-V[%d] weight = %d\n", arjvex[k], k, min);
	lowcost[k] = 0;		//表示将Vk纳入 U
	//查找邻接矩阵Vk行的各个权值，与lowcost的对应值进行比较，若更小则更新lowcost，并将k存入arjvex数组中
	for(int i = 1; i < G.numVertexes; i++) {
		if(lowcost[i] != 0 && G.arc[k][i] < lowcost[i]) {
			lowcost[i] = G.arc[k][i];
			arjvex[i] = k;
		}
	} 

}

int main() {
	MGraph *G = (MGraph *)malloc(sizeof(MGraph));
	CreateMGraph(G);
	MiniSpanTree_Prim(*G);
} 

/*
input:
	4 5
	a
	b
	c
	d
	0 1 2
	0 2 2
	0 3 7
	1 2 4
	2 3 8

output:
	V[0]-V[1] weight = 2
	V[0]-V[2] weight = 2
	V[0]-V[3] weight = 7
	最小总权值: 11
*/

时间复杂度O(n^2)

克鲁斯卡尔算法（Kruskal算法）

#include<bits/stdc++.h>
using namespace std;
#define MAXVEX 100
#define MAXEDGE 100
#define INF 65535
typedef char VertexType;
typedef int EdgeType;
//图的邻接矩阵存储结构的定义
typedef struct {
	VertexType vexs[MAXVEX];
	EdgeType arc[MAXVEX][MAXVEX];
	int numVertexes, numEdges;
}MGraph;

//边集数组Edge结构的定义
typedef struct {
	int begin;
	int end;
	int weight;
}Edge;

bool vis[100][100];

void CreateMGraph(MGraph *G) {
	int m, n, w;	//vm-vn的权重w
	scanf("%d %d", &G->numVertexes, &G->numEdges);

	for(int i = 0; i < G->numVertexes; i++) {
		getchar();
		scanf("%c", &G->vexs[i]);
	}

	for(int i = 0; i < G->numVertexes; i++) {
		for(int j = 0; j < G->numVertexes; j++) {
			if(i == j)	G->arc[i][j] = 0;
			else	G->arc[i][j] = INF;
		}
	}
	for(int k = 0; k < G->numEdges; k++) {
		scanf("%d %d %d", &m, &n, &w);
		G->arc[m][n] = w;
		G->arc[n][m] = G->arc[m][n];
	}
}

void MiniSpanTree_Kruskal(MGraph G) {
	int v1, v2, vs1, vs2;
	Edge edges[MAXEDGE];
	int parent[MAXVEX]; //标记各顶点所属的连通分量，用于判断边与边是否形成环路 

	//将邻接矩阵转换成按权值从小到大排序的边集数组
	/*

	*/
	int tmp = 0, m, n, ans;
	ans = (G.numVertexes*G.numVertexes) / 2 - G.numVertexes / 2;
	for(int k = 0; k < ans; k++) {
		int min = INF, i, j;
		for(i = 0; i < G.numVertexes; i++) {
			for(j = 0; j < G.numVertexes; j++) {
				if(!vis[i][j] && i < j && min > G.arc[i][j]) {
					min = G.arc[i][j];
					m = i;
					n = j;
				}
			}
		}
		if(G.arc[i][j] == INF)
			continue;
		edges[tmp].begin = m;
		edges[tmp].end = n;
		edges[tmp].weight = min;
		vis[m][n] = true;
		tmp++;
	}	

	//初始化为各顶点各自为一个连通分量
	for(int i = 0; i < G.numVertexes; i++)
		parent[i] = i;

	for(int i = 0; i < G.numEdges; i++) {
		//起点终点下标
		v1 = edges[i].begin;
		v2 = edges[i].end;
		//起点终点连通分量
		vs1 = parent[v1];
		vs2 = parent[v2];
		//边的两个顶点属于不同的连通分量，打印，将新来的连通分量更改为起始点的连通分量
		if(vs1 != vs2) {
			printf("V[%d]-V[%d] weight:%d\n", edges[i].begin, edges[i].end, edges[i].weight);
			for(int j = 0; j < G.numVertexes; j++) {
				if(parent[j] == vs2)	parent[j] = vs1;
			}
		}
	}
}

int main() {
	MGraph *G = (MGraph *)malloc(sizeof(MGraph));
	CreateMGraph(G);
	MiniSpanTree_Kruskal(*G);
} 

/*
input:
	4 5
	a
	b
	c
	d
	0 1 2
	0 2 2
	0 3 7
	1 2 4
	2 3 8

output:
	V[0]-V[1] weight = 2
	V[0]-V[2] weight = 2
	V[0]-V[3] weight = 7
	最小总权值: 11
*/

时间复杂度O(elog₂e) e为边数