最小生成树之Kruskal算法

对于城市交通问题，有一个要解决的问题是如何使n个城市之间在最节省经费的情况下建立交通路线，保证连通n个城市，在所有的线路中选择代价最小的。最小生成树就可以解决这一问题。

要构成最小生成树，必须满足一下两个条件

（1）尽可能选取权值最小的，但是不能构成回路。

（2）选取n-1条恰当的边连接往中的n个顶点。

Kruskal算法的基本思想：对于一个n个顶点的图，从权值最小的边开始，若它的添加不会与集合中已添加的边构成回路，则将该边添加到集合中。依次按照权值递增的顺序，选择合适的边进行添加，如此重复，直到加完n-1条边为止。

<pre name="code" class="cpp">#include<stdlib.h>
#include<string.h>
#include<time.h>
#define N 10			
struct Graph {
	char row[N];
	char col[N];
	int weight;
	struct Graph *next;
};
struct temp{
	char vexnum[N];
	int weight;
};
int k=0;
/*
 *创建链表存储图的顶点和权值
 */
struct Graph *create(int n){
	srand((unsigned)time(NULL));
	struct Graph *pHead=NULL;
	struct Graph *pEnd,*pNew;
	struct Graph *t;
	pEnd = pNew = (struct Graph *)malloc(sizeof(struct Graph));
	printf("Please input a pair of vexnums and a weight(stop when input '0'):\n");
	printf("vexnum1\tvexnum2\tweight\n");
	scanf("%s",&pNew->row);
	scanf("%s",&pNew->col);
	scanf("%d",&pNew->weight);
	while(1){
		k++;
		if(k == 1){
			pNew->next = pHead;
			pEnd = pNew;
			t = pHead = pNew;
		}
		else{
			pNew->next = NULL;
			pEnd->next = pNew;
			pEnd = pNew;
		}
		if(k == n*(n-1)/2){
			break;
		}
		pNew=(struct Graph *)malloc(sizeof(struct Graph));
		scanf("%s",&pNew->row);
		if(strcmp(pNew->row,"0") == 0){
			break;
		}
		scanf("%s",&pNew->col);
		scanf("%d",&pNew->weight);
	}
	return pHead;
}
/*
 *对输入的顶点进行判断，删掉重复的边
 */
Arrange(struct Graph *p){
	struct Graph *q,*s;
	s = p;
	q = p->next;
	while(1){
		while(q != NULL){
			if(strcmp(p->row,q->row) == 0 && strcmp(p->col,q->col) == 0
			|| strcmp(p->row,q->col) == 0 && strcmp(p->col,q->row) == 0){
				p->next = q->next;
				free(q);
				q = p->next;
				continue;
			}
			if(q == NULL)
				break;
			else
				q = q->next;
		}
		s = s->next;
		if(s == NULL)
			break;
		p = s;
		q = p->next;
		
	}
}
/*
 *对每条边的权值按照从小到大的顺序排序
 */
sort(struct Graph *New){  
    char row[N],col[N];  
	int weight;  
    struct Graph *End;  
    while(New->next!=NULL){  
        End = New->next;  
        while(End!=NULL){  
            if(New->weight>End->weight){  
                strcpy(row,New->row);  
                strcpy(New->row,End->row);  
                strcpy(End->row,row);  
                
                strcpy(col,New->col);  
                strcpy(New->col,End->col);  
                strcpy(End->col,col);
                
                weight = New->weight;  
                New->weight = End->weight;  
                End->weight = weight;
            }  
            End = End->next;  
        }  
        New = New->next;  
    }
}  
print(struct Graph *p){
	printf("创建的图为\n");
	while(p != NULL){
		printf("%s---%s\t",p->row,p->col);
		printf("%d\n",p->weight);
		p = p->next;
	}
}
/*
 *从最小的权值对应的边开始，若它的添加不会与集合中已添加的边构成回路，则将该边添加到集合中。
 */
found(struct Graph *p,int j){
	<pre name="code" class="java">int n,num = 0;
	int x=0,y=0;
	int count = 0;<pre name="code" class="java"><span style="white-space:pre">	</span>int temp = 0;

int rowCount = 0,colCount = 0;struct temp graph[k];printf("最小生成树的边为：\n");while(p != NULL){if(num == 0){printf("%s---%s\n",p->row,p->col);strcpy(graph[num].vexnum,p->row);strcpy(graph[num+1].vexnum,p->col);graph[num].count = graph[num+1].count = count;count ++;num += 2;p = p->next;continue;}for(n = 0;n < num;n ++){if(strcmp(p->row,graph[n].vexnum) == 0){rowCount = n;x=1;break;}}for(n = 0;n < num;n ++){if(strcmp(p->col,graph[n].vexnum) == 0){colCount = n;y=1;break;}}if(x == 0 && y == 0){strcpy(graph[num].vexnum,p->row);strcpy(graph[num+1].vexnum,p->col);graph[num].count = graph[num+1].count = count;num += 2;count ++;printf("%s---%s\n",p->row,p->col);p = p->next;} else if(x == 1 && y == 0){strcpy(graph[num].vexnum,p->col);graph[num].count = graph[rowCount].count;num++;printf("%s---%s\n",p->row,p->col);p = p->next;} else if(x == 0 && y == 1){strcpy(graph[num].vexnum,p->row);graph[num].count = graph[colCount].count;num++;printf("%s---%s\n",p->row,p->col);p = p->next;} else if(x == 1 && y == 1 && graph[rowCount].count != graph[colCount].count){

<span style="white-space:pre">			</span>temp = graph[colCount].count

for(n = 0 ; n < num ; n++){if(graph[n].count == temp){graph[n].count = graph[rowCount].count;}}printf("%s---%s\n",p->row,p->col);p = p->next;} else {x = y = 0;p = p->next;continue;}x = y = 0;

<pre name="code" class="java">	} 
}
main(){
	struct Graph *Head;
	int i;
	printf("Please input the number of the vexnum\n");
	scanf("%d",&i);
	while(i == 1 || i == 0){
		printf("Error, please input again\n");
		scanf("%d",&i);
	}
	Head = create(i);
	Arrange(Head);
	sort(Head);
	print(Head);
	found(Head,i);
}