/*
Name: Prime算法求最小生成树 (邻接表）
Copyright:
Author: 巧若拙
Date: 25/11/14 13:38
Description:
实现了 Prime算法求最小生成树 (邻接表）的普通算法和最小堆优化算法。
*/
#include<stdio.h>
#include<stdlib.h>
#define MAX 2000 //最大顶点数量
#define INFINITY 999999 //无穷大
typedef int VertexType; //顶点类型由用户自定义
typedef int EdgeType; //边上的权值类型由用户自定义
typedef struct EdgeNode{ //边表结点
int adjvex; //邻接点域，存储该顶点对应的下标
EdgeType weight; //权值，对于非网图可以不需要
struct EdgeNode *next; //链域，指向下一个邻接点
} EdgeNode;
typedef struct VertexNode{ //顶点表结点
VertexType data; //顶点域，存储顶点信息
EdgeNode *firstEdge; //边表头指针
} VertexNode;
typedef struct MinHeap{
int num;//存储顶点序号
int w; //存储顶点到最小生成树的距离
} MinHeap; //最小堆结构体
int map[MAX][MAX] = {0};//邻接矩阵存储图信息
int Locate(VertexNode *GL, int u, int v);//判断u，v是否是邻接点
void CreateGraph(VertexNode *GL, int n, int m);//创建邻接表图（人工图）
void CreateGraph_2(VertexNode *GL, int n, int m);//创建邻接表图（随机图）
void PrintGraph(VertexNode *GL, int n);//输出图
void DestroyGL(VertexNode *GL, int n);//销毁图并释放空间
void Prime(VertexNode *GL, int n, int v0);//Prime算法求最小生成树（原始版本）
void Prime_MinHeap(VertexNode *GL, int n, int v0);//Prime算法求最小生成树（优先队列版本）
void BuildMinHeap(MinHeap que[], int n);
void MinHeapSiftDown(MinHeap que[], int n, int pos);
void MinHeapSiftUp(MinHeap que[], int n, int pos);
void ChangeKey(MinHeap que[], int pos, int weight);//将第pos个元素的关键字值改为weight
int SearchKey(MinHeap que[], int pos, int weight);//查找最小堆中关键字值为k的元素下标，未找到则返回-1（非递归）
int ExtractMin(MinHeap que[]);//删除并返回最小堆中具有最小关键字的元素
int main()
{
int i, j, m, n, v0 = 0;
VertexNode GL[MAX];
printf("请输入顶点数量：");
scanf("%d", &n);
printf("\n请输入边数量：");
scanf("%d", &m);
CreateGraph(GL, n, m);//创建邻接矩阵图
PrintGraph(GL, n);//输出图
Prime(GL, n, v0);//Prime算法求最小生成树（原始版本）
Prime_MinHeap(GL, n, v0);//Prime算法求最小生成树（优先队列版本）
return 0;
}
void CreateGraph(VertexNode *GL, int n, int m)//创建一个图
{
int i, u, v, w;
EdgeNode *e;
for (i=0; i<n; i++)//初始化图
{
GL[i].data = i;
GL[i].firstEdge = NULL;
}
for (i=0; i<m; i++) //读入边信息（注意是无向图）
{
scanf("%d%d%d", &u, &v, &w);
e = (EdgeNode*)malloc(sizeof(EdgeNode)); //采用头插法插入边表结点
if (!e)
{
puts("Error");
exit(1);
}
e->next = GL[u].firstEdge;
GL[u].firstEdge = e;
e->adjvex = v;
e->weight = w;
e = (EdgeNode*)malloc(sizeof(EdgeNode)); //采用头插法插入边表结点
if (!e)
{
puts("Error");
exit(1);
}
e->next = GL[v].firstEdge;
GL[v].firstEdge = e;
e->adjvex = u;
e->weight = w;
}
}
void PrintGraph(VertexNode *GL, int n)//输出图
{
int i, j;
EdgeNode *e;
for (i=0; i<n; i++)
{
printf("%d: ", i);
e = GL[i].firstEdge;
while (e)
{
printf("<%d, %d> = %d ", i, e->adjvex, e->weight);
e = e->next;
}
printf("\n");
}
printf("\n");
}
void CreateGraph_2(VertexNode *GL, int n, int m)//创建邻接表图（随机图）
{
int i, j, u, v, w;
EdgeNode *e;
for (i=0; i<n; i++)//初始化图
{
GL[i].data = i;
GL[i].firstEdge = NULL;
}
for (i=1; i<n; i++)//确保是连通图
{
w = rand() % 100 + 1;
e = (EdgeNode*)malloc(sizeof(EdgeNode)); //采用头插法插入边表结点
if (!e)
{
puts("Error");
exit(1);
}
e->next = GL[0].firstEdge;
GL[0].firstEdge = e;
e->adjvex = i;
e->weight = w;
e = (EdgeNode*)malloc(sizeof(EdgeNode)); //采用头插法插入边表结点
if (!e)
{
puts("Error");
exit(1);
}
e->next = GL[i].firstEdge;
GL[i].firstEdge = e;
e->adjvex = 0;
e->weight = w;
}
m -= n - 1;
while (m > 0)
{
for (i=0; i<n; i++)
{
for (j=i+1; j<n; j++)
{
if (rand()%10 == 0) //有10%的概率出现边
{
if (!Locate(GL, i, j))
{
w = rand() % 100 + 1;
e = (EdgeNode*)malloc(sizeof(EdgeNode)); //采用头插法插入边表结点
if (!e)
{
puts("Error");
exit(1);
}
e->next = GL[i].firstEdge;
GL[i].firstEdge = e;
e->adjvex = j;
e->weight = w;
e = (EdgeNode*)malloc(sizeof(EdgeNode)); //采用头插法插入边表结点
if (!e)
{
puts("Error");
exit(1);
}
e->next = GL[j].firstEdge;
GL[j].firstEdge = e;
e->adjvex = i;
e->weight = w;
m--;
if (m == 0)
return;
}
}
}
}
}
}
int Locate(VertexNode *GL, int u, int v)//判断u，v是否是邻接点
{
EdgeNode *e = GL[u].firstEdge;
while (e)
{
if (e->adjvex == v)
return 1;
e = e->next;
}
return 0;
}
void DestroyGL(VertexNode *GL, int n)
{
int i;
EdgeNode *e, *q;
for (i=0; i<n; i++)
{
e = GL[i].firstEdge;
do
{
q = e->next;
free(e);
e = q;
} while (e);
GL[i].firstEdge = NULL;
}
}
void Prime(VertexNode *GL, int n, int v0)//Prime算法求最小生成树（原始版本）
{
int book[MAX] = {0}; //标记该顶点是否已经在路径中
int dic[MAX] = {0}; //存储顶点到最小生成树的距离
int adj[MAX] = {0}; //存储顶点在最小生成树树中的邻接点序号
int min, i, j, k;
EdgeNode *e;
for (i=0; i<n; i++) //初始化工作
{
dic[i] = INFINITY;
adj[i] = v0;
}
dic[v0] = 0;
book[v0] = 1;
for (i=0; i<n; i++) //每趟确定一个新顶点，共n趟
{
min = INFINITY;
k = v0;
for (j=0; j<n; j++)//找出离最小生成树最近的顶点k
{
if (book[j] == 0 && dic[j] < min)
{
min = dic[j];
k = j;
}
}
book[k] = 1;
printf("<%d, %d> = %d ", adj[k], k, dic[k]);
for (e=GL[k].firstEdge; e; e=e->next)//更新与顶点k的邻接点的dic[]值
{
j = e->adjvex;
if (book[j] == 0 && dic[j] > e->weight)
{
dic[j] = e->weight;
adj[j] = k;
}
}
}
min = 0;
for (i=0; i<n; i++) //输出各顶点在最小生成树中的邻接点及边的长度
{
// printf("<%d, %d> = %d\n", adj[i], i, dic[i]);
min += dic[i];
}
printf("最小生成树总长度（权值）为 %d\n", min);
}
void Prime_MinHeap(VertexNode *GL, int n, int v0)//Prime算法求最小生成树（优先队列版本）
{
int book[MAX] = {0}; //标记该城市是否已经在路径中
int dic[MAX] = {0}; //存储顶点到最小生成树的距离
int adj[MAX] = {0}; //存储顶点在最小生成树树中的邻接点序号
MinHeap que[MAX+1];//最小堆用来存储顶点序号和到最小生成树的距离
int min, i, j, k, pos;
EdgeNode *e;
que[0].num = n; //存储最小堆中的顶点数量
for (i=0; i<n; i++) //初始化工作
{
dic[i] = INFINITY;
adj[i] = v0;
que[i+1].num = i;
que[i+1].w = dic[i];
}
dic[v0] = 0;
book[v0] = 1;
BuildMinHeap(que, n);//构造一个最小堆
for (i=0; i<n; i++) //每趟确定一个新顶点，共n趟
{
k = ExtractMin(que);//删除并返回最小堆中具有最小关键字的元素
book[k] = 1; printf("<%d, %d> = %d ", adj[k], k, dic[k]);
for (e=GL[k].firstEdge; e; e=e->next)//更新与顶点k的邻接点的dic[]值
{
j = e->adjvex;
if (book[j] == 0 && dic[j] > e->weight)
{
pos = SearchKey(que, j, dic[j]);
dic[j] = e->weight;
adj[j] = k;
ChangeKey(que, pos, dic[j]);//将第pos个元素的关键字值改为weight
}
}
}
min = 0;
for (i=0; i<n; i++) //输出各顶点在最小生成树中的邻接点及边的长度
{
// printf("<%d, %d> = %d\n", adj[i], i, dic[i]);
min += dic[i];
}
printf("最小生成树总长度（权值）为 %d\n", min);
}
int ExtractMin(MinHeap que[])//删除并返回最小堆中具有最小关键字的元素
{
int pos = que[1].num;
que[1] = que[que[0].num--];
MinHeapSiftDown(que, que[0].num, 1);
return pos;
}
int SearchKey(MinHeap que[], int pos, int weight)//查找最小堆中关键字值为k的元素下标，未找到则返回-1（非递归）
{
int Stack[MAX] = {0};
int i = 1, top = -1;
while ((i <= que[0].num && que[i].w <= weight) || top >= 0)//类似先序遍历二叉树的方式查找
{
if (i <= que[0].num && que[i].w <= weight)
{
if (que[i].w == weight && que[i].num == pos)//权值与顶点序号都必须对应
return i;
Stack[++top] = i;//该结点入栈
i *= 2; //搜索左孩子
}
else
{
i = Stack[top--] * 2 + 1; //结点退栈并搜索右孩子
}
}
return -1;
}
void ChangeKey(MinHeap que[], int pos, int weight)//将第pos个元素的关键字值改为weight
{
if (weight < que[pos].w) //关键字值变小，向上调整最小堆
{
que[pos].w = weight;
MinHeapSiftUp(que, que[0].num, pos);
}
else if (weight > que[pos].w) //关键字值变大，向下调整最小堆
{
que[pos].w = weight;
MinHeapSiftDown(que, que[0].num, pos);
}
}
void MinHeapSiftDown(MinHeap que[], int n, int pos) //向下调整结点
{
MinHeap temp = que[pos];
int child = pos * 2; //指向左孩子
while (child <= n)
{
if (child < n && que[child].w > que[child+1].w) //有右孩子，且右孩子更小些，定位其右孩子
child += 1;
if (que[child].w < temp.w) //通过向上移动孩子结点值的方式，确保双亲小于孩子
{
que[pos] = que[child];
pos = child;
child = pos * 2;
}
else
break;
}
que[pos] = temp; //将temp向下调整到适当位置
}
void MinHeapSiftUp(MinHeap que[], int n, int pos) //向上调整结点
{
MinHeap temp = que[pos];
int parent = pos / 2; //指向双亲结点
if (pos > n) //不满足条件的元素下标
return;
while (parent > 0)
{
if (que[parent].w > temp.w) //通过向下移动双亲结点值的方式，确保双亲小于孩子
{
que[pos] = que[parent];
pos = parent;
parent = pos / 2;
}
else
break;
}
que[pos] = temp; //将temp向上调整到适当位置
}
void BuildMinHeap(MinHeap que[], int n)//构造一个最小堆
{
int i;
for (i=n/2; i>0; i--)
{
MinHeapSiftDown(que, n, i);
}
}