数据结构：最小生成树--Prim算法

                    最小生成树：Prim算法

最小生成树

    给定一无向带权图，顶点数是n，要使图连通只需n-1条边，若这n-1条边的权值和最小，则称有这n个顶点和n-1条边构成了图的最小生成树(minimum-cost spanning tree)。

Prim算法

    Prim算法是解决最小生成树的常用算法。它采取贪心策略，从指定的顶点开始寻找最小权值的邻接点。图G=，初始时S={V0}，把与V0相邻接，且边的权值最小的顶点加入到S。不断地把S中的顶点与V-S中顶点的最小权值边加入，直到所有顶点都已加入到S中。

算法说明

为了方便寻找最小权值的边，构建一最近边结构体CloseEdge：

//最近边
typedef struct closeedge_tag
{
	int adjvex; //邻接点
	int weight; //权值
}CloseEdge;

创建一数组CloseEdge closeedge[n];顶点u属于S，顶点v属于V-S，则closeedge[v].weight=min{weight(u,v)};closeedge[v].adjvex=u;另外设置一bool型的数组add，标记顶点i是否已加入S。结合closeedge和add即可得到当前最小权值边。每当有新的节点加入S时，则需更新closeedge。具体细节看代码。

实例

从V0开始

代码

类定义

#include  
#include
#include
using namespace std;
#define MAXWEIGHT 100
//边
typedef struct edge_tag
{
	int tail;
	int head;
}Edge;
//最近边
typedef struct closeedge_tag
{
	int adjvex; //邻接点
	int weight; //权值
}CloseEdge;
class Graph
{
private:
	//顶点数  
	int numV;
	//边数  
	int numE;
	//邻接矩阵  
	int **matrix;
public:
	Graph(int numV);
	//建图  
	void createGraph(int numE);
	//析构方法  
	~Graph();
	//Prim算法
	void Prim(int);
	int minEdgeVex(CloseEdge*, bool*);
	void updateCloseEdge(CloseEdge*, bool*, int);
	//打印邻接矩阵  
	void printAdjacentMatrix();
	//检查输入  
	bool check(int, int, int);
};

类实现

//构造函数，指定顶点数目
Graph::Graph(int numV)
{
	//对输入的顶点数进行检测
	while (numV <= 0)
	{
		cout << "顶点数有误！重新输入 ";
		cin >> numV;
	}
	this->numV = numV;
	//构建邻接矩阵，并初始化
	matrix = new int*[numV];
	int i, j;
	for (i = 0; i < numV; i++)
		matrix[i] = new int[numV];
	for (i = 0; i < numV; i++)
	for (j = 0; j < numV; j++)
	{
		if (i == j)
			matrix[i][i] = 0;
		else
			matrix[i][j] = MAXWEIGHT;
	}
}
void Graph::createGraph(int numE)
{
	/*
	对输入的边数做检测
	一个numV个顶点的有向图，最多有numV*(numV - 1)条边
	*/
	while (numE < 0 || numE > numV*(numV - 1))
	{
		cout << "边数有问题！重新输入 ";
		cin >> numE;
	}
	this->numE = numE;
	int tail, head, weight, i;
	i = 0;
	cout << "输入每条边的起点(弧尾)、终点(弧头)和权值" << endl;
	while (i < numE)
	{
		cin >> tail >> head >> weight;
		while (!check(tail, head, weight))
		{
			cout << "输入的边不正确！请重新输入 " << endl;
			cin >> tail >> head >> weight;
		}
		//Prim算法主要针对的是无向图
		matrix[tail][head] = weight;
		matrix[head][tail] = weight;
		i++;
	}
}
Graph::~Graph()
{
	int i;
	for (i = 0; i < numV; i++)
		delete[] matrix[i];
	delete[]matrix;
}
/*
Prim算法
求最小生成树
*/
void Graph::Prim(int vertex)
{
	//有numV个顶点的图的最小生成树有numV-1条边
	Edge *edges = new Edge[numV - 1];
	//标记顶点是否加入
	bool *add = new bool[numV];
	memset(add, 0, numV);
	//先把vertex加入
	add[vertex] = true;
	//最近边
	CloseEdge *closeedge = new CloseEdge[numV];
	int i;
	//初始化最近边
	for (i = 0; i < numV; i++)
	{
		closeedge[i].weight = matrix[vertex][i];
		if (!add[i] && matrix[vertex][i] > 0 && matrix[vertex][i] < MAXWEIGHT)
			closeedge[i].adjvex = vertex;
	}
	int v, count = 0;
	while (count < numV - 1)
	{
		//获取最近边的邻接点
		v = minEdgeVex(closeedge, add);
		add[v] = true;
		//把最小权值边依次加入数组edges
		edges[count].tail = closeedge[v].adjvex;
		edges[count].head = v;
		//更新最近边
		updateCloseEdge(closeedge, add, v);
		count++;
	}
	cout << "从顶点 " << vertex << " 开始，最小生成树的边是" << endl;
	for (i = 0; i < count; i++)
		cout << edges[i].tail << "---" << edges[i].head << endl;
	//释放空间
	delete[]edges;
	delete[]add;
	delete[]closeedge;
}
//从closeedge中寻找最小边的邻接顶点
int Graph::minEdgeVex(CloseEdge *closeedge, bool *add)
{
	int i, v, w; 
	v = 0;
	w = MAXWEIGHT;
	for (i = 0; i < numV ; i++)
	if (!add[i] && closeedge[i].weight < w)
	{
		w = closeedge[i].weight;
		v = i;
	}
	return v;
}
//顶点v的加入后，需要更新最近边
void Graph::updateCloseEdge(CloseEdge* closeedge, bool *add, int v)
{
	int i;
	for (i = 0; i < numV; i++)
	if (!add[i] && matrix[v][i] < closeedge[i].weight)
	{
		closeedge[i].adjvex = v;
		closeedge[i].weight = matrix[v][i];
	}
}
//打印邻接矩阵  
void Graph::printAdjacentMatrix()
{
	int i, j;
	cout.setf(ios::left);
	cout << setw(7) << " ";
	for (i = 0; i < numV; i++)
		cout << setw(7) << i;
	cout << endl;
	for (i = 0; i < numV; i++)
	{
		cout << setw(7) << i;
		for (j = 0; j < numV; j++)
			cout << setw(7) << matrix[i][j];
		cout << endl;
	}
}
bool Graph::check(int tail, int head, int weight)
{
	if ((tail == head) || tail < 0 || tail >= numV 
		|| head < 0 || head >= numV
		|| weight <= 0 || weight >= MAXWEIGHT)
		return false;
	return true;
}

主函数

int main()
{
	cout << "******Prim***by David***" << endl;
	int numV, numE;
	cout << "建图..." << endl;
	cout << "输入顶点数 ";
	cin >> numV;
	Graph graph(numV);
	cout << "输入边数 ";
	cin >> numE;
	graph.createGraph(numE);
	cout << endl << "Prim..." << endl;
	/*
	由于输出结果太长，不利于截图，故只打印一半的节点
	要想获得从所有节点开始的最小生成树，修改i的变化范围即可
	*/
	for (int i = 0; i < numV / 2; i++)
		graph.Prim(i);
	system("pause");
	return 0;
}

运行

完整代码下载：Prim算法

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