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，要使图连通仅仅需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经常使用算法。
它採取贪心策略，从指定的顶点開始寻找最小权值的邻接点。图G=<V,E>。初始时S={V0}。把与V0相邻接。且边的权值最小的顶点增加到S。
不断地把S中的顶点与V-S中顶点的最小权值边增加，直到全部顶点都已增加到S中。
算法说明
为了方便寻找最小权值的边，构建一近期边结构体CloseEdge：
//近期边
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是否已增加S。结合closeedge和add就可以得到当前最小权值边。每当有新的节点增加S时。则需更新closeedge。
详细细节看代码。
实例
从V0開始
代码
类定义
#include<iostream>
#include<iomanip>
#include<stack>
using namespace std;
#define MAXWEIGHT 100
//边
typedef struct edge_tag
{
int tail;
int head;
}Edge;
//近期边
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測
while (numV <= 0)
{
cout << "顶点数有误！又一次输入 ";
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)
{
/*
对输入的边数做检測
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。又一次输入 ";
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 << "输入的边不对！
请又一次输入 " << 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];
//标记顶点是否增加
bool *add = new bool[numV];
memset(add, 0, numV);
//先把vertex增加
add[vertex] = true;
//近期边
CloseEdge *closeedge = new CloseEdge[numV];
int i;
//初始化近期边
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)
{
//获取近期边的邻接点
v = minEdgeVex(closeedge, add);
add[v] = true;
//把最小权值边依次增加数组edges
edges[count].tail = closeedge[v].adjvex;
edges[count].head = v;
//更新近期边
updateCloseEdge(closeedge, add, v);
count++;
}
cout << "从顶点 " << vertex << " 開始。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的增加后，须要更新近期边
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;
/*
因为输出结果太长，不利于截图，故仅仅打印一半的节点
要想获得从全部节点開始的最小生成树，改动i的变化范围就可以
*/
for (int i = 0; i < numV / 2; i++)
graph.Prim(i);
system("pause");
return 0;
}执行
完整代码下载：Prim算法
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