C++ 实现无向图的最小生成树Prim算法(附完整代码)

Posted 2021-05-16 Wecccccccc

实现Prim算法，需设置两个辅助一维数组lowcost和closevertex。

1. 其中lowcost用来保存集合V-U中各顶点与集合U中各顶点构成的边中具有最小权值的边的权值；
2. 数组closevertex用来保存依附于该边的在集合U中的顶点。

过程:
假设初始状态时，U={u0}（u0为出发的顶点），这时有lowcost[0]=0，它表示顶点u0已加入集合U中，数组lowcost的其他各分量的值是顶点u0到其余各顶点所构成的直接边的权值。
然后不断选取权值最小的边（ui,uk）（ui∈U,uk∈V-U），每选取一条边，就将lowcost（k）置为0，表示顶点uk已加入集合U中。
由于顶点uk从集合V-U进入集合U后，这两个集合的内容发生了变化，就需依据具体情况更新数组lowcost和closevertex中部分分量的内容。
最后closevertex中即为所建立的最小生成树。

当无向网采用二维数组存储的邻接矩阵存储时，Prim算法的C++实现算法如下:

void MGraph::Min_Tree_Prim(int u)
{
	for (int i = 0; i < vertexnum; i++)
		if (i != u)
		{
			close_edge[i].adjvertex = u;
			close_edge[i].lowcost = arcs[u][i];
		}
	close_edge[u].lowcost = 0;
	int k;
	for (int i = 0; i < vertexnum - 1; i++)
	{
		int w = MAXW;
		for (int j = 0; j < vertexnum; j++)
		{
			if (close_edge[j].lowcost != 0 && close_edge[j].lowcost < w)
			{
				w = close_edge[j].lowcost;
				 k = j;
			}
		}
		close_edge[k].lowcost = 0;
		for (int j = 0;j<vertexnum;j++)
			if (arcs[k][j] < close_edge[j].lowcost)
			{
				close_edge[j].adjvertex = k;
				close_edge[j].lowcost = arcs[k][j];
			}
	}
	for (int i = 0; i < vertexnum; i++)
		if (i != u)
			cout << i << "->" << close_edge[i].adjvertex << "," << arcs[i][close_edge[i].adjvertex] << endl;
}

完整代码如下:

#include <iostream>
#include <queue>
using namespace std;
typedef int VertexType;
typedef int EdgeType;
const int MaxVertexNum = 30;
const int MAXW = 1e8;

class closEdge
{
	friend class MGraph;
private:
	int adjvertex;
	int lowcost;
};


class MGraph
{
public:
	MGraph()
	{
		CreatGraph();
	};
	void Min_Tree_Prim(int u);
	void CreatGraph();
	void Visit(int v);
	void BFS(int v);
	void BFStraverse();
	void DFStraverse();
	bool Ispath_BFS(int i, int j);//判断结点i和结点j之间是否有路径
	bool Ispath_DFS(int i, int j);//判断结点i和结点j之间是否有路径
	void Init_vis();
private:
	void DFS(int i, int j, bool &flag);
	void dfs_graph(int i);
	VertexType vertexs[MaxVertexNum];
	EdgeType arcs[MaxVertexNum][MaxVertexNum];
	int vertexnum;
	int edgenum;
	closEdge close_edge[MaxVertexNum];
	bool vis[MaxVertexNum];
};


void MGraph::Min_Tree_Prim(int u)
{
	for (int i = 0; i < vertexnum; i++)
		if (i != u)
		{
			close_edge[i].adjvertex = u;
			close_edge[i].lowcost = arcs[u][i];
		}
	close_edge[u].lowcost = 0;
	int k;
	for (int i = 0; i < vertexnum - 1; i++)
	{
		int w = MAXW;
		for (int j = 0; j < vertexnum; j++)
		{
			if (close_edge[j].lowcost != 0 && close_edge[j].lowcost < w)
			{
				w = close_edge[j].lowcost;
				 k = j;
			}
		}
		close_edge[k].lowcost = 0;
		for (int j = 0;j<vertexnum;j++)
			if (arcs[k][j] < close_edge[j].lowcost)
			{
				close_edge[j].adjvertex = k;
				close_edge[j].lowcost = arcs[k][j];
			}
	}
	for (int i = 0; i < vertexnum; i++)
		if (i != u)
			cout << i << "->" << close_edge[i].adjvertex << "," << arcs[i][close_edge[i].adjvertex] << endl;
}

void MGraph::CreatGraph()
{
	Init_vis();
	cout << "请输入图的顶点个数和边的条数" << endl;
	cin >> vertexnum >> edgenum;
	cout << "请依次输入按序号0到n顶点的中存储的信息" << endl;
	for (int i = 0; i < vertexnum; i++) cin >> vertexs[i];
	for (int i = 0; i < vertexnum; i++)
		for (int j = 0; j < vertexnum; j++)
			arcs[i][j] = MAXW;
	cout << "请输入边的信息(该图以有向图的邻接矩阵存储方式存储)" << endl;
	for (int i = 0; i < edgenum; i++)
	{
		int a1, a2, w;
		cout << "输入边<i,j>对应的顶点序号i,j,然后再输入该边的权值" << endl;
		cin >> a1 >> a2>>w;
		arcs[a1][a2] = w;
		arcs[a2][a1] = w;
	}
}

void MGraph::Init_vis()
{
	for (int i = 0; i < MaxVertexNum; i++) vis[i] = false;
}

void MGraph::Visit(int v)
{
	cout << vertexs[v] << " ";
}

void MGraph::BFS(int v)
{
	queue<int >q;
	q.push(v);
	vis[v] = true;
	while (q.size())
	{
		int t = q.front();
		Visit(t);
		q.pop();
		for (int i = 0; i < vertexnum; i++)
		{
			if (arcs[t][i] == 1 && vis[i] == false)
			{
				vis[i] = true;
				q.push(i);
			}
		}
	}
	cout << endl;
	Init_vis();
}

void MGraph::BFStraverse()
{
	queue<int >q;
	for (int i = 0; i < vertexnum; i++)
	{
		if (vis[i] == false)
		{
			vis[i] = true;
			q.push(i);
			while (q.size())
			{
				int t = q.front();
				Visit(t);
				q.pop();
				for (int j = 0; j < vertexnum; j++)
					if (arcs[t][j] == 1 && vis[j] == false)
					{
						vis[j] = true;
						q.push(j);
					}
			}
		}
	}
	cout << endl;
	Init_vis();
}


void MGraph::dfs_graph(int i)
{
	Visit(i);
	for (int j = 0; j < vertexnum; j++)
	{
		if (arcs[i][j] == 1 && vis[j] == false)
		{
			vis[j] = true;
			dfs_graph(j);
		}
	}
}

void MGraph::DFStraverse()
{
	for (int i = 0; i < vertexnum; i++)
	{
		if (vis[i] == false)
		{
			vis[i] = true;
			dfs_graph(i);
		}
	}
	cout << endl;
	Init_vis();
}




bool MGraph::Ispath_BFS(int i, int j)
{
	queue<int >q;
	vis[i] = true;
	q.push(i);
	while (q.size())
	{
		int t = q.front();
		q.pop();
		if (t == j)
		{
			Init_vis();
			return true;
		}
		for (int k = 0; k < vertexnum; k++)
		{
			if (arcs[t][k] == 1 && vis[k] == false)
			{
				vis[k] = true;
				q.push(k);
			}
		}
	}
	Init_vis();
	return false;
}



void MGraph::DFS(int i, int j, bool &flag)
{
	if (i == j)
	{
		flag = true;
		return;
	}
	for (int k = 0; k < vertexnum; k++)
	{
		if (arcs[i][k] == 1 && vis[k] == false)
		{
			vis[k] = true;
			DFS(k, j, flag);
			vis[k] = false;
		}
	}
}

bool MGraph::Ispath_DFS(int i, int j)
{
	bool flag = false;
	vis[i] = true;
	DFS(i, j, flag);
	Init_vis();
	if (flag) return true;
	else return false;
}


int main()
{
	MGraph g;
	int v;
	cin >> v;
	g.BFS(v);
	g.BFStraverse();
	g.DFStraverse();
	
	int n;
	cin >> n;
	g.Min_Tree_Prim(n);

	return 0;
}

测试结果: