给定一个图的数据结构G=(V,E)是一个具有含权边的连通无向图，它的一棵生成树(V,T)是G最为树的子图。若(V,T)满足T的所有边权值相加为最小值，那么这样的生成树称为最小（耗费）生成树。
计算一个给定加权连通图最小生成树的一个可行方法叫做PRim算法，它是从一个任一顶点开始生长生成树。其基本步骤如下：
设G=(V,E)，为了方便起见，V取整数集合{1，2，3，……，n}。建立两个集合X={1}，Y={2，3，……，n}。从顶点1开始生长一棵生成树，每次一条边。在每一步（循环）中，找出权值最小的边（x，y），这里x∈X，y∈Y。然后把y从Y中移至X中，并把这条边添加到当前最小生成树边集T中。当然，T最开始是空的。重复第3~4步，直至Y为空集。此时得到的边集T即为最小生成树的所有边集合。我们在上一节用邻接矩阵实现图的基础上来实现Prim算法。在上一节的基础上，添加了辅助类Edge用以表示边这种数据类型。另外在Graph类的基础上，添加了成员Edge *m_pEgde用以表示上述T集合。
下面先给出一个连通无向图的简单例子，我们将基于这个例子来做测试。 
下面先给出代码实现及测试代码：
#include<iostream>#include<cstdlib>#include<vector>using namespace std;/*节点类*/class Node{public: Node(char identifier = 0); char m_identifier; //顶点编号 bool m_isVisited; //顶点访问标志位：true表示已经被访问};Node::Node(char identifier){ m_identifier = identifier; m_isVisited = false;}/*边类，用于辅助实现生成最小生成树*/class Edge{public: Edge(int NodeIndexA = 0, int NodeIndexB = 0, int WeightValue = 0); int m_NodeIndexA, m_NodeIndexB; //边的两端点（索引），这里以无向图为例 int m_weightValue; //边的权值 bool m_selected; //选择标志位，true表示已被选择};Edge::Edge(int NodeIndexA, int NodeIndexB, int WeightValue){ m_NodeIndexA = NodeIndexA; m_NodeIndexB = NodeIndexB; m_weightValue = WeightValue; m_selected = false; //初始时未被选择}/*图类*/class Graph{public: Graph(int capacity); ~Graph(); int getGraphSize(); //获取当前图的大小 void resetNode(); //重置所有顶点的访问标志位为false，未访问 bool addNode(Node *pNode); //添加新顶点 bool addEdgeForUndirectedGraph(int row, int col, int val = 1); //添加边以构造无向图，val表示权值，默认权值1 bool addEdgeForDirectedGraph(int row, int col, int val = 1); //添加边以构造有向图，val表示权值，默认权值1 void printMatrix(); //打印邻接矩阵 void depthFirstTraverse(int nodeIndex); //深度优先遍历，指定第一个点 void widthFirstTraverse(int nodeIndex); //广度优先遍历，指定第一个点 void MSTPrim(int nodeIndex); //Prim算法求最小生成树，指定第一个点private: bool getValueOfEdge(int row, int col, int &val); //获取边权值 void widthFirstTraverseImplement(vector<int> preVec); //利用vector实现广度优先遍历 int getMinEdge(vector<Edge> edgeVec); //Prim算法辅助函数，用于在边集中选择权值最小的边 int m_iCapacity; //图容量，即申请的数组空间最多可容纳的顶点个数 int m_iNodeCount; //图的现有顶点个数 Node *m_pNodeArray; //存放顶点的数组 int *m_pMatrix; //为了方便，用一维数组存放邻接矩阵 Edge *m_pEgde; //边指针，存储最小生成树的边};Graph::Graph(int capacity){ m_iCapacity = capacity; m_iNodeCount = 0; m_pNodeArray = new Node[m_iCapacity]; m_pMatrix = new int[m_iCapacity*m_iCapacity]; for (int i = 0;i < m_iCapacity*m_iCapacity;i++) //初始化邻接矩阵 { m_pMatrix[i] = 0; } m_pEgde = new Edge[m_iCapacity - 1]; //最小生成树节点和边数量关系}Graph::~Graph(){ delete []m_pNodeArray; delete []m_pMatrix; delete []m_pEgde;}int Graph::getGraphSize(){ return m_iNodeCount;}void Graph::resetNode(){ for (int i = 0;i < m_iNodeCount;i++) { m_pNodeArray[i].m_isVisited = false; }}bool Graph::addNode(Node *pNode){ if (pNode == NULL) return false; m_pNodeArray[m_iNodeCount].m_identifier = pNode->m_identifier; m_iNodeCount++; return true;}bool Graph::addEdgeForUndirectedGraph(int row, int col, int val){ if (row < 0 || row >= m_iCapacity) return false; if (col < 0 || col >= m_iCapacity) return false; m_pMatrix[row*m_iCapacity + col] = val; m_pMatrix[col*m_iCapacity + row] = val; return true;}bool Graph::addEdgeForDirectedGraph(int row, int col, int val){ if (row < 0 || row >= m_iCapacity) return false; if (col < 0 || col >= m_iCapacity) return false; m_pMatrix[row*m_iCapacity + col] = val; return true;}void Graph::printMatrix(){ for (int i = 0;i < m_iCapacity;i++) { for (int k = 0;k < m_iCapacity;k++) cout << m_pMatrix[i*m_iCapacity + k] << " "; cout << endl; }}void Graph::depthFirstTraverse(int nodeIndex){ int value = 0; //访问第一个顶点 cout << m_pNodeArray[nodeIndex].m_identifier << " "; m_pNodeArray[nodeIndex].m_isVisited = true; //访问其他顶点 for (int i = 0;i < m_iCapacity;i++) { getValueOfEdge(nodeIndex, i, value); if (value != 0) //当前顶点与指定顶点连通 { if (m_pNodeArray[i].m_isVisited == true) //当前顶点已被访问 continue; else //当前顶点没有被访问，则递归 { depthFirstTraverse(i); } } else //没有与指定顶点连通 { continue; } }}void Graph::widthFirstTraverse(int nodeIndex){ //访问第一个顶点 cout << m_pNodeArray[nodeIndex].m_identifier << " "; m_pNodeArray[nodeIndex].m_isVisited = true; vector<int> curVec; curVec.push_back(nodeIndex); //将第一个顶点存入一个数组 widthFirstTraverseImplement(curVec);}void Graph::widthFirstTraverseImplement(vector<int> preVec){ int value = 0; vector<int> curVec; //定义数组保存当前层的顶点 for (int j = 0;j < (int)preVec.size();j++) //依次访问传入数组中的每个顶点 { for (int i = 0;i < m_iCapacity;i++) //传入的数组中的顶点是否与其他顶点连接 { getValueOfEdge(preVec[j], i, value); if (value != 0) //连通 { if (m_pNodeArray[i].m_isVisited==true) //已经被访问 { continue; } else //没有被访问则访问 { cout << m_pNodeArray[i].m_identifier << " "; m_pNodeArray[i].m_isVisited = true; //保存当前点到数组 curVec.push_back(i); } } } } if (curVec.size()==0) //本层次无被访问的点，则终止 { return; } else { widthFirstTraverseImplement(curVec); }}bool Graph::getValueOfEdge(int row, int col, int &val){ if (row < 0 || row >= m_iCapacity) return false; if (col < 0 || col >= m_iCapacity) return false; val = m_pMatrix[row*m_iCapacity + col]; return true;}void Graph::MSTPrim(int nodeIndex){ int value = 0; //存储当前边的权值 int edgeCount = 0; //已选出的边数量，用以判断算法终结 vector<int> nodeVec; //存储点（索引）集的数组 vector<Edge> edgeVec; //存储边的数组 cout << m_pNodeArray[nodeIndex].m_identifier << endl; nodeVec.push_back(nodeIndex); m_pNodeArray[nodeIndex].m_isVisited = true; while (edgeCount < m_iCapacity - 1) { int temp = nodeVec.back(); //将当前顶点索引复制给temp for (int i = 0;i < m_iCapacity;i++) //循环判断每一个顶点与当前顶点连接情况 { getValueOfEdge(temp, i, value); if (value != 0) //连通 { if (m_pNodeArray[i].m_isVisited == true) //已经被访问 continue; else //未被访问,则将边放入被选边集合 { Edge edge(temp, i, value); edgeVec.push_back(edge); } } } /*选择最小边*/ int edgeIndex = getMinEdge(edgeVec); if (edgeIndex == -1) { cout << "获取最小边失败，请重置后再试！" << endl; break; } edgeVec[edgeIndex].m_selected = true; //设置选择标志位为true，已被选择 cout << edgeVec[edgeIndex].m_NodeIndexA << "---" << edgeVec[edgeIndex].m_NodeIndexB<<" "; cout << edgeVec[edgeIndex].m_weightValue << endl; m_pEgde[edgeCount] = edgeVec[edgeIndex]; edgeCount++; /*寻找当前选择的最小边相连的下一个顶点*/ int nextNodeIndex = edgeVec[edgeIndex].m_NodeIndexB; nodeVec.push_back(nextNodeIndex); m_pNodeArray[nextNodeIndex].m_isVisited = true; cout << m_pNodeArray[nextNodeIndex].m_identifier << endl; } cout << "最小生成树计算完毕，如上所示！" << endl;}int Graph::getMinEdge(vector<Edge> edgeVec){ int minWeight = 0; //用于辅助选择最小权值边 int edgeIndex = 0; //用于存储最小边索引 int i = 0; /*找出第一条未被选择的边*/ for (;i < (int)edgeVec.size();i++) { if (edgeVec[i].m_selected == false) //当前边未被选择 { minWeight = edgeVec[i].m_weightValue; edgeIndex = i; break; } } if (minWeight == 0) //边集所有边被访问 { return -1; } for (;i < (int)edgeVec.size();i++) { if (edgeVec[i].m_selected == true) continue; else { if (minWeight > edgeVec[i].m_weightValue) { minWeight = edgeVec[i].m_weightValue; edgeIndex = i; } } } return edgeIndex;}int main(){ Graph *pGraph = new Graph(6); cout << "初始化顶点中……" << endl; Node *pNodeA = new Node('A'); Node *pNodeB = new Node('B'); Node *pNodeC = new Node('C'); Node *pNodeD = new Node('D'); Node *pNodeE = new Node('E'); Node *pNodeF = new Node('F'); cout << "添加顶点至图中……" << endl; pGraph->addNode(pNodeA); pGraph->addNode(pNodeB); pGraph->addNode(pNodeC); pGraph->addNode(pNodeD); pGraph->addNode(pNodeE); pGraph->addNode(pNodeF); pGraph->addEdgeForUndirectedGraph(0, 1, 6); pGraph->addEdgeForUndirectedGraph(0, 4, 5); pGraph->addEdgeForUndirectedGraph(0, 5, 1); pGraph->addEdgeForUndirectedGraph(1, 5, 2); pGraph->addEdgeForUndirectedGraph(1, 2, 3); pGraph->addEdgeForUndirectedGraph(2, 5, 8); pGraph->addEdgeForUndirectedGraph(2, 3, 7); pGraph->addEdgeForUndirectedGraph(3, 5, 4); pGraph->addEdgeForUndirectedGraph(3, 4, 2); pGraph->addEdgeForUndirectedGraph(4, 5, 9); cout << "邻接矩阵如下:" << endl; pGraph->printMatrix(); cout << endl << endl; cout << "深度优先遍历：" << endl; pGraph->depthFirstTraverse(0); cout << endl << endl; pGraph->resetNode(); cout << "广度优先遍历：" << endl; pGraph->widthFirstTraverse(0); cout << endl << endl; pGraph->resetNode(); cout << "最小生成树为：" << endl; pGraph->MSTPrim(0); cout << endl; system("pause");} 程序运行结果如下： 
经检验，可知结果正确。有兴趣的读者可以自己设计其他连通无向图做相关测试。最后给出两点注意事项：
程序测试时，记得调用相关函数前将顶点和边的访问状态重置，如上。Prim算法的时间复杂度是Θ（n*n）。新闻热点
疑难解答
图片精选
