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数据结构之最⼩⽣成树Prim算法普⾥姆算法介绍 普⾥姆（Prim）算法，是⽤来求加权连通图的最⼩⽣成树算法 基本思想：对于图G⽽⾔，V是所有顶点的集合；现在，设置两个新的集合U和T，其中U⽤于存放G的最⼩⽣成树中的顶点，T存放G的最⼩⽣成树中的边。
从所有uЄU，vЄ(V-U) (V-U表⽰出去U的所有顶点)的边中选取权值最⼩的边(u, v)，将顶点v加⼊集合U中，将边(u, v)加⼊集合T中，如此不断重复，直到U=V为⽌，最⼩⽣成树构造完毕，这时集合T中包含了最⼩⽣成树中的所有边。
代码实现1. 思想逻辑 （1）以⽆向图的某个顶点（A）出发，计算所有点到该点的权重值，若⽆连接取最⼤权重值#define INF (~(0x1<<31)) （2）找到与该顶点最⼩权重值的顶点（B），再以B为顶点计算所有点到改点的权重值，依次更新之前的权重值，注意权重值为0或⼩于当前权重值的不更新，因为1是⼀当找到最⼩权重值的顶点时，将权重值设为了0，2是会出现⽆连接的情况。
（3）将上述过程⼀次循环，并得到最⼩⽣成树。
2. Prim算法// Prim最⼩⽣成树void Prim(int nStart){int i = 0;int nIndex=0; // prim最⼩树的索引，即prims数组的索引char cPrims[MAX]; // prim最⼩树的结果数组int weights[MAX]; // 顶点间边的权值cPrims[nIndex++] = m_mVexs[nStart].data;// 初始化"顶点的权值数组"，// 将每个顶点的权值初始化为"第start个顶点"到"该顶点"的权值。
for (i = 0; i < m_nVexNum; i++){weights[i] = GetWeight(nStart, i);}for (i = 0; i < m_nVexNum; i ++){if (nStart == i){continue;}int min = INF;int nMinWeightIndex = 0;for (int k = 0; k < m_nVexNum; k ++){if (weights[k]!= 0 && weights[k] < min){min = weights[k];nMinWeightIndex = k;}}// 找到下⼀个最⼩权重值索引cPrims[nIndex++] = m_mVexs[nMinWeightIndex].data;// 以找到的顶点更新其他点到该点的权重值weights[nMinWeightIndex]=0;int nNewWeight = 0;for (int ii = 0; ii < m_nVexNum; ii++){nNewWeight = GetWeight(nMinWeightIndex, ii);// 该位置需要特别注意if (0 != weights[ii] && weights[ii] > nNewWeight){weights[ii] = nNewWeight;}}for (i = 1; i < nIndex; i ++){int min = INF;int nVexsIndex = GetVIndex(cPrims[i]);for (int kk = 0; kk < i; kk ++){int nNextVexsIndex = GetVIndex(cPrims[kk]);int nWeight = GetWeight(nVexsIndex, nNextVexsIndex);if (nWeight < min){min = nWeight;}}nSum += min;}// 打印最⼩⽣成树cout << "PRIM(" << m_mVexs[nStart].data <<")=" << nSum << ": ";for (i = 0; i < nIndex; i++)cout << cPrims[i] << "";cout << endl;}3. 全部实现#include "stdio.h"#include <iostream>using namespace std;#define MAX 100#define INF (~(0x1<<31)) // 最⼤值(即0X7FFFFFFF)class EData{public:EData(char start, char end, int weight) : nStart(start), nEnd(end), nWeight(weight){} char nStart;char nEnd;int nWeight;};// 边struct ENode{int nVindex; // 该边所指的顶点的位置int nWeight; // 边的权重ENode *pNext; // 指向下⼀个边的指针};struct VNode{char data; // 顶点信息ENode *pFirstEdge; // 指向第⼀条依附该顶点的边};// ⽆向邻接表class listUDG{public:listUDG(){};listUDG(char *vexs, int vlen, EData **pEData, int elen){m_nVexNum = vlen;m_nEdgNum = elen;// 初始化"邻接表"的顶点for (int i = 0; i < vlen; i ++){m_mVexs[i].data = vexs[i];m_mVexs[i].pFirstEdge = NULL;}char c1,c2;int p1,p2;ENode *node1, *node2;// 初始化"邻接表"的边for (int j = 0; j < elen; j ++){// 读取边的起始顶点和结束顶点p1 = GetVIndex(c1);p2 = GetVIndex(c2);node1 = new ENode();node1->nVindex = p2;node1->nWeight = pEData[j]->nWeight;if (m_mVexs[p1].pFirstEdge == NULL){m_mVexs[p1].pFirstEdge = node1;}else{LinkLast(m_mVexs[p1].pFirstEdge, node1);}node2 = new ENode();node2->nVindex = p1;node2->nWeight = pEData[j]->nWeight;if (m_mVexs[p2].pFirstEdge == NULL){m_mVexs[p2].pFirstEdge = node2;}else{LinkLast(m_mVexs[p2].pFirstEdge, node2);}}}~listUDG(){ENode *pENode = NULL;ENode *pTemp = NULL;for (int i = 0; i < m_nVexNum; i ++){pENode = m_mVexs[i].pFirstEdge;if (pENode != NULL){pTemp = pENode;pENode = pENode->pNext;delete pTemp;}delete pENode;}}void PrintUDG(){ENode *pTempNode = NULL;cout << "邻接⽆向表:" << endl;for (int i = 0; i < m_nVexNum; i ++){cout << "顶点:" << GetVIndex(m_mVexs[i].data)<< "-" << m_mVexs[i].data<< "->"; pTempNode = m_mVexs[i].pFirstEdge;while (pTempNode){cout <<pTempNode->nVindex << "->";pTempNode = pTempNode->pNext;}cout << endl;}}// Prim最⼩⽣成树void Prim(int nStart){int i = 0;int nIndex=0; // prim最⼩树的索引，即prims数组的索引char cPrims[MAX]; // prim最⼩树的结果数组int weights[MAX]; // 顶点间边的权值cPrims[nIndex++] = m_mVexs[nStart].data;// 初始化"顶点的权值数组"，// 将每个顶点的权值初始化为"第start个顶点"到"该顶点"的权值。
