数据结构【图】—024最小生成树
/*****************************普里姆（Prim）算法***************************/
/*
此为无向图
Prim算法思想很简单，依托临接矩阵
就是从顶点0开始，依次比较起始点到下一个点的最短路径，并将其更新
然后以新的点为起始点，再找到该点能够到达的下一个最短路径，
直到所有点都遍历完为止！
*/
根据程序，最小线段生成如下：
1 #include "000库函数.h" 2 3 #define MAXVEX 20 4 #define INFINITY 65535//视为无穷大 5 6 //临接矩阵的数据结构 7 struct MGraph { 8 int arc[MAXVEX][MAXVEX]; 9 int numVertexes, numEdges; 10 }; 11 12 //初始化临接矩阵 13 MGraph *CreateMGraph(MGraph *G) { 14 G->numEdges = 15; 15 G->numVertexes = 9; 16 17 for (int i = 0; i < G->numVertexes; i++)/* 初始化图 */ 18 { 19 for (int j = 0; j < G->numVertexes; j++) 20 { 21 if (i == j) 22 G->arc[i][j] = 0; 23 else 24 G->arc[i][j] = G->arc[j][i] = INFINITY; 25 } 26 } 27 28 G->arc[0][1] = 10; 29 G->arc[0][5] = 11; 30 G->arc[1][2] = 18; 31 G->arc[1][8] = 12; 32 G->arc[1][6] = 16; 33 G->arc[2][8] = 8; 34 G->arc[2][3] = 22; 35 G->arc[3][8] = 21; 36 G->arc[3][6] = 24; 37 G->arc[3][7] = 16; 38 G->arc[3][4] = 20; 39 G->arc[4][7] = 7; 40 G->arc[4][5] = 26; 41 G->arc[5][6] = 17; 42 G->arc[6][7] = 19; 43 44 for (int i = 0; i < G->numVertexes; i++) 45 { 46 for (int j = i; j < G->numVertexes; j++) 47 { 48 G->arc[j][i] = G->arc[i][j]; 49 } 50 } 51 return G; 52 } 53 54 //使用Prim算法生成最小树 55 void MiniSpanTree_Prim(MGraph *G) { 56 int min, i, j, k; 57 int adjvex[MAXVEX];//保存相关顶点的下标 58 int lowcost[MAXVEX]; // 保存相关顶点间边的权值 59 lowcost[0] = 0;//将v0顶点加入进来 60 adjvex[0] = 0;//初始化第一个顶点为0 61 for (int i = 1; i < G->numVertexes; ++i) { 62 lowcost[i] = G->arc[0][i];//先将v0能到其他点的距离记录下来 63 adjvex[i] = 0;//即到达每个点的起始点都为0点 64 } 65 66 for (int i = 1; i < G->numVertexes; ++i) { 67 min = INFINITY;//将最短路径设为无穷 68 j = 1; k = 0; 69 while (j < G->numVertexes) { 70 if (lowcost[j] != 0 && lowcost[j] < min) {//找到点0到下一个距离最短的值 71 min = lowcost[j];// 72 k = j;//记住最小值的点 73 } 74 ++j; 75 } 76 77 printf("(%d, %d)\n", adjvex[k], k); 78 lowcost[k] = 0;//重新以点k为起始点，然后继续寻找下一个最短路径 79 for (j = 1; j < G->numVertexes; ++j) { 80 if (lowcost[j] != 0 && G->arc[k][j] < lowcost[j]) {//找到下一个最短路劲 81 lowcost[j] = G->arc[k][j]; 82 adjvex[j] = k; 83 } 84 } 85 86 } 87 } 88 89 int T025(void){ 90 MGraph *G; 91 G = new MGraph;//初始化图 92 G = CreateMGraph(G); 93 MiniSpanTree_Prim(G); 94 95 return 0; 96 97 }
/************************克鲁斯卡尔（Kruskal）******************/
根据权值大小，生成权值表，然后根据权值表进行探索路径
1 #include "000库函数.h" 2 3 #define MAXVEX 20 4 #define INFINITY 65535//视为无穷大 5 6 //临接矩阵的数据结构 7 struct MGraph { 8 int arc[MAXVEX][MAXVEX]; 9 int numVertexes, numEdges; 10 }; 11 12 //初始化临接矩阵 13 MGraph *CreateMGraph(MGraph *G) { 14 G->numEdges = 15; 15 G->numVertexes = 9; 16 17 for (int i = 0; i < G->numVertexes; i++)/* 初始化图 */ 18 { 19 for (int j = 0; j < G->numVertexes; j++) 20 { 21 if (i == j) 22 G->arc[i][j] = 0; 23 else 24 G->arc[i][j] = G->arc[j][i] = INFINITY; 25 } 26 } 27 28 G->arc[0][1] = 10; 29 G->arc[0][5] = 11; 30 G->arc[1][2] = 18; 31 G->arc[1][8] = 12; 32 G->arc[1][6] = 16; 33 G->arc[2][8] = 8; 34 G->arc[2][3] = 22; 35 G->arc[3][8] = 21; 36 G->arc[3][6] = 24; 37 G->arc[3][7] = 16; 38 G->arc[3][4] = 20; 39 G->arc[4][7] = 7; 40 G->arc[4][5] = 26; 41 G->arc[5][6] = 17; 42 G->arc[6][7] = 19; 43 44 for (int i = 0; i < G->numVertexes; i++) 45 { 46 for (int j = i; j < G->numVertexes; j++) 47 { 48 G->arc[j][i] = G->arc[i][j]; 49 } 50 } 51 return G; 52 } 53 54 //寻找下一个顶点位置 55 int Find(vector<int>parent, int v) { 56 while (parent[v] > 0)v = parent[v]; 57 return v; 58 } 59 60 61 //使用MiniSpanTree_Kruskal生成最小树 62 void MiniSpanTree_Kruskal(MGraph *G) { 63 int n, m; 64 int k = 0; 65 vector<int> Parent(MAXVEX, 0);//定义一个数组，用来判断是否形成了环路 66 int Edge[MAXVEX][3];//权值表，存放起始点、终止点、权值 67 //构建权值表 68 for (int i = 0; i < G->numVertexes; ++i) { 69 for (int j = i + 1; j < G->numVertexes; ++j) { 70 if (G->arc[i][j] < INFINITY) { 71 Edge[k][0] = i; 72 Edge[k][1] = j; 73 Edge[k][2] = G->arc[i][j]; 74 k++; 75 } 76 } 77 } 78 //进行排序 79 for (int i = 0; i < k; ++i) { 80 int min = Edge[i][2]; 81 for (int j = i + 1; j < k; ++j) { 82 if (min > Edge[j][2]) { 83 min = Edge[j][2]; 84 for (int t = 0; t < 3; ++t) { 85 int temp; 86 temp = Edge[i][t]; 87 Edge[i][t] = Edge[j][t]; 88 Edge[j][t] = temp; 89 } 90 } 91 } 92 } 93 94 95 /*************************算法的核心*****************************/ 96 97 98 99 100 int adjvex[MAXVEX];//保存相关顶点的下标 101 int lowcost[MAXVEX]; // 保存相关顶点间边的权值 102 lowcost[0] = 0;//将v0顶点加入进来 103 adjvex[0] = 0;//初始化第一个顶点为0 104 for (int i = 1; i < G->numVertexes; ++i) { 105 lowcost[i] = G->arc[0][i];//先将v0能到其他点的距离记录下来 106 adjvex[i] = 0;//即到达每个点的起始点都为0点 107 } 108 109 for (int i = 0; i < k; ++i) {//循环每一个权值矩阵 110 n = Find(Parent, Edge[i][0]); 111 m = Find(Parent, Edge[i][1]); 112 if (n != m) {//不会形成环，可以使用 113 Parent[n] = m; 114 cout << Edge[i][0] << "," << Edge[i][1] << endl; 115 } 116 } 117 } 118 119 120 121 int T026(void) { 122 MGraph *G; 123 G = new MGraph;//初始化图 124 G = CreateMGraph(G); 125 MiniSpanTree_Kruskal(G); 126 127 return 0; 128 129 }
