kruskal算法基本思路:先对边按权重从小到大排序,先选取权重最小的一条边,如果该边的两个节点均为不同的分量,则加入到最小生成树,否则计算下一条边,直到遍历完所有的边。
prim算法基本思路:所有节点分成两个group,一个为已经选取的selected_node(为list类型),一个为candidate_node,首先任取一个节点加入到selected_node,然后遍历头节点在selected_node,尾节点在candidate_node的边,选取符合这个条件的边里面权重最小的边,加入到最小生成树,选出的边的尾节点加入到selected_node,并从candidate_node删除。直到candidate_node中没有备选节点(这个循环条件要求所有节点都有边连接,即边数要大于等于节点数-1,循环开始前要加入这个条件判断,否则可能会有节点一直在candidate中,导致死循环)。
#coding=utf-8 class Graph(object): def __init__(self, maps): self.maps = maps self.nodenum = self.get_nodenum() self.edgenum = self.get_edgenum() def get_nodenum(self): return len(self.maps) def get_edgenum(self): count = 0 for i in range(self.nodenum): for j in range(i): if self.maps[i][j] > 0 and self.maps[i][j] < 9999: count += 1 return count def kruskal(self): res = [] if self.nodenum <= 0 or self.edgenum < self.nodenum-1: return res edge_list = [] for i in range(self.nodenum): for j in range(i,self.nodenum): if self.maps[i][j] < 9999: edge_list.append([i, j, self.maps[i][j]])#按[begin, end, weight]形式加入 edge_list.sort(key=lambda a:a[2])#已经排好序的边集合 group = [[i] for i in range(self.nodenum)] for edge in edge_list: for i in range(len(group)): if edge[0] in group[i]: m = i if edge[1] in group[i]: n = i if m != n: res.append(edge) group[m] = group[m] + group[n] group[n] = [] return res def prim(self): res = [] if self.nodenum <= 0 or self.edgenum < self.nodenum-1: return res res = [] seleted_node = [0] candidate_node = [i for i in range(1, self.nodenum)] while len(candidate_node) > 0: begin, end, minweight = 0, 0, 9999 for i in seleted_node: for j in candidate_node: if self.maps[i][j] < minweight: minweight = self.maps[i][j] begin = i end = j res.append([begin, end, minweight]) seleted_node.append(end) candidate_node.remove(end) return res max_value = 9999 row0 = [0,7,max_value,max_value,max_value,5] row1 = [7,0,9,max_value,3,max_value] row2 = [max_value,9,0,6,max_value,max_value] row3 = [max_value,max_value,6,0,8,10] row4 = [max_value,3,max_value,8,0,4] row5 = [5,max_value,max_value,10,4,0] maps = [row0, row1, row2,row3, row4, row5] graph = Graph(maps) print('邻接矩阵为\n%s'%graph.maps) print('节点数据为%d,边数为%d\n'%(graph.nodenum, graph.edgenum)) print('------最小生成树kruskal算法------') print(graph.kruskal()) print('------最小生成树prim算法') print(graph.prim())
初始的图如下。
运行结果如下。
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