最小生成树Prim算法

Author: qiwsir

prim_algorithm.md

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+#问题
+
+无向图最小生成树的Prim算法
+
+#思路说明
+
+假设点A,B,C,D,E,F，两点之间有连线的，以及它们的距离分别是：(A-B:7);(A-D:5);(B-C:8);(B-D:9);(B-E:7);(C-E:5);(D-E:15);(D-F:6);(E-F:8);(E-G:9);(F-G:11)
+
+关于Prim算法的计算过程，参与维基百科的词条：[普里姆算法](http://zh.wikipedia.org/wiki/%E6%99%AE%E6%9E%97%E5%A7%86%E7%AE%97%E6%B3%95)
+
+将上述点与点关系以及两点之间距离（边长，有的文献中称之为权重）写成矩阵形式（在list中，每两个点及其之间的距离组成一个tuple)
+
+edges = [ ("A", "B", 7), 
+          ("A", "D", 5),
+          ("B", "C", 8), 
+          ("B", "D", 9), 
+          ("B", "E", 7), 
+          ("C", "E", 5),
+          ("D", "E", 15), 
+          ("D", "F", 6),
+          ("E", "F", 8), 
+          ("E", "G", 9),
+          ("F", "G", 11)
+        ]
+
+在下面的解决方法中，要计算出与已经选出的若干个点有相邻关系的点中，相应边长最短的点。这本质上是排序之后取出最小的，因为这种排序是动态的，如果用sorted或者list.sort()之类的方法对list排序，一则速度慢（python中的sort方法对大数据时不是很快),二则代码也长了。幸好python提供了一个非常好用的模块：heapq。这个模块是堆排序方法实现排序，并能够随时取出最小值。简化代码，更重要是提升了速度。
+
+就用这个来解决Prim算法问题了。
+
+#解决（Python）
+	
+	#! /usr/bin/env python
+	#coding:utf-8
+	
+	from collections import defaultdict
+	from heapq import *
+	
+	def prim( vertexs, edges ):
+	    adjacent_vertex = defaultdict(list)      #注意：defaultdict(list)必须以list做为变量，可以详细阅读：[collections.defaultdict](https://docs.python.org/2/library/collections.html#collections.defaultdict)
+	
+	    for v1,v2,length in edges:
+            adjacent_vertex[v1].append((length, v1, v2))
+            adjacent_vertex[v2].append((length, v2, v1))
+
+        """
+	    经过上述操作，将edges列表中各项归类成以某点为dictionary的key，其value则是其相邻的点以及边长。如下：
+	
+	    defaultdict(<type 'list'>, {'A': [(7, 'A', 'B'), (5, 'A', 'D')], 'C': [(8, 'C', 'B'), (5, 'C', 'E')], 'B': [(7, 'B', 'A'), (8, 'B', 'C'), (9, 'B', 'D'), (7, 'B', 'E')], 'E': [(7, 'E', 'B'), (5, 'E', 'C'), (15, 'E', 'D'), (8, 'E', 'F'), (9, 'E', 'G')], 'D': [(5, 'D', 'A'), (9, 'D', 'B'), (15, 'D', 'E'), (6, 'D', 'F')], 'G': [(9, 'G', 'E'), (11, 'G', 'F')], 'F': [(6, 'F', 'D'), (8, 'F', 'E'), (11, 'F', 'G')]})
+	
+	    """
+	
+	    mst = []        #存储最小生成树结果
+	
+	    chosed = set(vertexs[0]) 
+	
+	    """
+	    vertexs是顶点列表，vertexs = list("ABCDEFG")===>vertexs=['A', 'B', 'C', 'D', 'E', 'F', 'G']
+	    >> chosed=set(vertexs[0])
+	    >> chosed
+	    set(['A'])
+	    也就是，首先选一个点（这个点是可以任意选的），以这个点为起点，找其相邻点，以及最短边长。
+	
+	    """
+
+        #得到adjacent_vertexs_edges中顶点是'A'（nodes[0]='A')的相邻点list，即adjacent_vertexs['A']=[(7,'A','B'),(5,'A','D')]
+	
+	    adjacent_vertexs_edges = adjacent_vertex[vertexs[0]]  
+	    
+        #将usable_edges加入到堆中，并能够实现用heappop从其中动态取出最小值。关于heapq模块功能，参考python官方文档
+
+	    heapify(adjacent_vertexs_edges)
+	
+	    while adjacent_vertexs_edges:
+            #得到某个定点（做为adjacent_vertexs_edges的键）与相邻点距离（相邻点和边长/距离做为该键的值）最小值，并同时从堆中清除。
+	        w, v1, v2 = heappop(adjacent_vertexs_edges)     
+	        if v2 not in chosed:
+
+                #在used中有第一选定的点'A'，上面得到了距离A点最近的点'D',举例是5。将'd'追加到used中
+	            chosed.add(v2)                          
+
+	            mst.append((v1,v2,w))          #将v1,v2,w，第一次循环就是('A','D',5) append into mst
+	            
+                #再找与d相邻的点，如果没有在heap中，则应用heappush压入堆内，以加入排序行列 
+
+                for next_vertex in adjacent_vertex[v2]:                    
+	                if next_vertex[2] not in chosed:
+	                    heappush( adjacent_vertexs_edges,next_vertex)
+	    return mst
+	
+	
+	#test
+	vertexs = list("ABCDEFG")
+	edges = [ ("A", "B", 7), ("A", "D", 5),
+	          ("B", "C", 8), ("B", "D", 9), 
+	          ("B", "E", 7), ("C", "E", 5),
+	          ("D", "E", 15), ("D", "F", 6),
+	          ("E", "F", 8), ("E", "G", 9),
+	          ("F", "G", 11)]
+	
+	print "edges:",edges
+	print "prim:", prim( vertexs, edges )
+
+##运行结果
+
+edges: [('A', 'B', 7), ('A', 'D', 5), ('B', 'C', 8), ('B', 'D', 9), ('B', 'E', 7), ('C', 'E', 5), ('D', 'E', 15), ('D', 'F', 6), ('E', 'F', 8), ('E', 'G', 9), ('F', 'G', 11)]
+prim: [('A', 'D', 5), ('D', 'F', 6), ('A', 'B', 7), ('B', 'E', 7), ('E', 'C', 5), ('E', 'G', 9)]
+
+

prim_algorithm.py

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+#! /usr/bin/env python
+#coding:utf-8
+
+
+from collections import defaultdict
+from heapq import *
+
+
+def prim( vertexs, edges ):
+    adjacent_vertex = defaultdict(list)
+    for v1,v2,length in edges:
+        adjacent_vertex[v1].append((length, v1, v2))
+        adjacent_vertex[v2].append((length, v2, v1))
+
+    mst = []
+    chosed = set(vertexs[0]) 
+
+    adjacent_vertexs_edges = adjacent_vertex[vertexs[0]]  
+    
+    heapify(adjacent_vertexs_edges)
+
+    while adjacent_vertexs_edges:
+        w, v1, v2 = heappop(adjacent_vertexs_edges) 
+        if v2 not in chosed:
+            chosed.add(v2)                    
+            mst.append((v1,v2,w))   
+            for next_vertex in adjacent_vertex[v2]:            
+                if next_vertex[2] not in chosed:
+                    heappush( adjacent_vertexs_edges,next_vertex)
+
+    return mst
+
+
+#test
+vertexs = list("ABCDEFG")
+edges = [ ("A", "B", 7), ("A", "D", 5),
+          ("B", "C", 8), ("B", "D", 9), 
+          ("B", "E", 7), ("C", "E", 5),
+          ("D", "E", 15), ("D", "F", 6),
+          ("E", "F", 8), ("E", "G", 9),
+          ("F", "G", 11)]
+
+print "edges:",edges
+print "prim:", prim( vertexs, edges )