问题
给定一组男人和一组女人,每个人在心目中都对所有的异性有一个倾慕度排序,从最喜欢到最不喜欢依次排序1、2、3。现在给出问题,如何对这些男女进行配对使得在分配好后不出现偷情的现象。
算法
可以有男人优先和女人优先两种算法。以男人优先为例,为代码如下:
while 存在男人m是自由的且还没对每个女人都求过婚 选择这个男人m 令w是m的优先表中还没求过婚的最高排名的女人 if w是自由的 (m,w)变成约会状态 else w当前与m1约会 if w更偏爱m1而不爱m m保持自由 else w更偏爱m而不爱m1 (m,w)变成约会状态 m1变成自由 endif endif endwhile
需要特别注意的是!虽然从第9、10行来看,女人能够选择更好的伴侣。但从第3行来看,男人能够优先选择排名最高的女人。实际上,第9、10行只是为了保证婚姻的稳定性。
Python实现
# Homework 1, Problem 4: Implementing the Gale-Shapley algorithm # You are to code the function "find_stable_matching". DO NOT rename # the function. Details of the format of the input/output are given in # the function's comments. Edit this file and use blackboard to submit # the file along with your homework solutions. # Some basic testing for your code is provided below. DO NOT modify # these tests. Your code MUST pass these tests. Note, however, that # passing these tests does not guarantee that your algorithm is # correct. You should carry out more testing! # Use Python 2.x for your implementation def find_stable_matching(MP,WP): """ The input of this function is defined by two lists MP and WP Men and women are numbered 0 through n-1 MP[i] encodes the ith man's preferences. It is simply a list containing the numbers 0, 1, ... , n-1 in some order WP[i] encodes the ith woman's preferences. It is simply a list containing the numbers 0, 1,... , n-1 in some order The output is defined by a list of pairs of the form (i,j) indicating that man i is married to woman j Your output should be the man-optimal stable matching found by the GS-algorithm. """ # your code goes here! n=len(MP) isManFree=[True]*n isWomanFree=[True]*n #isManProposed=[[False]*n]*n isManProposed=[[False for i in range(n)] for j in range(n)] match=[(-1,-1)]*n while(True in isManFree): indexM=isManFree.index(True) if(False in isManProposed[indexM]): indexW=-1 for i in range(n): w=MP[indexM][i] if(not isManProposed[indexM][w]): indexW=w break isManProposed[indexM][indexW]=True if(isWomanFree[indexW]): #isManProposed[indexM][indexW]=True isWomanFree[indexW]=False isManFree[indexM]=False match[indexM]=(indexM,indexW) else: indexM1=-1 for j in range(n): if(match[j][1]==indexW): indexM1=j break if(WP[indexW].index(indexM)<WP[indexW].index(indexM1)): isManFree[indexM1]=True isManFree[indexM]=False match[indexM]=(indexM,indexW) print match return match def test1(): """ basic test for your code """ MP = [ [0,1], [1,0] ] WP = [ [0,1], [0,1] ] SM = find_stable_matching(MP,WP) assert ( (0,0) in SM and (1,1) in SM ) def test2(): """ basic test for your code """ MP = [ [0,1], [0,1] ] WP = [ [0,1], [0,1] ] SM = find_stable_matching(MP,WP) assert ( (0,0) in SM and (1,1) in SM ) def test3(): """ basic test for your code """ MP = [ [0,1], [0,1] ] WP = [ [1,0], [1,0] ] SM = find_stable_matching(MP,WP) assert ( (0,1) in SM and (1,0) in SM ) if __name__=="__main__": print "**********test1************" test1() print "**********test2************" test2() print "**********test3************" test3()
面向对象的Python实现请参见这里