稳定匹配 - Stable Matching

这篇文章将会对稳定匹配算法进行介绍及Python代码的实现，第一部分会针对稳定匹配的Gale-Shapley算法进行解析，第二部分就是用Python对该算法进行实现。

一、稳定匹配算法原理

１.１　介绍

稳定匹配（Stable Matching）问题就是假设现在有N个男生和N个女生跳舞选择伴侣，然后最开始的时候男、女生按照下面情况对彼此进行排序选择舞伴（见图１）：

• 每个男生都对女生按照最喜欢到最不喜欢进行排序；
• 同样的，女生也是按照最喜欢的到最不喜欢对男生进行排序。

算法目标：每个男都找到唯一一个女舞伴，反之亦如此，从而达到了所谓的稳定匹配。

演示步骤：

１.２　伪代码（Gale-Shapley Algorithm）

 1 # 首先初始化所有男生的状态为自由

 2 initialize each person to be free

 3

 4 # 当男生没有未曾被匹配过并且也没有向所有其他女生寻求舞伴过时不断循环

 5 while some man m is not yet matched:

 6     # 每个男生按照对女生的喜欢程度选择舞伴

 7     w := m's most favroite woman to whom he has not yet proposed

 8     # 如果女生未被匹配到，则与男生进行配对

 9     if w is also not yet matched:

10         w and m are paired

11     # 如果女生与已匹配的男生相比更喜欢当前的这个男生，则拆散重新匹配

12     elif w favors m to her current matched m':

13         w and m are paired and m' is dis-matched

14     # 否则该女生拒绝成为男生的舞伴

15     else:

16         w rejects m

17 # 返回所有匹配成功的舞伴对

18 return matched pairs

二、Python代码实现

# -*- encoding: UTF-8 -*-

import copy

# 男的所期望的对象

manPrefers = dict((m, prefs.split(', ')) for [m, prefs] in (line.rstrip().split(': ')

                                for line in open('men.txt')))

# 女的所期望的对象

womenPrefers = dict((m, prefs.split(', ')) for [m, prefs] in (line.rstrip().split(': ')

                                for line in open('women.txt')))

men = sorted(manPrefers.keys())

women = sorted(womenPrefers.keys())

# 定义检测函数检测匹配的伴侣是否稳定

def check(engaged):

    inverseengaged = dict((v,k) for k,v in engaged.items())

    for w, m in engaged.items():

        shelikes = womenPrefers[w]

        shelikesbetter = shelikes[:shelikes.index(m)]

        helikes = manPrefers[m]

        helikesbetter = helikes[:helikes.index(w)]

        for man in shelikesbetter:

            womenOftheMan = inverseengaged[man]

            manLoves = manPrefers[man]

            if manLoves.index(womenOftheMan) > manLoves.index(w):

                print("%s 和 %s 更喜欢彼此相比起他们当前的伴侣: %s 和 %s" % (w, man, m, womenOftheMan))

                return False

        for woman in helikesbetter:

            manOfTheWomen = engaged[woman]

            womanLoves = womenPrefers[woman]

            if womanLoves.index(manOfTheWomen) > womanLoves.index(m):

                print("%s 和 %s 更喜欢彼此相比起他们当前的伙伴：%s 和 %s" % (m, woman, w, manOfTheWomen))

                return False

    return True

def stableMatching():

    free_men = men[:]

    engaged  = {}

    manPref_temp = copy.deepcopy(manPrefers)

    womenPref_temp = copy.deepcopy(womenPrefers)

    while free_men:

        man = free_men.pop(0)

        manList = manPref_temp[man]

        woman = manList.pop(0)

        fiance = engaged.get(woman)

        if not fiance:

            engaged[woman] = man

            print("  %s 和 %s 成为伴侣" % (man, woman))

        else:

            womenList = womenPref_temp[woman]

            if womenList.index(fiance) > womenList.index(man):

                engaged[woman] = man

                print("  %s 舍弃 %s 而和 %s 成为伴侣" % (woman, fiance, man))

                if manPref_temp[fiance]:

                    free_men.append(fiance)

            else:

                if manList:

                    free_men.append(man)

    return engaged

if __name__ == '__main__':

    print('\n伴侣匹配:')

    engaged = stableMatching()

    print('\n伴侣匹配:')

    print('  ' + ',\n  '.join('%s 和 %s 成为伴侣' % couple for couple in sorted(engaged.items())))

    print()

    print('伴侣稳定性检测通过' if check(engaged) else '伴侣稳定性检测不通过')

    print('\n\n因交换而产生伴侣搭配错误')

    engaged[women[0]], engaged[women[1]] = engaged[women[1]], engaged[women[0]]

    for woman in women[:2]:

        print('  %s 现在和 %s 成为伴侣' % (woman, engaged[woman]))

    print()

    print('伴侣稳定性检测通过' if check(engaged) else '伴侣稳定性检测不通过')