Question

Create an implementation for the Gale-Shapley algorithm, determine its time complexity. you will need to write up your implementation and discuss the data structures that you select and the time complexity of your algorithm. You can use pseudocode, but more concrete

Answer 1

The Gale-Shapley algorithm can be explained by taking the example of Stable marriage problem:

The Stable Marriage Problem states that given N men and N women, where each person has ranked all members of the opposite sex in order of preference, marry the men and women together such that there are no two people of the opposite sex who would both rather have each other than their current partners. If there are no such people, all the marriages are “stable”.

Consider the following example.

Let there be two men m1 and m2 and two women w1 and w2.
Let m1‘s list of preferences be {w1, w2}
Let m2‘s list of preferences be {w1, w2}
Let w1‘s list of preferences be {m1, m2}
Let w2‘s list of preferences be {m1, m2}

The matching { {m1, w2}, {w1, m2} } is not stable because m1 and w1 would prefer each other over their assigned partners. The matching {m1, w1} and {m2, w2} is stable because there are no two people of the opposite sex that would prefer each other over their assigned partners.

It is always possible to form stable marriages from lists of preferences.

Following is Gale–Shapley algorithm to find a stable matching:

The idea is to iterate through all free men while there is any free man available. Every free man goes to all women in his preference list according to the order. For every woman he goes to, he checks if the woman is free, if yes, they both become engaged. If the woman is not free, then the woman chooses either says no to him or dumps her current engagement according to her preference list. So an engagement done once can be broken if a woman gets a better option.

Time Complexity of Gale-Shapley Algorithm is O(n²).

Pseudo-Code:

Initialize all men and women to free
while there exist a free man m who still has a woman w to propose to 
{
    w = m's highest ranked such woman to whom he has not yet proposed
    if w is free
       (m, w) become engaged
    else some pair (m', w) already exists
       if w prefers m to m'
          (m, w) become engaged
           m' becomes free
       else
          (m', w) remain engaged    
}

Describing data-structures of input and output:

Input is a 2D matrix of size (2*N)*N where N is number of women or men. Rows from 0 to N-1 represent preference lists of men and rows from N to 2*N – 1 represent preference lists of women. So men are numbered from 0 to N-1 and women are numbered from N to 2*N – 1. The output is list of married pairs.

Following is the implementation of the above algorithm in python. The bold text is the piece of code and the explanation is given as python comments:

# Number of Men or Women

N = 4

# This function returns true if woman 'w' prefers man 'm1' over man 'm'

def wPrefersM1OverM(prefer, w, m, m1):
#Check if w prefers m over her current engagment m1

for i in range(N):
#If m1 comes before m in lisr of w, then w prefers her current engagement, don't do anything
if (prefer[w][i] == m1):
return True
  
#If m comes before m1 in w's list,  then free her current engagement  and engage her with m
if(prefer[w][i] == m):
return False

#Prints stable matching for N boys and N girls.
#Boys are numbered as 0 to N-1.
#Girls are numbered as N to 2N-1.

def stableMarriage(prefer):
#Stores partner of women. This is our output array that stores pairing information.
    #The value of wPartner[i] indicates the partner assigned to woman N+i.
#Note that the woman numbers between N and 2*N-1. The value -1 indicates that (N+i)'th woman is free
wPartner = [-1 for i in range(N)]
    #An array to store availability of men.
    #If mFree[i] is false, then man 'i' is free,otherwise engaged.
    mFree = [False for i in range(N)]
    freeCount = N
  
    #While there are free men
    while(freeCount > 0):
        #Pick the first free man (we could pick any)
        m = 0
        while(m < N):
            if(mFree[m] == False):
                break
            m += 1
#One by one go to all women according to m's preferences. Here m is the picked free man
        i = 0
        while i < N and mFree[m] == False:
            w = prefer[m][i]
            #The woman of preference is free, w and m become partners (Note that the partnership maybe changed later).
            #So we can say they are engaged not married
            if (wPartner[w - N] == -1):
                wPartner[w - N] = m
                mFree[m] = True
                freeCount -= 1
            else:  
                #If w is not free, Find current engagement of w
                m1 = wPartner[w - N]
                #If w prefers m over her current engagement m1, then break the engagement between w and m1 and engage m with w.
                if (wPrefersM1OverM(prefer, w, m, m1) == False):
                    wPartner[w - N] = m
                    mFree[m] = True
                    mFree[m1] = False
            i += 1

            #End of Else
        #End of the for loop that goes to all women in m's list
    #End of the main while loop
  
    #Printing the solution
    print("Woman ", " Man")
    for i in range(N):
        print(i + N, "\t", wPartner[i])
  
#Driver Code
prefer = [[7, 5, 6, 4], [5, 4, 6, 7],
          [4, 5, 6, 7], [4, 5, 6, 7],
          [0, 1, 2, 3], [0, 1, 2, 3],
          [0, 1, 2, 3], [0, 1, 2, 3]]

stableMarriage(prefer)

OUTPUT: