gale shapley algorithm

8/13/2019 Gale Shapley Algorithm

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Gale-Shapley Algorithm

- for Stable Matching -


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The problem

Scenario: N couples want to adopt N kids. Each couple lists all the kids in the order of their preference. Each kid lists all the couples in the order of their preference. A couple cannot have 2 different kids on the same level of preference. A kid cannot have 2 different couples on the same level of preference.

What we have to do: Assign each couple a kid such that you a stable matching is achieved.


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Unstable Match

Couple A Kid A Couple B


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Unstable Match



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Unstable Match



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What is a stable match?

A couple C and a child K are in a stable mat least one of the following conditions is me C prefers Kover any other child that can be adopted

C prefers a different kid K1over Kbut K1is alreadyby C1and he prefers C1 overC.

In the result match there must not be a kid acouple that would prefer each other over thecurrent matches.


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How do we match the kids and the cou

Using Gale-Shapley Algorithm: Developed by Lloyd Shapley and David Gale.

Published 1962.

Also known as Algorithm of Happiness.


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How does it work?

1stpreference 2ndpreference 3rdpreference 4

Couple 1 Kid 3 Kid 2 Kid 4 K





Kid 1 Couple 2 Couple 3 Couple 1 C




Couple 2

Couple 3

Couple 4

Couple 1


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Couple 2

Couple 3

Couple 4

Couple 1 Kid 3


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How does it work?











Couple 2 Kid 2

Couple 3

Couple 4

Couple 1 Kid 3


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Couple 2 Kid 2

Couple 3 Kid 3

Couple 4

Couple 1 Kid 3


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Couple 2 Kid 2

Couple 3 Kid 3

Couple 4

Couple 1 Kid 3 Kid 2


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Couple 3 Kid 3

Couple 4



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Couple 3 Kid 3

Couple 4 Kid 4



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Couple 2 Kid 2 Kid 4 Kid 1

Couple 3 Kid 3

Couple 4 Kid 4



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The algorithm

Let Couplesbe the set of couples and Kidsbe the set of kids;

functionstableMatching{

Initialize all cCouples and kKids tofree

whilechildless couple Cwho still has a kid K to propose for adoption

{

K = C's highest ranked such child to whom he has not yet proposed

ifC is not adopted

(K, C) become a family

elsesome pair (C', K) already exists

ifK prefers C to C

(C, K) become a family

K' becomesfree for adoption

else

(C', K) remain together

}

}


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Running Time

Suppose we have N couples and N kids.

At each step from start to finish a couple maproposal to a kid.

Each proposal takes constant time O(1).

Since there are N couples and N kids => maxn*n possible proposals (a couple does not prtwice to the same kid).

Total time complexity is O(n*n).


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Other Application

Matching employees with employers.

Matching men and women on a Dating webs

Etc.

gale shapley algorithm

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