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Lecture 4 CSE 331 Sep 6, 2017
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Read the syllabus CAREFULLY!
No graded material will be handed back till you submit a signed form!
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Sign-up for mini projects
Deadline: Monday, Sep 25, 11:59pm me your group (=3) composition
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Separate Proof idea/proof details
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TA office hours finalized
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Makeup recitations TODAY, 4-5pm in Davis 338A
Tomorrow, 5-6pm in Davis 113A
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On matchings Mal Wash Simon Inara Zoe Kaylee
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A valid matching Mal Wash Simon Inara Zoe Kaylee
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Not a matching Mal Wash Simon Inara Zoe Kaylee
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Perfect Matching Mal Wash Simon Inara Zoe Kaylee
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Preferences Mal Wash Simon Inara Zoe Kaylee
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Instability Mal Wash Simon Inara Zoe Kaylee
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Even though BBT and JA are not very happy
A stable marriage Even though BBT and JA are not very happy
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Two stable marriages
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Stable Marriage problem
Input: M and W with preferences Output: Stable Matching Set of men M and women W Preferences (ranking of potential spouses) Matching (no polyandry/gamy in M X W) Perfect Matching (everyone gets married) m w m’ w’ Instablity Stable matching = perfect matching+ no instablity
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Questions/Comments?
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Two Questions Does a stable marriage always exist?
If one exists, how quickly can we compute one?
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Today’s lecture Naïve algorithm
Gale-Shapley algorithm for Stable Marriage problem
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Incremental algorithm to produce all n! prefect matchings?
The naïve algorithm Incremental algorithm to produce all n! prefect matchings? Go through all possible perfect matchings S n! matchings If S is a stable matching then Stop Else move to the next perfect matching
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Gale-Shapley Algorithm
David Gale Lloyd Shapley O(n3) algorithm
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Moral of the story… >
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Questions/Comments?
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Gale-Shapley Algorithm
Intially all men and women are free While there exists a free woman who can propose Let w be such a woman and m be the best man she has not proposed to w proposes to m If m is free (m,w) get engaged Else (m,w’) are engaged If m prefers w’ to w w remains free Else (m,w) get engaged and w’ is free Output the engaged pairs as the final output
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Preferences Mal Wash Simon Inara Zoe Kaylee
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GS algorithm: Firefly Edition
Mal Inara 1 2 3 4 5 6 Wash Zoe 1 Simon Kaylee
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Once a man gets engaged, he remains engaged (to “better” women)
Observation 1 Intially all men and women are free While there exists a free woman who can propose Let w be such a woman and m be the best man she has not proposed to w proposes to m If m is free (m,w) get engaged Once a man gets engaged, he remains engaged (to “better” women) Else (m,w’) are engaged If m prefers w’ to w w remains free Else (m,w) get engaged and w’ is free Output the engaged pairs as the final output
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If w proposes to m after m’, then she prefers m’ to m
Observation 2 Intially all men and women are free While there exists a free woman who can propose Let w be such a woman and m be the best man she has not proposed to w proposes to m If m is free If w proposes to m after m’, then she prefers m’ to m (m,w) get engaged Else (m,w’) are engaged If m prefers w’ to w w remains free Else (m,w) get engaged and w’ is free Output the engaged pairs as the final output
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Questions/Comments?
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Why bother proving correctness?
Consider a variant where any free man or free woman can propose Is this variant any different? Can you prove it?
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GS’ does not output a stable marriage
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