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Gale-Shapley Algorithm: Finding Stable Marriages

This lecture covers the Gale-Shapley algorithm, its run on an instance, and the proof of its correctness. The algorithm solves the stable marriage problem by matching men and women based on their preferences. Proofs and variations of the algorithm are explored.

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Gale-Shapley Algorithm: Finding Stable Marriages

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  1. Lecture 5 CSE 331 Sep 8, 2017

  2. HW 1 posted

  3. Take note of the many(!) notes

  4. Small changes We might tweak the autograder a bit over the weekend These changes should not affect your score Post questions on Piazza!

  5. Can you guess the correlation?

  6. Another comment Discomfort with proofs I will not cover proof basics in class Please read support pages and talk to us in person if you need help

  7. Lecture pace Mid-term

  8. Questions/Comments?

  9. 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 set S of engaged pairs as the final output

  10. Today’s agenda Run of GS algorithm on an instance Prove correctness of the GS algorithm

  11. Preferences Mal Inara Wash Zoe Simon Kaylee

  12. GS algorithm: Firefly Edition Inara Mal Zoe Wash 4 5 6 1 2 3 Kaylee Simon 4 5 6 1 2 3

  13. 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

  14. 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 set S of engaged pairs as the final output

  15. Questions/Comments?

  16. 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?

  17. GS’ does not output a stable marriage

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