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Lecture 6

This lecture delves into the Stable Marriage Problem, defining the inputs (men and women with preferences) and discussing the output: a stable matching with no polygamy. It explores key concepts like perfect matching, instability, and the existence of stable marriages. The presentation focuses on the Gale-Shapley algorithm, emphasizing its operations where women propose to men, resulting in engaged pairs. The lecture addresses two important questions regarding stable marriages and highlights the algorithm's effectiveness in finding solutions quickly.

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Lecture 6

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  1. Lecture 6 CSE 331 Sep 10, 2012

  2. Homeworks HW 1 posted online: see blog/piazza Pickup graded HW 0 in TA OHs

  3. Suggestions for Piazza Email them: team “at” piazza “dot” com

  4. Lecture pace Mid-term

  5. Online Office Hours Tomorrow: 9:30pm to 10:30pm

  6. 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 polygamy in M X W) Perfect Matching (everyone gets married) m w Instablity m’ w’ Stable matching = perfect matching+ no instablity

  7. Two Questions Does a stable marriage always exist? If one exists, how quickly can we compute one? Answer both Qs in +ve by the Gale-Shapley algorithm

  8. Gale-Shapley Algorithm (er, not Nobel prize winners, at least not yet) Women do all the proposing (different from the book) Everyone is in one of three states: free, engaged and married Step 1: A free woman w proposes to her most preferred man m. (m,w) get engaged General step: A free woman w proposes to her top unproposed man m.

  9. Questions/Comments?

  10. 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

  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

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