08 - Stable Matching, part 1

08 - Stable Matching, part 1 - i , w i )} n i+1 s.t. (1)...

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1/21/08 - Stable Matching, part 1 Lecture: Stable matching, part 1. Reading: Chapter 1.1; Chapters 2,3 (1) Formulating algorithmic problems as mathematically well- poised questions (2) Distinguishing “easy” from “hard” (3) General techniques for designing algorithms (4) Speci±c useful algorithms Roughly 1 homework per week 40% HW 3 problems per hw 2 prelims 2/21, 4/8 15+15 Final 5/14 30 Stable marriage (Gale-Shapley) Can we design a better matching system? i.e. one which is self-enforcing We say an assignment is stable if (1) No school prefers an un-admitted student to an admitted one unless un-admitted student is happier with his/her current current assignment (2) vice-versa Assume: n students men n schools women Each school admits man marries only one. Each student attends woman marries only one 3-1 3
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Each man m has a preference ordering of women. w < m w’ “m prefers w’ to w” Each woman w has a preference ordering of men m < w m’ Def. A stable marriage a set of ordered pairs {(m
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Unformatted text preview: i , w i )} n i+1 s.t. (1) Each man/woman is in exactly one pair. (2) pairs (m i , w i ), (m j , w j ) such that w j &lt; m j w i , m i &lt; w i m j First try. Local search. while (m i , w i ) s.t. w j &lt; mj wi m i &lt; wi m j replace these pairs with (w i , m j ), (m i , w j ) counterexample: A: X &gt; Z &gt; Y B: Z &gt; X &gt; Y C: X &gt; Y &gt; Z X: B &gt; A &gt; C Y: A &gt; B &gt; C Z: A &gt; B &gt; C Potential for innite loop! Proposal algorithm /* Init */ All men/women are free /* main */ while a free man who hasnt yet proposed to every woman m proposes to the highest-ranked w who didnt yet reject him if w is free (m, w) become engaged else w is engaged to m 3-2 if m &lt; w m m remains free else (m, w) become engaged m becomes free end Analysis Prop. Alg terminates after n 2 iterations of the loop Proof. In every loop iteration, a man proposes to a woman he never proposed to before. 3-3...
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08 - Stable Matching, part 1 - i , w i )} n i+1 s.t. (1)...

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