08 - Stable Matching, part 1

# 08 - Stable Matching, part 1 - i w i n i 1 s.t(1 Each...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

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 < m j w i , m i < w i m j First try. Local search. while ∃ (m i , w i ) s.t. w j < mj wi ⋀ m i < wi m j replace these pairs with (w i , m j ), (m i , w j ) counterexample: A: X > Z > Y B: Z > X > Y C: X > Y > Z X: B > A > C Y: A > B > C Z: A > B > C Potential for in±nite loop! Proposal algorithm /* Init */ All men/women are free /* main */ while ∃ a free man who hasn’t yet proposed to every woman m proposes to the highest-ranked w who didn’t yet reject him if w is free (m, w) become engaged else w is engaged to m’ 3-2 if m < 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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

08 - Stable Matching, part 1 - i w i n i 1 s.t(1 Each...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online