1. Lecture 5 CSE 331 Sep 11, 2015

2. Submit the form I’ll need confirmation in writing. No graded material will be handed back till I get this signed form from you!

3. Sign-up for mini projects Email me your group (=7) composition + your chosen algorithm

4. Homeworks HW 1 posted online: see piazza Pickup graded HW 0 starting Monday

5. Remember to read HW policies

6. Separate Proof idea/proof details Proof details with no proof idea will get you a zero for the entire problem

7. A valid matching Mal Wash Simon Inara Zoe Kaylee

8. Not a matching Mal Wash Simon Inara Zoe Kaylee

9. Perfect Matching Mal Wash Simon Inara Zoe Kaylee

10. Preferences Mal Wash Simon Inara Zoe Kaylee

11. Instability Mal Wash Simon Inara Zoe Kaylee

12. A stable marriage Even though BBT and JA are not very happy

13. Two stable marriages

14. Stable Marriage problem Set of men M and women W Matching (no polyandry/gamy in M X W) Perfect Matching (everyone gets married) Instablity m m w w m’m’w’w’ Preferences (ranking of potential spouses) Stable matching = perfect matching+ no instablity

15. Questions/Comments?

16. Two Questions Does a stable marriage always exist? If one exists, how quickly can we compute one?

17. Today’s lecture Naïve algorithm Gale-Shapley algorithm for Stable Marriage problem

18. Discuss: Naïve algorithm!

19. The naïve algorithm Go through all possible perfect matchings S If S is a stable matching then Stop Else move to the next perfect matching n! matchings Incremental algorithm to produce all n! prefect matchings?

20. Gale-Shapley Algorithm David Gale Lloyd Shapley O(n 3 ) algorithm

21. Moral of the story… >

22. Questions/Comments?

23. 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

24. Preferences Mal Wash Simon Inara Zoe Kaylee

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

26. 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 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 Once a man gets engaged, he remains engaged (to “better” women)

27. 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 (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 If w proposes to m after m’, then she prefers m’ to m

28. Questions/Comments?

29. 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?

30. GS’ does not output a stable marriage