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Stable Matching Examples

Stable Matching Examples. Ron K. Cytron http://www.cs.wustl.edu/~cytron/ Examples drawn from Knuths’ lecture in French: Marriages stables et leurs relations avec d’autres problemes combinatoires. 24 November 2008. Can’t we just guess?. #1. #2. #3. #4. #1. #2. #3. #4.

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Stable Matching Examples

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  1. Stable Matching Examples Ron K. Cytron http://www.cs.wustl.edu/~cytron/ Examples drawn from Knuths’ lecture in French: Marriages stables et leurs relations avec d’autres problemes combinatoires 24 November 2008

  2. Can’t we just guess? #1 #2 #3 #4 #1 #2 #3 #4

  3. Can’t we just guess? #1 #2 #3 #4 #1 #2 #3 #4 Aa

  4. Can’t we just guess? #1 #2 #3 #4 #1 #2 #3 #4 Aa Bb

  5. Can’t we just guess? #1 #2 #3 #4 #1 #2 #3 #4 Aa Bb Cc

  6. Can’t we just guess? #1 #2 #3 #4 #1 #2 #3 #4 Aa Bb Cc Dd

  7. Can’t we just guess? #1 #2 #3 #4 #1 #2 #3 #4 Aa Bb Cc Dd is not stable Do you see why?

  8. Can’t we just guess? #1 #2 #3 #4 #1 #2 #3 #4 Aa Bb Cc Dd is not stable Ab elope

  9. Let infidelity derive a solution? #1 #2 #3 #4 #1 #2 #3 #4 Aa Bb Cc Dd is not stable Ab elope

  10. Let infidelity derive a solution? Let Ab elope and….. #1 #2 #3 #4 #1 #2 #3 #4 Ab Ba Cc Dd is not stable Cb elope

  11. Let infidelity derive a solution? Let Cb elope and….. #1 #2 #3 #4 #1 #2 #3 #4 Ac Ba Cb Dd is not stable Dc elope

  12. Let infidelity derive a solution? Let Cb elope and….. #1 #2 #3 #4 #1 #2 #3 #4 Ad Ba Cb Dc is stable! How can we show this?

  13. Proof of no infidelity #1 #2 #3 #4 #1 #2 #3 #4 Ad Ba Cb Dc is stable!

  14. Proof of no infidelity #1 #2 #3 #4 #1 #2 #3 #4 Ad Ba Cb Dc is stable!

  15. Proof of no infidelity #1 #2 #3 #4 #1 #2 #3 #4 Ad Ba Cb Dc is stable! Algorithm?

  16. Your ideas? • Let infidelity rule and see what happens • Put top choices together and see • Compute average of X y y X and do something with that score • Favor one side with its preference, ignore the other side • Make smaller instances and try that • Start with worst matches and try to improve

  17. Iterate until stable?

  18. Iterate until stable? Ab elope

  19. Iterate until stable? Ab elope Cb elope

  20. Iterate until stable? Ab elope Cb elope Ca elope

  21. Iterate until stable? Ab elope Cb elope Ca elope Aa elope – back to start

  22. Try everything? • How many possibilities are there?

  23. How many possible stable matchings?

  24. How many possible stable matchings? Left group can decide to approach distinct people from the right group

  25. How many possible stable matchings? Each adjacent pair will decide between their first and second choices. For example, 1 and 2 ….

  26. How many possible stable matchings? Go with their first choices 1 and 2 are stable from the left Or…….

  27. How many possible stable matchings? Go with their second choices 1 and 2 are stable from the right

  28. How many possible stable matchings? n/2 groups can go 1 of 2 ways here Exponential number of possibilties: 2 n/2

  29. The algorithm

  30. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  31. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  32. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  33. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  34. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  35. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  36. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  37. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  38. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4

  39. Algorithm applied to first example #1 #2 #3 #4 #1 #2 #3 #4 C and D are unbreakable Little a could try for big A, but big A won’t go for it Little d could try for big B, but big B won’t go for it Stable!

  40. Desirable properties • Termination • Correctness • Everybody paired up • Stable arrangement • Fairness • Is it fair? • If not, who suffers?

  41. Fairness #1 #2 #3 #4 #1 #2 #3 #4

  42. Fairness #1 #2 #3 #4 #1 #2 #3 #4 Big letters have the advantage. Hospitals hosting residents.

  43. How about picking roommates? • Like stable matching, but instead of two populations • A B C … • a b c … • We have just one population: • A, B, C, … • Example: • A, B, C, D • Make any one pairing (such as AB) and the other paring (CD) is determined

  44. Roommates #1 #2 #3 We can cover all possibilities by trying A with C, then B, then D Let’s see what happens

  45. Roommates #1 #2 #3 Try A with C

  46. Roommates #1 #2 #3 Try A with C BC bail

  47. Roommates #1 #2 #3 Try A with B

  48. Roommates #1 #2 #3 Try A with B AC bail

  49. Roommates #1 #2 #3 Try A with D

  50. Roommates #1 #2 #3 No stable matching exists Try A with D AB bail

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