Class 15: Stable Matching

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

Transcript

1. Class 15: Stable Matching cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans | University of Virginia Storage fees on Exam 1 double after today! Can pick up after class today No class Thursday
2. Plan Stable Matching Gale-Shapley Algorithm Proving Correctness Secure Stable Matching No class Thursday PS6 is due Friday
3. Stable Matching
4. Formal Problem Definition Input: Output:
5. Formal Problem Definition Input: Output: Stable matching between and , , ≺% , ≺& A matching, = { , , , , . , . , … , 0 , 0 } is a stable matching if there is no pair (3 , 4 ) where 3 ≺67 4 and 4 ≺89 3 .
6. Stable Matching Anna Elsa Hans Kristoff
7. Stable Matching Anna Elsa Hans Kristoff Kristoff ≺:;;<Hans Kristoff ≺=>?<Hans Elsa ≺@<;?Anna Elsa ≺ABC?DEFFAnna
8. Stable Matches? Anna Elsa Hans Kristoff Kristoff ≺:;;<Hans Kristoff ≺=>?<Hans Elsa ≺@<;?Anna Elsa ≺ABC?DEFFAnna {(Anna, Hans), (Elsa, Kristoff)} {(Anna, Kristoff), (Elsa, Hans)}
9. Stable Matches? Anna Elsa Olaf Kristoff Olfa ≺:;;<Kristoff Kristoff ≺=>?<Olaf Anna ≺G><F Elsa Elsa ≺ABC?DEFFAnna {(Anna, Olaf), (Elsa, Kristoff)} {(Anna, Kristoff), (Elsa, Olaf)}
10. Gale- Shapley Algorithm Lloyd Shapley (1923-2016) accepting Nobel Prize (2012) (co-awarded to Alvin Roth)
11. David Gale (1921 – 2008)
12. Olfa ≺:;;< Kristoff Kristoff ≺=>?< Olaf Anna ≺G><F Elsa Elsa ≺ABC?DEFF Anna Implementing Gale-Shapley
14. Simplified Code (Non-Defensive) Do not write programs like this: assertions (especially) and comments are valuable (but don’t fit on slides)
16. Modeling the Program How should we model the program as a state machine?
18. = { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ }
19. = { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ } = , ⟶ g, g All cases:
20. = { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ } = , ⟶ g, g All cases: g = , , . , … , 3 + 1, … , 0 j is the 3 -ranked match for 3 Case 1:
21. = { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ } = , ⟶ g, g All cases: g = , , . , … , 3 + 1, … , 0 j is the 3 -ranked match for 3 Case 1: new pairing ∀l ∈ . l , j ∉ g = ∪ 3 , j Case 2: better pairing ∃l ∈ . l , j ∈ , j prefers 3 to j g = ∪ 3 , j Case 3: rejected proposal ∃l ∈ . l , j ∈ , j prefers j to 3 g =
22. Proving Termination For all valid inputs, gale_shapley eventually terminates (returns a pairing). Each step, one of the proposals counts increases by 1 (and all others stay the same). After .steps, must be , , … , so no further transitions possible.
23. Proving Correctness For all valid inputs, gale_shapley returns a stable matching of its inputs. A matching, = { , , , … , 0 , 0 } is a stable matching if there is no pair (3 , 4 ) where 3 ≺67 4 and 4 ≺89 3 .
24. For all valid inputs, gale_shapley returns a stable matching of its inputs. A matching, = { , , , … , 0 , 0 } is a stable matching if there is no pair (3 , 4 ) where 3 ≺67 4 and 4 ≺89 3 . = , ⟶ g, g All cases: g = , , . , …, 3 + 1, …, 0 jis the 3-ranked match for 3 Case 1: new pairing ∀l ∈ . l , j ∉ g = ∪ 3 , j Case 2: better pairing ∃l ∈ . l , j ∈ , j prefers 3 to j g = ∪ 3 , j Case 3: rejected proposal ∃l ∈ . l , j ∈ , j prefers j to 3 g = Left as problem for PS7
25. Charge No class Thursday Problem Set 6 due Friday