Matching and Stable Matching

Transcription

1 8/21/2017 Matching and Stable Matching CS 320 Fall 2017 Dr. Geri Georg Instructor A human with a beard A human who always wears a hat A human who likes to wear red shirts A human who only wears sandals A human who wears glasses CS320 Matching and Stable Matching 1 CS320 Matching and Stable Matchng 6 A Great Dane But they have preferences A Yorkshire Terrier A Norwegian Forest Kitten A Cockatoo A Parakeet CS320 Matching and Stable Matchng 11 CS320 Matching and Stable Matchng 12 1

2 Match them up!? Formulate problem. Our Approach Design algorithm. Prove correctness. Analyze complexity. Implement. CS320 Matching and Stable Matchng 13 CS320 Introduction Formulate the problem Example: Pets: P={p 1..p n } Humans: H={h 1..h n } Pets = P x H is set of all possible ordered pairs Humans = CS320 Matching and Stable Matchng 15 CS320 Matching and Stable Matchng 16 2

3 Possible Matchings Perfect Matchings CS320 Matching and Stable Matchng 17 CS320 Matching and Stable Matchng 18 Matching Requirements Every pet is matched with only 1 human; so every human has 1 pet. This is called a perfect matching. Instability Example Assume these preferences: : : We d also like a matching where no pet and no human prefer each other to the match they ended up with. If this happens it is called an instability. A stable matching is a perfect matching with no instabilities. : What about this matching? : Perfect? Yes. Stable? No. CS320 Matching and Stable Matchng 19 CS320 Matching and Stable Matchng 20 3

4 Matching algorithm 1 Free for all! Do your best to get your first choice You have 2 minutes. Matching algorithm 1 Free for all! How did it work? Was there any swapping that went on (i.e. across pairs that had already been matched up)? What were the conditions? CS320 Matching and Stable Matchng 21 CS320 Matching and Stable Matchng 22 Matching algorithm 1 Who got their first choice? Did anyone end up with their last choice? 2. Design the algorithm 1 every pet and every human start out free 2 while (some pet p is free and hasn t considered every human) 3 choose p 4 h = highest-ranked human in p s list who hasn t been considered 5 if h is free (p h) are in a trial 6 else if (h prefers p to current match p ) (p h) enter a trial and p becomes free 7 else h prefers p and p remains free 8 return the set S of matched pets and humans CS320 Matching and Stable Matchng 23 CS320 Matching and Stable Matchng 24 4

5 Matching algorithm 2 Gale Shapley round 1 pets got to consider humans How did it go? 3. Prove correctness Does it terminate? When? Who got their first choice? Did anyone end up with their last choice? CS320 Matching and Stable Matchng 25 CS320 Matching and Stable Matchng 26 1 every pet and every human start out free 2 while there is a pet p who is free and who hasn t considered every human 3 choose such a pet p 4 choose the highest-ranked human h in p s preference not yet considered 5 if h is free then (p h) are in a trial 6 else h is currently in a trial with p 7 if h prefers p to p then p remains free 8 else h prefers p to p and (p h) enter a trial and p becomes free 9 return the set S of matched pets and CS320 Matching and Stable Matchng 27 humans Correctness Does it terminate? Yes. When? When there are no free pets. Does it come up with a perfect matching? CS320 Matching and Stable Matchng 28 5

6 By contradiction: Assume the size of sets Pets and Humans is n. Assume there is a free pet p after the algorithm terminates. Since the loop terminated p had considered all the humans. If a human gets into a trial they stay in some trial until the algorithm ends so all n humans are in a trial. The set of trial pairs is a matching so there must be n pets also in trials. Since there are only n pets in total and p is not in a trial this is a contradiction. Correctness Does it terminate? Yes. When? When there are no free pets. Does it come up with a perfect matching? Yes. Does it come up with a stable matching? CS320 Matching and Stable Matchng 29 CS320 Matching and Stable Matchng 30 By contradiction: Assume a matching: S = {(p h ) (p h) } and an unstable pair: (p h) where p would prefer h to h and h would prefer p to p. Case 1: p never considered h This means p prefers h so (p h) is not an instability. Case 2: p did consider h This means: h rejected p by trading up to p and h prefers p over p so (p h) is not an instability. In either case (p h) is not an instability so this is a contradiction. CS320 Matching and Stable Matchng 31 Correctness Does it terminate? Yes. When? When there are no free pets. Does it come up with a perfect matching? Yes. Does it come up with a stable matching? Yes. CS320 Matching and Stable Matchng 32 6

7 4. Analyze complexity While proving termination we proved the worst case number of iterations is n 2 for sets of Pets and Humans of size n. What about the lower bound? Image Credits Great Dane: E1 zrhyv9ge3bhb0ioscemrvdr1wziaeb_1qvtmlwkj4khijesyphq1qb URHbLA0fAVSGwqEglqAVUZl4YuJxEY80_1PfIgVwSoSCTrPqWTCkSwfEQhPkZNOKkPB&tbo=u&sa=X&ved=0ahUKEwi3tL2lbHVAhVj44MKHXJWDCAQ9C8IHw&biw=1172&bih=799&dpr=1.15#imgrc=y3YOz4qQax0lNM: Yorkshire Terrier: terrier puppies/ Norwegian Forest Kitten: breeds encyclopedia.com/long haired cat breeds.html Cockatoo: australia/sulphur crested cockatoo photos/ Parakeet: Person with a Beard: Person wearing a Hat: People wearing Red shirts: red t shirts for men and women in india/ People wearing Sandals: scandal/ Person wearing Glasses: tvshows/harry potter/luna lovegood in astrange glasses wallpaper/download/ chalkformulation: tutoring online/online math tutoring for kids designroutes: today.org/orms 6 02/frnetwork.html DijAlgoProof: proof.html 3lineComplex: complexity analysis why its important impl: 0.1d/ods cpp/ CS320 Matching and Stable Matchng 33 CS320 Matching and Stable Matchng 34 7