MIT6_042JS10_lec15_sol

MIT6_042JS10_lec15_sol - Massachusetts Institute of...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 8 Prof. Albert R. Meyer revised March 8, 2010, 683 minutes Solutions to In-Class Problems Week 6, Mon. Problem 1. Four Students want separate assignments to four VI-A Companies. Here are their preference rankings: Student Albert: Rich: Megumi: Justin: Company AT&T: Bellcore: HP: Draper: Companies HP, Bellcore, AT&T, Draper AT&T, Bellcore, Draper, HP HP, Draper, AT&T, Bellcore Draper, AT&T, Bellcore, HP Students Justin, Albert, Megumi, Rich Megumi, Rich, Albert, Justin Justin, Megumi, Albert, Rich Rich, Justin, Megumi, Albert (a) Use the Mating Ritual to find two stable assignments of Students to Companies. Solution. Treat Students as Boys and the result is the following assignment: Student Albert: Rich: Megumi: Justin: Companies Bellcore AT&T HP Draper Rank in the original list 2 1 1 1 Treat Companies as Boys and the result is the following assignment: Company AT&T: Bellcore: HP: Draper: Students Albert Rich Megumi Justin Rank in the original list 2 2 2 2 (b) Describe a simple procedure to determine whether any given stable marriage problem has a unique solution, that is, only one possible stable matching. Creative Commons 2010, Prof. Albert R. Meyer . 2 Solutions to In-Class Problems Week 6, Mon. Solution. See if the Mating Ritual with Boys as suitors yields the same solution as the algorithm with Girls as suitors. These two marriage assignments are boy-optimal and boy-pessimal, respec- tive. Obviously, if every boy’s optimal and pessimal choices are the same, then every boy has an unique choice. The solution is unique. Problem 2. A preserved invariant of the Mating ritual is: For every girl, G , and every boy, B , if G is crossed off B ’s list, then G has a favorite suitor and she prefers him over B ....
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This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec15_sol - Massachusetts Institute of...

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