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Stable Marriage Problem. William Kozma Jr ECE 443/543. History. Gale-Shapley 1962, “College Admissions and the Stability of Marriage” Introduced both monogamous and polygamous versions Polygamous version used for years Assigned “job applicants” to “job positions”. The Problem.
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Stable Marriage Problem William Kozma Jr ECE 443/543
History • Gale-Shapley 1962, “College Admissions and the Stability of Marriage” • Introduced both monogamous and polygamous versions • Polygamous version used for years • Assigned “job applicants” to “job positions”
The Problem • Exists two equal-sized sets • Each element ranks all elements in other set • Match is one-to-one between sets • Matching must be stable • Cannot exists a member from each set which would rather be matched with each other, than with their current match
Example Alice Rank = {bob, adam} adam Rank = {Alice, Beth} Beth Rank = {bob, adam} bob Rank = {Alice, Beth}
Algorithm • [Morning] Each Boy goes to the Girl on the top of his list. • [Afternoon] Girl selects the highest ranking Boy visiting her. Sends rest home. • [Night] All Boys not selected cross the top Girl off their list.
Program: Input • Rankings imputed in data file • First line must contain size of each set • List Rankings file.txt 3 a A B C b A C B c B C A A b a c B b c a C a c b
Program: Data Structures • Allocate memory for, • 2 arrays [iSize x iSize]: hold rankings • 2 arrays [iSize]: list members of each set • 2 arrays [iSize]: current position on respective list • Initialize Boys position to 1 and Girls to -1
Program: Loop • [Morning] *already done in position array • [Afternoon] Each Girl searches Boy’s position on his list to see if corresponds to her. Sets her position to highest ranking Boy found. • [Night] Each Boy checks of he is listing in any of the Girls position. In not, increment his ranking.
Program: Loop (cont) • Initialize iCounter == 0 each loop iteration • iCounter++ if rejection occurs • Loop until iCounter == 0; • Result is Stable Matching
Program: Improvements • In “real-world” all Girls make decisions simultaneously • Since program is linear, allow Boys to visit more than 1 Girl per day • If a Boy is rejected, automatically send him to his next highest ranking Girl • May not always result in less days, but will never be worse.
