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} Beth Rank = {bob, adam} adam Rank = {Alice, Beth} 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 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 file.txt

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: Demo

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.

Program: Demo 2

Questions?