Adaptive stable marriage algorithms

Although it takes O(n2) worst-case time to solve a stable marriage problem instance with n men and n women, a trivial O(n) algorithm suffices if all men are known to have identical preference lists and all women also are known to have identical preference lists. Since real-world instances often involve men or women with similar but not necessarily identical preference lists, this motivates us to introduce the notion of an adaptive stable marriage algorithm --- an algorithm whose running time is of the form O(n + k), where k describes the aggregate amount of disagreement between the preference lists in our instance versus a pair of specified "consensus" preference lists, one for the men and one for women. The running time of an adaptive stable matching algorithm therefore gracefully scales from O(n2) in the worse case down to O(n) in the case where preference lists are all in close agreement. We show how the O(n+k) running time bound can be achieved if all women are known to have identical preference lists, leaving the case where both men and women can have non-identical but similar preference lists as an open question. We also show how this special case may serve as a good model for sports drafts.