MATCHING NUTS AND BOLTS IN O(N LOG N) TIME

Given a set of n nuts of distinct widths and a set of n bolts such tha t each nut corresponds to a unique bolt of the same width, how should we match every nut with its corresponding bolt by comparing nuts with bolts? (No comparison is allowed between two nuts or two bolts.) The p roblem can be naturally viewed as a variant of the classic sorting pro blem as follows. Given two lists of n numbers each such that one list is a permutation of the other, how should we sort the lists by compari sons only between numbers in different lists? We give an O(n log n)-ti me deterministic algorithm for the problem. This is optimal up to a co nstant factor and answers an open question posed by Alon et al. [Proce edings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, 19 94, pp. 690-696]. Moreover, when copies of nuts and bolts are allowed, our algorithm runs in optimal O(log n) time on n processors in Valian t's parallel comparison tree model. Our algorithm is based on the AKS sorting algorithm with substantial modifications.