Selection in monotone matrices and computing kth nearest neighbors

Abstract

We present an O((m + n)√n log n) time algorithm to select the k th smallest item from an m × n totally monotone matrix for any k ≤ ran. This is the first subquadratic algorithm for selecting an item from a totally monotone matrix. Our method also yields an algorithm for generalized row selection in monotone matrices of the same time complexity. Given a set S = {p1, …, pn} of n points in convex position and a vector k = {k1, …, kn}, we also present an O(n4/3 logO(1) n) algorithm to compute the kith nearest neighbor of pi for every i ≤ n; c is an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points of S are arbitrary, then the kith h nearest neighbor of pi, for all i ≤ n, can be computed in time O(n7/5 logc n), which also improves upon the previously best-known result.