Improved selection in totally monotone arrays

Abstract

This paper's main result is an O((√m lg m)(n lg n)+m lg n)-time algorithm for computing the kth smallest entry in each row of an mxn totally monotone array. (A two-dimensional array A = {a[i,j]} is totally monotone if for all i ₁< i₂ and j ₁<j₂, a[i₁,j₁]<a[i₁,j₂] implies a[i₂,j₁<a[i₂,j₂.) For large values of k (in particular, for k=[n/2]), this algorithm is significantly faster than the O(k(m+n))-time algorithm for the same problem due to Kravets and Park (1991). An immediate consequence of this result is an O(n^3/2 lg²n)-time algorithm for computing the kth nearest neighbor of each vertex of a convex n-gon. In addition to the main result, we also give an O(n lg m)-time algorithm for computing an approximate median in each row of an m× n totally monotone array; this approximate median is an entry whose rank in its row lies between [n/4] and [3n/4].