Covering many points with a small-area box

Abstract

Let P be a set of n points in the plane. We show how to find, for a given integer k >0, the smallest-area axis-parallel rectangle that covers k points of P in O(nk2logn+ n log2 n) time. We also consider the problem of, given a value α > 0, covering as many points of P as possible with an axis-parallel rectangle of area at most α. For this problem we give a probabilistic (1-ε)-approximation that works in near-linear time: In O((n/ε4) log3 n log(1/ε)) time we find an axis-parallel rectangle of area at most α that, with high probability, covers at least (1-ε)κ* points, where κ* is the maximum possible number of points that could be covered. © 2019, Macodrum library, Carleton University. All rights reserved.