Covering many points with a small-area box

Abstract

\u3cp\u3eLet 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(nk\u3csup\u3e2\u3c/sup\u3elogn+ n log\u3csup\u3e2\u3c/sup\u3e 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/ε\u3csup\u3e4\u3c/sup\u3e) log\u3csup\u3e3\u3c/sup\u3e 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.\u3c/p\u3