Translating a convex polygon to contain a maximum number of points

Abstract

Given a set S of n points in the plane and a convex polygon P with m vertices, we consider the problem of finding a translation of P that contains the maximum number of points in S. We present two different solutions. Our first algorithm uses standard line-sweep techniques and requires O( nklog( nm) + m) time where k is the maximum number of points contained. Our second algorithm requires O( nklog( mk) + m) time, which is the asymptotically fastest known algorithm for this problem. Both algorithms require optimal O( m + n) space. The algorithms also solve in the same running time the bichromatic variant of the problem, where we are given two point sets A and B and the goal is to maximize the number of points covered from A while minimizing the number of points covered from B.