Placing Two Axis-Parallel Squares to Maximize the Number of Enclosed Points

Abstract

Let [Formula: see text] be a set of [Formula: see text] input points in the plane. An algorithm is proposed to place a pair of axis-parallel unit squares, either intersecting with no points in the intersection zone or disjoint, together enclosing the maximum number of input points. The time and space complexities of the algorithm are both [Formula: see text]. In case the input points are allowed to lie inside the intersection zone of two intersecting axis-parallel unit squares, two such unit squares enclosing the maximum number of input points can be placed in [Formula: see text] time using [Formula: see text] space.