The reach of axis-aligned squares in the plane

Abstract

Given a set S of n points in the unit square U=[0,1]2, an axis aligned square r⊆U is anchored at S if a corner of r is in S, and empty if no point in S lies in the interior of r. The reachR(S) of S is the union of all anchored empty squares for S. The maximum area of a packing of U with anchored empty squares is bounded above by area(R(S)). We prove that area(R(S))≥12 for every nonempty finite set S⊂U, and this bound is the best possible. We also describe an algorithm that computes the region R(S) and its area in O(nlogn) time.