Covering Points by Isothetic Unit Squares

Given a set P of n points in R, we consider two related problems. Firstly, we study the problem of computing two isothetic unit squares which may be either disjoint or intersecting (having empty common zone) such that they together cover maximum number of points. The time and space complexities of the proposed algorithm for this problem are both O(n). We also study the problem of computing k disjoint isothetic unit squares, admitting sliceable k-partitions, which maximizes the sum of the points covered by them. Our proposed algorithm for this problem runs in time O(kn) and uses O(kn) space. To solve this problem, we propose an optimal O(n log n) time and O(n) space algorithm which computes O(n) isothetic unit squares each covering maximum number of points and having one side aligned with a point from P .