Extensions of the Maximum Bichromatic Separating Rectangle Problem

Abstract

An important topic in the field of geometric optimization is finding the largest rectangle separating two different points sets, which has significant applications in circuit design and data science. We consider some extensions of the maximum bichromatic separating rectangle (MBSR) problem. In one of the extensions, the optimal rectangle may include up to k outliers, where k is given as part of the input. This extension and is called MBSR with outliers or MBSR-O. In this paper, we improve the current known time bounds for MBSR-O from O(k7m logm+n) to O(k3m+km logm+n) using a clever staircase sweep approach. We also propose another extension, which is named MBSR among circles or MBSR-C and asks for the largest rectangle separating red points from blue unit circles. Our solution to MBSR-C is an O(m2+n)-time algorithm that involves an optimized scanning of all candidate circle arcs for locations of potential optimal solutions.