Extensions of the Maximum Bichromatic Separating Rectangle Problem

TL;DR

This paper extends the maximum bichromatic separating rectangle problem by introducing outliers and circle-based separation, providing improved algorithms and running time bounds for these new variants.

Contribution

It introduces two new variants of the MBSR problem, with improved algorithms and complexity bounds for each.

Findings

Improved running time for MBSR with outliers to O(k^3 m + m log m + n)

Developed an O(m^2 + n) algorithm for MBSR among circles

Extended the problem framework to handle outliers and circle separation scenarios

Abstract

In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in \cite{Armaselu-CCCG, Armaselu-arXiv}. One of the extensions, introduced in \cite{Armaselu-FWCG}, is called \textit{MBSR with outliers} or MBSR-O, and is a more general version of the MBSR problem in which the optimal rectangle is allowed to contain up to $k$ outliers, where $k$ is given as part of the input. For MBSR-O, we improve the previous known running time bounds of $O(k^7 m \log m + n)$ to $O(k^3 m + m \log m + n)$. The other extension is called \textit{MBSR among circles} or MBSR-C and asks for the largest axis-aligned rectangle separating red points from blue unit circles. For MBSR-C, we provide an algorithm that runs in $O(m^2 + n)$ time.