Maximum-width Axis-Parallel Empty Rectangular Annulus

TL;DR

This paper presents algorithms for finding the maximum-width axis-parallel empty rectangular annulus that contains all input points on its boundaries or inside, with improved efficiency in special cases.

Contribution

It introduces an $O(n^3)$ algorithm for the general problem and an $O(n ext{log} n)$ solution for a special case where the inner rectangle reduces to a point.

Findings

The general algorithm runs in $O(n^3)$ time.

The special case algorithm runs in $O(n ext{log} n)$ time.

The algorithms efficiently compute the maximum-width empty rectangular annulus.

Abstract

Given a set $P$ of $n$ points on $\mathbb R^{2}$, we address the problem of computing an axis-parallel empty rectangular annulus $A$ of maximum-width such that no point of $P$ lies inside $A$ but all points of $P$ must lie inside, outside and on the boundaries of two parallel rectangles forming the annulus $A$. We propose an $O(n^3)$ time and $O(n)$ space algorithm to solve the problem. In a particular case when the inner rectangle of an axis-parallel empty rectangular annulus reduces to an input point we can solve the problem in $O(n \log n)$ time and $O(n)$ space.