# On the Minimum-Area Rectangular and Square Annulus Problem

## TL;DR

This paper introduces algorithms for finding minimum-area rectangular and square annuli enclosing a set of points, improving previous results and addressing both fixed and arbitrary orientations.

## Contribution

It provides the first algorithms for minimum-area rectangular and square annuli in arbitrary orientations, with improved complexities over prior work.

## Key findings

- Algorithms with $O(n ext{log}^2 n)$ and $O(n ext{log} n)$ complexity for fixed orientation.
- New $O(n^3)$-time algorithms for arbitrary orientation.
- First algorithms for largest empty square problem in arbitrary orientation.

## Abstract

In this paper, we address the minimum-area rectangular and square annulus problem, which asks a rectangular or square annulus of minimum area, either in a fixed orientation or over all orientations, that encloses a set $P$ of $n$ input points in the plane. To our best knowledge, no nontrivial results on the problem have been discussed in the literature, while its minimum-width variants have been intensively studied. For a fixed orientation, we show reductions to well-studied problems: the minimum-width square annulus problem and the largest empty rectangle problem, yielding algorithms of time complexity $O(n\log^2 n)$ and $O(n\log n)$ for the rectangular and square cases, respectively. In arbitrary orientation, we present $O(n^3)$-time algorithms for the rectangular and square annulus problem by enumerating all maximal empty rectangles over all orientations. The same approach is shown to apply also to the minimum-width square annulus problem and the largest empty square problem over all orientations, resulting in $O(n^3)$-time algorithms for both problems. Consequently, we improve the previously best algorithm for the minimum-width square annulus problem by a factor of logarithm, and present the first algorithm for the largest empty square problem in arbitrary orientation. We also consider bicriteria optimization variants, computing a minimum-width minimum-area or minimum-area minimum-width annulus.