On the Minimum-Area Rectangular and Square Annulus Problem

04/15/2019
by   Sang Won Bae, et al.
0

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(nlog^2 n) and O(nlog 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
10/23/2020

Geometric Separability using Orthogonal Objects

Given a bichromatic point set P=R ∪ B of red and blue points, a separato...
research
11/29/2019

Empty Squares in Arbitrary Orientation Among Points

This paper studies empty squares in arbitrary orientation among a set P ...
research
11/15/2018

Maximum-Width Empty Square and Rectangular Annulus

An annulus is, informally, a ring-shaped region, often described by two ...
research
10/27/2022

Generalizing the German Tank Problem

The German Tank Problem dates back to World War II when the Allies used ...
research
11/18/2019

Minimum-Width Double-Strip and Parallelogram Annulus

In this paper, we study the problem of computing a minimum-width double-...
research
08/12/2019

A Natural Quadratic Approach to the Generalized Graph Layering Problem

We propose a new exact approach to the generalized graph layering proble...
research
07/26/2018

Computing optimal shortcuts for networks

We study augmenting a plane Euclidean network with a segment, called a s...

Please sign up or login with your details

Forgot password? Click here to reset