On the Minimum-Area Rectangular and Square Annulus Problem

Sang Won Bae
2020 Computational geometry  
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
more » ... dth 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 other geometric optimization problems with similar flavor, resulting in O (n 3 )-time algorithms. As a result, we improve the previously best algorithm for the minimum-width square annulus problem by a factor of logarithm. We also consider bicriteria optimization variants, computing a minimum-width minimum-area or minimum-area minimum-width annulus.
doi:10.1016/j.comgeo.2020.101697 fatcat:itttxkdykfbk5katebtc7jzzd4