Maximum-Width Empty Square and Rectangular Annulus

Bae, Sang Won; Baral, Arpita; Mahapatra, Priya Ranjan Sinha

doi:10.1007/978-3-030-10564-8_6

Cited by 2 publications

(2 citation statements)

References 18 publications

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“…They solved the problem in O(n 3 log n) time, and if the inner circle contains a fixed number of points then the problem can be solved in O(n log n) time. Maximum width empty annulus problems for axis parallel square and rectangles can be solved in O(n 3 ) and O(n 2 log n) time respectively [10]. Abravaya and Segal [2] studied the problem of locating maximum cardinality set of obnoxious facilities within a bounded rectangle in the Euclidean plane such that their pairwise distance is at least a given threshold.…”

Section: Related Workmentioning

confidence: 99%

Dispersing Facilities on Planar Segment and Circle Amidst Repulsion

Singireddy¹,

Basappa²

2022

Preprint

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In this paper we consider the problem of locating k obnoxious facilities (congruent disks of maximum radius) amidst n demand points (existing repulsive facility sites) ordered from left to right in the plane so that none of the existing facility sites are affected (no demand point falls in the interior of the disks). We study this problem in two restricted settings: (i) the obnoxious facilities are constrained to be centered on along a predetermined horizontal line segment pq, and (ii) the obnoxious facilities are constrained to lie on the boundary arc of a predetermined disk C. An (1 − )-approximation algorithm was given recently to solve the constrained problem inHere, for the problem in (i), we first propose an exact polynomial-time algorithm based on a binary search on all candidate radii computed explicitly. This algorithm runs in O((nk) 2 log (nk) + (n + k) log (nk)) time. We then show that using the parametric search technique of Megiddo [1]; we can solve the problem exactly in O((n + k) 2 ) time, which is faster than the latter. Continuing further, using the improved parametric technique we give an O(n log 2 n)-time algorithm for k = 2. We finally show that the above (1 − )-approximation algorithm of [6] can be easily adapted to solve the circular constrained problem of (ii) with an extra multiplicative factor of n in the running time.

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Section: Related Workmentioning

confidence: 99%

Dispersing Facilities on Planar Segment and Circle Amidst Repulsion

Singireddy¹,

Basappa²

2022

Preprint

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“…The author [7] recently showed that a minimum-width square annulus in arbitrary orientation can be computed in O(n 3 log n) time. Other variants such as the minimum-width annulus problem with outliers [6,8] and the maximum-width empty annulus problem [9,13] have also been studied.…”

Section: Introductionmentioning

confidence: 99%

On the Minimum-Area Rectangular and Square Annulus Problem

Bae¹

2019

Preprint

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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.

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Maximum-Width Empty Square and Rectangular Annulus

Cited by 2 publications

References 18 publications

Dispersing Facilities on Planar Segment and Circle Amidst Repulsion

Dispersing Facilities on Planar Segment and Circle Amidst Repulsion

On the Minimum-Area Rectangular and Square Annulus Problem

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