Online unit covering in Euclidean space

Document Type

Conference Proceeding

Publication Date

1-1-2018

Abstract

We revisit the online Unit Covering problem in higher dimensions: Given a set of n points in Rd, that arrive one by one, cover the points by balls of unit radius, so as to minimize the number of balls used. In this paper, we work in R d using Euclidean distance. The current best competitive ratio of an online algorithm, O(2 d d log d), is due to Charikar et al. (2004); their algorithm is deterministic. (I) We give an online deterministic algorithm with competitive ratio O(1.321 d ), thereby improving on the earlier record by an exponential factor. In particular, the competitive ratios are 5 for the plane and 12 for 3-space (the previous ratios were 7 and 21, respectively). For d = 3, the ratio of our online algorithm matches the ratio of the current best offline algorithm for the same problem due to Biniaz et al. (2017), which is remarkable (and rather unusual). (II) We show that the competitive ratio of every deterministic online algorithm (with an adaptive deterministic adversary) for Unit Covering in Rd under the L 2 norm is at least d + 1 for every d ≥ 1. This greatly improves upon the previous best lower bound, Ω(log d/ log log log d), due to Charikar et al. (2004). (III) We obtain lower bounds of 4 and 5 for the competitive ratio of any deterministic algorithm for online Unit Covering in R 2 and respectively R 3 ; the previous best lower bounds were both 3. (IV) When the input points are taken from the square or hexagonal lattices in R 2 , we give deterministic online algorithms for Unit Covering with an optimal competitive ratio of 3.

Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volume

11346 LNCS

First Page

609

Last Page

623

Digital Object Identifier (DOI)

10.1007/978-3-030-04651-4_41

ISSN

03029743

E-ISSN

16113349

ISBN

9783030046507

Link to Full Text

Share

COinS