Online euclidean spanners

Sujoy Bhore, Csaba D. Tóth

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

3 Scopus citations

Abstract

In this paper, we study the online Euclidean spanners problem for points in Rd. Given a set S of n points in Rd, a t-spanner on S is a subgraph of the underlying complete graph G = (S, (S2 )), that preserves the pairwise Euclidean distances between points in S to within a factor of t, that is the stretch factor. Suppose we are given a sequence of n points (s1, s2,..., sn) in Rd, where point si is presented in step i for i = 1,..., n. The objective of an online algorithm is to maintain a geometric t-spanner on Si = {s1,..., si} for each step i. The algorithm is allowed to add new edges to the spanner when a new point is presented, but cannot remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the weight of an optimum spanner. Here the weight of a spanner is the sum of all edge weights. First, we establish a lower bound of Ω(ε1 log n/log ε1) for the competitive ratio of any online (1 + ε)-spanner algorithm, for a sequence of n points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a (1 + ε)-spanner with competitive ratio O(ε1 log n/log ε1). Next, we design online algorithms for sequences of points in Rd, for any constant d ≥ 2, under the L2 norm. We show that previously known incremental algorithms achieve a competitive ratio O(ε(d+1) log n). However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of ε. We describe an online Steiner (1 + ε)-spanner algorithm with competitive ratio O(ε(1−d)/2 log n). As a counterpart, we show that the dependence on n cannot be eliminated in dimensions d ≥ 2. In particular, we prove that any online spanner algorithm for a sequence of n points in Rd under the L2 norm has competitive ratio Ω(f(n)), where limn→∞ f(n) = ∞. Finally, we provide improved lower bounds under the L1 norm: Ω(ε2/log ε1) in the plane and Ω(ε−d) in Rd for d ≥ 3.

Original languageEnglish
Title of host publication29th Annual European Symposium on Algorithms, ESA 2021
EditorsPetra Mutzel, Rasmus Pagh, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772044
DOIs
StatePublished - 1 Sep 2021
Externally publishedYes
Event29th Annual European Symposium on Algorithms, ESA 2021 - Vitual, Lisbon, Portugal
Duration: 6 Sep 20218 Sep 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume204
ISSN (Print)1868-8969

Conference

Conference29th Annual European Symposium on Algorithms, ESA 2021
Country/TerritoryPortugal
CityVitual, Lisbon
Period6/09/218/09/21

Keywords

  • (1 + ε)-spanner
  • Geometric spanner
  • Minimum weight
  • Online algorithm

ASJC Scopus subject areas

  • Software

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