Balancing degree, diameter and weight in Euclidean spanners

Shay Solomon, Michael Elkin

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

Abstract

In a seminal STOC'95 paper, Arya et al. [4] devised a construction that for any set S of n points in ℝd and any ε > 0, provides a (1 + ε)-spanner with diameter O(logn), weight O(log2 n)w(MST(S)), and constant maximum degree. Another construction of [4] provides a (1 + ε)-spanner with O(n) edges and diameter α(n), where α stands for the inverse-Ackermann function. Das and Narasimhan [12] devised a construction with constant maximum degree and weight O(w(MST(S))), but whose diameter may be arbitrarily large. In another construction by Arya et al. [4] there is diameter O(logn) and weight O(logn) w(MST(S)), but it may have arbitrarily large maximum degree. These constructions fail to address situations in which we are prepared to compromise on one of the parameters, but cannot afford it to be arbitrarily large. In this paper we devise a novel unified construction that trades between maximum degree, diameter and weight gracefully. For a positive integer k, our construction provides a (1 + ε)-spanner with maximum degree O(k), diameter O(logk n + α(k)), weight O(k logk n logn) w(MST(S)), and O(n) edges. For k = O(1) this gives rise to maximum degree O(1), diameter O(logn) and weight O(log2 n) w(MST(S)), which is one of the aforementioned results of [4]. For k = n 1/α(n) this gives rise to diameter O(α(n)), weight O(n1/α(n) (logn) α(n)) w(MST(S)) and maximum degree O(n 1/α(n)). In the corresponding result from [4] the spanner has the same number of edges and diameter, but its weight and degree may be arbitrarily large. Our construction also provides a similar tradeoff in the complementary range of parameters, i.e., when the weight should be smaller than log2 n, but the diameter is allowed to grow beyond logn.

Original languageEnglish
Title of host publicationAlgorithms, ESA 2010 - 18th Annual European Symposium, Proceedings
Pages48-59
Number of pages12
EditionPART 1
DOIs
StatePublished - 2010
Externally publishedYes
Event18th Annual European Symposium on Algorithms, ESA 2010 - Liverpool, United Kingdom
Duration: 6 Sep 20108 Sep 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume6346 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference18th Annual European Symposium on Algorithms, ESA 2010
Country/TerritoryUnited Kingdom
CityLiverpool
Period6/09/108/09/10

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