A 4-Approximation of the 2π/3 -MST

Stav Ashur, Matthew J. Katz

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


Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree (minimum) spanning trees, which have received significant attention. Let P be a set of n points in the plane, and let 0 < α< 2 π be an angle. An α -spanning tree (α -ST) of P is a spanning tree of the complete Euclidean graph over P, with the following property: For each vertex pi∈ P, the (smallest) angle that is spanned by all the edges incident to pi is at most α. An α -minimum spanning tree (α -MST) is an α -ST of P of minimum weight, where the weight of an α -ST is the sum of the lengths of its edges. In this paper, we consider the problem of computing an α -MST for the important case where α=2π/3. We present a 4-approximation algorithm, thus improving upon the previous results of Aschner and Katz and Biniaz et al., who presented algorithms with approximation ratios 6 and 16/3, respectively. To obtain this result, we devise an O(n)-time algorithm that, given any Hamiltonian path Π of P, constructs a 2π/3 -ST T of P, such that T ’s weight is at most twice that of Π and, moreover, T is a 3-hop spanner of Π. This latter result is optimal in the sense that for any ε> 0 there exists a polygonal path for which every 2π/3 -ST (of the corresponding set of points) has weight greater than 2 - ε times the weight of the path.

Original languageAmerican English
Title of host publicationAlgorithms and Data Structures - 17th International Symposium, WADS 2021, Proceedings
EditorsAnna Lubiw, Mohammad Salavatipour
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages15
ISBN (Print)9783030835071
StatePublished - 31 Jul 2021
Event17th International Symposium on Algorithms and Data Structures, WADS 2021 - Virtual, Online
Duration: 9 Aug 202111 Aug 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12808 LNCS


Conference17th International Symposium on Algorithms and Data Structures, WADS 2021
CityVirtual, Online


  • Bounded-angle spanning tree
  • Bounded-degree spanning tree
  • Hop-spanner

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science


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