# A 4-Approximation of the 2π/3 -MST

Stav Ashur, Matthew J. Katz

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

## Abstract

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 language American English Algorithms and Data Structures - 17th International Symposium, WADS 2021, Proceedings Anna Lubiw, Mohammad Salavatipour Springer Science and Business Media Deutschland GmbH 129-143 15 9783030835071 https://doi.org/10.1007/978-3-030-83508-8_10 Published - 31 Jul 2021 17th International Symposium on Algorithms and Data Structures, WADS 2021 - Virtual, OnlineDuration: 9 Aug 2021 → 11 Aug 2021

### Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 12808 LNCS

### Conference

Conference 17th International Symposium on Algorithms and Data Structures, WADS 2021 Virtual, Online 9/08/21 → 11/08/21

## Keywords

• 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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