TY - GEN

T1 - A 4-Approximation of the 2π/3 -MST

AU - Ashur, Stav

AU - Katz, Matthew J.

N1 - Funding Information:
M. Katz was supported by grant 1884/16 from the Israel Science Foundation.
Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021/7/31

Y1 - 2021/7/31

N2 - 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.

AB - 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.

KW - Bounded-angle spanning tree

KW - Bounded-degree spanning tree

KW - Hop-spanner

UR - http://www.scopus.com/inward/record.url?scp=85113509159&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-83508-8_10

DO - 10.1007/978-3-030-83508-8_10

M3 - Conference contribution

AN - SCOPUS:85113509159

SN - 9783030835071

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 129

EP - 143

BT - Algorithms and Data Structures - 17th International Symposium, WADS 2021, Proceedings

A2 - Lubiw, Anna

A2 - Salavatipour, Mohammad

PB - Springer Science and Business Media Deutschland GmbH

T2 - 17th International Symposium on Algorithms and Data Structures, WADS 2021

Y2 - 9 August 2021 through 11 August 2021

ER -