TY - UNPB

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

AU - Ashur, Stav

AU - Katz, Matthew J.

N1 - DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2020/10/22

Y1 - 2020/10/22

N2 - Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree spanning trees, which have received significant attention. Let P={p1,…,pn} be a set of n points in the plane, let Π be the polygonal path (p1,…,pn), and let 0<α<2π 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 simple 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 163, respectively. In order to obtain this result, we devise a simple O(n)-time algorithm for constructing a 2π3-ST\, Τ 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 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 spanning trees, which have received significant attention. Let P={p1,…,pn} be a set of n points in the plane, let Π be the polygonal path (p1,…,pn), and let 0<α<2π 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 simple 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 163, respectively. In order to obtain this result, we devise a simple O(n)-time algorithm for constructing a 2π3-ST\, Τ 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 has weight greater than 2−ε times the weight of the path.

U2 - 10.48550/arXiv.2010.11571

DO - 10.48550/arXiv.2010.11571

M3 - Preprint

BT - A 4-Approximation of the 2π/3-MST.

ER -