Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

Aschner, Rom; Katz, Matthew J.

doi:10.1007/s00453-015-0076-9

Cited by 14 publications

(7 citation statements)

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“…Notice that 2 is the best approximation ratio that one can hope for, since Biniaz et al [5] showed that for any α < π, the weight of an α-MST of a set of n points on the line, such that the distance between consecutive points is 1, is at least 2n − 3, whereas the weight of an MST is clearly n − 1. (This lower bound is also mentioned without a proof in [3]. )…”

Section: Introductionmentioning

confidence: 84%

“…Since there always exists a MST of P in which the degree of each vertex is at most 5, the interesting range for α is (0, 8π 5 ). The concept of bounded-angle (minimum) spanning tree (i.e., of an α-(M)ST) was introduced by Aschner and Katz [3], who arrived at it through the study of wireless networks with directional antennas. However, it is interesting in its own right.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, it is known (see [1,2,6]) that for any α ≥ π 3 , there always exists an α-ST of P . The next natural question is what is the status of the problem of computing an α-MST, for a given 'typical' angle α. Aschner and Katz [3] proved that (at least) for α = π and for α = 2π 3 the problem is NP-hard, and, therefore, it calls for efficient approximation algorithms. Obviously, the weight of an α-MST of P , for any angle α, is at least the weight of an MST of P , so if we develop an algorithm for constructing an α-ST, for some angle α, and prove that the weight of the trees constructed by the algorithm never exceeds some constant c times the weight of the corresponding MSTs, then we have a c-approximation algorithm for computing an α-MST.…”

Section: Introductionmentioning

confidence: 99%

“…That is, we are interested in an algorithm for computing a 'good' approximation of 2π 3 -MST, where by good we mean that the weight of the output 2π 3 -ST is not much larger than that of an MST (and thus of a 2π 3 -MST). Aschner and Katz [3] presented a 6-approximation algorithm for the problem. Subsequently, Biniaz et al [5] described an improved 16 3 -approximation algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…(Π is obtained by listing the vertices of P through an in-order traversal of MST(P ), where a vertex is added to the list when it is visited for the first time.) The algorithm of Aschner and Katz [3] operates on the path Π. It constructs a 2π 3 -ST of P from Π of weight at most 3ω(Π), and thus of weight at most 6ω(MST(P )).…”

Section: Introductionmentioning

confidence: 99%

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A 4-Approximation of the $\frac{2π}{3}$-MST

Ashur¹,

Katz²

2020

Preprint

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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 16 3 , respectively. In order to obtain this result, we devise a simple O(n)-time algorithm for constructing 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 2π3 -ST has weight greater than 2 − ε times the weight of the path. ACM Subject ClassificationTheory of computation → Computational geometry; Theory of computation → Design and analysis of algorithms Keywords and phrases bounded-angle spanning tree, bounded-degree spanning tree, hop-spanner Digital Object Identifier 10.4230/LIPIcs..2021.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%