Bounded-angle spanning tree: Modeling networks with angular constraints

Rom Aschner, Matthew J. Katz

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

8 Scopus citations

Abstract

We introduce a new structure for a set of points in the plane and an angle α, which is similar in flavor to a bounded-degree MST. We name this structure α-MST. Let P be a set of points in the plane and let 0 < α ≤ 2π be an angle. An α-ST of P is a spanning tree of the complete Euclidean graph induced by P, with the additional property that for each point p ∈ P, the smallest angle around p containing all the edges adjacent to p is at most α. An α-MST of P is then an α-ST of P of minimum weight. For α < π/3, an α-ST does not always exist, and, for α ≥ π/3, it always exists [1,2,9]. In this paper, we study the problem of computing an α-MST for several common values of α. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point p ∈ P, we associate a wedge W p of angle α and apex p. The goal is to assign an orientation and a radius rp to each wedge Wp, such that the resulting graph is connected and its MST is an α-MST. (We draw an edge between p and q if p ∈ Wq, q ∈ Wp, and |pq| ≤ r p, rq.) Unsurprisingly, the problem of computing an α-MST is NP-hard, at least for α = π and α = 2π/3. We present constant-factor approximation algorithms for α = π/2, 2π/3, π. One of our major results is a surprising theorem for α = 2π/3, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set P of 3n points in the plane and any partitioning of the points into n triplets, one can orient the wedges of each triplet independently, such that the graph induced by P is connected. We apply the theorem to the antenna conversion problem.

Original language English Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings Springer Verlag 387-398 12 PART 2 9783662439500 https://doi.org/10.1007/978-3-662-43951-7_33 Published - 1 Jan 2014 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, DenmarkDuration: 8 Jul 2014 → 11 Jul 2014

Publication series

Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) PART 2 8573 LNCS 0302-9743 1611-3349

Conference

Conference 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 Denmark Copenhagen 8/07/14 → 11/07/14

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