TY - JOUR

T1 - Bounded-Angle Spanning Tree

T2 - Modeling Networks with Angular Constraints

AU - Aschner, Rom

AU - Katz, Matthew J.

N1 - Funding Information:
A preliminary version of this paper appears in the Proceedings of ICALP’14 []. Work by R. Aschner was partially supported by the Lynn and William Frankel Center for Computer Sciences. Work by M. Katz was partially supported by Grant 1045/10 from the Israel Science Foundation. Work by M. Katz and R. Aschner was partially supported by Grant 2010074 from the United States—Israel Binational Science Foundation.
Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - 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≤ 2 π 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, where the weight of an α-ST is the sum of the lengths of its edges. For α< π/ 3 , an α-ST does not always exist, and, for α≥ π/ 3 , it always exists (Ackerman et al. in Comput Geom Theory Appl 46(3):213–218, 2013; Aichholzer et al. in Comput Geom Theory Appl 46(1):17–28, 2013; Carmi et al. in Comput Geom Theory Appl 44(9):477–485, 2011). 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 Wp 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| ≤ rp, rq). We prove that the problem of computing an α-MST is NP-hard, at least for α= π and α= 2 π/ 3 , and present constant-factor approximation algorithms for α= π/ 2 , 2 π/ 3 , π. One of our major results is a surprising theorem for α= 2 π/ 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 and to the orientation and power assignment problem.

AB - 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≤ 2 π 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, where the weight of an α-ST is the sum of the lengths of its edges. For α< π/ 3 , an α-ST does not always exist, and, for α≥ π/ 3 , it always exists (Ackerman et al. in Comput Geom Theory Appl 46(3):213–218, 2013; Aichholzer et al. in Comput Geom Theory Appl 46(1):17–28, 2013; Carmi et al. in Comput Geom Theory Appl 44(9):477–485, 2011). 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 Wp 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| ≤ rp, rq). We prove that the problem of computing an α-MST is NP-hard, at least for α= π and α= 2 π/ 3 , and present constant-factor approximation algorithms for α= π/ 2 , 2 π/ 3 , π. One of our major results is a surprising theorem for α= 2 π/ 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 and to the orientation and power assignment problem.

KW - Approximation algorithms

KW - Directional antennas

KW - Hop spanner

KW - Minimum spanning tree

KW - NP-hardness

KW - Power assignment

KW - Wireless networks

UR - http://www.scopus.com/inward/record.url?scp=84944567516&partnerID=8YFLogxK

U2 - 10.1007/s00453-015-0076-9

DO - 10.1007/s00453-015-0076-9

M3 - Article

AN - SCOPUS:84944567516

VL - 77

SP - 349

EP - 373

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 2

ER -