TY - JOUR

T1 - Connectivity guarantees for wireless networks with directional antennas

AU - Carmi, Paz

AU - Katz, Matthew J.

AU - Lotker, Zvi

AU - Rosén, Adi

PY - 2011/11

Y1 - 2011/11

N2 - We study a combinatorial geometric problem related to the design of wireless networks with directional antennas. Specifically, we are interested in necessary and sufficient conditions on such antennas that enable one to build a connected communication network, and in efficient algorithms for building such networks when possible. We formulate the problem by a set P of n points in the plane, indicating the positions of n transceivers. Each point is equipped with an α-degree directional antenna, and one needs to adjust the antennas (represented as wedges), by specifying their directions, so that the resulting (undirected) communication graph G is connected. (Two points p,qεP are connected by an edge in G, if and only if q lies in p's wedge and p lies in q's wedge.) We prove that if α=60°, then it is always possible to adjust the wedges so that G is connected, and that α≥60° is sometimes necessary to achieve this. Our proof is constructive and yields an O(nlogk) time algorithm for adjusting the wedges, where k is the size of the convex hull of P. Sometimes it is desirable that the communication graph G contain a Hamiltonian path. By a result of Fekete and Woeginger (1997) [8], if α=90°, then it is always possible to adjust the wedges so that G contains a Hamiltonian path. We give an alternative proof to this, which is interesting, since it produces paths of a different nature than those produced by the construction of Fekete and Woeginger. We also show that for any n and ε>0, there exist sets of points such that G cannot contain a Hamiltonian path if α=90°-ε.

AB - We study a combinatorial geometric problem related to the design of wireless networks with directional antennas. Specifically, we are interested in necessary and sufficient conditions on such antennas that enable one to build a connected communication network, and in efficient algorithms for building such networks when possible. We formulate the problem by a set P of n points in the plane, indicating the positions of n transceivers. Each point is equipped with an α-degree directional antenna, and one needs to adjust the antennas (represented as wedges), by specifying their directions, so that the resulting (undirected) communication graph G is connected. (Two points p,qεP are connected by an edge in G, if and only if q lies in p's wedge and p lies in q's wedge.) We prove that if α=60°, then it is always possible to adjust the wedges so that G is connected, and that α≥60° is sometimes necessary to achieve this. Our proof is constructive and yields an O(nlogk) time algorithm for adjusting the wedges, where k is the size of the convex hull of P. Sometimes it is desirable that the communication graph G contain a Hamiltonian path. By a result of Fekete and Woeginger (1997) [8], if α=90°, then it is always possible to adjust the wedges so that G contains a Hamiltonian path. We give an alternative proof to this, which is interesting, since it produces paths of a different nature than those produced by the construction of Fekete and Woeginger. We also show that for any n and ε>0, there exist sets of points such that G cannot contain a Hamiltonian path if α=90°-ε.

KW - Connectivity

KW - Directional antennas

KW - Polygonal paths

KW - Wireless networks

UR - http://www.scopus.com/inward/record.url?scp=79957542696&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2011.05.003

DO - 10.1016/j.comgeo.2011.05.003

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AN - SCOPUS:79957542696

SN - 0925-7721

VL - 44

SP - 477

EP - 485

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 9

ER -