Local construction of planar spanners in unit disk graphs with irregular transmission ranges

  • Authors:

  • Edgar Chávez;Stefan Dobrev;Evangelos Kranakis;Jaroslav Opatrny;Ladislav Stacho;Jorge Urrutia

  • Affiliations:

  • Escuela de Ciencias Fisico-Matemáticas de la, Universidad Michoacana de San Nicolás de Hidalgo, México;School of Information Technology and Engineering (SITE), University of Ottawa, Ottawa, Ontario, Canada;School of Computer Science, Carleton University, Ottawa, Ontario, Canada;Department of Computer Science, Concordia University, Montréal, Québec, Canada;Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada;Instituto de Matemáticas, Universidad Nacional Autónoma de México, México D.F., México

  • Venue:
  • LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
  • Year:
  • 2006

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Abstract

We give an algorithm for constructing a connected spanning subgraphs(panner) of a wireless network modelled as a unit disk graph with nodes of irregular transmission ranges, whereby for some parameter 0 r ≤ 1 the transmission range of a node includes the entire disk around the node of radius at least r and it does not include any node at distance more than one. The construction of a spanner is distributed and local in the sense that nodes use only information at their vicinity, moreover for a given integer k ≥ 2 each node needs only consider all the nodes at distance at most k hops from it. The resulting spanner has maximum degree at most 3 + $\frac{6}{\pi r}$ + $\frac{r+1}{r^{2}}$, when 0 r r = 1). Furthermore it is shown that the spanner is planar provided that the distance between any two nodes is at least $\sqrt{1-r^{2}}$. If the spanner is planar then for k ≥ 2 the sum of the Euclidean lengths of the edges of the spanner is at most $\frac{kr+1}{kr-1}$ times the sum of the Euclidean lengths of the edges of a minimum weight Euclidean spanning tree.