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Identifying and Locating–Dominating Codes in (Random) Geometric Networks

Published online by Cambridge University Press:  11 August 2009

T. MÜLLER  and
J.-S. SERENI
T. MÜLLER
Affiliation:
School of Mathematics, Tel Aviv University, Ramat Aviv 69978, Israel (e-mail: tobias@post.tau.ac.il)
J.-S. SERENI
Affiliation:
CNRS (LIAFA, Université Denis Diderot), Paris, France and Department of Applied Mathematics (KAM), Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic (e-mail: sereni@kam.mff.cuni.cz)
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Abstract

We model a problem about networks built from wireless devices using identifying and locating–dominating codes in unit disk graphs. It is known that minimizing the size of an identifying code is -complete even for bipartite graphs. First, we improve this result by showing that the problem remains -complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks, and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast to those concerning random graphs in the Erdős–Rényi model. Another well-studied class of codes is that of locating–dominating codes, which are less demanding than identifying codes. A locating–dominating code always exists, but minimizing its size is still -complete in general. We extend this result to our setting by showing that this question remains -complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)2/3+o(1) if r ≤ /2−ϵ is chosen such that nr2 → ∞, and of size n1+o(1) if nr2 ≪ lnn.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 6 , November 2009 , pp. 925 - 952
Copyright
Copyright © Cambridge University Press 2009

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