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Line-of-Sight Networks

Published online by Cambridge University Press:  01 March 2009

ALAN FRIEZE ,
JON KLEINBERG ,
R. RAVI  and
WARREN DEBANY
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213, USA (e-mail: alan@random.math.cmu.edu)
JON KLEINBERG
Affiliation:
Department of Computer Science, Cornell University, Ithaca NY 14853, USA (e-mail: kleinber@cs.cornell.edu)
R. RAVI
Affiliation:
Tepper School of Business, Carnegie Mellon University, Pittsburgh PA 15213, USA (e-mail: ravi+@andrew.cmu.edu)
WARREN DEBANY
Affiliation:
Information Grid Division, Air Force Research Laboratory/RIG, 525 Brooks Road, Rome, NY 13441-4505, USA (e-mail: warren.debany@rl.af.mil)
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Abstract

Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a central analytical tool in the wireless networking community.

For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and communication can only take place among nodes when they are close in space and when they have line-of-sight access to one another – consider, for example, urban settings or large indoor environments. In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the line-of-sight restrictions.

Here we propose a random-graph model incorporating both range limitations and line-of-sight constraints, and we prove asymptotically tight results for k-connectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as ‘streets’ and ‘avenues’ among a regularly spaced array of obstructions.) Further, we show that when the probability of node placement is a constant factor larger than the threshold for connectivity, near-shortest paths between pairs of nodes can be found, with high probability, by an algorithm using only local information. In addition to analysing connectivity and k-connectivity, we also study the emergence of a giant component, as well an approximation question, in which we seek to connect a set of given nodes in such an environment by adding a small set of additional ‘relay’ nodes.

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