Volume 7, issue 1 (2007)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Coverage in sensor networks via persistent homology

Vin de Silva and Robert Ghrist

Algebraic & Geometric Topology 7 (2007) 339–358
Abstract

We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris–Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications.

Keywords
Rips complex, Cech complex, persistent homology, sensor network, coverage
Mathematical Subject Classification 2000
Primary: 55M25, 93A15
Secondary: 55N35
References
Publication
Received: 25 November 2005
Revised: 29 January 2006
Accepted: 8 October 2006
Published: 25 April 2007
Authors
Vin de Silva
Department of Mathematics and Computer Science
Pomona College
Claremont CA 91711
USA
Robert Ghrist
Department of Mathematics and Coordinated Sciences Laboratory
University of Illinois
Urbana, IL 61801
USA