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.
"Coverage in sensor networks via persistent homology." Algebr. Geom. Topol. 7 (1) 339 - 358, 2007. https://doi.org/10.2140/agt.2007.7.339