Distributed computation of a sparse cover in sensor networks without location information

In this paper, we present a distributed algorithm for detecting redundancies in a sensor network with no location information. We demonstrate how, in the absence of localization devices, simplicial complexes and tools from computational homology can be used in providing valuable information on the properties of the cover. In particular, we capture the combinatorial relationships among the sensors by the means of the Rips complex, which is the generalization of the proximity graph of the network to higher dimensions. Our approach is based on computation of a certain generator of the second homology of the Rips complex of the network relative to the boundary. We formulate the problem of detecting redundant sensors as an optimization problem to compute the sparsest generator of the second relative homology classes. We also demonstrate how subgradient methods can be used in solving this optimization problem in a distributed manner. Finally, non-trivial simulations are provided that illustrate the performance of our algorithm.