controlledboundary - COORDINATE-FREE COVERAGE IN SENSOR...

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COORDINATE-FREE COVERAGE IN SENSOR NETWORKS WITH CONTROLLED BOUNDARIES VIA HOMOLOGY V. DE SILVA AND R. GHRIST A BSTRACT . We introduce tools from computational homology to verify cover- age in an idealized sensor network. Our methods are unique in that, while they are coordinate-free and assume no localization or orientation capabilities for the nodes, there are also no probabilistic assumptions. The key ingredient is the the- ory of homology from algebraic topology. We demonstrate the robustness of these tools by adapting them to a variety of settings, including static planar coverage, 3- d barrier coverage, and time-dependent sweeping coverage. We also give results on hole repair, error tolerance, optimal coverage, and variable radii. An overview of implementation is given. 1. I NTRODUCTION Sensor networks are an increasingly essential and pervasive feature of modern computation and automation [14]. Within this large topic of active and rapidly developing research, coverage problems are common. Such problems, involving gaps or holes in sensor networks, appear in a variety of settings relevant to ro- botics and networks: environmental sensing, communication and broadcasting, robot beacon navigation, surveillance, security, and warfare are common applica- tion domains. A specific example is as follows. Given a collection of nodes X in a bounded domain D of the plane, assume that each node can sense, broadcast to, or otherwise cover a region of fixed coverage radius about the node. The most basic form of coverage problem is the simple query: given the nodes, does the collection of coverage discs at X cover the domain D ? We provide a sufficiency criterion for coverage. We do not answer the problem of how the nodes should be placed in order to maximize coverage — nodes are assumed to be distributed a priori, yet not according to some fixed protocol. In particular, there are no assumptions about random distributions or densities. The coverage criterion we introduce is both computable and, at this time, centralized. We do not here demonstrate how to reduce the homological criteria of this paper to a distributed computation. VdS supported by DARPA # SPA 30759. RG supported by DARPA # HR0011-05-1-0008 and by NSF PECASE Grant # DMS - 0337713. 1
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2 V. DE SILVA & R. GHRIST 1.1. Assumptions. We assume a complete absence of localization capabilities. Nodes can determine neither distance nor direction. Only connectivity data between nodes is used. The only strong assumption we make is on the fence nodes set up along the boundary of the domain. This strong degree of control along the bound- ary is not strictly required (see § 6 of this paper and also [11]), but it simplifies the statements and proofs of theorems dramatically. A1: Nodes X broadcast their unique ID numbers. Each node can detect the identity of any node within broadcast radius r b .
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