Noisy source location on a line
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We study the problem of locating the source of an epidemic diffusion process from a sparse set of sensors, under noise. In a graph G=(V,E), an unknown source node v^* ∈ V is drawn uniformly at random, and unknown edge weights w(e) for e∈ E, representing the propagation delays along the edges, are drawn independently from a Gaussian distribution of mean 1 and variance σ^2. An algorithm then attempts to locate v^* by picking sensor (also called query) nodes s ∈ V and being told the length of the shortest path between s and v^* in graph G weighted by w. We consider two settings: static, in which all query nodes must be decided in advance, and sequential, in which each query can depend on the results of the previous ones. We characterize the query complexity when G is an n-node path. In the static setting, Θ(nσ^2) queries are needed for σ^2 ≤ 1, and Θ(n) for σ^2 ≥ 1. In the sequential setting, somewhat surprisingly, only Θ(loglog_1/σn) are needed when σ^2 ≤ 1/2, and Θ(loglog n)+O_σ(1) when σ^2 ≥ 1/2. This is the first mathematical study of source location under non-trivial amounts of noise.
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