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Coordinated search for an object hidden on the line

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We consider the coordinated search problem faced by two searchers who start together at zero and can move at speed one to find an object symmetrically distributed on the line. In particular we fully analyze the case of the negative exponential distribution given by the density f(x) = e−|x|μ/(2μ), μ > 0. The searchers wish to minimize the expected time to find the object and meet back together (with the object) at zero. We give necessary and sufficient conditions for the existence of an optimal search strategy when the target density is continuous and decreasing. We show that for the negative exponential distribution the optimal time is between 4.728μ and 4.729μ. A strategy with expected time in this interval begins with the searchers going in opposite directions and returning to the origin after searching up to successive distances 0.745μ, 2.11μ, 3.9μ, 6μ, 8.4μ,…. These results extend the theory of coordinated search to unbounded regions. It has previously been studied for objects hidden on a circle (by Thomas) and on an interval (by the author).

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