Revisiting the Problem of Searching on a Line

摘要

We revisit the problem of searching for a target at an unknown location on a line when given upper and lower bounds on the distance D that separates the initial position of the searcher from the target. Prior to this work, only asymptotic bounds were known for the optimal competitive ratio achievable by any search strategy in the worst case. We present the first tight bounds on the exact optimal competitive ratio achievable, parametrized in terms of the given range for D, along with an optimal search strategy that achieves this competitive ratio. We prove that this optimal strategy is unique and that it cannot be computed exactly in general. We characterize the conditions under which an optimal strategy can be computed exactly and, when it cannot, we explain how numerical methods can be used efficiently. In addition, we answer several related open questions and we discuss how to generalize these results to m rays, for any m ≥ 2.