The ultimate strategy to search on m rays?

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Abstract

We consider the problem of searching on m current rays for a target of unknown location. If no upper bound on the distance to the target is known in advance, then the optimal competitive ratio is 1+2mm/(m−1)m−1. We show that even if an upper bound of D on the distance to the target is known in advance, then the competitive ratio of any search strategy is at least 1+2mm/(m−1)m−1O(1/log2D). This is again optimal – but in a stricter sense.

In particular, this result implies the same lower bound for a robot searching for a target on infinite rays and finding it at a distance of D. To show that our lower bound is, indeed, optimal we construct a search strategy that achieves this ratio. Our strategy does not need to know an upper bound on the distance to the target in advance; it achieves a competitive ratio of 1+2mm/(m−1)m−1O(1/log2D) if the target is found at distance D.

Finally, we also present a linear time algorithm to compute the strategy that allows the robot to search the farthest for a given competitive ratio C.

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This research is supported by the DFG-Project “Diskrete Probleme”, No. Ot 64/8-3.