We consider the problem of searching on 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 . We show that even if an upper bound of on the distance to the target is known in advance, then the competitive ratio of any search strategy is at least . 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 . 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 if the target is found at distance .
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 .