The Ultimate Strategy to Search on m Rays?

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 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 1 O(1= log2 D) which is also optimal—but in a stricter sense. We also construct a search strategy that achieves this ratio. Astonishingly, our strategy works equally well for the unbounded case, that is, if the target is found at distance D from the starting point, then the competitive ratio is 1+2mm=(m 1)m 1 O(1= log2 D) and it is not necessary for our strategy to know an upper bound on D in advance.