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log
where log is used to denote the base-2 logarithm. We also give a strategy that obtains this ratio.
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doi:10.1016/S0304-3975(00)00144-4 
Copyright © 2001 Elsevier Science B.V. All rights reserved.
The ultimate strategy to search on m rays?*1
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Accepted 14 March 2000
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−1−O(1/log2 D). 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−1−O(1/log2 D) 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.
*1 This research is supported by the DFG-Project “Diskrete Probleme”, No. Ot 64/8-3.
Corresponding author; email: alopez-o@unb.ca; email: schuiere@informatik.uni-freiburg.de
