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 + 2m m/(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 searchst rategy is at least 1 + 2m m/(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 + 2m m/(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.
KeywordsSearch Strategy Optimal Strategy Recurrence Equation Competitive Ratio Positive Real Root
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