Searching with known error probability

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Abstract

We study the problem of interactive searching in a set of numbers using comparison queries, under the assumption that each answer can be erroneous with a constant probability p and that a given reliability 0 < q <1 of the result is required. The search is considered in three versions: continuous (the search space is the interval [0,1] and the unknown real x has to be found with a given accuracy 1/n), discrete bounded (x ε {1, ..., n}) and discrete unbounded (the unknown number n can be any positive integer). We prove that in all cases the search is feasible for any n and q iff p ≠1/2. For p ≠ 1/2 an O(logn) searching algorithm is given in the continuous case and O(log2n) algorithms in the discrete bounded and unbounded cases. For p < 1/3or p > 2/3, O(logn) algorithms are given in each version of search.

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This research was supported in part by the Natural Sciences and Engineering Research Council of Canada, Grant No. A8136.