Optimal binary search with two unreliable tests and minimum adaptiveness

Abstract

What is the minimum number of yes-no questions needed to find an m bit number x in the set S = {0,1,…,2m — 1} if up to ℓ answers may be erroneous/false ? In case when the (t + 1)th question is adaptivelyasked after receiving the answer to the tth question, the problem, posed by Ulam and Rényi, is a chapter of Berlekamp's theory of error-correcting communication with feedback. It is known that, with finitely many exceptions, one can find x asking Berlekamp's minimum number qℓ (m) of questions, i.e., the smallest integer q such that 2q ≥ 2m ((q/ℓ) + (q/ℓ-1+ + (q/2) + q + 1). At the opposite, nonadaptive extreme, when all questions are asked in a unique batch before receiving any answer, a search strategy with qℓ(m) questions is the same as an ℓ-error correcting code of length qℓ (m) having 2m codewords. Such codes in general do not exist for ℓ> 1. Focusing attention on the case I = 2, we shall show that, with the exception of m = 2 and m = 4, one can always find an unknown m bit number x ϵ S by asking q2(m) questions in two nonadaptive batches. Thus the results of our paper provide shortest strategies with as little adaptiveness/interaction as possible.