Binary Search with Errors and Variable Cost Queries

We consider the problem of identifying an unknown value x ∈ {1,2, . . . , n} by asking “Yes–No” questions about x. The goal is to minimize the number of questions required in the worst case, taking into account that no more than B questions may receive answer “Yes” and no more than E answers may be erroneous. We consider two versions of this problem: the discrete and the continuous version. In the discrete case x is a member of the finite set {1,2, . . . , n}; in the continuous case x is a member of the half-open real interval (0,1]. In the continuous case we will not in general be able to identify x exactly with a finite number of questions; rather we fix a size ε and then we compute the exact value of the minimal number of questions to get a subset of (0,1] of size ε which is known to contain x. The solution of the continuous version allows us to derive a lower bound for the minimal number of questions required for the discrete version of the problem.  1998 Elsevier Science B.V. All rights reserved.