Coping with errors in binary search procedures

We consider the problem of identifying an unknown value x ε{lunate} {1, 2,..., n} using only comparisons of x to constants when as many as E of the comparisons may receive erroneous answers. For a continuous analogue of this problem we show that there is a unique strategy that is optimal in the worst case. This strategy for the continuous problem is then shown to yield a strategy for the original discrete problem that uses log₂n + E · log₂log₂n + O(E · log₂E) comparisons in the worst case. This number is shown to be optimal even if arbitrary "Yes-No" questions are allowed. We show that a modified version of this search problem with errors is equivalent to the problem of finding the minimal root of a set of increasing functions. The modified version is then also shown to be of complexity log₂n + E · log₂log₂n + O(E · log₂E).