Coping with Errors in Binary Search Procedures (Preliminary Report)

We consider the problem of identifying an unknown value xε{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 log 2 n+E.log 2 log 2 n+O(E.log 2 E) 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 log 2 n+E.log 2 log 2 n+0(E.log 2 E).

Authors

• Ronald L. Rivest
• Albert R. Meyer
• Daniel J. Kleitman
• Karl Winklmann
• Joel Spencer

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