Delay Estimation for Truly Random Binary Sequences or How to Measure the Length of Rip van Winkle’s Sleep

Abstract

Given a sequence generated by a binary symmetric memoryless source and a delayed version of the same sequence, the problem is to determine the delay. As a measure of complexity we use the number of comparisons of two digits in the sequence. A straightforward exhaustive search would compare the sequences after having delayed one of them each of the N possible delay values. On the average, two bits are compared before a mismatch is discovered. Hence the exhaustive method requires on the order of 2N binary comparisons before all but one of the possible delay values are eliminated.

This paper constructs an algorithm that requires on the order of $\sqrt {N} {\log _{2}}N$ comparisons to determine the delay. It was previously known that at least $\sqrt {N}$ comparisons are needed on the average before the delay is determined.