On the distribution of runs of ones in binary strings

Abstract

In this paper, we derive the number of binary strings which contain, for a given i k , exactly i k runs of 1’s of length k in all possible binary strings of length n , 1 ≤ k ≤ n . Such a knowledge about the distribution pattern of runs of 1’s in binary strings is useful in many engineering applications — for example, data compression, bus encoding techniques to reduce crosstalk in VLSI chip design, computer arithmetic using redundant binary number system and design of energy-efficient communication schemes in wireless sensor networks by transformation of runs of 1’s into compressed information patterns, among others. We present, here, a generating function based approach to derive a solution to this counting problem. Our experimental results demonstrate that, for most commonly used file formats, the observed distributions of exactly i k runs of length k , 1 ≤ k ≤ n , closely follow the theoretically derived distributions, for a given n . For n = 8 , we find that the experimentally obtained values for most file formats agree within ± 5 % of the theoretically obtained values for all i k runs of length k , 1 ≤ k ≤ n . Also, the root mean square (RMS) values of these deviations across all file types studied in this paper are less than 5% for n = 8 . In view of these facts, the results presented in this paper could be useful in various application domains, like the ones mentioned above.