On the distribution of runs of ones in binary strings

In this paper, we derive the number of binary strings which contain, for a given ikik, exactly ikik runs of 1’s of length kk in all possible binary strings of length nn, 1≤k≤n1≤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 ikik runs of length kk, 1≤k≤n1≤k≤n, closely follow the theoretically derived distributions, for a given nn. For n=8n=8, we find that the experimentally obtained values for most file formats agree within ±5%±5% of the theoretically obtained values for all ikik runs of length kk, 1≤k≤n1≤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=8n=8. In view of these facts, the results presented in this paper could be useful in various application domains, like the ones mentioned above.