On the distribution of runs of ones in binary strings

Koushik Sinha, Bhabani P. Sinha

Research output: Contribution to journalArticle

13 Citations (Scopus)


In this paper, we derive the number of binary strings which contain, for a given ik, exactly ik 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 ik 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 ik 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.

Original languageEnglish
Pages (from-to)1816-1829
Number of pages14
JournalComputers and Mathematics with Applications
Issue number9
Publication statusPublished - 1 Nov 2009



  • Bernoulli's trial
  • Counting problem
  • Generating function,
  • Run distribution
  • Run statistics

ASJC Scopus subject areas

  • Modelling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics

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