### 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.

Original language | English |
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Pages (from-to) | 1816-1829 |

Number of pages | 14 |

Journal | Computers and Mathematics with Applications |

Volume | 58 |

Issue number | 9 |

DOIs | |

Publication status | Published - 1 Nov 2009 |

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### Keywords

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

### ASJC Scopus subject areas

- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computers and Mathematics with Applications*,

*58*(9), 1816-1829. https://doi.org/10.1016/j.camwa.2009.07.057