### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Computers and Mathematics with Applications*,

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

**On the distribution of runs of ones in binary strings.** / Sinha, Koushik; Sinha, Bhabani P.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 58, no. 9, pp. 1816-1829. https://doi.org/10.1016/j.camwa.2009.07.057

}

TY - JOUR

T1 - On the distribution of runs of ones in binary strings

AU - Sinha, Koushik

AU - Sinha, Bhabani P.

PY - 2009/11/1

Y1 - 2009/11/1

N2 - 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.

AB - 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.

KW - Bernoulli's trial

KW - Counting problem

KW - Generating function,

KW - Run distribution

KW - Run statistics

UR - http://www.scopus.com/inward/record.url?scp=70349108322&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70349108322&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2009.07.057

DO - 10.1016/j.camwa.2009.07.057

M3 - Article

VL - 58

SP - 1816

EP - 1829

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 9

ER -