On the distribution of runs of ones in binary strings

Koushik Sinha, Bhabani P. Sinha

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

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
Volume58
Issue number9
DOIs
Publication statusPublished - 1 Nov 2009
Externally publishedYes

Fingerprint

Strings
Binary
Numbering systems
Data compression
Crosstalk
Wireless sensor networks
Computer Arithmetic
Communication
Counting Problems
Number system
Data Compression
Engineering Application
Energy Efficient
Mean Square
Generating Function
Wireless Sensor Networks
Encoding
Chip
Deviation
Roots

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

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

In: Computers and Mathematics with Applications, Vol. 58, No. 9, 01.11.2009, p. 1816-1829.

Research output: Contribution to journalArticle

Sinha, Koushik ; Sinha, Bhabani P. / On the distribution of runs of ones in binary strings. In: Computers and Mathematics with Applications. 2009 ; Vol. 58, No. 9. pp. 1816-1829.
@article{19efdabf26274500ae5378a306c17340,
title = "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 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.",
keywords = "Bernoulli's trial, Counting problem, Generating function,, Run distribution, Run statistics",
author = "Koushik Sinha and Sinha, {Bhabani P.}",
year = "2009",
month = "11",
day = "1",
doi = "10.1016/j.camwa.2009.07.057",
language = "English",
volume = "58",
pages = "1816--1829",
journal = "Computers and Mathematics with Applications",
issn = "0898-1221",
publisher = "Elsevier Limited",
number = "9",

}

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

AN - SCOPUS:70349108322

VL - 58

SP - 1816

EP - 1829

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 9

ER -