Average value of sum of exponents of runs in a string

Kazuhiko Kusano, Wataru Matsubara, Akira Ishino, Ayumi Shinohara

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


A substring w[i.j] in w is called a repetition of period p if w[k] = w[k + p] for any i ≤ k ≤ j - p. Especially, a maximal repetition, which cannot be extended neither to left nor to right, is called a run. The ratio of the length of the run to its period, i.e. j - i + 1/ p, is called an exponent. The sum of exponents of runs in a string is of interest. The maximal value of the sum is still unknown, and the current upper bound is 2.9n given by Crochemore and Ilie, where n is the length of a string. In this paper we show a closed formula which exactly expresses the average value of it for any n and any alphabet size, and the limit of this value per unit length as n approaches infinity. For binary strings, the limit value is approximately 1.13103. We also show the average number of squares in a string of length n and its limit value.

Original languageEnglish
Pages (from-to)1135-1146
Number of pages12
JournalInternational Journal of Foundations of Computer Science
Issue number6
Publication statusPublished - 2009 Dec


  • Combinatorics on words
  • Repetition.
  • Run


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