TY - JOUR

T1 - Average value of sum of exponents of runs in a string

AU - Kusano, Kazuhiko

AU - Matsubara, Wataru

AU - Ishino, Akira

AU - Shinohara, Ayumi

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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.

AB - 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.

KW - Combinatorics on words

KW - Repetition.

KW - Run

UR - http://www.scopus.com/inward/record.url?scp=70749157066&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70749157066&partnerID=8YFLogxK

U2 - 10.1142/S0129054109007078

DO - 10.1142/S0129054109007078

M3 - Article

AN - SCOPUS:70749157066

VL - 20

SP - 1135

EP - 1146

JO - International Journal of Foundations of Computer Science

JF - International Journal of Foundations of Computer Science

SN - 0129-0541

IS - 6

ER -