On the maximal sum of exponents of runs in a string

Authors:
Maxime Crochemore;Marcin Kubica;Jakub Radoszewski;Wojciech Rytter;Tomasz Waleń
Affiliations:
King's College London, London, UK and Université Paris-Est, France;Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Warsaw, Poland;Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Warsaw, Poland;Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Warsaw and Dept. of Math. and Informatics, Copernicus University, Toruń, Poland;Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Warsaw, Poland
Venue:
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Year:
2010

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Abstract

A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition v with a period p such that 2p ≤ |v|. The exponent of a run is defined as |v|/p and is ≥ 2. We show new bounds on the maximal sum of exponents of runs in a string of length n. Our upper bound of 4.1 n is better than the best previously known proven bound of 5.6 n by Crochemore & Ilie (2008). The lower bound of 2.035 n, obtained using a family of binary words, contradicts the conjecture of Kolpakov & Kucherov (1999) that the maximal sum of exponents of runs in a string of length n is smaller than 2n.