On the maximal number of cubic runs in a string

Authors:
Maxime Crochemore;Costas Iliopoulos;Marcin Kubica;Jakub Radoszewski;Wojciech Rytter;Tomasz Waleń
Affiliations:
King's College London, London, UK;King's College London, London, UK;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, Poland;Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Warsaw, Poland
Venue:
LATA'10 Proceedings of the 4th international conference on Language and Automata Theory and Applications
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 maximal number of runs in a string of length n has been thoroughly studied, and is known to be between 0.944 n and 1.029 n. In this paper we investigate cubic runs, in which the shortest period p satisfies 3p≤|v|. We show the upper bound of 0.5 n on the maximal number of such runs in a string of length n, and construct an infinite sequence of words over binary alphabet for which the lower bound is 0.406 n.