Understanding maximal repetitions in strings

The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach, the stronger result concerning the linearity of the sum of exponents of all runs follows easily

Fewest repetitions in infinite binary words

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of the fact that there exists an infinite binary word which contains finitely many squares and simultaneously avoids words of exponent larger than 7/3. Our infinite word contains 12 squares, which is the smallest possible number of squares to get the property, and 2 factors of exponent 7/3. These are the only factors of exponent larger than 2. The value 7/3 introduces what we call the finite-repetition threshold of the binary alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive threshold

Foreword

Aprovats per la Comissió de Govern de 01-12-200

Quasiperiodicities in Fibonacci strings

We consider the problem of finding quasiperiodicities in a Fibonacci string. A factor u of a string y is a cover of y if every letter of y falls within some occurrence of u in y. A string v is a seed of y, if it is a cover of a superstring of y. A left seed of a string y is a prefix of y that it is a cover of a superstring of y. Similarly a right seed of a string y is a suffix of y that it is a cover of a superstring of y. In this paper, we present some interesting results regarding quasiperiodicities in Fibonacci strings, we identify all covers, left/right seeds and seeds of a Fibonacci string and all covers of a circular Fibonacci string.Comment: In Local Proceedings of "The 38th International Conference on Current Trends in Theory and Practice of Computer Science" (SOFSEM 2012

The Rightmost Equal-Cost Position Problem

LZ77-based compression schemes compress the input text by replacing factors in the text with an encoded reference to a previous occurrence formed by the couple (length, offset). For a given factor, the smallest is the offset, the smallest is the resulting compression ratio. This is optimally achieved by using the rightmost occurrence of a factor in the previous text. Given a cost function, for instance the minimum number of bits used to represent an integer, we define the Rightmost Equal-Cost Position (REP) problem as the problem of finding one of the occurrences of a factor which cost is equal to the cost of the rightmost one. We present the Multi-Layer Suffix Tree data structure that, for a text of length n, at any time i, it provides REP(LPF) in constant time, where LPF is the longest previous factor, i.e. the greedy phrase, a reference to the list of REP({set of prefixes of LPF}) in constant time and REP(p) in time O(|p| log log n) for any given pattern p

Identifying all abelian periods of a string in quadratic time and relevant problems

Abelian periodicity of strings has been studied extensively over the last years. In 2006 Constantinescu and Ilie defined the abelian period of a string and several algorithms for the computation of all abelian periods of a string were given. In contrast to the classical period of a word, its abelian version is more flexible, factors of the word are considered the same under any internal permutation of their letters. We show two O(|y|^2) algorithms for the computation of all abelian periods of a string y. The first one maps each letter to a suitable number such that each factor of the string can be identified by the unique sum of the numbers corresponding to its letters and hence abelian periods can be identified easily. The other one maps each letter to a prime number such that each factor of the string can be identified by the unique product of the numbers corresponding to its letters and so abelian periods can be identified easily. We also define weak abelian periods on strings and give an O(|y|log(|y|)) algorithm for their computation, together with some other algorithms for more basic problems.Comment: Accepted in the "International Journal of foundations of Computer Science

A fast implementation of the Boyer-Moore string matching algorithm

Manuscript, http://www-igm.univ-mlv.fr/~lecroq/articles/cl2008.pd

Finite-Repetition threshold for infinite ternary words

The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at most r(a). This notion was introduced in 1972 by Dejean who gave the exact values of r(a) for every alphabet size a as it has been eventually proved in 2009. The finite-repetition threshold for an a-letter alphabet refines the above notion. It is the smallest rational number FRt(a) for which there exists an infinite word whose finite factors have exponent at most FRt(a) and that contains a finite number of factors with exponent r(a). It is known from Shallit (2008) that FRt(2)=7/3. With each finite-repetition threshold is associated the smallest number of r(a)-exponent factors that can be found in the corresponding infinite word. It has been proved by Badkobeh and Crochemore (2010) that this number is 12 for infinite binary words whose maximal exponent is 7/3. We show that FRt(3)=r(3)=7/4 and that the bound is achieved with an infinite word containing only two 7/4-exponent words, the smallest number. Based on deep experiments we conjecture that FRt(4)=r(4)=7/5. The question remains open for alphabets with more than four letters. Keywords: combinatorics on words, repetition, repeat, word powers, word exponent, repetition threshold, pattern avoidability, word morphisms.Comment: In Proceedings WORDS 2011, arXiv:1108.341