String Periods in the Order-Preserving Model

The order-preserving model (op-model, in short) was introduced quite recently but has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(n log log n), O(n log^2 log n/log log log n), O(n log n) depending on the type of periodicity. In the most general variant the number of different periods can be as big as Omega(n^2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of such periods

Periodicity properties on partial words

The concept of periodicity has played over the years a centra1 role in the development of combinatorics on words and has been a highly valuable too1 for the design and analysis of algorithms. Fine and Wilf’s famous periodicity result, which is one of the most used and known results on words, has extensions to partia1 words, or sequences that may have a number of “do not know” symbols. These extensions fal1 into two categories: the ones that relate to strong periodicity and the ones that relate to weak periodicity. In this paper, we obtain consequences by generalizing, in particular, the combinatoria1 property that “for any word u over {a, b}, ua or ub is primitive,” which proves in some sense that there exist very many primitive partia1 words

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced

Fine and Wilf's theorem for k-abelian periods

Two words u and v are k-abelian equivalent if they contain the same number of occurrences of each factor of length at most k. This leads to a hierarchy of equivalence relations on words which lie properly in between the equality and abelian equality. The goal of this paper is to analyze Fine and Wilf&#39;s periodicity theorem with respect to these equivalence relations. Fine and Wilf&#39;s theorem tells exactly how long a word with two periods p and q can be without having the greatest common divisor of p and q as a period. Recently, the same question has been studied for abelian periods. In this paper we show that for k-abelian periods the situation is similar to the abelian case: In general, there is no bound for the lengths of such words, but the values of the parameters p, q and k for which the length is bounded can be characterized. In the latter case we provide nontrivial upper and lower bounds for the maximal lengths of such words. In some cases (e.g., for k = 2) we found the maximal length precisely.</p

String periods in the order-preserving model

In the order-preserving model, two strings match if they share the same relative order between the characters at the corresponding positions. This model is quite recent, but it has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(nlog⁡log⁡n), O(nlog2⁡log⁡n/log⁡log⁡log⁡n), O(nlog⁡n) depending on the type of periodicity. In the most general variant, the number of different op-periods can be as big as Ω(n2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of op-periods. In particular, we characterize the Fine–Wilf property for coprime op-periods. © 2019 Elsevier Inc.Supported by ISF grants no. 824/17 and 1278/16 and by an ERC grant MPM under the EU's Horizon 2020 Research and Innovation Programme (grant no. 683064).Supported by the Ministry of Science and Higher Education of the Russian Federation, project 1.3253.2017.A part of this work was done during the workshop StringMasters in Warsaw 2017 that was sponsored by the Warsaw Center of Mathematics and Computer Science. The authors thank the participants of the workshop, especially Hideo Bannai and Shunsuke Inenaga, for helpful discussions

Properties of Two-Dimensional Words

Combinatorics on words in one dimension is a well-studied subfield of theoretical computer science with its origins in the early 20th century. However, the closely-related study of two-dimensional words is not as popular, even though many results seem naturally extendable from the one-dimensional case. This thesis investigates various properties of these two-dimensional words. In the early 1960s, Roger Lyndon and Marcel-Paul Schutzenberger developed two famous results on conditions where nontrivial prefixes and suffixes of a one-dimensional word are identical and on conditions where two one-dimensional words commute. Here, the theorems of Lyndon and Schutzenberger are extended in the one-dimensional case to include a number of additional equivalent conditions. One such condition is shown to be equivalent to the defect theorem from formal languages and coding theory. The same theorems of Lyndon and Schutzenberger are then generalized to the two-dimensional case. The study of two-dimensional words continues by considering primitivity and periodicity in two dimensions, where a method is developed to enumerate two-dimensional primitive words. An efficient computer algorithm is presented to assist with checking the property of primitivity in two dimensions. Finally, borders in both one and two dimensions are considered, with some results being proved and others being offered as suggestions for future work. Another efficient algorithm is presented to assist with checking whether a two-dimensional word is bordered. The thesis concludes with a selection of open problems and an appendix containing extensive data related to one such open problem