Combinatorics and Algorithmics of Strings

Edited in cooperation with Robert MercaşStrings (aka sequences or words) form the most basic and natural data structure. They occur whenever information is electronically transmitted (as bit streams), when natural language text is spoken or written down (as words over, for example, the Latin alphabet), in the process of heredity transmission in living cells (through DNA sequences) or the protein synthesis (as sequence of amino acids), and in many more different contexts. Given this universal form of representing information, the need to process strings is apparent and is actually a core purpose of computer use. Algorithms to efficiently search through, analyze, (de-)compress, match, encode and decode strings are therefore of chief interest. Combinatorial problems about strings lie at the core of such algorithmic questions. Many such combinatorial problems are common in the string processing efforts in the different fields of application.http://drops.dagstuhl.de/opus/volltexte/2014/4552

Hardness of Detecting Abelian and Additive Square Factors in Strings

We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the 3SUM conjecture) of several problems related to finding Abelian square and additive square factors in a string. In particular, we conclude conditional optimality of the state-of-the-art algorithms for finding such factors. Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of an odd half-length, (b) computing centers of all Abelian square factors, (c) detecting an additive square factor in a length-$n$ string of integers of magnitude $n^{\mathcal{O}(1)}$, and (d) a problem of computing a double 3-term arithmetic progression (i.e., finding indices $i \ne j$ such that $(x_i+x_j)/2=x_{(i+j)/2}$) in a sequence of integers $x_1,\dots,x_n$ of magnitude $n^{\mathcal{O}(1)}$. Problem (d) is essentially a convolution version of the AVERAGE problem that was proposed in a manuscript of Erickson. We obtain a conditional lower bound for it with the aid of techniques recently developed by Dudek et al. [STOC 2020]. Problem (d) immediately reduces to problem (c) and is a step in reductions to problems (a) and (b). In conditional lower bounds for problems (a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it using several string gadgets that include arbitrarily long Abelian-square-free strings. Our reductions also imply conditional lower bounds for detecting Abelian squares in strings over a constant-sized alphabet. We also show a subquadratic upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].Comment: Accepted to ESA 202

Avoiding and Enforcing Repetitive Structures in Words

The focus of this thesis is on the study of repetitive structures in words, a central topic in the area of combinatorics on words. The results presented in the thesis at hand are meant to extend and enrich the existing theory concerning the appearance and absence of such structures. In the first part we examine whether these structures necessarily appear in infinite words over a finite alphabet. The repetitive structures we are concerned with involve functional dependencies between the parts that are repeated. In particular, we study avoidability questions of patterns whose repetitive structure is disguised by the application of a permutation. This novel setting exhibits the surprising behaviour that avoidable patterns may become unavoidable in larger alphabets. The second and major part of this thesis deals with equations on words that enforce a certain repetitive structure involving involutions in their solution set. Czeizler et al. (2009) introduced a generalised version of the classical equations u` Æ vmwn that were studied by Lyndon and Schützenberger. We solve the last two remaining and most challenging cases and thereby complete the classification of these equations in terms of the repetitive structures appearing in the admitted solutions. In the final part we investigate the influence of the shuffle operation on words avoiding ordinary repetitions. We construct finite and infinite square-free words that can be shuffled with themselves in a way that preserves squarefreeness. We also show that the repetitive structure obtained by shuffling a word with itself is avoidable in infinite words

Combinatorics and Algorithmics of Strings (Dagstuhl Seminar 14111)

Strings (aka sequences or words) form the most basic and natural data structure. They occur whenever information is electronically transmitted (as bit streams), when natural language text is spoken or written down (as words over, for example, the Latin alphabet), in the process of heredity transmission in living cells (through DNA sequences) or the protein synthesis (as sequence of amino acids), and in many more different contexts. Given this universal form of representing information, the need to process strings is apparent and is actually a core purpose of computer use. Algorithms to efficiently search through, analyze, (de-)compress, match, encode and decode strings are therefore of chief interest. Combinatorial problems about strings lie at the core of such algorithmic questions. Many such combinatorial problems are common in the string processing efforts in the different fields of application. The purpose of this seminar is to bring together researchers from different disciplines whose interests are string processing algorithms and related combinatorial problems on words. The two main areas of interest for this seminar are Combinatorics on Words and Stringology. This report documents the program and the outcomes of Dagstuhl Seminar 14111 "Combinatorics and Algorithmics of Strings"