arXiv subjects

Robert Mercaş

Publications and source records attributed to Robert Mercaş.

6 recordsLinked to original sources

arXiv2024

Ternary is Still Good for Parikh Matrices

The focus of this work is the study of Parikh matrices with emphasis on two concrete problems. In the first part of our presentation we show that a conjecture by Dick at al. in 2021 only stands in the case of ternary alphabets, while providing counterexamples for larger alphabets. In particular, we show that the only type of distinguishability in the case of 3-letter alphabets is the trivial one. The second part of the paper builds on the notion of Parikh matrices for projections of words, discussed initially in this work, and answers, once more in the case of a ternary alphabet, a question posed by Atanasiu et al. in 2022 with regards to the minimal Hamming distance in between words sharing a congruency class.

math.CO↗

arXiv2024

Subsequence Matching and Analysis Problems for Formal Languages

In this paper, we study a series of algorithmic problems related to the subsequences occurring in the strings of a given language, under the assumption that this language is succinctly represented by a grammar generating it, or an automaton accepting it. In particular, we focus on the following problems: Given a string $w$ and a language $L$, does there exist a word of $L$ which has $w$ as subsequence? Do all words of $L$ have $w$ as a subsequence? Given an integer $k$ alongside $L$, does there exist a word of $L$ which has all strings of length $k$, over the alphabet of $L$, as subsequences? Do all words of $L$ have all strings of length $k$ as subsequences? For the last two problems, efficient algorithms were already presented in [Adamson et al., ISAAC 2023] for the case when $L$ is a regular language, and efficient solutions can be easily obtained for the first two problems. We extend that work as follows: we give sufficient conditions on the class of input-languages, under which these problems are decidable; we provide efficient algorithms for all these problems in the case when the input language is context-free; we show that all problems are undecidable for context-sensitive languages. Finally, we provide a series of initial results related to a class of languages that strictly includes the regular languages and is strictly included in the class of context-sensitive languages, but is incomparable to the of class context-free languages; these results deviate significantly from those reported for language-classes from the Chomsky hierarchy.

cs.FL↗

arXiv2023

Sweep Complexity Revisited

We study the sweep complexity of DFA in one-way jumping mode answering several questions posed earlier. This measure is the number of times in the worst case that such machines have to return to the beginning of their input after having skipped some of the symbols. The class of languages accepted by these machines strictly includes the regular class and constant sweep complexity allows exactly the acceptance of regular languages. However, we show that there exist machines with higher than constant complexity still only accepting regular languages and that in general the sweep complexity of an automaton does not distinguish between accepting regular and non-regular languages. We establish separation results for asymptotic classes defined by this complexity measure and give a surprising exponential/logarithmic relation between factors of certain inputs which can be verified by such machines.

cs.FL↗

arXiv2016

A note on Thue games

In this work we improve on a result from~\cite{GryKosZma15}. In particular, we investigate the situation where a word is constructed jointly by two players who alternately append letters to the end of an existing word. One of the players (Ann) tries to avoid (non-trivial) repetitions, while the other one (Ben) tries to enforce them. We show a construction that is closer to the lower bound showed in~\cite{GryKozMic13} using entropy compression, and building on the probabilistic arguments based on a version of the Lovász Local Lemma from~\cite{Peg11}. We provide an explicit strategy for Ann to avoid (non-trivial) repetitions over a $7$-letter alphabet.

cs.DM↗

arXiv2015

Linear-Time Sequence Comparison Using Minimal Absent Words & Applications

Sequence comparison is a prerequisite to virtually all comparative genomic analyses. It is often realized by sequence alignment techniques, which are computationally expensive. This has led to increased research into alignment-free techniques, which are based on measures referring to the composition of sequences in terms of their constituent patterns. These measures, such as $q$-gram distance, are usually computed in time linear with respect to the length of the sequences. In this article, we focus on the complementary idea: how two sequences can be efficiently compared based on information that does not occur in the sequences. A word is an {\em absent word} of some sequence if it does not occur in the sequence. An absent word is {\em minimal} if all its proper factors occur in the sequence. Here we present the first linear-time and linear-space algorithm to compare two sequences by considering {\em all} their minimal absent words. In the process, we present results of combinatorial interest, and also extend the proposed techniques to compare circular sequences.

cs.DS↗

arXiv2015

A note on the avoidability of binary patterns with variables and reversals

In this note we present a characterisation of all unary and binary patterns that do not only contain variables, but also reversals of their instances. These types of variables were studied recently in either more general or particular cases. We show that the results are not surprising at all in the general case, and extend the avoidability of these patterns to enforce aperiodic words.

cs.FL↗