arXiv subjects

Gabriele Fici

Publications and source records attributed to Gabriele Fici.

At least 19 recordsLinked to original sources

arXiv2025

Dorst-Smeulders Coding for Arbitrary Binary Words

A binary word is Sturmian if the occurrences of each letter are balanced, in the sense that in any two factors of the same length, the difference between the number of occurrences of the same letter is at most 1. In digital geometry, Sturmian words correspond to discrete approximations of straight line segments in the Euclidean plane. The Dorst-Smeulders coding, introduced in 1984, is a 4-tuple of integers that uniquely represents a Sturmian word $w$, enabling its reconstruction using $|w|$ modular operations, making it highly efficient in practice. In this paper, we present a linear-time algorithm that, given a binary input word $w$, computes the Dorst-Smeulders coding of its longest Sturmian prefix. This forms the basis for computing the Dorst-Smeulders coding of an arbitrary binary word $w$, which is a minimal decomposition (in terms of the number of factors) of $w$ into Sturmian words, each represented by its Dorst-Smeulders coding. This coding could be leveraged in compression schemes where the input is transformed into a binary word composed of long Sturmian segments. Although the algorithm is conceptually simple and can be implemented in just a few lines of code, it is grounded in a deep analysis of the structural properties of Sturmian words.

cs.DM↗

arXiv2025

Unclustered BWTs of any Length over Non-Binary Alphabets

We prove that for every integer $n > 0$ and for every alphabet $\Sigma_k$ of size $k \geq 3$, there exists a necklace of length $n$ whose Burrows-Wheeler Transform (BWT) is completely unclustered, i.e., it consists of exactly $n$ runs with no two consecutive equal symbols. These words represent the worst-case behavior of the BWT for clustering, since the number of BWT runs is maximized. We also establish a lower bound on their number. This contrasts with the binary case, where the existence of infinitely many completely unclustered BWTs is still an open problem, related to Artin's conjecture on primitive roots.

cs.DM↗

arXiv2025

String Consensus Problems with Swaps and Substitutions

String consensus problems aim at finding a string that minimizes some given distance with respect to an input set of strings. In particular, in the Closest string problem, we are given a set of strings of equal length and a radius $d$. The objective is to find a new string that differs from each input string by at most $d$ substitutions. We study a generalization of this problem where, in addition to substitutions, swaps of adjacent characters are also permitted, each operation incurring a unit cost. Amir et al. showed that this generalized problem is NP-hard, even when only swaps are allowed. In this paper, we show that it is FPT with respect to the parameter $d$. Moreover, we investigate a variant in which the goal is to minimize the sum of distances from the output string to all input strings. For this version, we present a polynomial-time algorithm.

cs.DS↗

arXiv2025

Morphisms and BWT-run Sensitivity

We study how the application of injective morphisms affects the number $r$ of equal-letter runs in the Burrows-Wheeler Transform (BWT). This parameter has emerged as a key repetitiveness measure in compressed indexing. We focus on the notion of BWT-run sensitivity after application of an injective morphism. For binary alphabets, we characterize the class of morphisms that preserve the number of BWT-runs up to a bounded additive increase, by showing that it coincides with the known class of primitivity-preserving morphisms, which are those that map primitive words to primitive words. We further prove that deciding whether a given binary morphism has bounded BWT-run sensitivity is possible in polynomial time with respect to the total length of the images of the two letters. Additionally, we explore new structural and combinatorial properties of synchronizing and recognizable morphisms. These results establish new connections between BWT-based compressibility, code theory, and symbolic dynamics.

cs.FL↗

arXiv2025

Digital Convexity and Combinatorics on Words

An upward (resp. downward) digitally convex word is a binary word that best approximates from below (resp. from above) an upward (resp. downward) convex curve in the plane. We study these words from the combinatorial point of view, formalizing their geometric properties and highlighting connections with Christoffel words and finite Sturmian words. In particular, we study from the combinatorial perspective the operations of inflation and deflation on digitally convex words.

math.CO↗

arXiv2025

U-index: A Universal Indexing Framework for Matching Long Patterns

Text indexing is a fundamental and well-studied problem. Classic solutions either replace the original text with a compressed representation, e.g., the FM-index and its variants, or keep it uncompressed but attach some redundancy - an index - to accelerate matching. The former solutions thus retain excellent compressed space, but are slow in practice. The latter approaches, like the suffix array, instead sacrifice space for speed. We show that efficient text indexing can be achieved using just a small extra space on top of the original text, provided that the query patterns are sufficiently long. More specifically, we develop a new indexing paradigm in which a sketch of a query pattern is first matched against a sketch of the text. Once candidate matches are retrieved, they are verified using the original text. This paradigm is thus universal in the sense that it allows us to use any solution to index the sketched text, like a suffix array, FM-index, or r-index. We explore both the theory and the practice of this universal framework. With an extensive experimental analysis, we show that, surprisingly, universal indexes can be constructed much faster than their unsketched counterparts and take a fraction of the space, as a direct consequence of (i) having a lower bound on the length of patterns and (ii) working in sketch space. Furthermore, these data structures have the potential of retaining or even improving query time, because matching against the sketched text is faster and verifying candidates can be theoretically done in constant time per occurrence (or, in practice, by short and cache-friendly scans of the text). Finally, we discuss some important applications of this novel indexing paradigm to computational biology. We hypothesize that such indexes will be particularly effective when the queries are sufficiently long, and so demonstrate applications in long-read mapping.

cs.DS↗

arXiv2025

Generalized De Bruijn Words, Invertible Necklaces, and the Burrows-Wheeler Transform

We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles in the generalized de Bruijn graphs introduced in the early '80s in the context of network design. When the size of the alphabet is a prime $p$, we define invertible necklaces as those whose BWT-matrix is non-singular. We show that invertible necklaces of length $n$ correspond to normal bases of the finite field $F_{p^n}$, and that they form an Abelian group isomorphic to the Reutenauer group $RG_p^n$. Using known results in abstract algebra, we can make a bridge between generalized de Bruijn words and invertible necklaces. In particular, we highlight a correspondence between binary de Bruijn words of order $d+1$, binary necklaces of length $2^{d}$ having an odd number of $1$'s, invertible BWT matrices of size $2^{d}\times 2^{d}$, and normal bases of the finite field $F_{2^{2^{d}}}$.

math.CO↗

arXiv2024

The Shortest Interesting Binary Words

I will show that there exist two binary words (one of length 4 and one of length 6) that play a special role in many different problems in combinatorics on words. They can therefore be considered \textit{the shortest interesting binary words}. My claim is supported by the fact that these two words appear in dozens of papers in combinatorics on words.

math.CO↗

arXiv2024

Some Remarks on Palindromic Periodicities

We say a finite word $x$ is a palindromic periodicity if there exist two palindromes $p$ and $s$ such that $|x| \geq |ps|$ and $x$ is a prefix of the word $(ps)^ω= pspsps\cdots$. In this paper we examine the palindromic periodicities occurring in some classical infinite words, such as Sturmian words, episturmian words, the Thue-Morse word, the period-doubling word, the Rudin-Shapiro word, the paperfolding word, and the Tribonacci word, and prove a number of results about them. We also prove results about words with the smallest number of palindromic periodicities.

math.CO↗

arXiv2024

Maximal Closed Substrings

A string is closed if it has length 1 or has a nonempty border without internal occurrences. In this paper we introduce the definition of a \emph{maximal closed substring} (MCS), which is an occurrence of a closed substring that cannot be extended to the left nor to the right into a longer closed substring. MCSs with exponent at least $2$ are commonly called \emph{runs}; those with exponent smaller than $2$, instead, are particular cases of \emph{maximal gapped repeats}. We provide an algorithm that, given a string of length $n$ locates all MCSs the string contains in $\mathcal O(n\log n)$ time.

cs.DS↗

arXiv2023

Substring Complexity in Sublinear Space

Shannon's entropy is a definitive lower bound for statistical compression. Unfortunately, no such clear measure exists for the compressibility of repetitive strings. Thus, ad hoc measures are employed to estimate the repetitiveness of strings, e.g., the size $z$ of the Lempel-Ziv parse or the number $r$ of equal-letter runs of the Burrows-Wheeler transform. A more recent one is the size $γ$ of a smallest string attractor. Let $T$ be a string of length $n$. A string attractor of $T$ is a set of positions of $T$ capturing the occurrences of all the substrings of $T$. Unfortunately, Kempa and Prezza [STOC 2018] showed that computing $γ$ is NP-hard. Kociumaka et al. [LATIN 2020] considered a new measure of compressibility that is based on the function $S_T(k)$ counting the number of distinct substrings of length $k$ of $T$, also known as the substring complexity of $T$. This new measure is defined as $δ= \sup\{S_T(k)/k, k\geq 1\}$ and lower bounds all the relevant ad hoc measures previously considered. In particular, $δ\leq γ$ always holds and $δ$ can be computed in $\mathcal{O}(n)$ time using $Θ(n)$ working space. Kociumaka et al. showed that one can construct an $\mathcal{O}(δ\log \frac{n}δ)$-sized representation of $T$ supporting efficient direct access and efficient pattern matching queries on $T$. Given that for highly compressible strings, $δ$ is significantly smaller than $n$, it is natural to pose the following question: Can we compute $δ$ efficiently using sublinear working space? We address this algorithmic challenge by showing the following bounds to compute $δ$: $\mathcal{O}(\frac{n^3\log b}{b^2})$ time using $\mathcal{O}(b)$ space, for any $b\in[1,n]$, in the comparison model; or $\tilde{\mathcal{O}}(n^2/b)$ time using $\tilde{\mathcal{O}}(b)$ space, for any $b\in[\sqrt{n},n]$, in the word RAM model.

cs.DS↗

arXiv2022

Abelian Combinatorics on Words: a Survey

We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these results.

cs.DM↗

arXiv2022

Some Results on Digital Segments and Balanced Words

We exhibit combinatorial results on Christoffel words and binary balanced words that are motivated by their geometric interpretation as approximations of digital segments. We give a closed formula for counting the exact number of balanced words with $a$ zeroes and $b$ ones. We also study minimal non-balanced words.

cs.FL↗

arXiv2022

On the Lie complexity of Sturmian words

Bell and Shallit recently introduced the Lie complexity of an infinite word $s$ as the function counting for each length the number of conjugacy classes of words whose elements are all factors of $s$. They proved, using algebraic techniques, that the Lie complexity is bounded above by the first difference of the factor complexity plus one; hence, it is uniformly bounded for words with linear factor complexity, and, in particular, it is at most 2 for Sturmian words, which are precisely the words with factor complexity $n+1$ for every $n$. In this note, we provide an elementary combinatorial proof of the result of Bell and Shallit and give an exact formula for the Lie complexity of any Sturmian word.

cs.DM↗

arXiv2021

Properties of a Class of Toeplitz Words

We study the properties of the uncountable set of Stewart words. These are Toeplitz words specified by infinite sequences of Toeplitz patterns of the form $αβγ$, where $α,β,γ$ is any permutation of the symbols 0,1,?. We determine the critical exponent of the Stewart words, prove that they avoid the pattern $xxyyxx$, find all factors that are palindromes, and determine their subword complexity. An interesting aspect of our work is that we use automata-theoretic methods and a decision procedure for automata to carry out the proofs.

cs.FL↗

arXiv2021

Adaptive Learning of Compressible Strings

Suppose an oracle knows a string $S$ that is unknown to us and that we want to determine. The oracle can answer queries of the form "Is $s$ a substring of $S$?". In 1995, Skiena and Sundaram showed that, in the worst case, any algorithm needs to ask the oracle $σn/4 -O(n)$ queries in order to be able to reconstruct the hidden string, where $σ$ is the size of the alphabet of $S$ and $n$ its length, and gave an algorithm that spends $(σ-1)n+O(σ\sqrt{n})$ queries to reconstruct $S$. The main contribution of our paper is to improve the above upper-bound in the context where the string is compressible. We first present a universal algorithm that, given a (computable) compressor that compresses the string to $τ$ bits, performs $q=O(τ)$ substring queries; this algorithm, however, runs in exponential time. For this reason, the second part of the paper focuses on more time-efficient algorithms whose number of queries is bounded by specific compressibility measures. We first show that any string of length $n$ over an integer alphabet of size $σ$ with $rle$ runs can be reconstructed with $q=O(rle (σ+ \log \frac{n}{rle}))$ substring queries in linear time and space. We then present an algorithm that spends $q \in O(σg\log n)$ substring queries and runs in $O(n(\log n + \log σ)+ q)$ time using linear space, where $g$ is the size of a smallest straight-line program generating the string.

cs.DS↗

arXiv2020

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

A prefix normal word is a binary word with the property that no substring has more $1$s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length $n$ as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in $\Oh(\log^2 n)$ amortized time per word, while the best generation algorithm hitherto has $\Oh(n)$ running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages.

cs.DS↗

arXiv2020

Primitive Sets of Words

Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. We say that a submonoid $M$ generated by $k$ elements of $A^*$ is {\em $k$-maximal} if there does not exist another submonoid generated by at most $k$ words containing $M$. We call a set $X \subseteq A^*$ {\em primitive} if it is the basis of a $|X|$-maximal submonoid. This definition encompasses the notion of primitive word -- in fact, $\{w\}$ is a primitive set if and only if $w$ is a primitive word. By definition, for any set $X$, there exists a primitive set $Y$ such that $X \subseteq Y^*$. We therefore call $Y$ a {\em primitive root} of $X$. As a main result, we prove that if a set has rank $2$, then it has a unique primitive root. To obtain this result, we prove that the intersection of two $2$-maximal submonoids is either the empty word or a submonoid generated by one single primitive word. For a single word $w$, we say that the set $\{x,y\}$ is a {\em bi-root} of $w$ if $w$ can be written as a concatenation of copies of $x$ and $y$ and $\{x,y\}$ is a primitive set. We prove that every primitive word $w$ has at most one bi-root $\{x,y\}$ such that $|x|+|y|<\sqrt{|w|}$. That is, the bi-root of a word is unique provided the word is sufficiently long with respect to the size (sum of lengths) of the root. Our results are also compared to previous approaches that investigate pseudo-repetitions, where a morphic involutive function $θ$ is defined on $A^*$. In this setting, the notions of $θ$-power, $θ$-primitive and $θ$-root are defined, and it is shown that any word has a unique $θ$-primitive root. This result can be obtained with our approach by showing that a word $w$ is $θ$-primitive if and only if $\{w, θ(w)\}$ is a primitive set.

cs.FL↗