arXiv subjects

Estéban Gabory

Publications and source records attributed to Estéban Gabory.

6 recordsLinked to original sources

arXiv2026

Algorithms and Indexing Lower Bounds for Variable String Matching

A \emph{generalized degenerate string} (GD) is a sequence $T=T_1\dots T_n$ of nonempty finite sets of strings, called \emph{segments}, such that all strings in a segment have the same length. Given a solid pattern $P$, GD string matching asks whether $P$ occurs in $T$. Ascone et al. (WABI 2024) identified this as the main remaining boundary case in the fine-grained complexity of pattern matching on variable strings, between variants with near-linear algorithms and those with SETH-based quadratic lower bounds. We give a $\tilde{\mathcal O}(N\sqrt m)$-time algorithm, where $N$ is the total size of $T$ and $m=|P|$, placing GD matching on the subquadratic side of this boundary. We also study indexing. For elastic-degenerate strings (ED), which drop the equal-width restriction, Gibney (SPIRE 2020) obtained $\mathcal O(nm^2)$ query time after linear preprocessing. We adapt this index to GD strings, obtaining $\mathcal O(nm)$ query time. Conversely, under SETH, we rule out GD indices with polynomial preprocessing and query time $\mathcal O(n^{1-\varepsilon}m^{\mathcal O(1)}+m)$. Under the $k$-Clique conjecture, we further rule out combinatorial GD indices with query time $\mathcal O(n^{\mathcal O(1)}m^{1-\varepsilon}+m)$, and combinatorial ED indices with query time $\mathcal O(n^{\mathcal O(1)}m^{2-\varepsilon})$, matching the quadratic dependence on $m$ in Gibney's upper bound. Finally, under the OMv conjecture, we show that, after polynomial preprocessing of a string set and a pattern, active-prefix queries on a bit vector of length $m$ cannot be answered in $\mathcal O(m^{2-\varepsilon})$ time. Since these queries are the standard bottleneck in ED matching, improving indexed ED queries below $\mathcal O(n^{\mathcal O(1)}m^2)$ would require both non-combinatorial techniques and an approach that avoids using active-prefix queries as the main bottleneck.

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arXiv2025

Balancing Two-Dimensional Straight-Line Programs

We consider building, given a straight-line program (SLP) consisting of $g$ productions deriving a two-dimensional string $T$ of size $N\times N$, a structure capable of providing random access to any character of $T$. For one-dimensional strings, it is now known how to build a structure of size $\mathcal{O}(g)$ that provides random access in $\mathcal{O}(\log N)$ time. In fact, it is known that this can be obtained by building an equivalent SLP of size $\mathcal{O}(g)$ and depth $\mathcal{O}(\log N)$ [Ganardi, Jeż, Lohrey, JACM 2021]. We consider the analogous question for two-dimensional strings: can we build an equivalent SLP of roughly the same size and small depth? We show that the answer is negative: there exists an infinite family of two-dimensional strings of size $N\times N$ described by a 2D SLP of size $g$ such that any 2D SLP describing the same string of depth $\mathcal{O}(\log N)$ must be of size $Ω(g\cdot N/\log^{3}N)$. We complement this with an upper bound showing how to construct such a 2D SLP of size $\mathcal{O}(g\cdot N)$. Next, we observe that one can naturally define a generalization of 2D SLP, which we call 2D SLP with holes. We show that a known general balancing theorem by [Ganardi, Jeż, Lohrey, JACM 2021] immediately implies that, given a 2D SLP of size $g$ deriving a string of size $N\times N$, we can construct a 2D SLP with holes of depth $\mathcal{O}(\log N)$ and size $\mathcal{O}(g)$. This allows us to conclude that there is a structure of size $\mathcal{O}(g)$ providing random access in $\mathcal{O}(\log N)$ time for such a 2D SLP. Further, this can be extended (analogously as for a 1D SLP) to obtain a structure of size $\mathcal{O}(g \log^εN)$ providing random access in $\mathcal{O}(\log N/\log \log N)$ time, for any $ε>0$.

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arXiv2025

Unclustered BWTs of any Length over Non-Binary Alphabets

We prove that for every integer $n > 0$ and for every alphabet $Σ_k$ 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

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↗

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.

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arXiv2022

Elastic-Degenerate String Matching with 1 Error

An elastic-degenerate string is a sequence of $n$ finite sets of strings of total length $N$, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length $m$ in an ED text. This problem has recently received some attention by the combinatorial pattern matching community, culminating in an $\tilde{\mathcal{O}}(nm^{ω-1})+\mathcal{O}(N)$-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where $ω$ denotes the matrix multiplication exponent and the $\tilde{\mathcal{O}}(\cdot)$ notation suppresses polylog factors. In the $k$-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most $k$ errors. $k$-EDSM can be solved in $\mathcal{O}(k^2mG+kN)$ time, under edit distance, or $\mathcal{O}(kmG+kN)$ time, under Hamming distance, where $G$ denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, $G$ is only bounded by $N$, and so even for $k=1$, the existing algorithms run in $Ω(mN)$ time in the worst case. In this paper we show that $1$-EDSM can be solved in $\mathcal{O}((nm^2 + N)\log m)$ or $\mathcal{O}(nm^3 + N)$ time under edit distance. For the decision version, we present a faster $\mathcal{O}(nm^2\sqrt{\log m} + N\log\log m)$-time algorithm. We also show that $1$-EDSM can be solved in $\mathcal{O}(nm^2 + N\log m)$ time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from $1$-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the $k$-errata trees for indexing with errors [Cole et al., STOC 2004].

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