arXiv subjects

Lorenzo Carfagna

Publications and source records attributed to Lorenzo Carfagna.

5 recordsLinked to original sources

arXiv2026

Solving Square-Submatrix Equation Systems

We consider systems of submatrix equations, that is, sets of equality constraints over square submatrices of the input. By generalising the recursive algorithm of Gawrychowski et al. [Universal reconstruction of a string, Theoretical Computer Science 2020] to two dimensions, we obtain a linear-time procedure that finds a solution for any such input system. As an immediate by-product, this yields an optimal-time algorithm for decompressing any two-dimensional macro scheme based on copy operations of sub-squares.

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arXiv2025

New Entropy Measures for Tries with Applications to the XBWT

Entropy quantifies the number of bits required to store objects under certain given assumptions. While this is a well established concept for strings, in the context of tries the state-of-the-art regarding entropies is less developed. The standard trie worst-case entropy considers the set of tries with a fixed number of nodes and alphabet size. However, this approach does not consider the frequencies of the symbols in the trie, thus failing to capture the compressibility of tries with skewed character distributions. On the other hand, the label entropy [FOCS '05], proposed for node-labeled trees, does not take into account the tree topology, which has to be stored separately. In this paper, we introduce two new entropy measures for tries - worst-case and empirical - which overcome the two aforementioned limitations. Notably, our entropies satisfy similar properties of their string counterparts, thereby becoming very natural generalizations of the (simpler) string case. Indeed, our empirical entropy is closely related to the worst-case entropy and is reachable through a natural extension of arithmetic coding from strings to tries. Moreover we show that, similarly to the FM-index for strings [JACM '05], the XBWT of a trie can be compressed and efficiently indexed within our k-th order empirical entropy plus o(n) bits, with n being the number of nodes. Interestingly, the space usage of this encoding includes the trie topology and the upper-bound holds for every k sufficiently small, simultaneously. This XBWT encoding is always strictly smaller than the original one [JACM '09] and we show that in certain cases it is asymptotically smaller.

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arXiv2025

Indexing Tries within Entropy-Bounded Space

We study the problem of indexing and compressing tries using a BWT-based approach. Specifically, we consider a succinct and compressed representation of the XBWT of Ferragina et al.\ [FOCS '05, JACM '09] corresponding to the analogous of the FM-index [FOCS '00, JACM '05] for tries. This representation allows to efficiently count the number of nodes reached by a given string pattern. To analyze the space complexity of the above trie index, we propose a proof for the combinatorial problem of counting the number of tries with a given symbol distribution. We use this formula to define a worst-case entropy measure for tries, as well as a notion of k-th order empirical entropy. In particular, we show that the relationships between these two entropy measures are similar to those between the corresponding well-known measures for strings. We use these measures to prove that the XBWT of a trie can be encoded within a space bounded by our k-th order empirical entropy plus a o(n) term, with n being the number of nodes in the trie. Notably, as happens for strings, this space bound can be reached for every sufficiently small k simultaneously. Finally, we compare the space complexity of the above index with that of the r-index for tries proposed by Prezza [SODA '21] and we prove that in some cases the FM-index for tries is asymptotically smaller.

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arXiv2025

Generalization of Repetitiveness Measures for Two-Dimensional Strings

The problem of detecting and measuring the repetitiveness of one-dimensional strings has been extensively studied in data compression and text indexing. Our understanding of these issues has been significantly improved by the introduction of the notion of string attractor [Kempa and Prezza, STOC 2018] and by the results showing the relationship between attractors and other measures of compressibility. When the input data are structured in a non-linear way, as in two-dimensional strings, inherent redundancy often offers an even richer source for compression. However, systematic studies on repetitiveness measures for two-dimensional strings are still scarce. In this paper we extend to two or more dimensions the main measures of complexity introduced for one-dimensional strings. We distinguish between the measures $δ$ and $γ$, defined in terms of the substrings of the input, and the measures $g$, $g_{rl}$, and $b$, which are based on copy-paste mechanisms. We study the properties and mutual relationships between these two classes and we show that the two classes become incomparable for $d$-dimensional inputs as soon as $d\geq 2$. Moreover, we show that our grammar-based representation of a $d$-dimensional string of size $N$ enables direct access to any symbol in $O(\log N)$ time. We also compare our measures for two-dimensional strings with the 2D Block Tree data structure [Brisaboa et al., Computer J., 2024] and provide some insights for the design of future effective two-dimensional compressors.

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arXiv2024

The landscape of compressibility measures for two-dimensional data

In this paper we extend to two-dimensional data two recently introduced one-dimensional compressibility measures: the $γ$ measure defined in terms of the smallest string attractor, and the $δ$ measure defined in terms of the number of distinct substrings of the input string. Concretely, we introduce the two-dimensional measures $γ_{2D}$ and $δ_{2D}$, as natural generalizations of $γ$ and $δ$, and we initiate the study of their properties. Among other things, we prove that $δ_{2D}$ is monotone and can be computed in linear time, and we show that, although it is still true that $δ_{2D} \leq γ_{2D}$, the gap between the two measures can be $Ω(\sqrt{n})$ and therefore asymptotically larger than the gap between $γ$ and $δ$. To complete the scenario of two-dimensional compressibility measures, we introduce the measure $b_{2D}$ which generalizes to two dimensions the notion of optimal parsing. We prove that, somewhat surprisingly, the relationship between $b_{2D}$ and $γ_{2D}$ is significantly different than in the one-dimensional case. As an application of our results we provide the first analysis of the space usage of the two-dimensional block tree introduced in [Brisaboa et al., Two-dimensional block trees, The computer Journal, 2024]. Our analysis shows that the space usage can be bounded in terms of both $γ_{2D}$ and $δ_{2D}$. Finally, using insights from our analysis, we design the first linear time and space algorithm for constructing the two-dimensional block tree for arbitrary matrices.

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