Generalization of Repetitiveness Measures for Two-Dimensional Strings

Lorenzo Carfagna; Giovanni Manzini; Giuseppe Romana; Marinella Sciortino; Cristian Urbina

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arXiv · 2505.10680

Generalization of Repetitiveness Measures for Two-Dimensional Strings

Lorenzo Carfagna·Giovanni Manzini·Giuseppe Romana·Marinella Sciortino·Cristian Urbina

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Abstract

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 $\delta$ and $\gamma$, 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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Lorenzo Carfagna, Giovanni Manzini, Giuseppe Romana, Marinella Sciortino, Cristian Urbina. 2025-05-15. Generalization of Repetitiveness Measures for Two-Dimensional Strings. https://arxiv.org/abs/2505.10680

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