Generalization of Repetitiveness Measures for Two-Dimensional Strings

Detecting and measuring repetitiveness of strings is a problem that has been extensively studied in data compression and text indexing. When the data are structured in a non-linear way, as in two-dimensional strings, inherent redundancy offers a rich source for compression, yet systematic studies on repetitiveness measures are still lacking. In this paper, we extend to two dimensions the measures δ and γ, defined in terms of the submatrices of the input, as well as the measures g, g_rl, and b, which are based on copy-paste mechanisms. We study their properties and mutual relationships, and we show that the two classes of measures become incomparable when two-dimensional inputs are considered. We also compare our measures with the 2D Block Tree data structure [Brisaboa et al., Computer J., 2024], and provide some insights for the design of effective 2D compressors.