Compressibility measures for two-dimensional data
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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 study some 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≤γ_2D the gap between the two measures can be Ω(√(n)) for families of n× n matrices and therefore asymptotically larger than the gap in one-dimension. Finally, we use the measures γ_2D and δ_2D to 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, 2023].
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