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 $\gamma$ measure defined in terms of the smallest string attractor, and the $\delta$ measure defined in terms of the number of distinct substrings of the input string. Concretely, we introduce the two-dimensional measures $\gamma_{2D}$ and $\delta_{2D}$ as natural generalizations of $\gamma$ and $\delta$ and study some of their properties. Among other things, we prove that $\delta_{2D}$ is monotone and can be computed in linear time, and we show that, although it is still true that $\delta_{2D} \leq \gamma_{2D}$, the gap between the two measures can be $\Omega(\sqrt{n})$ for families of $n\times n$ matrices and therefore asymptotically larger than the gap in one-dimension. To complete the scenario of two-dimensional compressibility measures, we introduce also 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 $\gamma_{2D}$ is significantly different than in the one-dimensional case. As an application of the measures $\gamma_{2D}$ and $\delta_{2D}$ 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, 2023]. Finally, we present a linear time algorithm for constructing the two-dimensional block tree for arbitrary matrices, that is asymptotically faster than the (probabilistic) known solution which can only be used for binary matrices.