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. Finally, we use the measures $\gamma_{2D}$ and $\delta_{2D}$ 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].

翻译：本文将最近提出的两种一维压缩性度量扩展至二维数据：基于最小字符串吸引子定义的γ度量，以及基于输入字符串不同子串数量定义的δ度量。具体而言，我们引入二维度量γ₂ᴰ和δ₂ᴰ作为γ和δ的自然推广，并研究它们的若干性质。其中包括证明δ₂ᴰ具有单调性且可在线性时间内计算，并表明尽管仍有δ₂ᴰ ≤ γ₂ᴰ，但对于n×n矩阵族，两种度量之间的差距可达Ω(√n)，因此渐近大于一维情况下的差距。最后，我们利用γ₂ᴰ和δ₂ᴰ对[Brisaboa等，《二维块树》，The Computer Journal，2023]中提出的二维块树的空间使用量进行首次分析。