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 we initiate the study 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 $\Omega(\sqrt{n})$ and therefore asymptotically larger than the gap between $\gamma$ and $\delta$. To complete the scenario of two-dimensional compressibility measures, we introduce 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 our results 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, 2024]. Our analysis shows that the space usage can be bounded in terms of both $\gamma_{2D}$ and $\delta_{2D}$. Finally, using insights from our analysis, we design the first linear time and space algorithm for constructing the two-dimensional block tree for arbitrary matrices.

翻译：本文将最近提出的两种一维压缩度量——基于最小字符串吸引子的$\gamma$度量与基于输入字符串不同子串数量的$\delta$度量——推广至二维数据。具体地，我们引入$\gamma_{2D}$和$\delta_{2D}$作为$\gamma$与$\delta$的自然推广，并系统研究其性质。研究证明：$\delta_{2D}$具有单调性且可在线性时间内计算；尽管仍满足$\delta_{2D} \leq \gamma_{2D}$，但两者间的差距可达$\Omega(\sqrt{n})$，因此渐近大于$\gamma$与$\delta$的差距。为完善二维压缩度量体系，我们提出度量$b_{2D}$，将最优解析概念推广至二维。令人意外的是，$b_{2D}$与$\gamma_{2D}$的关系显著异于一维情形。作为应用，本文首次分析了文献[Brisaboa 等，二维块树，《计算机期刊》，2024]中二维块树的空间使用情况，证明其空间复杂度可通过$\gamma_{2D}$与$\delta_{2D}$双重界定。基于分析洞察，我们设计了首个针对任意矩阵的线性时空复杂度二维块树构建算法。