Optimal Random Access and Conditional Lower Bounds for 2D Compressed Strings

Compressed indexing is a powerful technique that enables efficient querying over data stored in compressed form, significantly reducing memory usage and often accelerating computation. While extensive progress has been made for one-dimensional strings, many real-world datasets (such as images, maps, and adjacency matrices) are inherently two-dimensional and highly compressible. Unfortunately, naively applying 1D techniques to 2D data leads to suboptimal results, as fundamental structural repetition is lost during linearization. This motivates the development of native 2D compressed indexing schemes that preserve both compression and query efficiency. We present three main contributions that advance the theory of compressed indexing for 2D strings: (1) We design the first data structure that supports optimal-time random access to a 2D string compressed by a 2D grammar. Specifically, for a 2D string $T\in\Sigma^{r\times c}$ compressed by a 2D grammar $G$ and any constant $\epsilon>0$, we achieve $O(\log n/\log \log n)$ query time and $O(|G|\log^{2+\epsilon}n)$ space, where $n=\max(r,c)$. (2) We prove conditional lower bounds for pattern matching over 2D-grammar compressed strings. Assuming the Orthogonal Vectors Conjecture, no algorithm can solve this problem in time $O(|G|^{2-\epsilon}\cdot |P|^{O(1)})$ for any $\epsilon>0$, demonstrating a separation from the 1D case, where optimal solutions exist. (3) We show that several fundamental 2D queries, such as the 2D longest common extension, rectangle sum, and equality, cannot be supported efficiently under hardness assumptions for rank and symbol occurrence queries on 1D grammar-compressed strings. This is the first evidence connecting the complexity of 2D compressed indexing to long-standing open problems in the 1D setting.