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

2025-10-22

AI Summary

Two-dimensional strings (e.g., images, maps, adjacency matrices) resist efficient processing by standard one-dimensional compressed indexes. Method: We propose the first two-dimensional grammar-based compression data structure supporting optimal-time random access. Our approach models 2D strings via 2D grammars, integrates divide-and-conquer with compressed string complexity theory, and—crucially—introduces the Orthogonal Vectors Conjecture to establish conditional lower bounds. Contributions/Results: (i) We rigorously prove a conditional lower bound for 2D pattern matching, exposing a fundamental computational gap from its 1D counterpart; (ii) we achieve $O(log n / log log n)$-time random access with space $O(|G|log^{2+varepsilon} n)$, where $|G|$ is the grammar size; (iii) we provide the first rigorous computational hardness assumptions underpinning multiple 2D query problems, establishing a foundational framework for conditional complexity in two dimensions.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Data transparency and provenance

Abstract

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 $TinSigma^{r imes c}$ 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.

Problem

• Developing efficient random access for 2D compressed strings using grammars
• Establishing conditional lower bounds for 2D pattern matching problems
• Connecting 2D compressed indexing complexity to fundamental 1D problems

Innovation

• Data structure for optimal random access to 2D grammar-compressed strings
• Conditional lower bounds for 2D pattern matching under hardness assumptions
• Connection between 2D query complexity and open 1D problems