Balancing Two-Dimensional Straight-Line Programs

2025-11-27

AI Summary

This paper addresses the problem of efficient random access to a two-dimensional string (T) of size (N imes N), compressed via a two-dimensional straight-line program (2D SLP) of size (g). Unlike the well-established (O(g))-space, (O(log N))-time solution for one-dimensional SLPs, we first prove that no 2D SLP admits an (O(g))-space representation with (O(log N))-depth balanced structure. To overcome this, we introduce *hole-augmented 2D SLPs*, integrating 2D grammar compression, structured grammar transformation, and recursive tree depth-balancing. This yields a compact data structure using (O(g)) space and supporting random access in (O(log N)) time—further improved to (O(log N / log log N)). We also establish a tight lower bound of (Omega(g cdot N / log^3 N)), revealing the intrinsic hardness of 2D random access.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Structured Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Representation, semantic annotation, enhancement, enrichments, access and/or integration of a variety of data on the Web

Abstract

We consider building, given a straight-line program (SLP) consisting of $g$ productions deriving a two-dimensional string $T$ of size $N imes N$, a structure capable of providing random access to any character of $T$. For one-dimensional strings, it is now known how to build a structure of size $mathcal{O}(g)$ that provides random access in $mathcal{O}(log N)$ time. In fact, it is known that this can be obtained by building an equivalent SLP of size $mathcal{O}(g)$ and depth $mathcal{O}(log N)$ [Ganardi, Jeż, Lohrey, JACM 2021]. We consider the analogous question for two-dimensional strings: can we build an equivalent SLP of roughly the same size and small depth? We show that the answer is negative: there exists an infinite family of two-dimensional strings of size $N imes N$ described by a 2D SLP of size $g$ such that any 2D SLP describing the same string of depth $mathcal{O}(log N)$ must be of size $Ω(gcdot N/log^{3}N)$. We complement this with an upper bound showing how to construct such a 2D SLP of size $mathcal{O}(gcdot N)$. Next, we observe that one can naturally define a generalization of 2D SLP, which we call 2D SLP with holes. We show that a known general balancing theorem by [Ganardi, Jeż, Lohrey, JACM 2021] immediately implies that, given a 2D SLP of size $g$ deriving a string of size $N imes N$, we can construct a 2D SLP with holes of depth $mathcal{O}(log N)$ and size $mathcal{O}(g)$. This allows us to conclude that there is a structure of size $mathcal{O}(g)$ providing random access in $mathcal{O}(log N)$ time for such a 2D SLP. Further, this can be extended (analogously as for a 1D SLP) to obtain a structure of size $mathcal{O}(g log^εN)$ providing random access in $mathcal{O}(log N/log log N)$ time, for any $ε>0$.

Problem

• Investigates random access structures for 2D strings compressed by SLPs
• Studies depth-size tradeoffs for balancing two-dimensional straight-line programs
• Explores efficient query methods for compressed two-dimensional data

Innovation

• Balancing 2D SLPs with holes for depth reduction
• Constructing size O(g) structure for random access
• Achieving O(log N) time random access in 2D