Balancing Two-Dimensional Straight-Line Programs

TL;DR

This paper explores the complexity of building efficient random access structures for two-dimensional straight-line programs (SLPs), revealing limitations and proposing a balanced structure with near-optimal size and access time.

Contribution

It demonstrates the inherent size limitations for balanced 2D SLPs and introduces a balanced 2D SLP with holes that achieves efficient random access.

Findings

Balanced 2D SLPs of size O(g) enable O(log N) access time.

Any balanced 2D SLP of depth O(log N) must be of size at least Ω(g·N/log^3 N).

A generalized balanced structure with holes achieves near-optimal size and access time.

Abstract

We consider building, given a straight-line program (SLP) consisting of $g$ productions deriving a two-dimensional string $T$ of size $N\times N$, 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, Je\.z, 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 $N\times N$ 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 $\Omega(g\cdot N/\log^{3}N)$. We complement this with an upper bound showing how to construct such a 2D SLP of size $\mathcal{O}(g\cdot N)$. 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, Je\.z, Lohrey, JACM 2021] immediately implies that, given a 2D SLP of size $g$ deriving a string of size $N\times N$, 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 $\mathcal{O}(g \log^{\epsilon}N)$ providing random access in $\mathcal{O}(\log N/\log \log N)$ time, for any $\epsilon >0$.