Compression by Contracting Straight-Line Programs

TL;DR

This paper introduces contracting straight-line programs (SLPs), a new class of balanced grammars that enable efficient string compression, navigation, and finger search operations with improved time complexities and linear space requirements.

Contribution

The authors define contracting SLPs, show how to transform any SLP into this form efficiently, and demonstrate their applications in compressed tree navigation and finger search problems.

Findings

Transform SLPs into contracting SLPs in linear time with constant rule length.

Achieve $O( ext{log } d)$ time for child navigation in compressed trees.

Improve finger search time to $O( ext{log } d + ext{log}^{(t)} N)$, surpassing previous methods.

Abstract

In grammar-based compression a string is represented by a context-free grammar, also called a straight-line program (SLP), that generates only that string. We refine a recent balancing result stating that one can transform an SLP of size $g$ in linear time into an equivalent SLP of size $O(g)$ so that the height of the unique derivation tree is $O(\log N)$ where $N$ is the length of the represented string (FOCS 2019). We introduce a new class of balanced SLPs, called contracting SLPs, where for every rule $A \to \beta_1 \dots \beta_k$ the string length of every variable $\beta_i$ on the right-hand side is smaller by a constant factor than the string length of $A$. In particular, the derivation tree of a contracting SLP has the property that every subtree has logarithmic height in its leaf size. We show that a given SLP of size $g$ can be transformed in linear time into an equivalent contracting SLP of size $O(g)$ with rules of constant length. We present an application to the navigation problem in compressed unranked trees, represented by forest straight-line programs (FSLPs). We extend a linear space data structure by Reh and Sieber (2020) by the operation of moving to the $i$-th child in time $O(\log d)$ where $d$ is the degree of the current node. Contracting SLPs are also applied to the finger search problem over SLP-compressed strings where one wants to access positions near to a pre-specified finger position, ideally in $O(\log d)$ time where $d$ is the distance between the accessed position and the finger. We give a linear space solution where one can access symbols or move the finger in time $O(\log d + \log^{(t)} N)$ for any constant $t$ where $\log^{(t)} N$ is the $t$-fold logarithm of $N$. This improves a previous solution by Bille, Christiansen, Cording, and G{\o}rtz (2018) with access/move time $O(\log d + \log \log N)$.