A Factor-5 Conversion to Internal Collage Systems

2026-10-03

AI Summary

This study addresses the limitation that the previously established nine-fold size bound during the conversion from general collage systems to internal collage systems is excessively high, thereby constraining compression efficiency. To overcome this, the work proposes an improved top-down conversion algorithm incorporating three core techniques: truncation normalization, repetition decomposition, and monotonicity of structural reachability. By leveraging a deterministic unit-cost random access machine model with O(m²) time complexity, the approach integrates grammar-based compression with structural transformation to achieve optimization. The primary contribution is a rigorous proof demonstrating that the minimum size of an internal collage system for any string does not exceed five times that of its minimal general collage system. This result significantly reduces the compression blow-up ratio, providing new theoretical foundations for efficient structural transformations in data compression.

Abstract

A collage system is a grammar-based compression model that extends straight-line programs with repetition and substring truncation. Internal collage systems additionally require every nonterminal to be structurally reachable from the start symbol. Migita, Uehata, and I (CPM 2026) showed that any collage system of size $m$ can be converted into an internal one generating the same string with size at most $9m$, and left the improvement of this constant as an open problem. We show that a simple refinement of their top-down conversion reduces the bound to $5m$. The proof combines three elementary ideas: canonicalization of generated truncations that remain aligned with a target endpoint; a repetition decomposition that absorbs every complete copy of the repetition base into a maximal core; and monotonicity of structural reachability, which prevents an input truncation rule from both becoming structurally reachable and later acting as a hidden target at which endpoint alignment is lost. The conversion runs in deterministic $O(m^2)$ worst-case time in the stated unit-cost random-access machine model. Consequently, for every string, the minimum size of an internal collage system is at most five times the minimum size of a general collage system.

Problem

• collage system
• internal collage system
• grammar-based compression
• size bound
• conversion

Innovation

• collage system
• grammar-based compression
• internal collage system
• structural reachability
• repetition decomposition