String Representation in Suffixient Set Size Space

2026-04-05

AI Summary

This study resolves a long-standing open problem concerning the attainability of the string repetitiveness measure χ(w): whether there always exists a string representation of size O(χ(w)). To this end, we introduce the Substring Equation System (SES), a novel theoretical framework, and combine it with combinatorial structures such as suffix-complete sets to construct the first compression scheme capable of representing any string w within O(χ(w)) space. Our work not only establishes, for the first time, the attainability of the χ measure but also provides a new modeling paradigm for string compression that leverages structural regularities in repetitive strings.

Abstract

Repetitiveness measures quantify how much repetitive structure a string contains and serve as parameters for compressed representations and indexing data structures. We study the measure $χ$, defined as the size of the smallest suffixient set. Although $χ$ has been studied extensively, its reachability, whether every string $w$ admits a string representation of size $O(χ(w))$ words, has remained an important open problem. We answer this question affirmatively by presenting the first such representation scheme. Our construction is based on a new model, the substring equation system (SES), and we show that every string admits an SES of size $O(χ(w))$.

Problem

• string representation
• suffixient set
• repetitiveness measure
• reachability
• compressed representation

Innovation

• suffixient set
• substring equation system
• string repetitiveness
• compressed representation
• χ measure