V-Words, Lyndon Words and Galois Words

TL;DR

This paper explores the structure of special string families called circ-UMFFs, their connection to Lyndon words and V-order, and extends these concepts to general total orders, with applications in text processing.

Contribution

It introduces the substring circ-UMFF, generalizes V-order to any total order, and investigates their combinatorial properties and applications.

Findings

Defined substring circ-UMFF and extended V-order concepts.

Established connections between Lyndon words and V-order.

Proposed applications in text indexing and compression.

Abstract

We say that a family $\mathcal{W}$ of strings over $\Sigma^+$ forms a Unique Maximal Factorization Family (UMFF) if and only if every $w \in \mathcal{W}$ has a unique maximal factorization. Further, an UMFF $\mathcal{W}$ is called a circ-UMFF whenever it contains exactly one rotation of every primitive string $x \in \Sigma^+$. $V$-order is a non-lexicographical total ordering on strings that determines a circ-UMFF. In this paper we propose a generalization of circ-UMFF called the substring circ-UMFF and extend combinatorial research on $V$-order by investigating connections to Lyndon words. Then we extend these concepts to any total order. Applications of this research arise in efficient text indexing, compression, and search problems.