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V-Words, Lyndon Words and Galois Words

Jacqueline W. Daykin, Neerja Mhaskar, W. F. Smyth

Abstract

We say that a family $\mathcal{W}$ of strings over $Σ^+$ 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 $x \in Σ^+$. $V$-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.

V-Words, Lyndon Words and Galois Words

Abstract

We say that a family of strings over forms a Unique Maximal Factorization Family (UMFF) if and only if every has a unique maximal factorization. Further, an UMFF is called a circ-UMFF whenever it contains exactly one rotation of every primitive string . -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 -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.
Paper Structure (12 sections, 24 theorems, 13 equations, 2 figures, 2 algorithms)

This paper contains 12 sections, 24 theorems, 13 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $V$-order
  4. UMFF and circ-UMFF theory
  5. $V$-words, Lyndon Words and circ-UMFFs
  6. Substring circ-UMFF: Generalization of circ-UMFF
  7. Galois words
  8. Binary Border-free Galois words
  9. UMFF to circ-UMFF construction
  10. Concluding comments
  11. Open Problems
  12. Acknowledgements

Key Result

Theorem 2

CFL58 Any nonempty string $\hbox{\boldmath $x$}$ can be written uniquely as a product $\hbox{LF}_{\hbox{\boldmath $x$}} = \hbox{\boldmath $x$} = \hbox{\boldmath $u$}_1 \hbox{\boldmath $u$}_2 \cdots \hbox{\boldmath $u$}_k$ of $k \ge 1$ Lyndon words, with $(\hbox{\boldmath $u$}_1 \ge \hbox{\boldmath $

Figures (2)

  • Figure 1: $929 \prec 922911$
  • Figure 2: $unique \succ equitant$

Theorems & Definitions (67)

  • Definition 1
  • Theorem 2
  • Definition 3: $V$-order DaD96
  • Example 4
  • Example 5
  • Definition 6: $V$-form DaD96DD03DDS11DDS13
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Definition 10: $V$-word DD03
  • ...and 57 more