Combinatorics of Unique Maximal Factorization Families (UMFFs)

  • Authors:

  • David E. Daykin;Jacqueline W. Daykin;W.F. (Bill) Smyth

  • Affiliations:

  • Department of Mathematics University of Reading, UK;Department of Computer Science Royal Holloway & King's College, University of London, UK. E-mail: J.Daykin@cs.rhul.ac.uk, jackie.daykin@kcl.ac.uk;Algorithms Research Group, Department of Computing & Software, McMaster University, Hamilton ON L8S 4K1, Canada and Digital Ecosystems and Business Intelligence Institute Curtin University, GPO Bo ...

  • Venue:
  • Fundamenta Informaticae - Special Issue on Stringology
  • Year:
  • 2009

Quantified Score

Hi-index 0.00

Visualization

Abstract

Suppose a set W of strings contains exactly one rotation (cyclic shift) of every primitive string on some alphabet Σ. Then W is a circ-UMFF if and only if every word in Σ$^+$ has a unique maximal factorization over W. The classic circ-UMFF is the set of Lyndon words based on lexicographic ordering (1958). Duval (1983) designed a linear sequential Lyndon factorization algorithm; a corresponding PRAMparallel algorithmwas described by J. Daykin, Iliopoulos and Smyth (1994). Daykin and Daykin defined new circ-UMFFs based on various methods for totally ordering sets of strings (2003), and further described the structure of all circ-UMFFs (2008). Here we prove new combinatorial results for circ-UMFFs, and in particular for the case of Lyndon words. We introduce Acrobat and Flight Deck circ-UMFFs, and describe some of our results in terms of dictionaries. Applications of circ-UMFFs pertain to structured methods for concatenating and factoring strings over ordered alphabets, and those of Lyndon words are wide ranging and multidisciplinary.