Language and Automata Theory and Applications

Volume 5457 of the series Lecture Notes in Computer Science pp 176-187

How Many Holes Can an Unbordered Partial Word Contain?

  • Francine Blanchet-SadriAffiliated withDepartment of Computer Science, University of North Carolina
  • , Emily AllenAffiliated withDepartment of Mathematical Sciences, Carnegie Mellon University
  • , Cameron ByrumAffiliated withDepartment of Mathematics, University of Mississippi
  • , Robert MercaşAffiliated withGRLMC, Universitat Rovira i Virgili

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Partial words are sequences over a finite alphabet that may have some undefined positions, or “holes,” that are denoted by \(\ensuremath{\diamond}\)’s. A nonempty partial word is called bordered if one of its proper prefixes is compatible with one of its suffixes (here \(\ensuremath{\diamond}\) is compatible with every letter in the alphabet); it is called unbordered otherwise. In this paper, we investigate the problem of computing the maximum number of holes a partial word of a fixed length can have and still fail to be bordered.