License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.52
URN: urn:nbn:de:0030-drops-144921
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14492/
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Gruber, Hermann ; Holzer, Markus

Optimal Regular Expressions for Palindromes of Given Length

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LIPIcs-MFCS-2021-52.pdf (0.6 MB)

Abstract

The language P_n (P̃_n, respectively) consists of all words that are palindromes of length 2n (2n-1, respectively) over a fixed binary alphabet. We construct a regular expression that specifies P_n (P̃_n, respectively) of alphabetic width 4⋅ 2ⁿ-4 (3⋅ 2ⁿ-4, respectively) and show that this is optimal, that is, the expression has minimum alphabetic width among all expressions that describe P_n (P̃_n, respectively). To this end we give optimal expressions for the first k palindromes in lexicographic order of odd and even length, proving that the optimal bound is 2n+4(k-1)-2 S₂(k-1) in case of odd length and 2n+3(k-1)-2 S₂(k-1)-1 for even length, respectively. Here S₂(n) refers to the Hamming weight function, which denotes the number of ones in the binary expansion of the number n.

BibTeX - Entry

@InProceedings{gruber_et_al:LIPIcs.MFCS.2021.52,
  author =	{Gruber, Hermann and Holzer, Markus},
  title =	{{Optimal Regular Expressions for Palindromes of Given Length}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{52:1--52:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14492},
  URN =		{urn:nbn:de:0030-drops-144921},
  doi =		{10.4230/LIPIcs.MFCS.2021.52},
  annote =	{Keywords: regular expression, descriptional complexity, lower bound, upper bound, recurrence, sum of digits}
}

Keywords: regular expression, descriptional complexity, lower bound, upper bound, recurrence, sum of digits
Collection: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue Date: 2021
Date of publication: 18.08.2021

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