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# Optimal Regular Expressions for Palindromes of Given Length

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LIPIcs.MFCS.2021.52.pdf
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## Acknowledgements

We would like to thank the anonymous reviewers for their valuable suggestions.

## Cite As

Hermann Gruber and Markus Holzer. Optimal Regular Expressions for Palindromes of Given Length. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.MFCS.2021.52

## 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Regular languages
• Mathematics of computing → Nonlinear equations
##### Keywords
• regular expression
• descriptional complexity
• lower bound
• upper bound
• recurrence
• sum of digits

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