### Abstract

We consider the problem of reverse-engineering the Lyndon tree, i.e., given a full binary ordered tree T with n leaves as input, we are to compute a string w of length n of which Lyndon tree is isomorphic to the input tree T. Hereby we call such a string a solution string. Although the problem is easily solvable in linear time for binary alphabets and unbounded-size alphabets, it is not known how to efficiently find the smallest alphabet size for a solution string. In this paper, we show several new observations concerning this problem. Namely, we show that: 1) For any positive integer n, there exists a full binary ordered tree T with n leaves, s.t. the smallest alphabet size of a solution string for T is ⌊ [Formula presented] ⌋+1. 2) For any full binary ordered tree T with n leaves, there exists a solution string w over an alphabet of size at most ⌊ [Formula presented] ⌋+1. 3) For any full binary ordered tree T, there exists a solution string w over an alphabet of size at most h+1, where h is the height of T. 4) For any complete binary ordered tree T with 2^{k} leaves, there exists a solution string w over an alphabet of size at most 4.

Original language | English |
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Pages (from-to) | 131-143 |

Number of pages | 13 |

Journal | Theoretical Computer Science |

Volume | 792 |

DOIs | |

Publication status | Published - Nov 5 2019 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)