On the size of the smallest alphabet for Lyndon trees

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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 2k leaves, there exists a solution string w over an alphabet of size at most 4.

Original languageEnglish
Pages (from-to)131-143
Number of pages13
JournalTheoretical Computer Science
Volume792
DOIs
Publication statusPublished - Nov 5 2019

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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