TY - JOUR

T1 - On the size of the smallest alphabet for Lyndon trees

AU - Nakashima, Yuto

AU - Takagi, Takuya

AU - Inenaga, Shunsuke

AU - Bannai, Hideo

AU - Takeda, Masayuki

PY - 2019/11/5

Y1 - 2019/11/5

N2 - 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.

AB - 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.

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U2 - 10.1016/j.tcs.2018.06.044

DO - 10.1016/j.tcs.2018.06.044

M3 - Article

AN - SCOPUS:85049994829

VL - 792

SP - 131

EP - 143

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -