# On the size of the smallest alphabet for Lyndon trees

### 抄録

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.

元の言語 英語 131-143 13 Theoretical Computer Science 792 https://doi.org/10.1016/j.tcs.2018.06.044 出版済み - 11 5 2019

### Fingerprint

Binary trees
Ordered Trees
Strings
Binary Tree
Leaves
Reverse engineering
Reverse Engineering
Linear Time
Isomorphic
Binary
Integer

### All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Computer Science(all)

### これを引用

：: Theoretical Computer Science, 巻 792, 05.11.2019, p. 131-143.

@article{9ebdf9b60c254b4b83ac71430829aa20,
title = "On the size of the smallest alphabet for Lyndon trees",
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.",
author = "Yuto Nakashima and Takuya Takagi and Shunsuke Inenaga and Hideo Bannai and Masayuki Takeda",
year = "2019",
month = "11",
day = "5",
doi = "10.1016/j.tcs.2018.06.044",
language = "English",
volume = "792",
pages = "131--143",
journal = "Theoretical Computer Science",
issn = "0304-3975",
publisher = "Elsevier",

}

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.

UR - http://www.scopus.com/inward/record.url?scp=85049994829&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049994829&partnerID=8YFLogxK

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 -