On prefix normal words and prefix normal forms

被引:9
|
作者
Burcsi, Peter [1 ]
Fici, Gabriele [2 ]
Liptak, Zsuzsanna [3 ]
Ruskey, Frank [4 ]
Sawada, Joe [5 ]
机构
[1] Eotvos Lorand Univ, Dept Comp Algebra, Budapest, Hungary
[2] Univ Palermo, Dipartimento Matemat & Informat, I-90133 Palermo, Italy
[3] Univ Verona, Dipartimento Informat, I-37100 Verona, Italy
[4] Univ Victoria, Dept Comp Sci, Victoria, BC V8W 2Y2, Canada
[5] Univ Guelph, Sch Comp Sci, Guelph, ON N1G 2W1, Canada
基金
加拿大自然科学与工程研究理事会;
关键词
Prefix normal words; Prefix normal forms; Binary languages; Binary jumbled pattern matching; Pre-necklaces; Lyndon words; Enumeration; ALGORITHMS;
D O I
10.1016/j.tcs.2016.10.015
中图分类号
TP301 [理论、方法];
学科分类号
081202 ;
摘要
A 1-prefix normal word is a binary word with the property that no factor has more 1s than the prefix of the same length; a 0-prefix normal word is defined analogously. These words arise in the context of indexed binary jumbled pattern matching, where the aim is to decide whether a word has a factor with a given number of is and 0s (a given Parikh vector). Each binary word has an associated set of Parikh vectors of the factors of the word. Using prefix normal words, we provide a characterization of the equivalence class of binary words having the same set of Parikh vectors of their factors. We prove that the language of prefix normal words is not context-free and is strictly contained in the language of pre-necklaces, which are prefixes of powers of Lyndon words. We give enumeration results on pnw(n), the number of prefix normal words of length n, showing that, for sufficiently large n, 2(n-4 root nlgn) <= pnw(n) <= 2(n-lgn+1). For fixed density (number of 1s), we show that the ordinary generating function of the number of prefix normal words of length n and density d is a rational function. Finally, we give experimental results on pnw(n), discuss further properties, and state open problems. (C) 2016 Elsevier B.V. All rights reserved.
引用
收藏
页码:1 / 13
页数:13
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