Herman Z. Q. Chen ; Sergey Kitaev ; Torsten Mütze ; Brian Y. Sun - On universal partial words

dmtcs:2205 - Discrete Mathematics & Theoretical Computer Science, May 31, 2017, Vol. 19 no. 1 - https://doi.org/10.23638/DMTCS-19-1-16
On universal partial wordsArticle

Authors: Herman Z. Q. Chen ; Sergey Kitaev ; Torsten Mütze ; Brian Y. Sun

    A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer.

    Volume: Vol. 19 no. 1
    Section: Combinatorics
    Published on: May 31, 2017
    Accepted on: May 5, 2017
    Submitted on: May 31, 2017
    Keywords: Mathematics - Combinatorics,Computer Science - Formal Languages and Automata Theory,Computer Science - Information Theory

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