Two strings at Hamming distance 1 cannot be both quasiperiodic

Amihood Amir; Costas S. Iliopoulos; Jakub Radoszewski

doi:10.1016/j.ipl.2017.08.005

Two strings at Hamming distance 1 cannot be both quasiperiodic

Amihood Amir, Costas S. Iliopoulos, Jakub Radoszewski

Department of Computer Science

King's College London

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

We present a generalization to quasiperiodicity of a known fact from combinatorics on words related to periodicity. A string is called periodic if it has a period which is at most half of its length. A string w is called quasiperiodic if it has a non-trivial cover, that is, there exists a string c that is shorter than w and such that every position in w is inside one of the occurrences of c in w. It is a folklore fact that two strings that differ at exactly one position cannot be both periodic. Here we prove a more general fact that two strings that differ at exactly one position cannot be both quasiperiodic. Along the way we obtain new insights into combinatorics of quasiperiodicities.

Original language	English
Pages (from-to)	54-57
Number of pages	4
Journal	Information Processing Letters
Volume	128
DOIs	https://doi.org/10.1016/j.ipl.2017.08.005
State	Published - Dec 2017

Bibliographical note

Publisher Copyright:
© 2017 Elsevier B.V.

Funding

The authors thank Maxime Crochemore, Solon P. Pissis, and Wojciech Rytter for helpful discussions. We also thank an anonymous referee whose suggestions helped to simplify the proof of Theorem 2 . Amihood Amir was partially supported by the ISF grant 571/14 and the Royal Society . Costas S. Iliopoulos was partially supported by the Onassis Foundation and the Royal Society .

Funders	Funder number
Royal Society
Israel Science Foundation	571/14
Alexander S. Onassis Public Benefit Foundation

Keywords

Combinatorial problems
Combinatorics on words
Cover
Quasiperiodicity
Seed

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10.1016/j.ipl.2017.08.005

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AB - We present a generalization to quasiperiodicity of a known fact from combinatorics on words related to periodicity. A string is called periodic if it has a period which is at most half of its length. A string w is called quasiperiodic if it has a non-trivial cover, that is, there exists a string c that is shorter than w and such that every position in w is inside one of the occurrences of c in w. It is a folklore fact that two strings that differ at exactly one position cannot be both periodic. Here we prove a more general fact that two strings that differ at exactly one position cannot be both quasiperiodic. Along the way we obtain new insights into combinatorics of quasiperiodicities.

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KW - Seed

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ER -

Two strings at Hamming distance 1 cannot be both quasiperiodic

Abstract

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Keywords

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