Closest periodic vectors in Lp spaces

The problem of finding the period of a vector V is central to many applications. Let V′ be a periodic vector closest to V under some metric. We seek this V′, or more precisely we seek the smallest period that generates V′. In this paper we consider the problem of finding the closest periodic vector...

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Vydané v:	Theoretical computer science Ročník 533; s. 26 - 36
Hlavní autori:	Amir, Amihood, Eisenberg, Estrella, Levy, Avivit, Lewenstein, Noa
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Elsevier B.V 08.05.2014
Predmet:	Approximate periodicity Closest vector String algorithms String algorithms Closest vector Approximate periodicity
ISSN:	0304-3975, 1879-2294
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Shrnutí:	The problem of finding the period of a vector V is central to many applications. Let V′ be a periodic vector closest to V under some metric. We seek this V′, or more precisely we seek the smallest period that generates V′. In this paper we consider the problem of finding the closest periodic vector in Lp spaces. The measures of “closeness” that we consider are the metrics in the different Lp spaces. Specifically, we consider the L1,L2 and L∞ metrics. In particular, for a given n-dimensional vector V, we develop O(n2) time algorithms (a different algorithm for each metric) that construct the smallest period that defines such a periodic n-dimensional vector V′. We call that vector the closest periodic vector of V under the appropriate metric. We also show (three) O˜(n) time constant approximation algorithms for the period of the approximate closest periodic vector.
ISSN:	0304-3975 1879-2294
DOI:	10.1016/j.tcs.2014.03.019