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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Bibliographic Details
Published in:Theoretical computer science Vol. 533; pp. 26 - 36
Main Authors: Amir, Amihood, Eisenberg, Estrella, Levy, Avivit, Lewenstein, Noa
Format: Journal Article
Language:English
Published: Elsevier B.V 08.05.2014
Subjects:
Approximate periodicity
Closest vector
String algorithms
String algorithms
Closest vector
Approximate periodicity
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary: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