Closest periodic vectors in l

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 L p spaces. The measures of "closeness" that we consider are the metrics in the different L p spaces. Specifically, we consider the L 1 , L 2 and L ∞ metrics. In particular, for a given n -dimensional vector V , we develop O (n 2) 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 logn ) time constant approximation algorithms for the (appropriate) period of the closest periodic vector.