Closest Periodic Vectors In L-P 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 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-infinity 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 log n) time constant approximation algorithms for the (appropriate) period of the closest periodic vector.