Multidimensional Period Recovery

Abstract

Multidimensional data are widely used in real-life applications. Intel’s new brand of SSDs, called 3D XPoint, is an example of three-dimensional data. Motivated by a structural analysis of multidimensional data, we introduce the multidimensional period recovery problem, defined as follows. The input is a d-dimensional text array, with dimensions $n_1 \times n_2 \times \dots \times n_d$, that contains corruptions, while the original text without the corruptions is periodic. The goal is then to report the period of the original text. We show that, if the number of corruptions is at most $\left\lfloor \frac{1}{2 + \epsilon }\left\lfloor \frac{n_1}{p_1}\right\rfloor \cdots \left\lfloor \frac{n_d}{p_d}\right\rfloor \right\rfloor$, where $\epsilon > 0$ and $p_1 \times \cdots \times p_d$ are the period’s dimensions, then the amount of possible period candidates is $O(\log N)$, where $N = \varPi _{i=1}^{d}n_i$. The independency of this bound of the number of dimensions is a surprising key contribution of this paper. We present an $O(\varPi _{i=1}^{d} n_i \varPi _{i=1}^{d} \log n_i)$ algorithm for any constant dimension d (linear time up to logarithmic factor), to report these candidates. The tightness of the bound on the number of errors enabling a small size candidate set is demonstrated by showing that if the number of errors is equal to $\left\lfloor \frac{1}{2}\left\lfloor \frac{n_1}{p_1}\right\rfloor \cdots \left\lfloor \frac{n_d}{p_d}\right\rfloor \right\rfloor$, a family of texts with $\varTheta (N)$ period candidates can be constructed for any dimension $d \ge 2$.