Periodicity and Cyclic Shifts via Linear Sketches

Abstract

We consider the problem of identifying periodic trends in data streams. We say a signal ${\mathbf a} \in \ensuremath{\mathbb{R}} ^n$ is p-periodic if a _i  = a _i + p for all i ∈ [n − p]. Recently, Ergün et al. [4] presented a one-pass, O(polylog n)-space algorithm for identifying the smallest period of a signal. Their algorithm required ${\mathbf a}$ to be presented in the time-series model, i.e., a _i is the ith element in the stream. We present a more general linear sketch algorithm that has the advantages of being applicable to a) the turnstile stream model, where coordinates can be incremented/decremented in an arbitrary fashion and b) the parallel or distributed setting where the signal is distributed over multiple locations/machines. We also present sketches for (1 + ε) approximating the ℓ₂ distance between ${\mathbf a}$ and the nearest p-periodic signal for a given p. Our algorithm uses O(ε ^− 2 polylog n) space, comparing favorably to an earlier time-series result that used $O(\epsilon^{-5.5} \sqrt{p} polylog n)$ space for estimating the Hamming distance to the nearest p-periodic signal. Our last periodicity result is an algorithm for estimating the periodicity of a sequence in the presence of noise. We conclude with a small-space algorithm for identifying when two signals are exact (or nearly) cyclic shifts of one another. Our algorithms are based on bilinear sketches [10] and combining Fourier transforms with stream processing techniques such as ℓ _p sampling and sketching [13, 11].