Abstract. We consider the problem of identifying periodic trends in data streams. We say a signal a ∈ R n is p-periodic if ai = ai+p for all i ∈ [n − p]. Recently, Ergün et al.  presented a one-pass, O(polylog n)space algorithm for identifying the smallest period of a signal. Their algorithm required a to be presented in the time-series model, i.e., ai 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+ɛ) approximating the ℓ2 distance between a and the nearest p-periodic signal for a given p. Our algorithm uses O(ɛ −2 polylog n) space, comparing favorably to an earlier time-series result that used O(ɛ −5.5√ 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  and combining Fourier transforms with stream processing techniques such as ℓp sampling and sketching [11, 13].
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