Periodicity in Streams

Periodicity and Cyclic Shifts via Linear Sketches

“…In contrast, an earlier paper by Ergün et al [4] presented and an algorithm using O( −5.5 √ p polylog n) space for estimating the Hamming distance to the nearest p-periodic signal. They also present a single-pass, O(polylog n)-space algorithm for computing period(a) in the time-series model.…”

“…The algorithm is based on sampling in the Fourier domain and was actually inspired by Shor's algorithm for quantum factorization [17]. There is no analog in the recent Ergün et al [4] paper but an earlier result [5] in the combinatorial property-testing model can be applied in the streaming setting if we may use O( √ n polylog n) space. We conclude with a simple sketch algorithm for the related problem of identifying when two sequences are cyclic shifts of one another.…”

“…In contrast, Ergün and coauthors [4,5] considered a measure based on the Hamming distance, D 0 p (a) ≡ min y∈Pp,n ∆(a, y) where ∆(a, y) = |{i ∈ [n] : a i = y i }|. While different measures are suited to different applications, many of our algorithms can also be applied to approximate the Hamming distance, at least in the permutation model.…”

“…Previous periodicity work assumes that the stream is the signal, e.g., the stream 1, 2, 3, 4 defines the signal a = [1,2,3,4]. However, we wish to consider a more general setting.…”

Sliding Suffix Tree

“…This approach is used by the well-known grep command. Instead of matching the pattern characters with the text, we could use a rolling hash and compare the pattern's fingerprint with the fingerprint of the current sliding window [6][7][8]. String matching is also used in digital forensics, where we typically match multiple regular expressions on massive amounts of data, which involves multiple streams and parallelism [9][10][11].…”