Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
The space complexity of approximating the frequency moments
Journal of Computer and System Sciences
Near-optimal sparse fourier representations via sampling
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Identifying Representative Trends in Massive Time Series Data Sets Using Sketches
VLDB '00 Proceedings of the 26th International Conference on Very Large Data Bases
On the Complexity of Determining the Period of a String
COM '00 Proceedings of the 11th Annual Symposium on Combinatorial Pattern Matching
An improved data stream summary: the count-min sketch and its applications
Journal of Algorithms
Stable distributions, pseudorandom generators, embeddings, and data stream computation
Journal of the ACM (JACM)
Data streams: algorithms and applications
Foundations and Trends® in Theoretical Computer Science
Declaring independence via the sketching of sketches
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Periodicity testing with sublinear samples and space
ACM Transactions on Algorithms (TALG)
Exact and Approximate Pattern Matching in the Streaming Model
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
An optimal algorithm for the distinct elements problem
Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
1-pass relative-error Lp-sampling with applications
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On the exact space complexity of sketching and streaming small norms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Homomorphic fingerprints under misalignments: sketching edit and shift distances
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We consider the problem of identifying periodic trends in data streams. We say a signal a ∈ Rn is p-periodic if ai = ai+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 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+ε) approximating the l2 distance between 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(ε-5.5 √ppolylon 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 lp sampling and sketching [13, 11].