TY - GEN
T1 - Universal sketches for the frequency negative moments and other decreasing streaming sums
AU - Braverman, Vladimir
AU - Chestnut, Stephen R.
N1 - Publisher Copyright:
© Vladimir Braverman and Stephen R. Chestnut;licensed under Creative Commons License CC-BY.
PY - 2015/8/1
Y1 - 2015/8/1
N2 - Given a stream with frequencies fd, for d ∈ [n], we characterize the space necessary for approximating the frequency negative moments Fp = Σ |fd|p, where p < 0 and the sum is taken over all items d ∈ [n] with nonzero frequency, in terms of n, ∈, and m = Σ |fd|. To accomplish this, we actually prove a much more general result. Given any nonnegative and nonincreasing function g, we characterize the space necessary for any streaming algorithm that outputs a (1±ε)-approximation to Σ g(|fd|), where again the sum is over items with nonzero frequency. The storage required is expressed in the form of the solution to a relatively simple nonlinear optimization problem, and the algorithm is universal for (1±ε)-approximations to any such sum where the applied function is nonnegative, nonincreasing, and has the same or smaller space complexity as g. This partially answers an open question of Nelson (IITK Workshop Kanpur, 2009).
AB - Given a stream with frequencies fd, for d ∈ [n], we characterize the space necessary for approximating the frequency negative moments Fp = Σ |fd|p, where p < 0 and the sum is taken over all items d ∈ [n] with nonzero frequency, in terms of n, ∈, and m = Σ |fd|. To accomplish this, we actually prove a much more general result. Given any nonnegative and nonincreasing function g, we characterize the space necessary for any streaming algorithm that outputs a (1±ε)-approximation to Σ g(|fd|), where again the sum is over items with nonzero frequency. The storage required is expressed in the form of the solution to a relatively simple nonlinear optimization problem, and the algorithm is universal for (1±ε)-approximations to any such sum where the applied function is nonnegative, nonincreasing, and has the same or smaller space complexity as g. This partially answers an open question of Nelson (IITK Workshop Kanpur, 2009).
KW - Data streams
KW - Frequency moments
KW - Negative moments
UR - https://www.scopus.com/pages/publications/84958541050
U2 - 10.4230/LIPIcs.APPROX-RANDOM.2015.591
DO - 10.4230/LIPIcs.APPROX-RANDOM.2015.591
M3 - Conference contribution
AN - SCOPUS:84958541050
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 591
EP - 605
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015
A2 - Garg, Naveen
A2 - Jansen, Klaus
A2 - Rao, Anup
A2 - Rolim, Jose D. P.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015
Y2 - 24 August 2015 through 26 August 2015
ER -