The l_2 tracking problem is the task of obtaining a streaming algorithm that, given access to a stream of items a_1,a_2,a_3,... from a universe [n], outputs at each time t an estimate to the l_2 norm of the frequency vector f^{(t)}in R^n (where f^{(t)}_i is the number of occurrences of item i in the stream up to time t). The previous work [Braverman-Chestnut-Ivkin-Nelson-Wang-Woodruff, PODS 2017] gave a streaming algorithm with (the optimal) space using O(epsilon^{-2}log(1/delta)) words and O(epsilon^{-2}log(1/delta)) update time to obtain an epsilon-accurate estimate with probability at least 1-delta. We give the first algorithm that achieves update time of O(log 1/delta) which is independent of the accuracy parameter epsilon, together with the nearly optimal space using O(epsilon^{-2}log(1/delta)) words. Our algorithm is obtained using the Count Sketch of [Charilkar-Chen-Farach-Colton, ICALP 2002].
@InProceedings{chou_et_al:LIPIcs.APPROX-RANDOM.2019.2, author = {Chou, Chi-Ning and Lei, Zhixian and Nakkiran, Preetum}, title = {{Tracking the l\underline2 Norm with Constant Update Time}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {2:1--2:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.2}, URN = {urn:nbn:de:0030-drops-112175}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.2}, annote = {Keywords: Streaming algorithms, Sketching algorithms, Tracking, CountSketch} }
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