Tracking the ℓ_2 Norm with Constant Update Time
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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)∈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, FOCS 2017] gave an streaming algorithm with (the optimal) space using O(ϵ^-2(1/δ)) words and O(ϵ^-2(1/δ)) update time to obtain an ϵ-accurate estimate with probability at least 1-δ. We give the first algorithm that achieves update time of O( 1/δ) which is independent of the accuracy parameter ϵ, together with the optimal space using O(ϵ^-2(1/δ)) words. Our algorithm is obtained using the Count Sketch of [Charilkar-Chen-Farach-Colton, ICALP 2002].
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