Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows
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We study the distinct elements and ℓ_p-heavy hitters problems in the sliding window model, where only the most recent n elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and ℓ_p-heavy hitters that are nearly optimal in both n and ϵ. Applying our new composable histogram framework, we provide an algorithm that outputs a (1+ϵ)-approximation to the number of distinct elements in the sliding window model and uses Ø1/ϵ^2 n1/ϵ n+1/ϵ^2 n bits of space. For ℓ_p-heavy hitters, we provide an algorithm using space O(1/ϵ^p^2 n( n+1/ϵ)) for 0<p< 2, improving upon the best-known algorithm for ℓ_2-heavy hitters (Braverman et al., COCOON 2014), which has space complexity O(1/ϵ^4^3 n). We also show complementing nearly optimal lower bounds of Ω(1/ϵ^2 n+1/ϵ^2 n) for distinct elements and Ω(1/ϵ^p^2 n) for ℓ_p-heavy hitters, both tight up to O( n) and O(1/ϵ) factors.
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