Streaming Algorithms with Large Approximation Factors
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We initiate a broad study of classical problems in the streaming model with insertions and deletions in the setting where we allow the approximation factor α to be much larger than 1. Such algorithms can use significantly less memory than the usual setting for which α = 1+ϵ for an ϵ∈ (0,1). We study large approximations for a number of problems in sketching and streaming and the following are some of our results. For the ℓ_p norm/quasinorm x_p of an n-dimensional vector x, 0 < p ≤ 2, we show that obtaining a (n)-approximation requires the same amount of memory as obtaining an O(1)-approximation for any M = n^Θ(1). For estimating the ℓ_p norm, p > 2, we show an upper bound of O(n^1-2/p (log n log M)/α^2) bits for an α-approximation, and give a matching lower bound, for almost the full range of α≥ 1 for linear sketches. For the ℓ_2-heavy hitters problem, we show that the known lower bound of Ω(k log nlog M) bits for identifying (1/k)-heavy hitters holds even if we are allowed to output items that are 1/(α k)-heavy, for almost the full range of α, provided the algorithm succeeds with probability 1-O(1/n). We also obtain a lower bound for linear sketches that is tight even for constant probability algorithms. For estimating the number ℓ_0 of distinct elements, we give an n^1/t-approximation algorithm using O(tloglog M) bits of space, as well as a lower bound of Ω(t) bits, both excluding the storage of random bits.
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