Deterministically Estimating Data Stream Frequencies

We consider updates to an n-dimensional frequency vector of a data stream, that is, the vector f is updated coordinate-wise by means of insertions or deletions in any arbitrary order. A fundamental problem in this model is to recall the vector approximately, that is to return an estimate f̂ of f such that ∣f̂i − fi∣ < ∥f∥p, for every i = 1, 2, . . . , n, where is an accuracy parameter and p is the index of the lp norm used to calculate the norm ∥f∥p. This problem, denoted by ApproxFreqp( ), is fundamental in data stream processing and is used to solve a number of other problems, such as heavy hitters, approximating range queries and quantiles, approximate histograms, etc.. Suppressing poly-logarithmic factors in n and ∥f∥1, for p = 1 the problem is known to have ̃(1/ ) randomized space complexity [2, 4] and ̃(1/ ) deterministic space complexity[6, 7]. However, the deterministic space complexity of this problem for any value of p > 1 is not known. In this paper, we show that the deterministic space complexity of the problem ApproxFreqp( ) is ̃(n 2−2/p/ ) for 1 < p < 2, and (n) for p ≥ 2.