Lower bound for estimating frequency for update data streams

Author Sumit Ganguly



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Sumit Ganguly

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Sumit Ganguly. Lower bound for estimating frequency for update data streams. In Sublinear Algorithms. Dagstuhl Seminar Proceedings, Volume 8341, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008) https://doi.org/10.4230/DagSemProc.08341.4

Abstract

We consider general update streams, where, the stream is a sequence of updates of the form $(index, i, v)$, where, $i in {1,2 ldots, n}$ and $v in {-1,+1}$, signifying deletion or insertion, respectively of an instance of $i$. The frequency of $i in {1,2,ldots, n}$ is given as the sum of the updates to $i$, that is, 
$f_i(sigma) = sum_{(index,i,v) in sigma} v $. The $n$-dimensional vector $f(sigma)$ with $i$th coordinate $f_i(sigma)$  is called the frequency vector of the stream $sigma$. We consider the problem of finding an n-dimensional integer vector $hat{f}(sigma)$ that estimates the frequency vector $f(sigma)$ of an input  stream $sigma$ in the following sense: 


orm{hat{f} (sigma)- f(sigma)} le epsilon 
orm{f(sigma)}_p

For $p=1$ and $2$, there are randomized algorithms known with space bound $	ilde{O}(epsilon^{-p})$. A space lower bound of $Omega(epsilon^{-1} log (nepsilon))$ is also known. However, the  deterministic space upper bound is $	ilde{O}(epsilon^{-2})$. 

In this work, we present a deterministic space lower bound of $Omega(n^{2-2/p}epsilon^{-2} log |{sigma}|)$, for $1le p < 2$ and $1/4 le epsilon = Omega(n^{1/2-1/p})$. For $p ge 2$, we show an $Omega(n)$ space lower bound for all $epsilon < 1/4$.

The results are obtained using a new characterization of data stream computations, that show that any uniform computation over a data stream may be viewed as an appropriate linear map.

Subject Classification

Keywords
  • Data stream
  • lower bound
  • frequency estimation
  • stream automata
  • linear map

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