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Lower bounds on frequency estimation of data streams (extended abstract). (English) Zbl 1142.68331
Hirsch, Edward A. (ed.) et al., Computer science – theory and applications. Third international computer science symposium in Russia, CSR 2008 Moscow, Russia, June 7–12, 2008. Proceedings. Berlin: Springer (ISBN 978-3-540-79708-1/pbk). Lecture Notes in Computer Science 5010, 204-215 (2008).
Summary: We consider a basic problem in the general data streaming model, namely, to estimate a vector \(f\in \mathbb Z^n\) that is arbitrarily updated (i.e., incremented or decremented) coordinate-wise. The estimate \(\hat f\in \mathbb Z^n\) must satisfy \(\|\hat f - f\|_{\infty} \leq \epsilon\| f\|_1\), that is, \(\forall i(|\hat f_i - f_i| \leq \epsilon\| f\|_1)\). It is known to have \(\tilde{O}(\epsilon^{-1})\) randomized space upper bound, \(\Omega (\epsilon ^{ - 1} \log (\epsilon n))\) space lower bound and deterministic space upper bound of \(\tilde{\Omega}(\epsilon^{-2})\) bits. We show that any deterministic algorithm for this problem requires space \(\Omega(\epsilon^{-2}(\log \| f\|_1)(\log n)(\log^{-1}(\epsilon^{-1})))\) bits.
For the entire collection see [Zbl 1136.68005].
68P05 Data structures
68Q25 Analysis of algorithms and problem complexity
68Q45 Formal languages and automata
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