Distributed Streams Algorithms for Sliding Windows

Abstract

This paper presents algorithms for estimating aggregate functions over a "sliding window" of the <i>N</i> most recent data items in one or more streams. Our results include:&lt;ol&gt;<li>For a <i>single stream</i>, we present the first &#949;-approximation scheme for the number of 1's in a sliding window that is optimal in both worst case time and space. We also present the first &#949; for the sum of integers in [<i>0</i>..<i>R</i>] in a sliding window that is optimal in both worst case time and space (assuming <i>R</i> is at most polynomial in <i>N</i>). Both algorithms are deterministic and use only logarithmic memory words.</li><li>In contrast, we show that <i>an </i> deterministic algorithm that estimates, to within a small constant relative error, the number of 1's (or the sum of integers) in a sliding window over the <i>union of distributed streams</i> requires &OHgr;(<i>N</i>) space.</li><li> We present the first randomized (&#949;,&sgr;)-approximation scheme for the number of 1's in a sliding window over the <i>union of distributed streams</i> that uses only logarithmic memory words. We also present the first (&#949;,&sgr;)-approximation scheme for the number of distinct values in a sliding window over distributed streams that uses only logarithmic memory words.</li>&lt;/olOur results are obtained using a novel family of synopsis data structures.

DOI: 10.1007/s00224-004-1156-4

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@article{Gibbons2002DistributedSA, title={Distributed Streams Algorithms for Sliding Windows}, author={Phillip B. Gibbons and Srikanta Tirthapura}, journal={Theory of Computing Systems}, year={2002}, volume={37}, pages={457-478} }