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Tracking set-expression cardinalities over continuous update streams
Authors:Email author" target="_blank">Sumit?GangulyEmail author  Minos?Garofalakis  Rajeev?Rastogi
Affiliation:(1) Bell Laboratories, Lucent Technologies, 600 Mountain Avenue, NJ 07974 Murray Hill, USA
Abstract:There is growing interest in algorithms for processing and querying continuous data streams (i.e., data seen only once in a fixed order) with limited memory resources. In its most general form, a data stream is actually an update stream, i.e., comprising data-item deletions as well as insertions. Such massive update streams arise naturally in several application domains (e.g., monitoring of large IP network installations or processing of retail-chain transactions). Estimating the cardinality of set expressions defined over several (possibly distributed) update streams is perhaps one of the most fundamental query classes of interest; as an example, such a query may ask ldquowhat is the number of distinct IP source addresses seen in passing packets from both router R 1 and R 2 but not router R 3?rdquo. Earlier work only addressed very restricted forms of this problem, focusing solely on the special case of insert-only streams and specific operators (e.g., union). In this paper, we propose the first space-efficient algorithmic solution for estimating the cardinality of full-fledged set expressions over general update streams. Our estimation algorithms are probabilistic in nature and rely on a novel, hash-based synopsis data structure, termed rdquo2-level hash sketchrdquo. We demonstrate how our 2-level hash sketch synopses can be used to provide low-error, high-confidence estimates for the cardinality of set expressions (including operators such as set union, intersection, and difference) over continuous update streams, using only space that is significantly sublinear in the sizes of the streaming input (multi-)sets. Furthermore, our estimators never require rescanning or resampling of past stream items, regardless of the number of deletions in the stream. We also present lower bounds for the problem, demonstrating that the space usage of our estimation algorithms is within small factors of the optimal. Finally, we propose an optimized, time-efficient stream synopsis (based on 2-level hash sketches) that provides similar, strong accuracy-space guarantees while requiring only guaranteed logarithmic maintenance time per update, thus making our methods applicable for truly rapid-rate data streams. Our results from an empirical study of our synopsis and estimation techniques verify the effectiveness of our approach.Received: 20 October 2003, Accepted: 16 April 2004, Published online: 14 September 2004Edited by: J. Gehrke and J. Hellerstein.Sumit Ganguly: sganguly@cse.iitk.ac.in Current affiliation: Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, India
Keywords:Data streams  Set expressions  Data synopses  Approximate query processing  Randomized algorithms
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