Recent years have seen growing interest in effective algorithms for summarizing and querying massive, high-speed data streams. Randomized sketch synopses provide accurate approximations for general-purpose summaries of the streaming data distribution (e.g., wavelets). The focus of existing work has typically been on minimizing space requirements of the maintained synopsis - however, to effectively support high-speed data-stream analysis, a crucial practical requirement is to also optimize: (1) the update time for incorporating a streaming data element in the sketch, and (2) the query time for producing an approximate summary (e.g., the top wavelet coefficients) from the sketch. Such time costs must be small enough to cope with rapid stream-arrival rates and the real-time querying requirements of typical streaming applications (e.g., ISP network monitoring). With cheap and plentiful memory, space is often only a secondary concern after query/update time costs.In this paper, we propose the first fast solution to the problem of tracking wavelet representations of one-dimensional and multi-dimensional data streams, based on a novel stream synopsis, the Group-Count Sketch (GCS). By imposing a hierarchical structure of groups over the data and applying the GCS, our algorithms can quickly recover the most important wavelet coefficients with guaranteed accuracy. A tradeoff between query time and update time is established, by varying the hierarchical structure of groups, allowing the right balance to be found for specific data stream. Experimental analysis confirms this tradeoff, and shows that all our methods significantly outperform previously known methods in terms of both update time and query time, while maintaining a high level of accuracy.
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