Content Summary | There is growing interest in algorithms for processing and
querying continuous data streams (i.e., data that is 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 (perhaps, distributed) update streams is perhaps one
of the most fundamental query classes of interest; as an example,
such a query may ask “what is the number of distinct
IP source addresses seen in passing packets from both router
R1 and R2 but not router R3?”. Earlier work has 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 “2-level hash sketch”. 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
small space and small processing time per update. 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. Preliminary
experimental results verify the effectiveness of our approach. | en |