Database management systems face the challenge of dealing with massive data distributions which arrive at high speeds while there is only small storage available for managing and mining them. Emerging data stream management systems approach this problem by summarizing and mining the distributions using samples or sketches. However, data distributions can be “viewed” in different ways. For example, a data stream of integer values can be viewed either as the forward distribution f(x), ie., the number of occurrences of x in the stream, or as its inverse, f-1(i), which is the number of items that appear i times. While both such “views” are equivalent in stored data systems, over data streams that entail approximations, they may be significantly different. In other words, samples and sketches developed for the forward distribution may be ineffective for summarizing or mining the inverse distribution. Yet, many applications such as IP traffic monitoring naturally rely on mining inverse distributions.
We formalize the problems of managing and mining inverse distributions and show provable differences between summarizing the forward distribution vs the inverse distribution. We present methods for summarizing and mining inverse distributions of data streams: they rely on a novel technique to maintain a dynamic sample over the stream with provable guarantees which can be used for variety of summarization tasks (building quantiles or equidepth histograms) and mining (anomaly detection: finding heavy hitters, and measuring the number of rare items), all with provable guarantees on quality of approximations and time/space used by our streaming methods. We also complement our analytical and algorithmic results by presenting an experimental study of the methods over network data streams.
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