A Fast Algorithm for Approximate Quantiles in High Speed Data Streams

We present a fast algorithm for computing approx-imate quantiles in high speed data streams with deter-ministic error bounds. For data streams of size N where N is unknown in advance, our algorithm par-titions the stream into sub-streams of exponentially increasing size as they arrive. For each sub-stream which has a xed size, we compute and maintain a multi-level summary structure using a novel algorithm. In order to achieve high speed performance, the algo-rithm uses simple block-wise merge and sample oper-ations. Overall, our algorithms for xed-size streams and arbitrary-size streams have a computational cost of O(N log ( 1 log N)) and an average per-element update cost of O(log log N) if is xed.

Effective computation of biased quantiles over data streams

Skew is prevalent in many data sources such as IP traffic streams. To continually summarize the distribution of such data, a high-biased set of quantiles (e.g., 50th, 90th and 99th percentiles) with finer error guarantees at higher ranks (e.g., errors of 5, 1 and 0.1 percent, respectively) is more useful than uniformly distributed quantiles (e.g., 25th, 50th and 75th percentiles) with uniform error guarantees. In this paper, we address the following two problems. First, can we compute quantiles with finer error guarantees for the higher ranks of the data distribution effectively, using less space and computation time than computing all quantiles uniformly at the finest error? Second, if specific quantiles and their error bounds are requested a priori, can the necessary space usage and computation time be reduced? We answer both questions in the affirmative by formalizing them as the “high-biased ” and the “targeted ” quantiles problems, respectively, and presenting algorithms with provable guarantees, that perform significantly better than previously known solutions for these problems. We implemented our algorithms in the Gigascope data stream management system, and evaluated alternate approaches for maintaining the relevant summary structures. Our experimental results on real and synthetic IP data streams complement our theoretical analyses, and highlight the importance of lightweight, non-blocking implementations when maintaining summary structures over high-speed data streams.