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.

Rejoinder: Monitoring Networked Applications With Incremental Quantile Estimation

Rejoinder: Monitoring Networked Applications With Incremental Quantile Estimation [arXiv:0708.0302]Comment: Published at http://dx.doi.org/10.1214/088342306000000592 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

SQUAD: Combining Sketching and Sampling Is Better than Either for Per-item Quantile Estimation

Latency quantiles measurements are essential as they often capture the user's utility. For example, if a video connection has high tail latency, the perceived quality will suffer, even if the average and median latencies are low. In this work, we consider the problem of approximating the per-item quantiles. Elements in our stream are (ID, latency) tuples, and we wish to track the latency quantiles for each ID. Existing quantile sketches are designed for a single number stream (e.g., containing just the latency). While one could allocate a separate sketch instance for each ID, this may require an infeasible amount of memory. Instead, we consider tracking the quantiles for the heavy hitters (most frequent items), which are often considered particularly important, without knowing them beforehand. We first present a simple sampling algorithm that serves as a benchmark. Then, we design an algorithm that augments a quantile sketch within each entry of a heavy hitter algorithm, resulting in similar space complexity but with a deterministic error guarantee. Finally, we present SQUAD, a method that combines sampling and sketching while improving the asymptotic space complexity. Intuitively, SQUAD uses a background sampling process to capture the behaviour of the latencies of an item before it is allocated with a sketch, thereby allowing us to use fewer samples and sketches. Our solutions are rigorously analyzed, and we demonstrate the superiority of our approach using extensive simulations