Stream sampling for variance-optimal estimation of subset sums

Edith Cohen*, Nick Duffield, Haim Kaplan, Carsten Lund, Mikkel Thorup

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

From a high volume stream of weighted items, we want to maintain a generic sample of a certain limited size k that we can later use to estimate the total weight of arbitrary subsets. This is the classic context of on-line reservoir sampling, thinking of the generic sample as a reservoir. We present an efficient reservoir sampling scheme, VAROPTk, that dominates all previous schemes in terms of estimation quality. VAROPTk provides variance optimal unbiased estimation of subset sums. More precisely, if we have seen n items of the stream, then for any subset size m, our scheme based on k samples minimizes the average variance over all subsets of size m. In fact, the optimality is against any off-line scheme with k samples tailored for the concrete set of items seen. In addition to optimal average variance, our scheme provides tighter worst-case bounds on the variance of particular subsets than previously possible. It is efficient, handling each new item of the stream in O(log k) time, which is optimal even on the word RAM. Finally, it is particularly well suited for combination of samples from different streams in a distributed setting.

Original languageEnglish
Title of host publicationProceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherAssociation for Computing Machinery (ACM)
Pages1255-1264
Number of pages10
ISBN (Print)9780898716801
DOIs
StatePublished - 2009
Event20th Annual ACM-SIAM Symposium on Discrete Algorithms - New York, NY, United States
Duration: 4 Jan 20096 Jan 2009

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Conference

Conference20th Annual ACM-SIAM Symposium on Discrete Algorithms
Country/TerritoryUnited States
CityNew York, NY
Period4/01/096/01/09

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