NDLI: Maintaining Stream Statistics over Sliding Windows

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Motwani, Rajeev Indyk, Piotr Datar, Mayur Gionis, Aristides
Copyright Year	2002
Abstract	We consider the problem of maintaining aggregates and statistics over data streams, with respect to the last N data elements seen so far. We refer to this model as the sliding window model. We consider the following basic problem: Given a stream of bits, maintain a count of the number of 1's in the last N elements seen from the stream. We show that, using $O(\frac{1}{\epsilon} \log^2 N)$ bits of memory, we can estimate the number of 1's to within a factor of $1 + \epsilon$. We also give a matching lower bound of $\Omega(\frac{1}{\epsilon}\log^2 N)$ memory bits for any deterministic or randomized algorithms. We extend our scheme to maintain the sum of the last N positive integers and provide matching upper and lower bounds for this more general problem as well. We also show how to efficiently compute the Lp norms ($p \in [1,2]$) of vectors in the sliding window model using our techniques. Using our algorithm, one can adapt many other techniques to work for the sliding window model with a multiplicative overhead of $O(\frac{1}{\epsilon}\log N)$ in memory and a $1 +\epsilon$ factor loss in accuracy. These include maintaining approximate histograms, hash tables, and statistics or aggregates such as sum and averages.
Starting Page	1794
Ending Page	1813
Page Count	20
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539701398363
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6
Volume Number	31
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2002-01-01
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity data streams Data structures General Approximation algorithms approximation algorithms sliding windows statistics
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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