eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-06-22
11:1
11:14
10.4230/LIPIcs.SWAT.2016.11
article
Efficient Summing over Sliding Windows
Ben Basat, Ran
Einziger, Gil
Friedman, Roy
Kassner, Yaron
This paper considers the problem of maintaining statistic aggregates over the last W elements of a data stream. First, the problem of counting the number of 1's in the last W bits of a binary stream is considered. A lower bound of Omega(1/epsilon + log(W)) memory bits for Wepsilon-additive approximations is derived. This is followed by an algorithm whose memory consumption is O(1/epsilon + log(W)) bits, indicating that the algorithm is optimal and that the bound is tight. Next, the more general problem of maintaining a sum of the last W integers, each in the range of {0, 1, ..., R}, is addressed. The paper shows that approximating the sum within an additive error of RW epsilon can also be done using Theta(1/epsilon + log(W)) bits for epsilon = Omega(1/W). For epsilon = o(1/W), we present a succinct algorithm which uses B(1 + o(1)) bits, where B = Theta(W*log(1/(W*epsilon))) is the derived lower bound. We show that all lower bounds generalize to randomized algorithms as well. All algorithms process new elements and answer queries in O(1) worst-case time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol053-swat2016/LIPIcs.SWAT.2016.11/LIPIcs.SWAT.2016.11.pdf
Streaming
Statistics
Lower Bounds