TY - GEN

T1 - Efficient summing over sliding windows

AU - Basat, Ran Ben

AU - Einziger, Gil

AU - Friedman, Roy

AU - Kassner, Yaron

N1 - Publisher Copyright:
© Ran Ben-Basat, Gil Einziger, Roy Friedman, and Yaron Kassner.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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 Ω(1/ε+log W) memory bits for Wε-additive approximations is derived. This is followed by an algorithm whose memory consumption is O(1/ε + logW) 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 RWε can also be done using Θ(1/ε + logW) bits for ε = Ω(1/W). For ε = o(1/W), we present a succinct algorithm which uses B·(1 + o(1)) bits, where B = Θ(W log (1/Wε)) 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.

AB - 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 Ω(1/ε+log W) memory bits for Wε-additive approximations is derived. This is followed by an algorithm whose memory consumption is O(1/ε + logW) 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 RWε can also be done using Θ(1/ε + logW) bits for ε = Ω(1/W). For ε = o(1/W), we present a succinct algorithm which uses B·(1 + o(1)) bits, where B = Θ(W log (1/Wε)) 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.

KW - Lower bounds

KW - Statistics

KW - Streaming

UR - http://www.scopus.com/inward/record.url?scp=85012005516&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SWAT.2016.11

DO - 10.4230/LIPIcs.SWAT.2016.11

M3 - Conference contribution

AN - SCOPUS:85012005516

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 11.1-11.14

BT - 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016

A2 - Pagh, Rasmus

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016

Y2 - 22 June 2016 through 24 June 2016

ER -