Efficient Summing over Sliding Windows

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 W{\epsilon}-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}(Wlog(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.Comment: A shorter version appears in SWAT 201

Counting Distinct Objects over Sliding Windows

Aggregation against distinct objects has been in-volved in many real applications with the presence of duplicates, including real-time monitoring mov-ing objects. In this paper, we investigate the prob-lem of counting distinct objects over sliding windows with arbitrary lengths. We present novel, time and space efficient, one scan algorithms to continuously maintain a sketch so that the counting can be ap-proximately conducted with a relative error guaran-tee ∈ in the presence of object duplicates. Efficient query algorithms have also been developed by ef-fectively utilizing the skyband property. Moreover, the proposed techniques may be immediately applied to the range counting aggregation and heavy hitter problem against distinct elements. A comprehensive performance study demonstrates that our algorithms can support real-time computation against high speed data streams. © 2010, Australian Computer Society, Inc