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 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.