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 +logW ) memory bits forW -additive approximations is derived. This is followed by an algorithm whose memory consumption is O( 1 + 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 RW can also be done using Θ( 1 + logW ) bits for = Ω ( 1 W ) . For = o ( 1 W ) , we present a succinct algorithm which uses B · (1 + o(1)) bits, where B = Θ ( W log ( 1 W )) 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. 1998 ACM Subject Classification E.1 [Data Structures] Lists, stacks, and queues