Succinct Summing over Sliding Windows

This paper considers the problem of estimating the sum the last W elements of a stream of integers in {0, 1, ... , R}. Specifically, we study the memory requirements for computing a RW epsilon-additive approximation for the window's sum. We derive a lower bound of W log left perpendicular1/2W epsilon + 1right perpendicular bits when epsilon <= 1/2W and show a matching succinct algorithm that uses (1 + o(1)) (W log left perpendicular1/2W epsilon + 1right perpendicular bits. Next, we prove a (1 - o(1))epsilon(-1)/2 bits lower bound when epsilon = omega (W-1) <^> epsilon = o(log(-1) W) and provide a succinct algorithm that requires (1 + o(1))epsilon(-1)/2 bits. We show that when epsilon = Omega (log(-1) W) any solution to the problem must consume at least (1 - o(1)) . (epsilon(-1)/2 + log W) bits, while our algorithm needs (1 + o(1)) . (epsilon(-1)/2 + 2 log W) bits. Finally, we show that our lower bounds generalize to randomized algorithms as well, while our algorithms are deterministic and can process elements and answer queries in O(1) worst-case time.