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 R Wε -additive approximation for the window’s sum. We derive a lower bound of Wlog⌊1/2Wε + 1⌋ bits when ε≤ 1/2W and show a matching succinct algorithm that uses (1+o(1)) ( Wlog⌊1/2Wε + 1⌋) bits. Next, we prove a (1-o(1)) ε ^-1 /2 bits lower bound when ε =ω( W^-1) ∧ε =o(log ^-1W) and provide a succinct algorithm that requires (1+o(1)) ε ^-1 /2 bits. We show that when ε =( log ^-1W) any solution to the problem must consume at least (1-o(1))·( ε ^-1 /2+log W) bits, while our algorithm needs (1+o(1))·( ε ^-1 /2+2log 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.