Succinct Summing over Sliding Windows

Abstract

This paper considers the problem of estimating the sum the last $W$ elements of a stream of integers in $\left\{ 0,1,\ldots , R \right\}$. Specifically, we study the memory requirements for computing a $R W\varepsilon$-additive approximation for the window’s sum. We derive a lower bound of $W\log \left\lfloor {\frac{1}{2W\varepsilon } + 1}\right\rfloor$ bits when $\varepsilon \le 1/2W$ and show a matching succinct algorithm that uses $(1+o(1)) \left( {W\log \left\lfloor {\frac{1}{2W\varepsilon } + 1}\right\rfloor }\right)$ bits. Next, we prove a $(1-o(1)) \varepsilon ^{-1} /2$ bits lower bound when $\varepsilon =\omega \left( {W^{-1}}\right) \wedge \varepsilon =o(\log ^{-1}W)$ and provide a succinct algorithm that requires $(1+o(1)) \varepsilon ^{-1} /2$ bits. We show that when $\varepsilon =\varOmega \left( {\log ^{-1}W}\right)$ any solution to the problem must consume at least $(1-o(1))\cdot \left( {{ \varepsilon ^{-1} /2}+\log W}\right)$ bits, while our algorithm needs $(1+o(1))\cdot \left( {{ \varepsilon ^{-1} /2}+2\log W}\right)$ 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.