Efficient Network Measurements through Approximated Windows

Many networking applications require timely access to recent network measurements, which can be captured using a sliding window model. Maintaining such measurements is a challenging task due to the fast line speed and scarcity of fast memory in routers. In this work, we study the efficiency factor that can be gained by approximating the window size. That is, we allow the algorithm to dynamically adjust the window size between W and W (1 + τ) where τ is a small positive parameter. For example, consider the basic summing problem of computing the sum of the last W elements in a stream whose items are integers in {0, 1 . . . , R}, where R = poly(W ). While it is known that Ω(W logR) bits are needed in the exact window model, we show that approximate windows allow an exponential space reduction for constant τ . Specifically, we present a lower bound of Ω(τ−1 log(RWτ)) bits for the basic summing problem. Further, an (1 + ) multiplicative approximation of this problem requires Ω(log (W/ )+log logR) bits for constant τ . Additionally, for RW additive approximations, we show an Ω(τ−1 log b1 + τ/ c+ log (W/ )) lower bound . For all three settings, we provide memory optimal algorithms that operate in constant time. Finally, we demonstrate the generality of the approximated window model by applying it to counting the number of distinct flows in a sliding window over a network stream. We present an algorithm that solves this problem while requiring asymptotically less space than previous sliding window methods when τ = O(1).