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 isa challenging task due to the fast line speed and scarcity of fast memory in routers. Inthis work, we study the efficiency factor that can be gained by approximating the windowsize. That is, we allow the algorithm to dynamically adjust the window size between W andW (1 + τ) where τ is a small positive parameter. For example, consider the basic summingproblem of computing the sum of the last W elements in a stream whose items are integersin {0, 1 . . . , R}, where R = poly(W ). While it is known that Ω(W logR) bits are neededin the exact window model, we show that approximate windows allow an exponential spacereduction for constant τ .Specifically, we present a lower bound of Ω(τ−1 log(RWτ)) bits for the basic sum-ming problem. Further, an (1 + ) multiplicative approximation of this problem requiresΩ(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 pro-vide memory optimal algorithms that operate in constant time. Finally, we demonstratethe generality of the approximated window model by applying it to counting the numberof distinct flows in a sliding window over a network stream. We present an algorithm thatsolves this problem while requiring asymptotically less space than previous sliding windowmethods when τ = O(1).