The area of distributed monitoring requires tracking the value of a function of distributed data as new observations are made. An important case is when attention is restricted to only a recent time period, such as the last hour of readings-the sliding window case. In this paper, we introduce a novel paradigm for handling such monitoring problems, which we dub the “forward/backward” approach. This view allows us to provide optimal or near-optimal solutions for several fundamental problems, such as counting, tracking frequent items, and maintaining order statistics. The resulting protocols improve on previous work or give the first solutions for some problems, and operate efficiently in terms of space and time needed. Specifically, we obtain optimal O((k)/(ε) log(εn/k)) communication per window of n updates for tracking counts and heavy hitters with accuracy ε across k sites; and near-optimal communication of O((k)/(ε) log2(1/ε) log(n/k)) for quantiles. We also present solutions for problems such as tracking distinct items, entropy, and convex hull and diameter of point sets.
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