We study what we callfunctional monitoringproblems. We havekplayers each tracking their inputs, sayA_{j}(t) up until timet, and communicating with a central coordinator. The coordinator's task is to compute a given functionfover the union of the inputs U_{i}A_{j}(t),continuouslyat all timest. The goal is to minimize the number of bits communicated between the players and the coordinator. A simple example is whenfis simply the sum, and the coordinator is required to alert when the sum of a distributed set of values exceeds a given threshold. Of interest is the approximate version where the coordinator outputs 1 iff>=τ and 0 isf<=τ-ε. This defines the (k,f,τ,ε) distributed, functional monitoring problem.Functional monitoring problems are fundamental in distributed systems, in particular sensor networks, where we must minimize communication; they also connect to problems in communication complexity, communication theory, and signal processing. Yet few formal bounds are known for functional monitoring.

We give upper and lower bounds for the (

k,f,τ,ε) problem for foundationalf's. In particular, we study frequency moments (F_{0},F_{1},F_{2}). We show that for the simplest cases, the cost of monitoring a function can be the same as the cost of its one-time computation. However, forF_{2}we show a carefully constructed multi-round algorithm which uses “sketch summaries” at multiple levels of detail and solves the (k,F_{2},τ,ε) problem with communicationO(k^{2}/ε+k^{3/2}/ε^{3}). Since frequency moment estimation is central to other problems, our results have immediate applications to histograms, wavelet computations, and others. Our algorithmic techniques are likely to be useful for other functional monitoring problems.

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