Functional monitoring without monotonicity

Abstract

The notion of distributed functional monitoring was recently introduced by Cormode, Muthukrishnan and Yi to initiate a formal study of the communication cost of certain fundamental problems arising in distributed systems, especially sensor networks. In this model, each of k sites reads a stream of tokens and is in communication with a central coordinator, who wishes to continuously monitor some function f of σ, the union of the k streams. The goal is to minimize the number of bits communicated by a protocol that correctly monitors f(σ), to within some small error. As in previous work, we focus on a threshold version of the problem, where the coordinator's task is simply to maintain a single output bit, which is 0 whenever f(σ)≤τ(1-ε) and 1 whenever f(σ)≥τ. Following Cormode et al., we term this the (k,f,τ,ε) functional monitoring problem. In previous work, some upper and lower bounds were obtained for this problem, with f being a frequency moment function, e.g., F 0, F 1, F 2. Importantly, these functions are monotone. Here, we further advance the study of such problems, proving three new classes of results. First, we provide nontrivial monitoring protocols when f is either H, the empirical Shannon entropy of a stream, or any of a related class of entropy functions (Tsallis entropies). These are the first nontrivial algorithms for distributed monitoring of non-monotone functions. Second, we study the effect of non-monotonicity of f on our ability to give nontrivial monitoring protocols, by considering f=F p with deletions allowed, as well as f=H. Third, we prove new lower bounds on this problem when f=F p , for several values of p. © 2009 Springer Berlin Heidelberg.