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.

Work supported in part by an NSF CAREER Award CCF-0448277 and NSF grant EIA-98-02068.