Effective Computations on Sliding Windows

Abstract

In the streaming model, elements arrive sequentially and can be observed only once. Maintaining statistics and aggregates is an important and nontrivial task in this model. These tasks become even more challenging in the sliding windows model, where statistics must be maintained only over the most recent $n$ elements. In their pioneering paper, Datar et al. [SIAM J. Comput., 31 (2002), pp. 1794-1813] presented the exponential histogram, an effective method for estimating statistics on sliding windows. In this paper we present a novel smooth histogram method that is more general and achieves stronger bounds than the exponential histogram. In particular, the smooth histogram method improves the approximation error rate obtained via exponential histograms. Furthermore, the smooth histogram method not only captures and improves multiple previous results on sliding windows but also extends the class of functions that can be approximated on sliding windows. In particular, we provide the first approximation algorithms for the following functions: $L_p$ norms, frequency moments, the length of the increasing subsequence, and the geometric mean.