The Sliding-Window Computation Model and Results

Abstract

We present some results related to small space computation over sliding windows in the data-stream model. Most research in the data-stream model, including results presented in some of the other chapters, assume that all data elements seen so far in the stream are equally important and synopses, statistics or models that are built should reflect the entire data set. However, for many applications this assumption is not true, particularly those that ascribe more importance to recent data items. One way to discount old data items and only consider recent ones for analysis is the sliding-window model: Data elements arrive at every instant; each data element expires after exactly N time steps; and, the portion of data that is relevant to gathering statistics or answering queries is the set of last N elements to arrive. The sliding window refers to the window of active data elements at a given time instant and window size refers to N. This chapter presents a general technique, called the Exponential Histogram (EH) technique, that can be used to solve a wide variety of problems in the sliding-window model; typically problems that require us to maintain statistics. We will showcase this technique through solutions to basic counting problems, as well as other applications.