Algorithms for data streams

Efficient processing over massive data sets has taken an increased importance in the last few decades due to the growing availability of large volumes of data in a variety of applications in computational sciences. In particular, monitoring huge and rapidly changing streams of data that arrive online has emerged as an important data management problem: Relevant applications include analyzing network traffic, online auctions, transaction logs, telephone call records, automated bank machine operations, and atmospheric and astronomical events. For these reasons, the streaming model has recently received a lot of attention. This model differs from computation over traditional stored data sets since algorithms must process their input by making one or a small number of passes over it, using only a limited amount of working memory. The streaming model applies to settings where the size of the input far exceeds the size of the main memory available and the only feasible access to the data is by making one or more passes over it. Typical streaming algorithms use space at most polylogarithmic in the length of the input stream and must have fast update and query times. Using sublinear space motivates the design for summary data structures with small memory footprints, also known as synopses [34]. Queries are answered using information provided by these synopses, and it may be impossible to produce an exact answer. The challenge is thus to produce high quality approximate answers, that is, answers with confidence bounds on the possible error: Accuracy guarantees are typically made in terms of a pair of user-specified parameters, ε and δ, meaning that the error in answering a query is within a factor of 1 + ε of the true answer with probability at least 1 − δ. The space and update time will depend on these parameters and the goal is to limit this dependence as much as possible. Major progress has been achieved in the last 10 years in the design of streaming algorithms for several fundamental data sketching and statistics problems, for which several different synopses have been proposed. Examples include number of distinct