Abstract. Nowadays, we have witnessed the widely recognized phenomenon of high speed data streams. Various statistics computation over data streams is often required by many applications, including processing of relational type queries, data mining and high speed network management. In this paper, we provide survey for three important kinds of aggregate computations over data streams: frequency moment, frequency count and order statistic. 1

Aggregation against distinct objects has been involved in many real applications with the presence of duplicates, including real-time monitoring moving objects. In this paper, we investigate the problem of counting distinct objects over sliding windows with arbitrary lengths. We present novel, time and space efficient, one scan algorithms to continuously maintain a sketch so that the counting can be approximately conducted with a relative error guarantee ɛ in the presence of object duplicates. Efficient query algorithms have also been developed by effectively utilizing the skyband property. Moreover, the proposed techniques may be immediately applied to the range counting aggregation and heavy hitter problem against distinct elements. A comprehensive performance study demonstrates that our algorithms can support real-time computation against high speed data streams. 1

Many scenarios, such as network analysis, utility monitoring, and financial applications, generate massive streams of data. These streams consist of millions or billions of simple updates every hour, and must be processed to extract the information described in tiny pieces. These notes provide an introduction to (and set of references for) data stream algorithms, and some of the techniques that have been developed over recent years to help mine the data while avoiding drowning in these massive flows of information. 1

A fundamental problem in data management is to draw and maintain a sample of a large data set, for approximate query answering, selectivity estimation, and query planning. With large, streaming data sets, this problem becomes particularly difficult when the data is shared across multiple distributed sites. The main challenge is to ensure that a sample is drawn uniformly across the union of the data while minimizing the communication needed to run the protocol on the evolving data. At the same time, it is also necessary to make the protocol lightweight, by keeping the space and time costs low for each participant. In this paper, we present communication-efficient protocols for continuously maintaining a sample (both with and without replacement) from k distributed streams. These apply to the case when we want a sample from the full streams, and to the sliding window cases of only the W most recent elements, or arrivals within the last w time units. We show that our protocols are optimal (up to logarithmic factors), not just in terms of the communication used, but also the time and space costs for each participant. 1