Given a dataset P and a preference function f , a top-k query retrieves the k tuples in P with the highest scores according to f . Even though the problem is well-studied in conventional databases, the existing methods are inapplicable to highly dynamic environments involving numerous longrunning queries. This paper studies continuous monitoring of top-k queries over a fixed-size window W of the most recent data. The window size can be expressed either in terms of the number of active tuples or

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... time units. We propose a general methodology for top-k monitoring that restricts processing to the sub-domains of the workspace that influence the result of some query. To cope with high stream rates and provide fast answers in an on-line fashion, the data in W reside in main memory. The valid records are indexed by a grid structure, which also maintains book-keeping information. We present two processing techniques: the first one computes the new answer of a query whenever some of the current top-k points expire; the second one partially precomputes the future changes in the result, achieving better running time at the expense of slightly higher space requirements. We analyze the performance of both algorithms and evaluate their efficiency through extensive experiments. Finally, we extend the proposed framework to other query types and a different data stream model.