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We consider the problem of maintaining frequency counts for items occurring frequently in the union of multiple distributed data streams. Naive methods of combining approximate frequency counts from multiple nodes tend to result in excessively large data structures that are costly to transfer among nodes. To minimize communication requirements, the degree of precision maintained by each node while counting item frequencies must be managed carefully. We introduce the concept of a precision gradient for managing precision when nodes are arranged in a hierarchical communication structure. We then study the optimization problem of how to set the precision gradient so as to minimize communication, and provide optimal solutions that minimize worst-case communication load over all possible inputs. We then introduce a variant designed to perform well in practice, with input data that does not conform to worstcase characteristics. We verify the effectiveness of our approach empirically using real-world data, and show that our methods incur substantially less communication than naive approaches while providing the same error guarantees on answers.In addition, we extend techniques for maintaining frequency counts of high-frequency items in one or more streams by making them time-sensitive. Time-sensitivity is achieved by associating weights with items that decay exponentially with time. We analyze the error bounds and worst-case space bounds for the extended algorithms.

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