## Improved range-summable random variable construction algorithms

Range-summable universal hash functions, also known as range-summable random variables, are binary-valued hash functions which can efficiently hash single values as well as ranges of values from the domain. They have found several applications in the area of data stream processing where they are used to construct sketches - small-space summaries of the input sequence. We present two new constructions of range-summable universal hash functions on n-bit strings, one based on Reed-Muller codes which gives k-universal hashing using O(n log k) space arid time for point operations and O(n 2 1og k) for range operations, and another based on a new subcode of the second-order Reed-Muller code, which gives 5-universal hashing using O(n) space, O(n log 3 n) time for point operations, and O(n 3) time for range operations. We also present a new sketch data structure using the new hash functions which improves several previous results.

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*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms*, 840–849.

*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms*, July 1, 2005, 840–49.

*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms*, July 2005, pp. 840–49.