Pseudo-random number generation for sketch-based estimations

The exact computation of aggregate queries, like the size of join of two relations, usually requires large amounts of memory (constrained in data-streaming) or communication (constrained in distributed computation) and large processing times. In this situation, approximation techniques with provable guarantees, like sketches, are one possible solution. The performance of sketches depends crucially on the ability to generate particular pseudo-random numbers. In this article we investigate both theoretically and empirically the problem of generating k-wise independent pseudo-random numbers and, in particular, that of generating 3- and 4-wise independent pseudo-random numbers that are fast range-summable (i.e., they can be summed in sublinear time). Our specific contributions are: (a) we provide a thorough comparison of the various pseudo-random number generating schemes; (b) we study both theoretically and empirically the fast range-summation property of 3- and 4-wise independent generating schemes; (c) we provide algorithms for the fast range-summation of two 3-wise independent schemes, BCH and extended Hamming; and (d) we show convincing theoretical and empirical evidence that the extended Hamming scheme performs as well as any 4-wise independent scheme for estimating the size of join of two relations using AMS sketches, even though it is only 3-wise independent. We use this scheme to generate estimators that significantly outperform state-of-the-art solutions for two problems, namely, size of spatial joins and selectivity estimation.