Balls and Bins: Smaller Hash Families and Faster Evaluation

by Laura Elisa Celis, Omer Reingold, Gil Segev, and Udi Wieder

Oded's comments

I find it quite strange that these construction were not discovered twenty years ago, when almost k-wise independence generators and space-bounded generators were greatly improved and utilized. Still, this comment is not meant to diminish this work, but rather to say that it feels classic (or "from the book" or "just so right and nice"). In each of the two constructions, the authors "just" do the right thing (which is easy to say in retrospect, but two decades of failures to "do it right" just illustrate the difference between retrospect and a priori).

An afterthought (Aug 19th, 2011): The iterative construction method used in the 1st construction fits nicely in the gradual improvement paradigm (surveyed in my article Bravely, Moderately).

The original abstract

A fundamental fact in the analysis of randomized algorithm is that when $n$ balls are hashed into $n$ bins independently and uniformly at random, with high probability each bin contains at most $O(\log n / \log \log n)$ balls. In various applications, however, the assumption that a truly random hash function is available is not always valid, and explicit functions are required.

In this paper we study the size of families (or, equivalently, the description length of their functions) that guarantee a maximal load of $O(\log n / \log \log n)$ with high probability, as well as the evaluation time of their functions. Whereas such functions must be described using $\Omega( \log n)$ bits, the best upper bound was formerly $O(\log^2 n / \log \log n)$ bits, which is attained by $O(\log n / \log \log n)$-wise independent functions. Traditional constructions of the latter offer an evaluation time of $O(\log n / \log \log n)$, which according to Siegel's lower bound [FOCS '89] can be reduced only at the cost of significantly increasing the description length.

We construct two families that guarantee a maximal load of $O(\log n / \log \log n)$ with high probability. Our constructions are based on two different approaches, and exhibit different trade-offs between the description length and the evaluation time. The first construction shows that $O(\log n / \log \log n)$-wise independence can in fact be replaced by ``gradually increasing independence'', resulting in functions that are described using $O(\log n \log \log n)$ bits and evaluated in time $O(\log n \log \log n)$. The second construction is based on derandomization techniques for space-bounded computations combined with a tailored construction of a pseudorandom generator, resulting in functions that are described using $O(\log^{3/2} n)$ bits and evaluated in time $O(\sqrt{\log n})$. The latter can be compared to Siegel's lower bound stating that $O(\log n / \log \log n)$-wise independent functions that are evaluated in time $O(\sqrt{\log n})$ must be described using $\Omega(2^{\sqrt{\log n}})$ bits

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