NDLI: Balls and Bins: Smaller Hash Families and Faster Evaluation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Reingold, Omer Segev, Gil Celis, L. Elisa Wieder, Udi
Copyright Year	2013
Abstract	A fundamental fact in the analysis of randomized algorithms 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 $O(\log n/\log\log n)$-wise independent functions offer an evaluation time of $O(\log n/\log\log n)$, which according to Siegel's lower bound [A. Siegel, Proceedings of the $30$th Annual IEEE Symposium on Foundations of Computer Science, 1989, pp. 20--25] 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 $O(\sqrt{\log n})$. Our second construction 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.
Starting Page	1030
Ending Page	1050
Page Count	21
File Format	PDF
ISSN	00975397
DOI	10.1137/120871626
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3 (Special Section on the Forty-Second Annual ACM Symposium on Theory of Computing (STOC 2010))
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-05-21
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity pseudorandom generators hash functions balls and bins Analysis of algorithms
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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