Dispersing Hash Functions

Rasmus Pagh

December 2000


A new hashing primitive is introduced: dispersing hash functions. A family of hash functions $F$ is dispersing if, for any set $S$ of a certain size and random $h\in F$, the expected value of $\vert S\vert-\vert h[S]\vert$ is not much larger than the expectancy if $h$ had been chosen at random from the set of all functions.

We give tight, up to a logarithmic factor, upper and lower bounds on the size of dispersing families. Such families previously studied, for example universal families, are significantly larger than the smallest dispersing families, making them less suitable for derandomization. We present several applications of dispersing families to derandomization (fast element distinctness, set inclusion, and static dictionary initialization). Also, a tight relationship between dispersing families and extractors, which may be of independent interest, is exhibited.

We also investigate the related issue of program size for hash functions which are nearly perfect. In particular, we exhibit a dramatic increase in program size for hash functions more dispersing than a random function

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