Balanced families of perfect hash functions and their applications

The construction of perfect hash functions is a well-studied topic. In this article, this concept is generalized with the following definition. We say that a family of functions from [<i>n</i>] to [<i>k</i>] is a Δ-balanced (<i>n,k</i>)-family of perfect hash functions if for every <i>S</i> ⊆ [<i>n</i>], | <i>S</i> |=<i>k</i>, the number of functions that are 1-1 on <i>S</i> is between <i>T</i>/Δ and Δ <i>T</i> for some constant <i>T</i>>0. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 1-1 on <i>S</i>, for each <i>S</i> of size <i>k</i>. In the new notion of balanced families, we require the number of 1-1 functions to be almost the same (taking Δ to be close to 1) for every such <i>S</i>. Our main result is that for any constant Δ > 1, a Δ-balanced (<i>n,k</i>)-family of perfect hash functions of size 2<sup><i>O</i>(<i>k</i> log log <i>k</i>)</sup> log <i>n</i> can be constructed in time 2<sup><i>O</i>(<i>k</i> log log <i>k</i>)</sup> <i>n</i> log <i>n</i>. Using the technique of color-coding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial-time algorithm for approximating both the number of simple paths of length <i>k</i> and the number of simple cycles of size <i>k</i> for any <i>k</i> ≤ <i>O</i>(log <i>n</i>/log log log <i>n</i>) in a graph with <i>n</i> vertices. The approximation is up to any fixed desirable relative error.

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