Balanced families of perfect hash functions and their applications

Abstract

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 [ n ] to [ k ] is a δ-balanced ( n,k )-family of perfect hash functions if for every S ⊆ [ n ], | S |= k , the number of functions that are 1-1 on S is between T /δ and δ T for some constant T >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 S , for each S of size k . 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 S . Our main result is that for any constant δ > 1, a δ-balanced ( n,k )-family of perfect hash functions of size 2 O ( k log log k ) log n can be constructed in time 2 O ( k log log k ) n log n . 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 k and the number of simple cycles of size k for any k ≤ O (log n /log log log n ) in a graph with n vertices. The approximation is up to any fixed desirable relative error.