The construction of perfect hash functions is a well-studied topic. In this paper, 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 ST 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 2O(k log log k) log n can be constructed in time 2O(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. © Springer-Verlag Berlin Heidelberg 2007.
CITATION STYLE
Alon, N., & Gutner, S. (2007). Balanced families of perfect hash functions and their applications. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4596 LNCS, pp. 435–446). Springer Verlag. https://doi.org/10.1007/978-3-540-73420-8_39
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