Balanced families of perfect hash functions and their applications

Noga Alon, Shai Gutner

Research output: Chapter in Book/Report/Conference proceedingConference contribution

9 Scopus citations

Abstract

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.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 34th International Colloquium, ICALP 2007, Proceedings
PublisherSpringer Verlag
Pages435-446
Number of pages12
ISBN (Print)3540734198, 9783540734192
DOIs
StatePublished - 2007
Event34th International Colloquium on Automata, Languages and Programming, ICALP 2007 - Wroclaw, Poland
Duration: Jul 9 2007Jul 13 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4596 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other34th International Colloquium on Automata, Languages and Programming, ICALP 2007
CountryPoland
CityWroclaw
Period7/9/077/13/07

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

Keywords

  • Approximate counting of subgraphs
  • Color-coding
  • Perfect hashing

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  • Cite this

    Alon, N., & Gutner, S. (2007). Balanced families of perfect hash functions and their applications. In Automata, Languages and Programming - 34th International Colloquium, ICALP 2007, Proceedings (pp. 435-446). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4596 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-540-73420-8_39