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On the structure and complexity of infinite sets with minimal perfect hash functions

  • Judy Goldsmith
  • Lane A. Hemachandra
  • Kenneth Kunen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 560)

Abstract

This paper studies the class of infinite sets that have minimal perfect hash functions—one-to-one onto maps between the sets and Σ*—computable in polynomial time. We show that all standard NP-complete sets have polynomialtime computable minimal perfect hash functions, and give a structural condition, E=Σ 2 E , sufficient to ensure that all infinite NP sets have polynomial-time computable minimal perfect hash functions. On the other hand, we present evidence that some infinite NP sets, and indeed some infinite P sets, do not have polynomial-time computable minimal perfect hash functions: if an infinite NP set A has polynomial-time computable perfect minimal hash functions, then A has an infinite sparse NP subset, yet we construct a relativized world in which some infinite NP sets lack infinite sparse NP subsets. This world is built upon a result that is of interest in its own right; we determine optimally—with respect to any relativizable proof technique—the complexity of the easiest infinite sparse subsets that infinite P sets are guaranteed to have.

Keywords

Hash Function Ranking Function Random Oracle Polynomial Hierarchy Relativize World 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Judy Goldsmith
    • 1
  • Lane A. Hemachandra
    • 2
  • Kenneth Kunen
    • 3
  1. 1.Dept of Computer ScienceUniversity of ManitobaWinnipegCanada
  2. 2.Department of Computer ScienceUniversity of RochesterRochester
  3. 3.Department of Computer SciencesUniversity of WisconsinMadison

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