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

Abstract

This paper studies the class of infinite sets that have minimal perfect hash functions one-to-one onto maps between the sets and E·-computable in polynomial time. We show that all standard NP-complete sets have polynomial-time computable minimal per fect hash functions, and give a structural condition sufficient to ensure that all infinite NP sets have polynomial-time computable minimal perfect hash functions: If E = Ef, then 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.

On Sets with Efficient Implicit Membership Tests

“…Indeed, the study of exactly which sets are simple under various operations other than membership testing is a newly emergent theme in theoretical computer science. Though it has long been known that many complex problems can be efficiently approximately solved, current research efforts show that fundamental algorithmic operations-data compression, perfect hashing, and enumeration-can be efficiently performed on many complex sets [Hem90,GHK,HHS89,HHSY]. The present paper shows that even sets of extremely high complexity may have polynomial-time algorithms for implicitly testing membership.…”