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 E2 , 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.
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Work done in part while at Dartmouth College. Research supported by the National Science Foundation under grant RII-9003056.
Research supported in part by a Hewlett-Packard Corporation equipment grant and by the National Science Foundation under grant CCR-8809174/CCR-8996198 and a Presidential Young Investigator Award.
Research supported in part by the National Science Foundation under grant DMS-8501521.
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References
L. Adleman. Time, space, and randomness. Technical Report MIT/LCS/TM-131, MIT, Cambridge, MA, April 1979.
E. Allender and R. Rubinstein. P-printable sets. SIAM Journal on Computing, 17(6):1193–1202, 1988.
T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal on Computing, 4(4):431–442, 1975.
L. Berman and J. Hartmanis. On isomorphisms and density of NP and other complete sets. SIAM Journal on Computing, 6(2):305–322, 1977.
R. Book. Bounded query machines: On NP and PSPACE. Theoretical Computer Science, 15:27–39, 1981.
R. Book and C. Wrathall. Bounded query machines: On NP() and NPQUERY(). Theoretical Computer Science, 15:41–50, 1981.
C. Chang and C. Chang. An ordered minimal perfect hashing scheme with single parameter. Information Processing Letters, 27:79–83, 1988.
C. Chang. The study of an ordered minimal perfect hashing. Communications of the ACM, 27(4):384–387, 1984.
M. Garey and D. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979.
S. Gordon. A mathematical formulation of the requirements of an order preserving hash function. In Computing and Information: Proceedings of the 1989 International Conference on Computing and Information, pages 64–68. Canadian Scholars' Press Inc., 1989.
A. Goldberg and M. Sipser. Compression and ranking. In Proceedings of the 17th ACM Symposium on Theory of Computing, pages 440–448, 1985.
L. Hemachandra. The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3):299–322, 1989.
L. Hemachandra. Algorithms from complexity theory: Polynomial-time operations for complex sets. In Proceedings of the 1990 SIGAL International Symposium on Algorithms, pages 221–231. Springer-Verlag Lecture Notes in Computer Science #450, August 1990.
L. Hemachandra, A. Hoene, D. Siefkes, and P. Young. On sets polynomially enumerable by iteration. Theoretical Computer Science, 80(2):203–226, 1991.
J. Hartmanis, N. Immerman, and V. Sewelson. Sparse sets in NP-P: EXP-TIME versus NEXPTIME. Information and Control, 65(2/3):159–181, 1985.
S. Homer and W. Maass. Oracle dependent properties of the lattice of NP sets. Theoretical Computer Science, 24:279–289, 1983.
L. Hemachandra and S. Rudich. On the complexity of ranking. Journal of Computer and System Sciences, 41(2):251–271, 1990.
D. Huynh. The complexity of ranking simple languages. Mathematical Systems Theory, 23:1–20, 1990.
J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities. Theoretical Computer Science, 34:17–32, 1984.
D. Joseph and P. Young. Some remarks on witness functions for non-polynomial and non-complete sets in NP. Theoretical Computer Science, 39:225–237, 1985.
S. Mahaney. Sparse sets and reducibilities. In R. Book, editor, Studies in Complexity Theory, pages 63–118. John Wiley and Sons, 1986.
S. Mahaney and P. Young. Reductions among polynomial isomorphism types. Theoretical Computer Science, 39:207–224, 1985.
C. Papadimitriou and M. Yannakakis. The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences, 28(2):244–259, 1984.
U. Schöning. A low and a high hierarchy in NP. Journal of Computer and System Sciences, 27:14–28, 1983.
G. Tardos. Query complexity, or why is it difficult to separate NPA ∩ coNPA from PA by random oracles A. Combinatorica, 9, 1989.
L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5:20–23,1976.
L. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410–421, 1979.
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Goldsmith, J., Hemachandra, L.A., Kunen, K. (1991). On the structure and complexity of infinite sets with minimal perfect hash functions. In: Biswas, S., Nori, K.V. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1991. Lecture Notes in Computer Science, vol 560. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54967-6_70
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DOI: https://doi.org/10.1007/3-540-54967-6_70
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