Abstract
In the previous chapters we studied mostly subclasses of the class NC. As we have stressed already, NC is meant to stand for problems with feasible highly parallel solutions (meaning very efficient runtime using a reasonable number of processors). In the theory of sequential algorithms, the class P = DTIME(n O (1)) is meant to capture the notion of problems with feasible sequential algorithms (meaning reasonable runtime on a sequential machine). Every feasible highly parallel solution can be used to obtain a feasible sequential solution (NC ⊆ P). The question whether every problem with a feasible sequential algorithm also has a feasible parallel algorithm was discussed in detail in Sect. 4.6.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliographic Remarks
J. L. Balcâzar, R. V. Book, and U. Schöning. The polynomialtime hierarchy and sparse oracles. Journal of the ACM, 33: 603–617, 1986.
U. Schöning. Complexity and Structure. Lecture Notes in Com- puter Science 211, Springer-Verlag, Berlin, 1986.
R. Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55: 40–56, 1982.
J. Simon. On Some Central Problems in Computational Complexity. PhD thesis, Cornell University, 1975.
K. W. Wagner. The complexity of combinatorial problems with succinct input representation. Acta Informatica, 23: 325–356, 1986.
U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer, and K. W. Wagner. On the power of polynomial time bit-reductions. In Proceedings 8th Structure in Complexity Theory, p. 200–207. IEEE Computer Society Press, 1993.
H. Caussinus, P. McKenzie, D. Thérien, and H. Vollmer. Nondeterministic NC1 computation. Journal of Computer and System Sciences, 57: 200–212, 1998.
J.-Y. Cai and L. Hemachandra. On the power of parity polynomial time. In Proceedings 6th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 349, p. 229–239, Springer-Verlag, Berlin, 1989.
R. Beigel, J. Gill, and U. Hertrampf. Counting classes: thresholds, parity, mods, and fewness. In Proceedings 7th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 415, p. 49–57, Springer-Verlag, Berlin, 1990.
E. Allender. The permanent requires large uniform circuits. Chicago Journal of Theoretical Computer Science, 1999. To appear. A preliminary version appeared as: A note on uniform circuit lower bounds for the counting hierarchy, in Proceedings 2nd Computing and Combinatorics Conference, Lecture Notes in Computer Science 1090, p. 127–135, Springer-Verlag, Berlin, 1996.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Vollmer, H. (1999). Polynomial Time and Beyond. In: Introduction to Circuit Complexity. Texts in Theoretical Computer Science An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03927-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-662-03927-4_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08398-3
Online ISBN: 978-3-662-03927-4
eBook Packages: Springer Book Archive