Complexity classes

P-complete

In computational complexity theory, a decision problem is P-complete (complete for the complexity class P) if it is in P and every problem in P can be reduced to it by an appropriate reduction. The notion of P-complete decision problems is useful in the analysis of: * which problems are difficult to parallelize effectively, * which problems are difficult to solve in limited space. specifically when stronger notions of reducibility than polytime-reducibility are considered. The specific type of reduction used varies and may affect the exact set of problems. Generically, reductions stronger than polynomial-time reductions are used, since all languages in P (except the empty language and the language of all strings) are P-complete under polynomial-time reductions. If we use NC reductions, that is, reductions which can operate in polylogarithmic time on a parallel computer with a polynomial number of processors, then all P-complete problems lie outside NC and so cannot be effectively parallelized, under the unproven assumption that NC ≠ P. If we use the stronger log-space reduction, this remains true, but additionally we learn that all P-complete problems lie outside L under the weaker unproven assumption that L ≠ P. In this latter case the set P-complete may be smaller. (Wikipedia).

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Related pages

Extended Euclidean algorithm | Integer factorization | Boolean circuit | Decision problem | Maximum weight matching | Complete (complexity) | Context-free grammar | Lambda calculus | Conway's Game of Life | Greatest common divisor | Type inference | Unary numeral system | Sparse language | Circuit Value Problem | Graph theory | NC (complexity) | L (complexity) | Boolean satisfiability problem | Turing machine | Graph isomorphism | Horn-satisfiability | Log-space reduction | Time complexity | Type theory | Computational complexity theory | Horn clause | P (complexity) | Reduction (complexity) | Linear programming