In computational complexity theory, a complexity class is a set of problems of related complexity. A typical complexity class has a definition of the form: :the set of problems that can be solved by abstract machine M using Big O notation(f(''n'')) of resource R (''n'' is the size of the input) For example, the class NP (complexity) is the set of decision problems that can be solved by a non-deterministic Turing machine in polynomial time, while the class PSPACE is the set of decision problems that can be solved by a deterministic Turing machine in polynomial space. Some complexity classes are sets of function problems, such as FP (complexity). Many complexity classes can be characterized in terms of the mathematical logic needed to express them; see descriptive complexity. ==See also== * List of complexity classes Complexity classes
I'm surprised there aren't pages for linear time, log(n) time, or even constant time. They're not as intriguing as P vs. NP, but many important (albeit ordinary) tasks require knowing about the distinctions between these complexity classes, eg. when to use linked lists vs. arrays or vectors. Any thoughts? :These are only really useful in the analysis of algorithms with regard to particular machine models; for example, most sorting algorithms assume that you have random access, that the input has a particular structure, and that a comparison or exchange is a unit operation. In complexity theory, we try to avoid any such assumptions by using classes that are invariant over a large class of abstract machines, like P, NP, and L. :For example, anything you can do on a random access machine in polynomial time, you can also do on a Turing machine (with or without multiple tapes) in polynomial time. If the input is a graph, you can specify it in adjacency matrix form, adjacency list form, or incidence matrix form - in these coarse classes, it doesn't matter. You can't say that about the classes you mention. In fact, a Turing machine can't do much of anything useful in log time, although a random access machine can. :This might be (should be?) discussed somewhere in the encyclopedia. User:Dcoetzee 22:36, 28 Apr 2005 (UTC)
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Complexity class
In computational complexity theory, a complexity class is a set of problems of related complexity. A typical complexity class has a definition of the form: :the set of problems that can be solved by abstract machine M using Big O notation(f(''n'')) of resource R (''n'' is the size of the input) For example, the class NP (complexity) is the set of decision problems that can be solved by a non-deterministic Turing machine in polynomial time, while the class PSPACE is the set of decision problems that can be solved by a deterministic Turing machine in polynomial space. Some complexity classes are sets of function problems, such as FP (complexity). Many complexity classes can be characterized in terms of the mathematical logic needed to express them; see descriptive complexity. ==See also== * List of complexity classes Complexity classes
Complexity class
I'm surprised there aren't pages for linear time, log(n) time, or even constant time. They're not as intriguing as P vs. NP, but many important (albeit ordinary) tasks require knowing about the distinctions between these complexity classes, eg. when to use linked lists vs. arrays or vectors. Any thoughts? :These are only really useful in the analysis of algorithms with regard to particular machine models; for example, most sorting algorithms assume that you have random access, that the input has a particular structure, and that a comparison or exchange is a unit operation. In complexity theory, we try to avoid any such assumptions by using classes that are invariant over a large class of abstract machines, like P, NP, and L. :For example, anything you can do on a random access machine in polynomial time, you can also do on a Turing machine (with or without multiple tapes) in polynomial time. If the input is a graph, you can specify it in adjacency matrix form, adjacency list form, or incidence matrix form - in these coarse classes, it doesn't matter. You can't say that about the classes you mention. In fact, a Turing machine can't do much of anything useful in log time, although a random access machine can. :This might be (should be?) discussed somewhere in the encyclopedia. User:Dcoetzee 22:36, 28 Apr 2005 (UTC)
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