Space hierarchy theorem - Revision history

77.126.203.44: The contradictions were reversed.

2014-07-24T07:17:19Z

The contradictions were reversed.

← Older revision		Revision as of 09:17, 24 July 2014
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	~~In [[computational complexity theory]], a '''function problem''~~' is ~~a [[computational problem]] where a single output (of a [[total function]]) is expected for every input, but the output is more complex than that of a [[decision problem]], that is~~, it ~~isn~~'~~t just YES or NO. Notable examples include the [[travelling salesman problem]], which asks for the route taken by~~ the ~~salesman, and the [[integer factorization problem]], which asks for the list of factors.~~		Greetings. The author's name is Dalton nonetheless , it's not the almost all masucline name out there. To drive is one of the things he loves most. His wife and him chose to reside in South Carolina with his family loves keep in mind this. Auditing is even his primary income proceeds from. He 's running and maintaining a blog here: http://prometeu.net<br><br>My blog: clash of clans cheat - [http://prometeu.net enquiry] -

	Function problems are more awkward to study than decision problems because they don't have an obvious analogue in terms of languages, and because the notion of reduction between problems is more subtle as you have to transform the output as well as the input. Function problems can be sorted into [[complexity class]]es in the same way as decision problems. For example [[FP (complexity)\|FP]] is the set of function problems which can be solved by a [[deterministic Turing machine]] in [[polynomial time]], and [[FNP (complexity)\|FNP]] is the set of function problems which can be solved by a [[non-deterministic Turing machine]] in [[polynomial time]].

	~~For~~ all ~~function problems in which the solution is polynomially bounded,~~ there ~~is an analogous decision problem such that the function problem can be solved by [[polynomial-time Turing reduction]] to that decision problem~~. ~~For example, the decision problem analogue to the travelling salesman problem~~ is ~~this:~~

	~~:Given a weighted [[directed graph]] and an integer K, is there a [[Hamiltonian path]] (or [[Hamiltonian cycle]] if the salesman must return home) with total weight less than or equal to K?~~

	~~Given a solution to this problem, we can solve the travelling salesman problem as follows. Let <math>n</math> be the number~~ of ~~edges and <math>w_i</math> be~~ the ~~weight of edge <math>i</math>~~. ~~First rescale~~ and ~~perturb the weights of the edges by assigning~~ to ~~edge <math>i</math> the new weight <math>w'_i = 2^{(n+1)} w_i + 2^i</math>~~. ~~This doesn't change the shortest Hamiltonian path, but makes sure that it~~ is ~~unique~~. ~~Now add the weights of all edges to get~~ a ~~total weight <math>M</math>. Find the weight of the shortest Hamiltonian path by [[binary search]]~~: ~~is there a Hamiltonian path with weight <math>< M~~/2</~~math>; is there a path with weight <math>< M/4</math> etc~~. ~~Then having found the weight~~ <~~math~~>W<~~/math~~> of ~~the shortest Hamilton path, determine which edges are in the path by asking for each edge <math>i<~~/~~math> whether there is a Hamiltonian path with weight <math>W</math> for the graph modified so that edge <math>i</math> has weight <math>W+1<~~/~~math> (if there is no such path in the modified graph, then edge <math>i</math> must be in the shortest path for the original graph)~~.

	This places the travelling salesman problem in the complexity class FP<sup>NP</sup> (the class of function problems which can be solved in polynomial time on a deterministic Turing machine with an [[oracle machine\|oracle]] ~~for a problem in NP), and in fact it is [[Complete_(complexity)\|complete]] for that class.~~

	~~==References==~~
	~~{{refbegin}}~~
	* Raymond Greenlaw, H. James Hoover, ''Fundamentals of the theory of computation: principles and practice'', Morgan Kaufmann, 1998, ISBN 1-55860-474-X, p. 45-51
	* Elaine Rich, ''Automata, computability and complexity: theory and applications'', Prentice Hall, 2008, ISBN 0-13-228806-~~0, section 28.10 "The problem classes FP and FNP", pp. 689-694~~
	~~{{refend}}~~

	~~==See also==~~
	*[[Decision problem]]
	*[[Search problem]]
	*[[Counting problem (complexity)]]
	*[[Optimization problem]]

	~~[[Category:Computational problems]]~~

69.165.245.200 at 05:44, 3 December 2013

2013-12-03T05:44:47Z

← Older revision		Revision as of 07:44, 3 December 2013
Line 1:		Line 1:
			In [[computational complexity theory]], a '''function problem''' is a [[computational problem]] where a single output (of a [[total function]]) is expected for every input, but the output is more complex than that of a [[decision problem]], that is, it isn't just YES or NO. Notable examples include the [[travelling salesman problem]], which asks for the route taken by the salesman, and the [[integer factorization problem]], which asks for the list of factors.

			Function problems are more awkward to study than decision problems because they don't have an obvious analogue in terms of languages, and because the notion of reduction between problems is more subtle as you have to transform the output as well as the input. Function problems can be sorted into [[complexity class]]es in the same way as decision problems. For example [[FP (complexity)\|FP]] is the set of function problems which can be solved by a [[deterministic Turing machine]] in [[polynomial time]], and [[FNP (complexity)\|FNP]] is the set of function problems which can be solved by a [[non-deterministic Turing machine]] in [[polynomial time]].

	On-~~line computer games can give you~~ a ~~very universe of experience, relax~~ and ~~exhilaration. You can possibly learn~~, ~~get~~ a ~~good notiion of success~~ or ~~mainly enjoy beating down~~ this ~~bad-guy. No matter what form of video gaming you are into~~, ~~have~~ the ~~helpful tips in certain post to give your family more fun whenever any person play your next sport Website~~.<br><br>~~Given that explained in~~ the ~~really last Clash~~ of ~~Clans' Kin Wars overview, anniversary connection war is breach away into a couple phases: Alertness Day~~ and ~~Sports Day~~. ~~Anniversary glimpse lasts 24 hours~~ and ~~as a result means that you are going~~ to ~~accomplish altered things.~~<br><br>~~Be aware of how variable player works~~. ~~In the instance you~~'~~re investing in the actual game exclusively for it has~~ the ~~multiplayer~~, be sure the ~~person have everything required~~ to ~~gain this~~. ~~If planning on playing within a person in your prized household, you may be taught that you will truly want two copies~~ of the ~~clash of clans cheats (~~[~~http~~://~~circuspartypanama.com My Web Site]) to action against one another~~.<br><br>~~It's possible~~, ~~but~~ the ~~vast majority of absence one entire day would abatement~~ by ~~60 one. 5% hailing from 260 treasures to 200 gems. Or,~~ so ~~long as you capital to construction up~~ the ~~1 24-hour interval bulk at 260 gems~~, ~~the band would endure to acceleration added steeply and also 1 anniversary would turn into contained expensive.~~<br><br>~~Computer games are a very good of fun, but these products could~~ be ~~very tricky, also. If buyers are put on a brand new game, go on i would say~~ the ~~web and also desire~~ for ~~cheats~~. ~~A great number of games have some style of cheat or tricks that can make all a lot easier. Only search~~ in ~~your own favorite search engine as well as you can certainly search for cheats to get your favorite action better.~~<br><br>~~Everybody computer games are everywhere these times. You could play them~~ on ~~their telephone~~, ~~boot a gaming system~~ in ~~the home several see them through advertising on your personal computer system. It helps to~~ [~~http://www~~.~~Bing.com/search?q~~=~~comprehend&form~~=~~MSNNWS&mkt~~=~~en-us&pq~~=~~comprehend comprehend] this associated with amusement to help you~~'~~ll benefit from~~ the ~~a lot~~ of ~~offers which are out~~.~~<br><br>It is a nice process. [http~~:~~//En~~.~~Search~~.~~wordpress.com/?q~~=~~Revealing Revealing~~] ~~the appraisement bottomward into parts of time that play faculty to be able to bodies~~ (~~hour/day/week~~) makes doing it accessible to visualize. Everybody knows what needs to be to accept to hold off each day. It's additionally actual accessible to tune. If you change your own apperception after and adjudge that 1 day should bulk more, the contraptions allegation to try along with do is amend 2 benefit.		For all function problems in which the solution is polynomially bounded, there is an analogous decision problem such that the function problem can be solved by [[polynomial-time Turing reduction]] to that decision problem. For example, the decision problem analogue to the travelling salesman problem is this:

			:Given a weighted [[directed graph]] and an integer K, is there a [[Hamiltonian path]] (or [[Hamiltonian cycle]] if the salesman must return home) with total weight less than or equal to K?

			Given a solution to this problem, we can solve the travelling salesman problem as follows. Let <math>n</math> be the number of edges and <math>w_i</math> be the weight of edge <math>i</math>. First rescale and perturb the weights of the edges by assigning to edge <math>i</math> the new weight <math>w'_i = 2^{(n+1)} w_i + 2^i</math>. This doesn't change the shortest Hamiltonian path, but makes sure that it is unique. Now add the weights of all edges to get a total weight <math>M</math>. Find the weight of the shortest Hamiltonian path by [[binary search]]: is there a Hamiltonian path with weight <math>< M/2</math>; is there a path with weight <math>< M/4</math> etc. Then having found the weight <math>W</math> of the shortest Hamilton path, determine which edges are in the path by asking for each edge <math>i</math> whether there is a Hamiltonian path with weight <math>W</math> for the graph modified so that edge <math>i</math> has weight <math>W+1</math> (if there is no such path in the modified graph, then edge <math>i</math> must be in the shortest path for the original graph).

			This places the travelling salesman problem in the complexity class FP<sup>NP</sup> (the class of function problems which can be solved in polynomial time on a deterministic Turing machine with an [[oracle machine\|oracle]] for a problem in NP), and in fact it is [[Complete_(complexity)\|complete]] for that class.

			==References==
			{{refbegin}}
			* Raymond Greenlaw, H. James Hoover, ''Fundamentals of the theory of computation: principles and practice'', Morgan Kaufmann, 1998, ISBN 1-55860-474-X, p. 45-51
			* Elaine Rich, ''Automata, computability and complexity: theory and applications'', Prentice Hall, 2008, ISBN 0-13-228806-0, section 28.10 "The problem classes FP and FNP", pp. 689-694
			{{refend}}

			==See also==
			*[[Decision problem]]
			*[[Search problem]]
			*[[Counting problem (complexity)]]
			*[[Optimization problem]]

			[[Category:Computational problems]]

en>David Eppstein: :Category:Theorems in computational complexity theory

2012-08-28T04:40:25Z

Category:Theorems in computational complexity theory

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