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| {{Refimprove|date=October 2009}}
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| In [[computational complexity theory]], '''DSPACE''' or '''SPACE''' is the [[computational resource]] describing the resource of [[memory space]] for a [[deterministic Turing machine]]. It represents the total amount of memory space that a "normal" physical computer would need to solve a given [[computational problem]] with a given [[algorithm]]. It is one of the most well-studied complexity measures, because it corresponds so closely to an important real-world resource: the amount of physical [[Computer storage|computer memory]] needed to run a given program.
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| ==Complexity classes==
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| The measure '''DSPACE''' is used to define [[complexity class]]es, sets of all of the [[decision problem]]s which can be solved using a certain amount of memory space. For each function ''f''(''n''), there is a [[complexity class]] '''SPACE(''f''(''n''))''', the set of [[decision problem]]s which can be solved by a [[deterministic Turing machine]] using space ''O''(''f''(''n'')). There is no restriction on the amount of [[computation time]] which can be used, though there may be restrictions on some other complexity measures (like [[Alternation (complexity)|alternation]]).
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| Several important complexity classes are defined in terms of '''DSPACE'''. These include:
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| * '''[[regular language|REG]]''' = '''DSPACE'''(''O''(1)), where '''REG''' is the class of [[regular language]]s. In fact, '''REG''' = '''DSPACE'''(''o''(log log ''n'')) (that is, Ω(log log ''n'') space is required to recognize any nonregular language).
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| * '''[[L (complexity)|L]]''' = '''DSPACE'''(''O''(log ''n''))
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| * '''[[PSPACE]]''' = <math>\bigcup_{k\in\mathbb{N}} \mbox{DSPACE}(n^k)</math>
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| * '''[[EXPSPACE]]''' = <math>\bigcup_{k\in\mathbb{N}} \mbox{DSPACE}(2^{n^k})</math>
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| ==Machine models==
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| '''DSPACE''' is traditionally measured on a [[deterministic Turing machine]]. Several important space complexity classes are [[sublinear]], that is, smaller than the size of the input. Thus, "charging" the algorithm for the size of the input, or for the size of the output, would not truly capture the memory space used. This is solved by defining the [[multi-string Turing machine with input and output]], which is a standard multi-tape Turing machine, except that the input tape may never be written-to, and the output tape may never be read from. This allows smaller space classes, such as [[L (complexity)|L]] (logarithmic space), to be defined in terms of the amount of space used by all of the work tapes (excluding the special input and output tapes).
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| Since many symbols might be packed into one by taking a suitable power of the alphabet, for all ''c'' ≥ 1 and ''f'' such that ''f''(''n'') ≥ ''1'', the class of languages recognizable in ''c f''(''n'') space is the same as the class of languages recognizable in ''f''(''n'') space. This justifies usage of [[big O notation]] in the definition.
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| ==Hierarchy Theorem==
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| The [[space hierarchy theorem]] shows that, for every [[space-constructible function]] <math>f: \mathbb{N} \to \mathbb{N}</math>, there exists some language L which is decidable in space <math>O(f(n))</math> but not in space <math>o(f(n))</math>. See '''[[space hierarchy theorem]]''' for details.
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| ==Relation with other complexity classes==
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| '''DSPACE''' is the deterministic counterpart of '''[[NSPACE]]''', the class of [[memory space]] on a [[nondeterministic Turing machine]]. By [[Savitch's theorem]],<ref name=AB86>Arora & Barak (2009) p.86</ref> we have that
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| <blockquote>
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| <math>\mbox{DSPACE}[s(n)] \subseteq \mbox{NSPACE}[s(n)] \subseteq \mbox{DSPACE}[(s(n))^2].</math>
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| </blockquote>
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| '''[[NTIME]]''' is related to DSPACE in the following way. For any [[time constructible]] function ''t''(''n''), we have
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| :<math>\mbox{NTIME}(t(n)) \subseteq \mbox{DSPACE}(t(n))</math>. | |
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| ==References==
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| {{reflist}}
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| * {{cite book | zbl=1193.68112 | last1=Arora | first1=Sanjeev | authorlink1=Sanjeev Arora | last2=Barak | first2=Boaz | title=Computational complexity. A modern approach | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-42426-4 }}
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| ==External links==
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| {{ComplexityZoo|DSPACE(''f''(''n''))|D#dspace}}.
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| {{ComplexityClasses}}
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| {{DEFAULTSORT:Dspace}}
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| [[Category:Computational resources]]
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| [[Category:Complexity classes]]
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I'm Mickie and I live in Totnes Valley.
I'm interested in Hotel Administration, Sailing and Arabic art. I like travelling and watching Breaking Bad.
Feel free to visit my page; 4Inkjets 15 off coupon code