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| In [[computational complexity theory|computational complexity]] and [[communication complexity]] theories the '''decision tree model''' is the [[model of computation]] or [[communication]] in which an algorithm or communication process is considered to be basically a [[decision tree]], i.e., a sequence of branching operations based on comparisons of some quantities, the comparisons being assigned the unit computational cost.
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. Word - Press also provides protection against spamming, as security is a measure issue. After activating, you will find their website link and get the activation code from their website. <br><br>Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. This plugin allows a blogger get more Facebook fans on the related fan page. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. <br><br>In the event you loved this article and you would want to receive more information about [http://xyz.ms/wordpress_dropbox_backup_379550 backup plugin] please visit our own web site. Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Our skilled expertise, skillfulness and excellence have been well known all across the world. Article Source: Stevens works in Internet and Network Marketing. <br><br>The disadvantage is it requires a considerable amount of time to set every thing up. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. I don't want that for my visitors and I'm quite sure they don't either. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. You can check out the statistics of page of views for your web pages using free tools that are available on the internet. |
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| The branching operations are called "tests" or "queries". In this setting the algorithm in question may be viewed as a computation of a [[Boolean function]] <math>f: \{0,1\}^n \rightarrow \{0,1\}</math> where the input is a series of queries and the output is the final decision. Every query is dependent on previous queries.
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| Several variants of decision tree models may be considered, depending on the complexity of the operations allowed in the computation of a single comparison and the way of branching.
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| Decision trees models are instrumental in establishing [[lower bound]]s for [[computational complexity]] for certain classes of computational problems and algorithms: the lower bound for [[worst-case analysis|worst-case computational complexity]] is proportional to the largest depth among the decision trees for all possible inputs for a given computational problem. The computation complexity of a problem or an algorithm expressed in terms of the decision tree model is called '''decision tree complexity''' or '''query complexity'''.
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| ==Classification by query computational complexity==
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| ===Simple decision tree ===
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| The model in which every decision is based on the comparison of two numbers within constant time is called simply a decision tree model. It was introduced to establish computational complexity of [[sorting]] and searching.<ref>"Data structures and algorithms, by [[Alfred V. Aho]], [[John E. Hopcroft]], [[Jeffrey D. Ullman]]</ref> | |
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| The simplest illustration of this lower bound technique is for the problem of finding the smallest number among ''n'' numbers using only comparisons. In this case the decision tree model is a [[binary tree]]. Algorithms for this searching problem may result in ''n'' different outcomes (since any of the ''n'' given numbers may turn out to be the smallest one). It is known that the depth of a binary tree with ''n'' [[Leaf node|leaves]] is at least <math>\log n</math>, which gives a lower bound of <math>\Omega(\log n)</math> for the searching problem. However this lower bound is known to be slack, since the following simple argument shows that at least ''n'' - 1 comparisons are needed: Before the smallest number can be determined, every number except the smallest must "lose" (compare greater) in at least one comparison.
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| Along the same lines the lower bound of <math>\Omega(n \log n)</math> for [[sorting]] may be proved. In this case, the existence of numerous comparison-sorting algorithms having this time complexity, such as [[mergesort]] and [[heapsort]], demonstrates that the bound is tight.
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| ===Linear decision tree ===
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| Linear decision trees, just like the simple decision trees, make a branching decision based on a set of values as input. As opposed to binary decision trees, linear decision trees have three output branches. A linear function <math>f(x_1, \dots, x_i)</math> is being tested and branching decisions are made based on the sign of the function (negative, positive, or 0).
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| Geometrically, <math>f(x)</math> defines a hyperplane in '''R'''<sup>''n''</sup>. A set of input values reaching any particular nodes represents the intersection of the half-planes defined by the nodes.
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| ===Algebraic decision tree===
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| Algebraic decision trees are a generalization of linear decision trees to allow test functions to be polynomials of degree ''d''. Geometrically, the space is divided into semi-algebraic sets (a generalization of hyperplane). The evaluation of the complexity is more difficult.
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| ==Classification by query computational model ==
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| ===Deterministic decision tree===
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| If the output of a decision tree is <math>f(x)</math>, for all <math>x\in \{0,1\}^n </math>, the decision tree is said to "compute" <math>f</math>. The depth of a tree is the maximum number of queries that can happen before a leaf is reached and a result obtained. '''<math>D(f)</math>''', the '''deterministic decision tree''' complexity of <math>f</math> is the smallest depth among all deterministic decision trees that compute <math>f</math>.
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| ===Randomized decision tree===
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| One way to define a '''randomized decision tree''' is to add additional nodes to the tree, each controlled by a probability <math>p_i</math>. Another equivalent definition is to select a whole decision tree at the beginning from a set of decision trees based on a probability distribution. Based on this second definition, the complexity of the randomized tree is defined as the greatest depth among all the trees associated with probabilities greater than 0. '''<math>R_2(f)</math>''' is defined as the complexity of the lowest-depth randomized decision tree whose result is <math>f(x)</math> with probability at least <math>2/3</math> for all <math>x\in \{0,1\}^n </math> (i.e., with bounded 2-sided error).
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| '''<math>R_2(f)</math>''' is known as the [[Monte Carlo algorithm|Monte Carlo]] randomized decision-tree complexity, because the result is allowed to be incorrect with bounded two-sided error. The [[Las Vegas algorithm|Las Vegas]] decision-tree complexity '''<math>R_0(f)</math>''' measures the ''expected'' depth of a decision tree that must be correct (i.e., has zero-error). There is also a one-sided bounded-error version known as '''<math>R_1(f)</math>'''.
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| ===Nondeterministic decision tree===
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| The nondeterministic decision tree complexity of a function is known more commonly as the [[Certificate (complexity)|certificate complexity]] of that function. It measures the number of input bits that a nondeterministic algorithm would need to look at in order to evaluate the function with certainty. | |
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| ===Quantum decision tree===
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| The quantum decision tree complexity '''<math>Q_2(f)</math>''' is the depth of the lowest-depth quantum decision tree that gives the result <math>f(x)</math> with probability at least <math>2/3</math> for all <math>x\in \{0,1\}^n </math>. Another quantity, '''<math>Q_E(f)</math>''', is defined as the depth of the lowest-depth quantum decision tree that gives the result <math>f(x)</math> with probability 1 in all cases (i.e. computes <math>f</math> exactly). <math>Q_2(f)</math> and <math>Q_E(f)</math> are more commonly known as '''quantum query complexities''', because the direct definition of a quantum decision tree is more complicated than in the classical case. Similar to the randomized case, we define <math>Q_0(f)</math> and <math>Q_1(f)</math>.
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| ==Relationship between different models==
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| It follows immediately from the definitions that for all <math>n</math>-bit Boolean functions <math>f</math>,
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| <math>Q_2(f) \leq R_2(f) \leq R_1(f) \leq R_0(f) \leq D(f) \leq n</math>, and <math>Q_2(f) \leq Q_E(f) \leq D(f) \leq n</math>.
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| Blum and Impagliazzo,<ref name="BlumImpagliazzo 1995">{{cite conference | last = Blum | first=M. | coauthors=Impagliazzo, R. | title=Generic oracles and oracle classes | year=1987 | booktitle=Proceedings of 18th IEEE FOCS | pages=118–126}}</ref> Hartmanis and Hemachandra,<ref name="HartmanisHemachandra">{{Citation | last1=Hartmanis | first1=J. | last2 = Hemachandra | first2=L. | year = 1987 | contribution=One-way functions, robustness, and non-isomorphism of NP-complete sets | title=Technical Report DCS TR86-796, Cornell University}}</ref> and Tardos<ref name="Tardos">{{cite journal | last=Tardos | first=G. | title=Query complexity, or why is it difficult to separate ''NP''<sup>''A''</sup> ∩ ''coNP''<sup>''A''</sup> from ''P''<sup>''A''</sup> by random oracles ''A''? | journal=Combinatorica | year=1989 | pages=385–392|volume=9 | doi=10.1007/BF02125350}}</ref> independently discovered that <math>D(f) \leq R_0(f)^2</math>. [[Noam Nisan]] found that the Monte Carlo randomized decision tree complexity is also polynomially related to deterministic decision tree complexity: <math>D(f) = O(R_2(f)^3)</math>.<ref name="Nisan">{{cite conference | last=Nisan | first=N. | authorlink = Noam Nisan | title=CREW PRAMs and decision trees | year=1989 | booktitle=Proceedings of 21st ACM STOC | pages=327–335}}</ref> (Nisan also showed that <math>D(f) = O(R_1(f)^2)</math>.) A corollary of this result is that <math>R_0(f) = O(R_2(f)^3)</math>. This inequality may be loose, however; no example is known of even a super-linear separation between <math>R_0(f)</math> and <math>R_2(f)</math>.<ref name="Santha 1995">{{Citation | last = Santha | first=Miklos | title=On the Monte Carlo Boolean Decision Tree Complexity of Read-Once Formulae | year=1995 |url=http://www.lri.fr/~santha/Papers/s95.ps.gz}} (ps format)</ref>
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| The quantum decision tree complexity <math>Q_2(f)</math> is also polynomially related to <math>D(f)</math>. Midrijanis showed that <math>D(f) = O(Q_E(f)^3)</math>,<ref name="Midrijanis">{{Citation | last=Midrijanis | first=Gatis | year = 2004 | contribution= Exact quantum query complexity for total Boolean functions | title = arXiv:quant-ph/0403168 | arxiv=quant-ph/0403168}}</ref><ref name="Midrijanis2">{{Citation | last=Midrijanis | first=Gatis | year = 2005 | contribution= On randomized and quantum query complexities | title = arXiv:quant-ph/0501142 | arxiv=quant-ph/0501142}}</ref> improving a quartic bound due to Beals et al.<ref name="Beals">{{cite journal | last=Beals | first=R. | coauthors=Buhrman, H.; Cleve, R.; Mosca, M.; de Wolf, R. | year = 2001 | title= Exact quantum query complexity for total Boolean functions | journal =Journal of ACM | volume=47 | pages=778–797}}</ref> Beals et al. also showed that <math>D(f) = O(Q_2(f)^6)</math>, and this is still the best known bound. However, the largest known gap between deterministic and quantum query complexities is only quadratic. A quadratic gap is achieved for the [[Grover's algorithm|OR function]]; <math>D(OR_n) = n</math> while <math>Q_2(OR_n) = \Theta(\sqrt{n})</math>.
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| It is important to note that these polynomial relationships are valid only for ''total'' Boolean functions. For [[Total function|partial Boolean functions]], that have a domain a subset of <math>\{0,1\}^n</math>, an exponential separation between <math>Q_E(f)</math> and <math>D(f)</math> is possible; the first example of such a problem was discovered by [[Deutsch-Jozsa algorithm|Deutsch and Jozsa]]. The same example also gives an exponential separation between <math>R_2(f)</math> and <math>D(f)</math>.
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| These relationships can be summarized by the following inequalities, which are true up to constant factors:<ref name="Midrijanis2" />
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| *<math>D(f) \leq R_0(f)^2</math><ref name="BlumImpagliazzo 1995" /><ref name="HartmanisHemachandra" /><ref name="Tardos" />
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| *<math>D(f)\leq R_1(f)R_2(f)</math><ref name="Nisan" />
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| *<math>D(f) \leq R_2(f)^3</math><ref name="Nisan" />
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| *<math>R_0(f) \leq R_2(f)^2 \log N</math><ref name="Midrijanis2" />
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| *<math>D(f) \leq Q_2(f)^6</math><ref name="Beals" />
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| *<math>D(f) \leq Q_E(f)^2Q_2(f)^2</math><ref name="Beals" />
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| *<math>D(f) \leq Q_1(f)^2Q_2(f)^2</math><ref>H. Buhrman, R. Cleve, R. de Wolf, and Ch. Zalka. Bounds for Small-Error and Zero-Error Quantum Algorithms. In 40th IEEE Symposium on Foundations of Computer Science (FOCS'99), pp.358-368. cs.CC/9904019, 1999.</ref>
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| *<math>R_0(f) \leq Q_1(f)Q_2(f)^2 \log N</math><ref>S. Aaronson. Quantum Certificate Complexity. IEEE Conference on Computational Complexity, pp. 171-178, 2003.</ref>
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| *<math>D(f) \leq Q_1(f)Q_2(f)^2</math><ref name="Midrijanis2" />
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| ==References==
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| {{reflist}}
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| ===Surveys===
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| * {{Citation
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| |last1 = Buhrman
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| |first1=Harry
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| |last2=de Wolf
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| |first2=Ronald
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| |title=Complexity Measures and Decision Tree Complexity: A Survey
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| |journal=Theoretical Computer Science
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| |volume=288
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| |number=1
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| |pages=21–43
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| |year=2002
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| |doi=10.1016/S0304-3975(01)00144-X
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| |url=http://homepages.cwi.nl/~rdewolf/publ/qc/dectree.pdf}}
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| {{DEFAULTSORT:Decision Tree Model}}
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| [[Category:Computational complexity theory]]
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| [[Category:Models of computation]]
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| [[Category:Decision trees]]
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