en>JoeSlote: Made wording clearer in intro

2014-12-12T22:54:17Z

Made wording clearer in intro

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inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

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	~~{{no footnotes\|date=February 2013}}~~		In front of enjoying a brand fresh clash of clans hack tool, see the hack book. In the event you loved this article and you wish to receive more details concerning [http://prometeu.net hack clash of clans] please visit our own web page. Most games possess a book would like buy individually. For you to think about doing this guidance and studying it a person begin play, or even while you are playing. In doing this manner, you can be getting the most out of your gameplay.<br><br>
	In ~~[[computer science]]~~, the ~~'''Akra–Bazzi method''', or '''Akra–Bazzi theorem''', is used~~ to ~~analyze the asymptotic behavior of the mathematical~~ [~~[recurrence relation\|recurrences]] that appear in the analysis~~ of ~~[[divide and conquer algorithm\|divide and conquer~~]~~] [[algorithm]]s where the sub-problems have substantially different sizes~~. ~~It is~~ a ~~generalization of the well-known [[master theorem]]~~, ~~which assumes that~~ the ~~sub-problems have equal size~~.

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	~~The Akra–Bazzi method applies~~ to ~~recurrence formulas of~~ the ~~form~~

	~~:<math>T(x)=g(x) + \sum_{i=1}^k a_i T(b_i x + h_i(x))\qquad \text{for }x \geq x_0.</math>~~

	~~The conditions for usage are:~~

	* sufficient base cases are provided
	* <math>a_i</math> and ~~<math>b_i</math> are constants for all i~~
	* <math>a_i > 0</math> for all i
	* <math>0 < b_i < 1</math> for all i
	* <math>\left\|g(x)\right\| \in O(x^c)</math>, where ''~~c'' is~~ a ~~constant and ''O'' notates~~ [~~[Big O notation]]~~
	* <math>\left\| h_i(x) \right\| \in O\left(\frac{x}{(\log x)^2}\right)</~~math> for all i~~
	* <math>x_0</~~math> is a constant~~

	~~The asymptotic behavior of T(x) is found by determining the value of p for which <math>\sum_{i~~=~~1}^k a_i b_i^~~p = ~~1</math> and plugging that value into the equation~~

	~~:<math>T(x) \~~in ~~\Theta \left( x^p\left( 1+\int_1^x \frac{g(u)}{u^{p+1}}du \right)\right)</math>~~

	~~(see [[Big O notation\|Θ]]). Intuitively~~, ~~<math>h_i(x)</math> represents~~ a small ~~perturbation in~~ the ~~index of T~~. ~~By noting that~~ <~~math~~>~~\lfloor b_i x \rfloor = b_i x + (\lfloor b_i x \rfloor - b_i x)</math> and that <math>\lfloor b_i x \rfloor - b_i x~~<~~/math~~> ~~is always between 0 and 1~~, ~~<math>h_i(x)</math> can be used to ignore~~ the ~~[[floor function]]~~ in ~~the index~~. ~~Similarly~~, ~~one can also ignore~~ the ~~[[ceiling function]]~~. ~~For example, <math>T(n) = n + T \left(\frac{1}{2} n \right)</math>~~ and ~~<math>T(n) = n + T \left(\left\lfloor \frac{1}{2} n \right\rfloor \right)</math> will,~~ as ~~per the Akra–Bazzi theorem~~, ~~have~~ the ~~same asymptotic behavior~~.

	~~== An example ==~~
	~~Suppose~~ <~~math~~>~~T(n)~~<~~/math~~> ~~is defined as 1 for integers <math>0 \leq n \leq 3<~~/~~math>~~ and ~~<math>n^2 + \frac{7}{4} T \left( \left\lfloor \frac{1}{2} n \right\rfloor \right) + T \left( \left\lceil \frac{3}{4} n \right\rceil \right)</math> for integers <math>n > 3</math>~~. ~~In applying the Akra–Bazzi method~~, ~~the first step is~~ to ~~find the value~~ of ~~p for which <math>\frac{7}{4} \left(\frac{1}{2}\right)^p + \left(\frac{3}{4} \right)^p = 1</math>. In~~ this ~~example, ''p'' = 2~~. ~~Then, using the formula, the asymptotic behavior can be determined as follows:~~

	:<~~math~~>
	~~\begin{align}~~
	~~T(x) & \in \Theta \left( x^p\left( 1+\int_1^x \frac{g(u)}{u^{p+1}}\,du \right)\right) \\~~
	~~& = \Theta \left( x^2 \left( 1+\int_1^x \frac{u^2}{u^3}\,du \right)\right) \\~~
	~~& = \Theta(x^2(1 + \ln x)) \\~~
	~~& = \Theta(x^2 \log x).~~
	~~\end{align}~~
	<~~/math~~>

	~~== Significance ==~~
	~~The Akra–Bazzi method~~ is ~~more useful than most other techniques for determining asymptotic behavior~~ because ~~it covers such~~ a ~~wide variety of cases. Its primary application is the approximation of the [[Run time (program lifecycle phase)\|runtime]] of many divide~~-~~and~~-~~conquer algorithms~~. ~~For example, in~~ the ~~[[merge sort]]~~, ~~the number~~ of ~~comparisons required in the worst case~~, ~~which is roughly proportional to its runtime~~, ~~is given recursively as~~ <~~math~~>~~T(1) = 0~~<~~/math~~> and

	~~:<math>T(n) = T\left(\left\lfloor \frac{1}{2} n \right\rfloor \right) + T\left(\left\lceil \frac{1}{2} n \right\rceil \right) + n~~ - ~~1</math>~~

	~~for integers <math>n > 0</math>, and~~ can ~~thus be computed using~~ the ~~Akra–Bazzi method~~ to be <~~math~~>~~\Theta(n \log n)~~<~~/math~~>.

	~~== References ==~~
	*Mohamad Akra, ~~Louay Bazzi: On the solution~~ of ~~linear recurrence equations. ''Computational Optimization and Applications'' '''10'''(~~2~~):195–210, 1998~~.
	*Tom Leighton: [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.1636 Notes on Better Master Theorems for ~~Divide~~-~~and~~-~~Conquer Recurrences], Manuscript. Massachusetts Institute of Technology, 1996, 9 pages.~~
	*[http://www.mpi-~~inf~~.~~mpg~~.~~de/~mehlhorn/DatAlg2008/NewMasterTheorem~~.~~pdf Proof and application on few examples]~~

	~~{{DEFAULTSORT:Akra-Bazzi Method}}~~
	~~[[Category:Asymptotic analysis]]~~
	~~[[Category:Theorems in discrete mathematics]]~~
	~~[[Category:Recurrence relations]]~~
	~~[[Category:Bazzi family]]~~

193.136.24.142: /* Definition */

2014-01-08T10:37:45Z

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134.100.29.29: /* See also */

2012-06-20T11:17:46Z

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en>JoeSlote: Made wording clearer in intro

en>Wavelength: inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

193.136.24.142: /* Definition */

134.100.29.29: /* See also */