Master Theorem

Notes maitre dhotel Theorem Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Spring 2006 Computer comprehension & technology 235 Introduction to Discrete Mathematics cse235@cse.unl.edu cover Theorem I Notes When analyzing algorithms, think that we b arly care about the asymptotic behavior. Recursive algorithms are no di?erent. Rather than solve exactly the counter recounting associated with the hail of an algorithm, it is enough to give an asymptotic characterization. The main calamus for doing this is the rule theorem. Master Theorem II Notes Theorem (Master Theorem) permit T (n) be a monotonically change magnitude function that satis?es T (n) = aT ( n ) + f (n) b T (1) = c where a ? 1, b ? 2, c > 0. If f (n) ? ?(nd ) ? if ? ?(nd ) ?(nd logarithm n) if T (n) = ? ?(nlogb a ) if where d ? 0, accordingly a < bd a = bd a > bd Master Theorem Pitfalls Notes You cannot use the Master Theorem if T (n) is not monotone, ex: T (n) = sin n f (n) is not a polynomial, ex: T (n) = 2T ( n ) + 2n 2 ? b cannot be expressed as a constant, ex: T (n) = T ( n) Note here, that the Master Theorem does not solve a recurrence relation. Does the base cause bear a concern? Master Theorem Example 1 Notes permit T (n) = T n 2 + 1 n2 + n. What are the parameters? 2 a = 1 b = 2 d = 2 Therefore which nog down? Since 1 < 22 , baptismal font 1 applies.

olibanum we finish that T (n) ? ?(nd ) = ?(n2 ) Master Theorem Example 2 Notes ? Let T (n) = 2T n 4 + n + 42. What are the parameters? a = 2 b = 4 d = 1 2 Therefo re which condition? Since 2 = 4 2 , case 2 a! pplies. Thus we conclude that ? T (n) ? ?(nd log n) = ?( n log n) 1 Master Theorem Example 3 Notes Let T (n) = 3T n 2 + 3 n + 1. What are the parameters? 4 a = 3 b = 2 d = 1 Therefore which condition? Since 3 > 21 , case 3 applies. Thus we conclude that T (n) ? ?(nlogb a ) = ?(nlog2 3 ) Note that log2 3 ? 1.5849 . . .. undersurface we say that T (n) ? ?(n1.5849 ) ? twenty-five percent Condition...If you want to get a full essay, order it on our website: OrderCustomPaper.com

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