Master Theorem

Notes maitre dhotel Theorem Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry form 2006 Computer comprehension & engineer 235 Introduction to Discrete Mathematics cse235@cse.unl.edu repress Theorem I Notes When analyzing algorithms, recall that we b arely sustentation about the asymptotic behavior. algorithmic algorithms are no di?erent. earlier than solve exactly the comeback relation associated with the hail of an algorithm, it is seemly to give an asymptotic characterization. The primary(prenominal) tool for doing this is the rule theorem. cut through Theorem II Notes Theorem ( eclipse Theorem) permit T (n) be a mo nononically 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 recordarithm n) if T (n) = ? ?(nlogb a ) if where d ? 0, whence a < bd a = bd a > bd  pilot 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 issue relation. Does the base faux pas substantiate a concern? Master Theorem theoretical account 1 Notes Let T (n) = T n 2 + 1 n2 + n. What are the parameters? 2 a = 1 b = 2 d = 2 then which peg bolt down? Since 1 < 22 , case 1 applies.

consequently we final stage 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 thence which condition? Since 2 = 4 2 , case 2 applies. 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 ) ? Fourth Condition...If you want to issue forth a full essay, rate it on our website: Orderessay

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