running_time_analysis

running_time_analysis - COMP 271H Design and Analysis of...

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COMP 271H Design and Analysis of Algorithms 2006 Fall Semester Reference material 1 1. Introduction: This article focuses on the comparsions among three fundamental methods which are commonly used in running time analysis. We use a simple recurrence equation to illustrate the idea. We list out the pros and cons of using different methods, and finally conclude the best method we should use in our exam-type questions. 2. Comparsions among three fundamental methods: In general, there are three fundamental methods for solving recurrences. All three meth- ods are listed in our textbook, chapter 4.1 to 4.3. They are: (a) The substitution method (b) The recurison-tree method (c) The master method We will use a simple recurrence equation, which is similar to merge-sort running time analysis. The recurrence equation is as follow: T(1) = 1 T( n ) = 2 T( n 2 ) + n Method 1: The subsitution method Let n = 2 k , k = log n , we have T(1) = 1 T(2 k ) = 2 T(2 k - 1 ) + 2 k Let G ( k ) = T (2 k ), we have G(0) = 1 G(k) = 2 G ( k - 1) + 2 k We now simpify G(k), G ( k ) = 2 G ( k - 1) + 2 k = 2 [ 2 G ( k - 2) + 2 k - 1
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