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4.3-1 Use the substitution method to show that each of the following recurrences defined on the reals has the asymptotic solution specified: a. T(n) = T(n − 1) + n has solution T(n) = O(n²). b. T(n) = T(n/2) + (1) has solution T(n) = O(lgn). c. T(n) = 2T (n/2) + n has solution T(n) = (n lgn).

4.3-1 Use the substitution method to show that each of the following recurrences defined on the reals has the asymptotic solution specified: a. T(n) = T(n − 1) + n has solution T(n) = O(n²). b. T(n) = T(n/2) + (1) has solution T(n) = O(lgn). c. T(n) = 2T (n/2) + n has solution T(n) = (n lgn).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**4.3-1** Use the substitution method to show that each of the following recurrences defined on the reals has the asymptotic solution specified: a. \( T(n) = T(n-1) + n \) has solution \( T(n) = O(n^2) \). b. \( T(n) = T(n/2) + \Theta(1) \) has solution \( T(n) = O(\log n) \). c. \( T(n) = 2T(n/2) + n \) has solution \( T(n) = \Theta(n \log n) \). d. \( T(n) = 2T(n/2 + 17) + n \) has solution \( T(n) = O(n \log n) \). e. \( T(n) = 2T(n/3) + \Theta(n) \) has solution \( T(n) = \Theta(n) \). f. \( T(n) = 4T(n/2) + \Theta(n) \) has solution \( T(n) = \Theta(n^2) \).
Transcribed Image Text:**4.3-1** Use the substitution method to show that each of the following recurrences defined on the reals has the asymptotic solution specified: a. \( T(n) = T(n-1) + n \) has solution \( T(n) = O(n^2) \). b. \( T(n) = T(n/2) + \Theta(1) \) has solution \( T(n) = O(\log n) \). c. \( T(n) = 2T(n/2) + n \) has solution \( T(n) = \Theta(n \log n) \). d. \( T(n) = 2T(n/2 + 17) + n \) has solution \( T(n) = O(n \log n) \). e. \( T(n) = 2T(n/3) + \Theta(n) \) has solution \( T(n) = \Theta(n) \). f. \( T(n) = 4T(n/2) + \Theta(n) \) has solution \( T(n) = \Theta(n^2) \).
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