chap3-solutions - Selected Solutions for Chapter 3 Growth...

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Selected Solutions for Chapter 3: Growth of Functions Solution to Exercise 3.1-2 To show that .n C a/ b D ‚.n b / , we want to find constants c 1 ; c 2 ; n 0 > 0 such that 0 ± c 1 n b ± .n C a/ b ± c 2 n b for all n ² n 0 . Note that n C a ± n C j a j ± 2n when j a j ± n , and n C a ² n N j a j ² 1 2 n when j a j ± 1 2 n . Thus, when n ² 2 j a j , 0 ± 1 2 n ± n C a ± 2n : Since b > 0 , the inequality still holds when all parts are raised to the power b : 0 ± ± 1 2 n ² b ± .n C a/ b ± .2n/ b ; 0 ± ± 1 2 ² b n b ± .n C a/ b ± 2 b n b : Thus, c 1 D .1=2/ b , c 2 D 2 b , and n 0 D 2 j a j satisfy the definition. Solution to Exercise 3.1-3 Let the running time be T .n/ . T.n/ ² O.n 2 / means that T .n/ ² f .n/ for some function f .n/ in the set O.n 2 / . This statement holds for any running time T.n/ , since the function g.n/ D 0 for all n is in O.n 2 / , and running times are always nonnegative. Thus, the statement tells us nothing about the running time.
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3-2 Selected Solutions for Chapter 3: Growth of Functions Solution to Exercise 3.1-4 2 n C 1 D O.2 n / , but 2
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This note was uploaded on 04/20/2010 for the course IE ie200 taught by Professor . during the Spring '10 term at 카이스트, 한국과학기술원.

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chap3-solutions - Selected Solutions for Chapter 3 Growth...

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