07 - Fall 2009 CS131 Combinatorial Structures Homework 7...

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Fall 2009 CS131 – Combinatorial Structures Homework 7 Homework 7, due Nov 3 You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check ex- ercises of the easier kind in the book, try to solve them, too! Problem 1 (CLRS 3.1-4) . (a) (5pts) Is 2 n + 1 = O (2 n )? Solution. 2 n + 1 = O (2 n ) since it is 2 · 2 n . (b) (5pts) Is 2 2 n = O (2 n )? Solution. The function 2 2 n is not O (2 n ) since 2 2 n /2 n = 2 n →∞ as n →∞ . Problem 2. For the following pairs of functions, show which one grows faster and why. (a) (5pts) 4 log n or n 2 ? Solution. They are equal, since 4 log n = n log4 = n 2 . (b) (5pts) ( f ( n /2)) 3 or ( f ( n /3)) 2 where f ( n ) = 2 n ? Solution. ( f ( n /2)) 3 = 2 n · (3/2) ± 2 n · (2/3) = ( f ( n /3)) 2 . (c) (5pt) 1 + 3 + 3 2 +···+ 3 n or 3 n + 1 + 3 n + 2 +···+ 3 2 n ? Solution. We have 1 + 3 + 3 2 +···+ 3 n ² 3 n + 1 + 3 n + 2 +···+ 3 2 n . Indeed, a geometric series has the same rate of growth as its largest term, so 1
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This note was uploaded on 10/15/2011 for the course MATHS 100 taught by Professor Fredphelps during the Fall '11 term at Jordan University of Science & Tech.

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07 - Fall 2009 CS131 Combinatorial Structures Homework 7...

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