Assignment_01_soluti - CS1102S Data Structures and Algorithms Assignment 01 Algorithm Analysis Solution 1 Exercise 2.1 on page 50 Order the

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CS1102S Data Structures and Algorithms Assignment 01: Algorithm Analysis – Solution 1. Exercise 2.1 on page 50: Order the following functions by growth rate: N , N , N 1 . 5 , N 2 , N log N , N log log N , N log 2 N , N log( N 2 ), 2 /N , 2 N , 2 N/ 2 , 37, N 2 log N , N 3 . Indicate which functions grow at the same rate and show why this is the case. Answer: 2 /N < 37 < N < N < N log log N < N log N N log( N 2 ) < N log 2 N < N 1 . 5 < N 2 < N 2 log N < N 3 < 2 N/ 2 < 2 N (1) The only two functions that grow at the same rate are N log N and N log( N 2 ): N log( N 2 ) = 2 N log N = Θ( N log N ) (2) For all other functions, the ordering is strict. In particular the following functions do not grow at the same rate: 2 N/ 2 n = Θ(2 n ) , as: lim N →∞ 2 N/ 2 2 N = lim N →∞ 2 N/ 2 2 N/ 2 2 N/ 2 = lim N →∞ 1 2 N/ 2 = 0 (3) N log 2 N = N [log N ] 2 n = N log log N (4) N 1 . 5 n = Θ( N log 2 N ) , as lim N →∞ N 1 . 5 N log
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This note was uploaded on 06/09/2011 for the course CS 11025 taught by Professor Nil during the Spring '11 term at National University of Singapore.

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Assignment_01_soluti - CS1102S Data Structures and Algorithms Assignment 01 Algorithm Analysis Solution 1 Exercise 2.1 on page 50 Order the

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