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Unformatted text preview: University of Toronto ECE345: Algorithms and Data Structures Solutions to Midterm Examination (Fall 2008) 1. (a) False. 2 3 n +1 = 2 3 · 3 n = (2 3 ) 3 n = 8 3 n , which is clearly not O (2 3 n ) (compare the limits as n → ∞ ). (b) True. Take the limit of n 2 / n 3 lg n as n → ∞ . We get lg n n , which tends to 0. Therefore, n 2 = O ( n 3 lg n ). (c) Base case: n = 1 holds easily. Induction Hypothesis: Assume statement holds for n . Induction Step: Consider statement for n +1. The LHS of the statement becomes equal to ( 1) n +1 parenleftBig n ( n +1) 2 parenrightBig + ( 1) n +2 ( n +1) 2 by Induction Hypothesis. This is equal to ( 1) n +2 ( k ( k +1) 2 +( k +1) 2 ) = ( 1) n +2 ( ( k +1)( k +2) 2 ), which is the LHS of the statement for n + 1. Thus, the statement holds for n + 1. (d) 65 / \ 22 77 / \ 5 37 / 2 \ 4 (e) Master theorem doesn’t apply here. Draw recursion tree. At each level, do Θ( n ) work. Number of levels on one side is log 5 / 4 n = Θ(lg n ), on the other side is log...
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This note was uploaded on 01/29/2012 for the course ECE 345 taught by Professor Veneris during the Fall '10 term at University of Toronto.
 Fall '10
 Veneris
 Algorithms, Data Structures

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