# HW1 - points 6 Use a recursion tree to give an...

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Fall 2011 - CPS 130 Assignment I: Asymptotes and Recurrences Department of Computer Science - Duke University Due on Tuesday 20 th Sep in Class 1 Explain why the statement, ‘The running time of algorithm A is at least O ( n 2 ) is mean- ingless. (5 points) 2 Prove n ! = ω (2 n ) and n ! = o ( n n ). (10 points) 3 Is the function d log( n ) e ! polynomially bounded? Prove your claim. (15 points) 4 Which is asymptotically larger: log(log * n ) or log * (log n )? Prove your claim. (20 points) 5 Solve the recurrence T ( n ) = 2 T ( n ) + 1 by making a change of variables. Your solu- tion should be asymptotically tight. Do not worry about whether values are integral. (10
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Unformatted text preview: points) 6 Use a recursion tree to give an asymptotically tight solution to the recurrence T ( n ) = T ( n-a ) + T ( a ) + cn , where a ≤ 1 and c > 0 are constants. Express your answer in terms of T ( a ). (10 points) 7 Use the master method to give tight asymptotic bounds for the following recurrences (15 points) a. T ( n ) = 4 T ( n/ 2) + n b. T ( n ) = 4 T ( n/ 2) + n 2 c. T ( n ) = 4 T ( n/ 2) + n 3 8 Use the master method to show that the solution to the binary-search recurrence T ( n ) = T ( n/ 2) + Θ(1) is T ( n ) = Θ(log n ). (15 points) 1...
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## This document was uploaded on 01/17/2012.

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