MIT6_042JF10_assn08

MIT6_042JF10_assn08 - 6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk Problem Set 8 Problem 1[25 points Find bounds

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6.042/18.062J Mathematics for Computer Science October 28, 2010 Tom Leighton and Marten van Dijk Problem Set 8 Problem 1. [25 points] Find Θ bounds for the following divide-and-conquer recurrences. Assume T (1) = 1 in all cases. Show your work. (a) [5pts] T ( n ) = 8 T ( n/ 2 ± ) + n (b) [5pts] T ( n ) = 2 T ( n/ 8 ± + 1 /n ) + n (c) [5pts] T ( n ) = 7 T ( n/ 20 ± ) + 2 T ( n/ 8 ± ) + n (d) [5pts] T ( n ) = 2 T ( n/ 4 ± + 1) + n 1 / 2 (e) [5pts] T ( n ) = 3 T ( n/ 9 + n 1 / 9 ) + 1 Problem 2. [30 points] It is easy to misuse induction when working with asymptotic notation. False Claim If T (1) = 1 and T ( n ) = 4 T ( n/ 2) + n Then T(n) = O(n). False Proof We show this by induction. Let P ( n ) be the proposition that T ( n ) = O ( n ). Base Case : P (1) is true because T (1) = 1 = O (1). Inductive Case : For n 1, assume that P ( n 1) ,...,P (1) are true. We then have that T ( n ) = 4 T ( n/ 2) + n = 4 O ( n/ 2) + n = O ( n ) And we are done. (a)
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This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.

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MIT6_042JF10_assn08 - 6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk Problem Set 8 Problem 1[25 points Find bounds

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