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# hwk3 - remaining elements(a Write down the recursive...

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Fall 2011 CMSC 351: Homework 3 Clyde Kruskal Due at the start of class Wednesday, September 28, 2011. Problem 1. (a) Use the iteration method to solve the following recurrence exactly: T ( n ) = 5 T ( n/ 2) + 2 n - 1 where T (1) = 7. Assume n is a power of 2. (b) Justify your answer using mathematical induction. (c) Solve the same recurrence using the “Master Theorem”. Problem 2. (a) Use the iteration method to solve the following recurrence exactly: T ( n ) = T ( n - 1) + 2 n - 1 where n is a natural number and T (1) = 1. (b) Justify your answer using mathematical induction. (c) State in English what you have proved. Make your statement as short, clean, clear, and accurate as possible. Problem 3. Bubble Sort can be thought of as a recursive algorithm as follows: Bubble the largest element to the bottom of the list (to be sorted). Recursively sort the remaining elements.

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Unformatted text preview: remaining elements. (a) Write down the recursive version of Bubble Sort in psuedocode. (b) Derive a recurrence for the exact number of comparisons the algorithm uses. (c) Solve the recurrence (any way you like). Simplify as much as possible. “Master Theorem”: T ( n ) = b aT ( n/b ) + cn d n > 1 f n = 1 implies T ( n ) = p f + c ab-d-1 P n log b a-cn d ab-d-1 = b Θ( n log b a ) a > b d Θ( n d ) a < b d n d ( f + c log b n ) = Θ( n d log b n ) a = b d . Summing solutions: If T ( n ) = b aT ( n/b ) + ∑ c i n d i n > 1 f n = 1 then we can just sum the solutions of each recurrence: T i ( n ) = b aT i ( n/b ) + c i n d i n > 1 n = 1 and add in fn log b a for the contribution from the leaves. 2...
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hwk3 - remaining elements(a Write down the recursive...

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