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final-practice

# final-practice - Introduction to Algorithms Massachusetts...

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Introduction to Algorithms May 10, 2005 Massachusetts Institute of Technology 6.046J/18.410J Professors Charles E. Leiserson and Ronald L. Rivest Practice Final Exam—From Fall 2004 Practice Final Exam—From Fall 2004 Problem 1. Recurrences (4 parts) [8 points] For each of the recurrences below, do the following: Give the solution using Θ -notation. You need not provide a proof or other justification. Name a recursive algorithm we’ve seen during the term whose running time is described by that recurrence. (a) T ( n ) = T ( n/ 2) + Θ(1) (b) T ( n ) = 2 T ( n/ 2) + Θ( n ) (c) T ( n ) = T ( n/ 5) + T (7 n/ 10) + Θ( n ) (d) T ( n ) = 7 T ( n/ 2) + Θ( n 2 ) Problem 2. Design Techniques and Data Structures (5 parts) [10 points] For each of the following design techniques and data structures, name an algorithm covered this term that uses it. (a) Divide and conquer: (b) Dynamic programming: (c) Greedy: (d) Binary search tree: (e) FIFO queue: Problem 3. True or False, and Justify (12 parts) [84 points] Circle T or F for each of the following statements to indicate whether the statement is true or false, respectively. If the statement is correct, briefly state why. If the statement is wrong, explain why. The more content you provide in your justification, the higher your grade, but be brief. Your justification is worth more points than your true-or-false designation. (a) T F If f ( n ) is asymptotically positive, then f ( n ) + o ( f ( n )) = Θ( f ( n )) . (b) T F An adversary can provide an input to randomized quicksort that will elicit 1 its Θ( n 2 ) worst-case running time.

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