prob1 - CS38 Spring 2009 Introduction to Algorithms Prof...

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CS38, Spring 2009 Introduction to Algorithms Prof. Alexei Kitaev Problem set #1 Due Friday April 10, 2009 at 5pm in the box in the VLSI lab. Problems 1. Exercises: a) (4 points) Sort the following functions in order of their asymptotic growth (i.e., the O -relation means “ ”, the Ω-relation means ‘ ”, and Θ means “=”: n, n log n, (log 2 n ) n , (ln n ) n , n ln n , 2 n , n ! , 2 n n , ln( n !) , ln 2 n n . You may use the Stirling approximation: n ! 2 πn ( n/e ) n . b) (3 points) Solve the recurrence T ( n ) = 3 T ( n/ 2) + O ( n 2 ). c) (3 points) Solve the recurrence T ( n ) = T ( n/ 3) + T (2 n/ 3) + O ( n ). 2. CLRS problem 2-2 (Correctness of bubblesort): a) (1 point) b) (4 points) c) (4 points) d) (1 point) 3. Let a sequence S = ( x 1 , . . . , x n ) be sorted. A corresponding sequence A = ( x f (1) , . . . , x f ( n ) ) is called “ k -almost sorted” if | f ( i ) i | ≤ k for all i = 1 , . . . , n . a) (10 points) Find an algorithm that sorts a k -almost sorted sequence in time O ( n log k ). You don’t need to reason using pseudocode; just give a higher-level correctness proof. Hint: Use merge-sort as a subroutine in your algorithm. b) (5 points) Prove the lower bound Ω( n log k ). Hint: Use the standard lower bound Ω( n log n
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