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Unformatted text preview: Design and Analysis of Computer Algorithms Midterm Exam Solution 13:1015:00 pm Monday, April 14, 2008 Name: ID #: This is a Close Book examination. Only an A4 cheating sheet belonging to you is acceptable. You can write your answers in English or Chinese. Maximum Score Problem 1 24 24 Problem 2 16 16 Problem 3 10 10 Problem 4 10 10 Problem 5 15 15 Problem 6 15 15 Problem 7 10 10 Total 100 100 1 1. (24 pts) Please consider the following equalities. Mark by T(=true) or F(=False) each of the following statements. You don’t need to prove it. (1) ∑ n i =1 √ i = O ( n 3 2 ) (2) n n = O (2 n ) (3) n ! = O ( n n ) (4) n 2 log n = Θ( n 2 ) (5) n 2 / log n = Θ( n 2 ) (6) 33 n 3 + 4 n 2 = Ω( n 2 ) Question Answer (1) T (2) F (3) T (4) F (5) F (6) T 2. (16 pts) Suppose A is a problem with a wellknown lower bound Ω( T ( n )) and B is a problem of which the lower bound is unknown. However, we can find a O( n )time reduction from A to B . Mark by T(=true) or F(=False) each of the following statements: (1) If B can be solved in O( n 2 ), then A can be solved in O( n 2 ). (2) If A can be solved in O( n 2 ), then B can be solved in O( n 2 ). (3) If T ( n ) = log n , then B has an Ω(log n ) time lower bound. (4) If B has an Ω(log n ) time lower bound, then T ( n ) = log n . Question Answer (1) T (2) F (3) F (4) F 2 3. (10 pts) Use a recursion tree to determine a good asymptotic upper bound on the recurrence T ( n ) = T ( n 3 ) + T ( 2 n 3 ) + O ( n ) . Furthermore, use the substitution method to verify your answer. Sol: The recursion tree is in Figure 1. Summing up all the terms on the righthand side of the tree, we can guess c ( n /3) log 3/2 n c (2 n /3) cn cn c n Total: O( n log n ) c ( n /9) c (2 n /9) c (2 n /9) c (4 n /9) Figure 1: The recursion tree for the recurrence relation in problem 4.Figure 1: The recursion tree for the recurrence relation in problem 4....
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This note was uploaded on 05/11/2010 for the course COMPUTER S 301 taught by Professor . during the Spring '10 term at Kadir Has Üniversitesi.
 Spring '10
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 Algorithms

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