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midterm2007-fall

# midterm2007-fall - Data Structures and Algorithms(I...

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Data Structures and Algorithms (I) close-book midterm exam November 30, 2007 You may answer the questions in any order. Dishonest behaviors and attempts will be punished most seriously. When you are asked to justify, prove, or disprove your answers, you may directly use anything that we have shown in class in a “black-box” manner. Problem 1 (20 points) Let f ( n ) and g ( n ) be two positive functions. We say that f ( n ) = O ( g ( n )) if there exist two positive constants c and n 0 such that f ( n ) c · g ( n ) holds for any n n 0 . Let us define a “new” asymptotic notation ξ by saying that f ( n ) = ξ ( g ( n )) if f ( n ) = O ( g ( n )) and g ( n ) = O ( f ( n )) . Prove or disprove the statement that f ( n ) = ξ ( g ( n )) implies f ( n ) log 2 g ( n ) f ( n ) = ξ ( g ( n )) . Problem 2 (15 points) Prove the following theorem. Let a 1 and b > 1 be constants. Let f ( n ) be a positive function. Let T ( n ) be a positive monotonically increasing function defined on nonnegative integers by the recurrence T ( n ) = 0 if n = 0 1 if n = 1 a · T ( n b ) + f ( n ) if n > 1 .

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