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PracticeProblems

# PracticeProblems - UML CS 91.404 Practice Problems and...

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Unformatted text preview: UML CS 91.404 Practice Problems and Solutions 1. (20 points) What can you conclude? Given: a) (10 points) Can we conclude from statements (1)-(4) that Why or why not? Either prove or provide a counterexample. SOLUTION: The statement is FALSE. Counterexample: n n f n n f lg lg ) ( ) ( 3 2 = = 1 of 6 Ω ∈ 3 2 1 ) ( ) 2 n n f ) 7 lg lg 3 ( ) ( ) 3 3- ∈ n O n f ? )) ( ( ) ( 3 2 n f O n f ∈ Θ ∈ n n f 4 log 1 16 1 ) ( ) 1 ) 7 lg lg 3 ( ) ( ) 4 4- Ω ∈ n n f 3 1 n 7 lg lg 3- n n 4 log 16 1 f 2 (n) f 4 (n) f 3 (n) f 1 (n) UML CS 91.404 b) (10 points) Can we conclude from statements (1)-(4) that Why or why not? Either prove or provide a counterexample. SOLUTION: The statement is TRUE. Proof: First, observe that: Now, and n n 4 log 16 1 7 lg lg 3 Ω ∈- transitivity of Ω yields: n n f 4 log 4 16 1 ) ( Ω ∈ Applying the definition of Θ and using transpose symmetry yields: Applying transitivity of Ω again yields: )) ( ( ) ( 1 4 n f n f Ω ∈ 2 of 6 ? )) ( ( ) ( 1 4 n f n f Ω ∈ )) ( ( 16 1 16 1 ) ( 16 1 ) ( 1 log log 1 log 1 4 4 4 n f O n f n f n n n Ω ∈ ⇒ ∈ ⇒ Θ ∈ ) 7 lg lg 3 ( ) ( 4- Ω ∈ n n f 2 log log 2 log 2 log 1 4 1 4 1 4 1 16 1 2 4 4 4 4 n n n n n = = = = UML CS 91.404 2. (20 points) Recurrence In this problem, you will find a tight upper bound on the closed-form solution for the following recurrence: = + = 1 1 1 log 7 5 ) ( 7 n n n n T n T That is, find the smallest function ) ( n f that you can, such that )) ( ( ) ( n f O n T ∈ . You may assume that n is a power of 7....
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PracticeProblems - UML CS 91.404 Practice Problems and...

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