mid1solnsFall2010

mid1solnsFall2010 - CMPS 101 Algorithms and Abstract Data...

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CMPS 101 Algorithms and Abstract Data Types Fall 2010 Midterm Exam 1 Solutions 1. (20 Points) Prove that )) ( ) ( ( )) ( ( )) ( ( n g n f O n g O n f O = . In other words, if )) ( ( ) ( 1 n f O n h = and )) ( ( ) ( 2 n g O n h = , then )) ( ) ( ( ) ( ) ( 2 1 n g n f O n h n h = . Proof: Assume )) ( ( ) ( 1 n f O n h = and )) ( ( ) ( 2 n g O n h = . Then there exist positive constants 2 1 2 1 , , , n n c c such that for all 1 n n : ) ( ) ( 0 1 1 n f c n h , and for all 2 n n : ) ( ) ( 0 2 2 n g c n h . Let ) , max( 2 1 0 n n n = . Then both inequalities hold when 0 n n . Multiply the two inequalities to get ) ( ) ( ) ( ) ( 0 2 1 2 1 n g n f c c n h n h . Let 2 1 c c c = . Observe that c , and 0 n are positive, and ) ( ) ( ) ( ) ( 0 2 1 n g n cf n h n h for all 0 n n , showing that )) ( ) ( ( ) ( ) ( 2 1 n g n f O n h n h = , as required. /// 2. (20 Points) Use Stirling’s formula, to prove that ( ) n n n log ) ! log( Θ = . Proof:

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This note was uploaded on 11/10/2010 for the course COMPUTER S 101 taught by Professor Patricktantalo during the Fall '10 term at University of California, Santa Cruz.

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mid1solnsFall2010 - CMPS 101 Algorithms and Abstract Data...

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