# f06-hwex - CS 473U Undergraduate Algorithms Fall 2006...

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CS 473U Homework 0 (due September 1, 2006) Fall 2006 Please put your answers to problems 1 and 2 on the same page. 1. Sort the functions listed below from asymptotically smallest to asymptotically largest, indi- cating ties if there are any. Do not turn in proofs , but you should probably do them anyway, just for practice. To simplify your answers, write f ( n ) g ( n ) to mean f ( n ) = o ( g ( n )) , and write f ( n ) g ( n ) to mean f ( n ) = Θ( g ( n )) . For example, the functions n 2 , n, ( n 2 ) , n 3 could be sorted either as n n 2 ( n 2 ) n 3 or as n ( n 2 ) n 2 n 3 . lg n ln n n n n lg n n 2 2 n n 1 /n n 1+1 / lg n lg 1000 n 2 lg n ( 2) lg n lg 2 n n 2 (1 + 1 n ) n n 1 / 1000 H n H n 2 H n H 2 n F n F n/ 2 lg F n F lg n In case you’ve forgotten: lg n = log 2 n = ln n = log e n lg 3 n = (lg n ) 3 = lg lg lg n . The harmonic numbers: H n = n i =1 1 /i ln n + 0 . 577215 . . . The Fibonacci numbers: F 0 = 0 , F 1 = 1 , F n = F n - 1 + F n - 2 for all n 2 2. Solve the following recurrences. State tight asymptotic bounds for each function in the form Θ( f ( n )) for some recognizable function f ( n ) . Proofs are not required; just give us the list of answers. Don’t turn in proofs , but you should do them anyway, just for practice. Assume reasonable but nontrivial base cases.
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