# s04-hwex - CS 373U Combinatorial Algorithms Spring 2004...

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CS 373U Homework 0 (due January 28, 2004) Spring 2004 1. Sort the functions in each box from asymptotically smallest to asymptotically largest, indi- cating ties if there are any. You do not need to turn in proofs (in fact, please don’t turn in proofs), but you should do them anyway, just for practice. Don’t merge the lists together. 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 . (a) 2 lg n 2 lg n 2 lg n 2 lg n lg 2 n lg 2 n lg 2 n lg 2 n lg n 2 lg n 2 lg n 2 lg n 2 lg 2 n lg 2 n lg 2 n lg n 2 (b) lg( n !) lg( n !) lg( n !) (lg n )! ( lg n )! (lg n )! [Hint: Use Stirling’s approximation for factorials: n ! n n +1 / 2 /e n ] 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. You do not need to turn in proofs (in fact, please don’t turn in proofs), but you should do them anyway, just for practice. Assume reasonable but nontrivial base cases. If your solution requires specific base cases, state them! Extra credit will be awarded for more exact solutions.
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