f05-hwex - CS 473G Combinatorial Algorithms Fall 2005...

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CS 473G Homework 0 (due September 1, 2005) Fall 2005 1. Solve the following recurrences. State tight asymptotic bounds for each function in the form Θ( f ( n )) for some recognizable function f ( n ). 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! (a) A ( n ) = 2 A ( n/ 4) + n (b) B ( n ) = max n/ 3 <k< 2 n/ 3 ( B ( k ) + B ( n - k ) + n ) (c) C ( n ) = 3 C ( n/ 3) + n/ lg n (d) D ( n ) = 3 D ( n - 1) - 3 D ( n - 2) + D ( n - 3) (e) E ( n ) = E ( n - 1) 3 E ( n - 2) [Hint: This is easy!] (f) F ( n ) = F ( n - 2) + 2 /n (g) G ( n ) = 2 G ( ( n + 3) / 4 - 5 n/ lg n + 6 lg lg n ) + 7 8 n - 9 - lg 10 n/ lg lg n + 11 lg * n - 12 (h) H ( n ) = 4 H ( n/ 2) - 4 H ( n/ 4) + 1 [Hint: Careful!] (i) I ( n ) = I ( n/ 2) + I ( n/ 4) + I ( n/ 8) + I ( n/ 12) + I ( n/ 24) + n (j) J ( n ) = 2 n · J ( n ) + n [Hint: First solve the secondary recurrence j ( n ) = 1 + j ( n ) .] 2. Penn and Teller agree to play the following game. Penn shuffles a standard deck 1 of playing cards so that every permutation is equally likely. Then Teller draws cards from the deck, one at a time without replacement, until he draws the three of clubs (3
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