HW1AOA - a T n = 1 for n = 4 and T n = √ nT √ n n for...

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CS583 - Homework 1 (Due Tuesday, February 15th) Jana Koˇ seck´ a 1. 2.3-4 To what asymptotically, does the recurrence evaluate ? 2. Consider pairs of functions f 1 and f 2 . For each pairs establish and show whether f 1 = O ( f 2 ), f 1 = Ω( f 2 ), and f 1 = Θ( f 2 ). (It is not enough just to state the result. You need to write down why it is the case.) a) n k , c n b) 2 n , 3 3 / 2 c) n lgn, n d) ( n + 1) 2 , n 2 3. Prove or disprove a) 2 n = O (2 n + 1) b) 2 2 n = O ( n !) c) max ( n 2 , 100 n 3 ) = Ω( n 3 ) d) if f ( n ) = Ω( g ( n )) and g ( n ) = Ω( h ( n )) then h ( n ) = O ( f ( n )) 4. Solve the following recurrence relations by the method of your choice
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Unformatted text preview: a) T ( n ) = 1 for n = 4 and T ( n ) = √ nT ( √ n ) + n for n > 4 b) T ( n ) = aT ( n-1) + bn for n ≥ 2 and T (1) = 1 c) T ( n ) = 1 for n = 1 and T ( n ) = 3 T ( n 2 ) + n log n for n > 1 5. Argue that the solution to the recurrence T ( n ) = T ( n/ 3) + T (2 n/ 3) + cn is Ω( n lg n ) by appealing to the recursion tree. (For more practice and examples, read chapter 4.4 in third edition and 4.2 in second edition of the book)....
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