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Unformatted text preview: CMPS 101 Summer 2009 Homework Assignment 3
1. (1 Point) The last exercise in the handout entitled Some Common Functions. 4n 2n Use Stirling's formula to prove that = Θ n n . 2. (2 Points) (Exercise 1 from the induction handout) n(n + 1) ∑ i 3 = 2 . Do this twice: i =1 a. (1 Point) using form IIa of the induction step. b. (1 Point) using form IIb of the induction step. Prove that for all n ≥ 1 : n 2 3. (1 Point) Exercise 2 from the induction handout) Define S (n) for n ∈ Z + by the recurrence:
0 if n = 1 S ( n) = S (n / 2) + 1 if n ≥ 2 Prove that S (n) ≥ lg( n) for all n ≥ 1 , and hence S (n) = Ω(lg n) . 4. (2 Points) a. (1 Point) Let f (n) be a positive, increasing function that satisfies f (n / 2) = Θ( f (n)) . Show that ∑ f (i) = Θ(nf (n)) .
i =1 n (Hint: follow the Example on page 4 of the handout on asymptotic growth rates in which it is proved that ∑i
i =1 n k = Θ(n k +1 ) for any positive integer k.) b. (1 Point) Use the result in part (a) to deduce that log(n!) = Θ(n log(n)) . 5. (1 Point) Let T (n) be defined by the recurrence formula
1 T ( n) = 2 T ( n / 2) + n n =1 n≥2 42 n , and hence T (n) = O (n 2 ) . (Hint: follow Example 3 on page 3 of the 3 handout on induction proofs.) Show that ∀n ≥ 1 : T (n) ≤ 1 ...
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This note was uploaded on 12/03/2009 for the course CS CS101 taught by Professor Agoreback during the Spring '09 term at American College of Gastroenterology.
- Spring '09