1. Assignment 1: due 1/13/16 Geometric sum Prove by induction on integers that

2. Assignment 2: due 1/20/16 Prove by induction on integers that recurrence T(2 k ) = 2 if k=1 T(2 k ) = 2T(2 k-1 ) + 2 k if k>1 has solution T(2 k ) = k2 k Base case k=1 true by definition. Show that also true for k=2.

3. Assignment 3: due 1/22/16 1)Ex 3.1-1 text p52 if f(n) and g(n) are asymptotically non-negative, show that max(f(n),g(n))=  (f(n)+g(n)) hints: how are f(n) and g(n) related to max(f(n),g(n))? how is max(f(n),g(n)) related to f(n)+g(n)? 2) Use Stirling’s approximation (Eq(3.18) text p57) to give an informal proof that lg(n!)=  (nlgn). Show all steps.

4. Cpt S 350 Spring 2016 [All problems are from Cormen et al, 3 rd Edition] Homework Assignment 4: due 1/25/16 1. Use Stirling’ approximation, Eq(3.18) p 57, to show that n! = o(n n ) and n! =  (2 n ) hint: use limits as n goes to infinity 2. prob 3-2 p 61 justify your answers to a through f Example: A=n k, B=c n, k>0, c>1, is A = O(B), o(B),  (B),  (B),  (B), or none of these and why?

5. CptS 350 Spring 2016 [All problems are from Cormen et al, 3nd Edition] Homework Assignment 5: due 2/5/16 1.ex A.1-3 p 1149 hint: use derviatives 2.ex A.1-6 p 1149 hint: let g k (n)=O(f k (n)) and use definition of Big O

6. CptS 350 Spring 2016 [All problems are from Cormen et al, 3nd Edition] Homework Assignment 6: due 2/8/16 1.ex A.2-1 p 1156 (hint: use integration) 2.Find upper and lower bounds on using bound each term

7. Assignment 7: due 2/10/16 Prove by substitution that T(n)=2T(n/2)+  (n) has T(n)=  (nlgn) as an asymptotic solution. Prove by substitution that T(n)=8T(n/2)+  (n 2 ) has T(n) =  (n 3 ) as an asymptotic solution. Prove by substitution that T(n) = 3T(n/4)+  (n 2 ) has T(n) =  (n 2 ) as an asymptotic solution. Show constraints on c and n.

8. CptS 350 Spring 2016 [All problems are from Cormen et al, 3 rd Edition] Homework Assignment 8: due 2/12/16 ex 4.4-7 on page 93 (tree analysis and substitution) Remember! Tight bound means 

9. CptS 350 Spring 2016 [All problems are from Cormen et al, 3 rd Edition] Homework Assignment 9: due 2/24/16 1.Show by substitution that T(n)=T(floor(n/3))+T(ceiling(2n/3)) +  (n) has asymptotic solution T(n)=  (nlgn) 2. problem 4-3f on page 108 by tree analysis and substitution method for tight bounds.

10. CptS 350 Spring 2016 [All problems are from Cormen et al, 3 rd Edition] Homework Assignment 10: due 2/26/16 Show by substitution that T(n)=T(n-2)+  (n 2 ) has asymptotic solution T(n)=  (n 3 )

11. Cpt S 350 Spring 2016 [All problems are from Cormen et al, 3rd Edition] Homework Assignment 11: due 3/4/16 Problems 4-1a, c and e on page 107 by master method