Question

2. Suppose that, in a divide-and-conquer algorithm, we always divide an instance of size n of a problem into 5 sub-instances of size n/3, and the dividing and combining steps take a time in O'nun). Write a recurrence equation for the running time T(n), and solve the equation for T(n). Your solution should only involve integers and the following constants: log, 5, 73 (hint: other constants that might appear are 1,5 and 3). Your answer should be in the form T(n)= n—_T(1)+ndn( _ _)+n/n (_ where the elements in the blanks are constants. Your goal is to find the power of n in the first term, the constant in parentheses in the second term, and the constant in parentheses as well as the exponent in the third term. Use the “tear-down” approach to solve this recurrence. You must show your work in order to receive credit for your answer. (10 pts)

Suppose that, in a divide-and-conquer algorithm, we always divide an instance of size n of a problem into 5 sub-instances of size n/3, and the dividing and combining steps take a time

in Θ(n n). Write a recurrence equation for the running time T (n) , and solve the

equation for T (n)

Answer 1

TO Second replace ment Per la (n)=5[571432) tagjai 19)=571%29751961974 = 5(34173) *C71383 + 011/33973% - Toh) = 57 (193) + 5 1/2 77 $123 geroago, 1930) + 5* (19)* 56919- 57 (3) 7 n 445 +57(3): 44[4 ] 7/2 ² 0.9622 <1; & ratio <li

> T()- 5442) +23/2017 Decreasing Geeometric peries] = 5(3x) + nova [at] approx overall Bummation of Decreasing Geometrie series. Tin = 577 (2 k) + 232 + 3/2 na gk take logoz both sides- logn = logezk 1x = logg go Tin) = 51839T (L) + 15 (1) + nun (4) Tin) - nº I1) + n/h (9tn/ (1)