Question

Question 6 (20 points) Solve the following recurrences using the Master Theorem. T(n) = 2T (3/4)+1 T(n) = 2T (n/4) + va 7(n) = 2T (n/4) +n T(n) = 2T (3/4) + n

Answer 1

Master theorem is a way to solve recurrences.It works for the recurrences which are in the form:

T(n) = x * T(n/y) + f(n) where x >= 1 and y > 1

There are following three cases:
1. If f(n) = Θ(n^c) where c < Log_y(X) then T(n) = Θ(n^Logy(x))

2. If f(n) = Θ(n^c) where c = Log_y(X) then T(n) = Θ(n^cLog n)

3.If f(n) = Θ(n^c) where c > Log_y(X) then T(n) = Θ(f(n))

The solution for the above questions is as follows:

In Tin) = 274" /u) +! comparing the above quafion using the master Theorem with the equation : Ting Ianly) + f(n) [i:2 01 Sicce fin):1-nºs Hose celoguing this using case + I(): 0(n.9,") - 06.109.) - 0(K) = 86) 12) 719) 2ten14) + ñ comparing the above equation using the master theorem with the equation :- 10): x + I(19) + f (0) e since rcm)Snant olm) 189= 109. 1.5 Meme Cozlogy this using case 2 7(n) = Orniugn) = O(n+10ga) - O(In ingin)

13). Tin) = 2T (nu) tn comparing the above equation using the master thoosen with the equation: rin: X * Ilnly) + fin) Since fin) = n =n's ocm) x=2 y: 4 2 - st 1 < C 4 logo logi - 444 Hene c> 109, thens using case 3 Tin) = OFFEN)) = O(n) lu) T(N= 21 (nu) un comparing the above equation using the master theorem using the equation: In : x* tanly) + f(n) x=2 1 y=4 Since Finanon) logga 1ogle = < Hent c> logo this using case 3 Ten) = 0 (f(n)) = O(n)