Question

3. Solve the follwoing recurrences using the master method. (a) T(n) = 4T (n/2) + navn. (8 pt) (b) T(n) = 2T (n/4) + n. (8 pt) (c) T(n) = 7T(n/2) +n?. (8 pt)

Answer 1

Masters theorem for Dividing functions. TCN) - QTCN/b) + fch) a>,! f(n) = 0 (sk login) bol case 1: if log ask then TC) = 0 Cologo) case 2% if tog a ak if p>-1 och togeth if par o C nk loglogn) if pa-1 ocak) case 3: if loga <k af P70 o Cnt login if p<0 ocak

5. O @ T(n) = 4 T (7/2) + un T(h) = 4T()/2) + 6 login a=4, b=2, K-5, P=0 log a = log 4 = 2 loga <k case 3 - log, ack and p=0 so TCh) = OCH login) o cost login) och focus

(5) TC) - 2TC/G) + b TCH) = 2T (1/4) + 2 loger a=2, b=4., K= ½, P=0. log, a = 10g 2 = 0.5 K=0.5 case ar log a = k and p>-1 so TCh) = och logeth) TCn) = o Chh2 log tih) o (un logn)

© Th) = ат(nl) ) the Тcn) = атау, ) th? log°, а: 2, :2, К-2, P: 0 log a. 109,7 - 2- 80 93 .. ( 109 a >k case : teg g>К. Тcy) = 0 ,9) - ес хоя) |