Question

(1) (1) (a) (14 pts.) Solve the following recurrence relation with the method of the charac- teristic equation: T(n) = 4T(n/2) + (n/2), for n > 1, n a power of 2 T(1) = 1 Determine the coefficients. (b) (1 PT.) What is the big O) order of the solution as a function of n? (c) (5 PTS.) Verify your solution by substituting back in the recurrence relation. (ii) (10 PTS.) Solve using the method of the characteristic equation to find the big O) order of T(n) if n is a power of 3 and if T(n) = 4T(n/3) + n(n-1) log, n, for n > 1

Answer 1

solation & 1) ů) (a) Given TCU) = 1 & T(D) FEAT(1/2) +(1/2)* Where nolin a power of 2 Let us consider n=k , K31 So, T(n = 4 T(012) +(1/2) T) = @JP+44(7)+()] +16T Ceny 167(9/4) =-() +re[(3)(23) = 3 +64T = *(4/2) + 4* + ( 7 ) The process starts when ak reaches a so, k = login > T(n) = logn (4) + (22 jt qt) - TCĐ) = log(3 +(*) T(N) = logo un? (6) T(n) = 7 logn tor order =0 (or logn )

LC) Veribication : T(0) - n1(2) + (0/2)2 check for na 2 T()) = pr(21+( 21 ) T(2) = NT (1) +iYorp T(5) = (1) 47 - 04 tionis NOW ac@ording to solution usca) T(n) = or logntor TO)- 692 + 2 = 1+4=5 Corzoz

AS PER CHEGG RULES I HAVE DONE ONLY (i) (a),(b) &(c) ANSWER

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