Question

Discrete mathematics question

Can you please answe the following question?

Please show your answer clearly.

et n be a positive integer. Use the Master Theorem to obtain the big-O class for the functions that satisfy the following recurrences. (a) (4 points) g(n) -4g(n/2)+ n b) (4 points) (n) 2f (n/3) 0(n)

Answer 1

Master Theorem
T(n) = aT(n/b) + f(n)
If f(n) = Θ(n^c) where c < Logb(a) then T(n) = Θ(n^Logb(a))
If f(n) = Θ(n^c) where c = Logb(a) then T(n) = Θ((n^c)(Log(n)))
If f(n) = Θ(n^c) where c > Logb(a) then T(n) = Θ(f(n))

a)
g(n) = 4g(n/2)+n^2
So, from the definition of masters theorem
a=4, b=2 and f(n)=n^2 then c=2
Logb(a)=Log2(4)=2
Where Logb(a)=c
So From the above values we can say that, 
So, T(n) = Θ((n^c)(Log(n)))
T(n) = Θ((n^2)(Log(n)))

b)
f(n) = 2g(n/3)+O(n)
So, from the definition of masters theorem
a=2, b=3 and f(n)=O(n) then c=1
Logb(a)=Log3(2)=0.63092975357146
where c > Logb(a)
So From the above values we can say that, 
So, T(n) = Θ(f(n))
T(n) = Θ(n)

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