解决递归问题T（n）= 4T（n / 4）+ 3log n

Question

I'm really getting frustrated about solving the Recurrence above. I was trying to solve it by using the Master Method, but I just didn't get it done...

I'm having a recursive algorithm that takes 3log n times (three binary searches) to identify four sub-problems, each with with a size of n/4, and then solves them individually until n is smaller than some constant given by input. So I got this recurrence as a result:

T(n) = 4*T(n/4) + 3*log(n)

Base-Case if n < c (c = some constant given by program input):

T(n) = 1

I'm trying to find the asymptotic running time of my recursive program, and wanted to solve it by using the master theorem. Can anybody tell me if it's possible to use the master theorem with this recurrence, and if yes, which case of the master theorem is it?

All help is appreciated, thanks.

Answer 1

T(n) = O(n) , because a logarithm of 4 base 4 is 1 and 3 * log(n) is O(n ^ 0.5) (0.5 < 1). It corresponds to the first case of the Master theorem as described here .