Q: Solving the following recurrence: T(n) = 8T(n/8) + n log n

Question

I am trying to solve the following recurrence:

T(n) = 8T(n/8) + n* log n. 

I currently have done the following but am not sure if I am on the right track:

1. T(n)= 8 T(n/8) + n log n;
2. T(n)= 8^2 T(n/8^2) + n log (n/8) + n log n
3. T(n)= 8^3 T(n/8^3) + n log (n/8^2) + n log (n/8) + n log n

So the general formula came up for me as:

T(n)= 8^k T(n/8^k) + n log(n* n/8 * n/8^2 * ... * n/8^k).

And i am not sure how to continue this. I tried to rewrite the log as n^k / 8^(k*(k+1)/2) , but I still don't see the solution.

Answer 1

You are almost there. Set k = log_8(n) then you can solve for T(n)