Question

I have the following recursion: T(n) = 2*T(n/4) + T(n/2) + n and I need to know the exact equation, I know Master theorem won’t help me, and the iteration seems to be wrong…

Please tell me how to do it in general for such recursions.
Thanks in advance.

Hey all, thanks for replying I need complexity. I need to understand how to solve such problems.

Answer 1

T(n) = O(nlogn) and W(nlogn)

To prove that, by definition of O, we need to find constants n0 and c such that:
for every n>=n0, T(n)<=cnlogn.

We will use induction on n to prove that T(n)<=cnlogn for all n>=n0

Let’s skip the base case for now… (we’ll return later)

Hipothesis: We assume that for every k<n, T(k)<=cklogk

Thesis: We want to prove that T(n)<=cnlogn

But, T(n)=2T(n/4)+T(n/2)+n

Using the hipothesis we get:

T(n)<=2(c(n/4)log(n/4))+c(n/2)log(n/2)+n=cnlogn + n(1-3c/2)

So, taking c>=2/3 would prove our thesis, because then T(n)<=cnlogn

Now we need to prove the base case:

We will take n0=2 because if we take n0=1, the logn would be 0 and that wouldn’t work with our thesis. So our base cases would be n=2,3,4. We need the following propositions to be true:

T(2) <= 2clog2

T(3) <= 3clog3

T(4) <= 4clog4

So, by taking c=max{2/3, T(2)/2, T(3)/3log3, T(4)/8} and n0=2, we would be finding constants c and n0 such that for every natural n>=n0, T(n)<=cnlogn

The demonstration for T(n) = W(nlogn) is analog.

So basically, in these cases where you can’t use the Masther Theorem, you need to ‘guess’ the result and prove it by induction.

For more information on these kind of demonstrations, refer to ‘Introduction to Algorithms’