Question

Given Equation: T(n) = T(n/4) + T(n/2) + n^2

Model of Tree:

              T(n)                 -- Level 1
             /    \
       T(n/4)       T(n/2)         -- Level 2
        /   \       /    \
 T(n/16)  *T(n/8) T(n/4)  *T(n/8)  -- Level 3

From Lecture of MIT Algorithm Class: http://www.youtube.com/watch?v=whjt_N9uYFI

Minute: 38:53

Question: How, What and Why 3rd level becomes n/8 ? What is the explicit equation to create recursion tree?

This isn’t the homework question by the way.

Answer 1

The original tree is this:

  T(n)    =   n^2  ->  T(n/4)
                   ->  T(n/2)

When you expand the first branch, you make a substitution n = n/4 so you get:

  T(n/4)  =   (n/4)^2  ->  T((n/4)/4)
                       ->  T((n/4)/2)

          =   (n/4)^2  ->  T(n/16)
                       ->  T(n/8)

and similarly for the second branch, n = n/2

  T(n/2)  =   (n/2)^2  ->  T(n/8)
                       ->  T(n/4)

so substituting these expansions back into T(n) you get

  T(n)    =   (n^2)    ->  (n/4)^2   ->  T(n/16)
                                     ->  T(n/8)
                       ->  (n/2)^2   ->  T(n/8)
                                     ->  T(n/4)