Use the recursion tree method to find a closed form solution to T(n) = 4T(n/4) +...

Question

Question

Use the recursion tree method to find a closed form solution to T(n) = 4T(n/4) + n.

engineering Computer-Science

Answer 1

Answer #1

Solution for the problem is provided below, please comment if any doubts:

The recursion tree for the recursion “T(n) = 4T(n/4) + n”. is provided below,

From the recursion tree,

The depth of the tree will be the depth when T(1) become at leaves.
The n is divided by 4 each step, thus depth will be “log₄(n)”.
Total number of leaf nodes at the last level will be = 4 ^log4(n)
The summation work needed to be done at each stage is,
- n, 4*n/4, 16*n/16, …. = n, n, n….. log₄(n) times = n* log₄(n)
Now the total operation to be done = n* log₄(n) + 4 ^log4(n)
In closed form = n* log₄(n) + 4 ^log4(n)= Θ(n log(n))

Thus the closed form solution of the given recursion is Θ(n log(n)).

Answer 2

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