Question

*algorithm analysis and design*

Solve the following recurrence relation T(n) = Tỉn/2) + 1 Using: 1-Recurrence Tree. 2-Master Therom.

Answer 1

`Hey,

Note: If you have any queries related the answer please do comment. I would be very happy to resolve all your queries.

a)

10022 ... | ビーニー # ollos

Just like the picture above shows.

For a recurrence T(n)=aT(n/b)+f(n), first you should draw a a-ary tree. In this problema=1a=1, so we draw a 1-ary tree.

Then we calculate the depth of the tree, which is logbn

After that, we calculate the contribution of every level.

So, it is theta(log2(n))

b)

Your recurrence is

T(n) = T(n / 2) + O(1)

Since the Master Theorem works with recurrences of the form

T(n) = aT(n / b) + n^c

In this case you have

• a = 1
• b = 2
• c = 0

Since c = log_ba (since 0 = log₂ 1), you are in case two of the Master Theorem, which solves to Θ(n^c log n) = Θ(n⁰ log n) = Θ(log n).

Kindly revert for any queries

Thanks.