## Master Theorem Examples Solved Assignment

I'm trying to solve the recurrence relation: $$T(n) = T(\sqrt n) + \Theta(\lg \lg n)$$

My first step was to let $m = \lg n$, making the above: $$T(2^m) = T(2^{m\cdot 1/2}) + \Theta(\lg m)$$

If $S(m) = T(2^m)$, then $$S(m) = S(m/2) + \Theta(\lg m)$$

This is an easier recurrence to solve. If I try and use the Master Theorem, I calculate $n^{\log_b a}$ where $a=1$ and $b=2$ to be $n^0=1$. It seems like this might be case 2 of the Master Theorem where $f(n)= \Theta(n^{\log_b a})$.

For $S(m)$, $\Theta(n^{\log_b a})= \Theta(1)$. But $f(m) = \lg m$. Therefore $f(n) \neq \Theta(n^{\log_b a})$. So it doesn't seem case 2 applies.

If I use case 1 or 3 of the Master Theorem, I have to be sure that $f(n)$ is polynomially smaller or larger than $n^{log_b{a}}$, that is to say $f(n)/n^{\log_b a} \le n^\varepsilon$ for some $\varepsilon > 0$. However, $\lg m/1$ does not meet this requirement either.

There's a solution posted to this problem on the MIT OpenCourseWare website that claims that you may use case-2 of the Master's Theorem. It's

(http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-046j-introduction-to-algorithms-sma-5503-fall-2005/assignments/ps1sol.pdf) as problem 1-2d.

I don't see how the Master Theorem applies to the recurrence $S(m)$. In fact I'm not comfortable with this definition of "polynomially" larger, since traditionally polynomials must have integer exponents. If anyone could shed some light on the matter it would be greatly appreciated!

Thanks!

algorithmsasymptoticsrecurrence-relationsrecursive-algorithms

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