Recurrence relation and Time Complexity closed

Question

What is the time complexity of the following recurrence relation and step to derive the same

$T(n) = T(\sqrt{n}) + log(logn)$

Answer 1

Take $n=2^m$ then

$T(2^m)=T(2^{m/2})+\log m$

$\implies S(m)=S(m/2)+\log m$

Solve using master's theorem .$S(m)=O(\log ^2 m)=T(2^m)$

Put $m=logn \implies T(n)=(\log \log n)^2$

Comments

How did you derive the 3rd step, m = $2^{m}$ is understod but how $2^{m/2}$ written as m/2

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We are not replacing $2^m \;by\; m$  but we are changing function T to S.

$T(2^m)\implies$  function of m.    (S(m))

$T(2^{m/2})\implies$  function of m/2.    (S(m/2))

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can you please tell me $log^2(logn)$ and $(loglogn)^2$both are same??

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Yes..same