Algorithms - Solving the recurrence $T(n) = \sqrt{n} T \left(\sqrt n \right) + n$

Asked Feb 04 '15 at 01:54

Active Feb 04 '15 at 01:54

Viewed 705 times

I have been trying to solve the recurrence

$T(n) = \sqrt{n} T \left(\sqrt n \right) + n$

for some time now. I only know substitution, recursion trees, and the master method (though it doesn't apply here, so i'd prefer help in those forms. I also attempted to change variables but got nowhere.

My plan was to use a recursion tree, but I seem to be stuck, as this recurrence isn't in the usual form $T(n)= aT(n/b)+f(n)$. However, I do know that the first level of the tree has the cost of $cn$, and that the next level down has $\sqrt n$ nodes. My main concern is how I can express the next level's recurrence, which I got to be $T(\sqrt n)=\sqrt nT(\sqrt(\sqrt n))+\sqrt n$, though I feel like that is wrong.

asked Feb 04 '15 at 01:54

user2789945

Does $T(n)=n\log_2\log_2 n$ work? – Empy2 Feb 04 '15 at 02:01
Hint: Use variable substitution $n=2^m$ to get $T(2^m)=2^{m/2}T(2^{m/2})+2^m$, and set $T'(m)=T(2^m)$ to solve for $T'$. – Mario Carneiro Feb 04 '15 at 02:04
@Michael How did you come up with that? – user2789945 Feb 04 '15 at 02:09
The first log turns $\sqrt{n}$ into $m/2$. The second turns $m/2$ into $k-1$. – Empy2 Feb 04 '15 at 03:09

Algorithms - Solving the recurrence $T(n) = \sqrt{n} T \left(\sqrt n \right) + n$

0 Answers0