Solving $T(n) = 7T(n/2) + n^2 + \log(n)$

Asked Feb 03 '15 at 21:40

Active Feb 03 '15 at 21:40

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I'm working on a problem set for my algorithms course and we're asked to solve the recurrence relation

$$ T(n) = 7T(n/2) + n^2 + \log(n). $$

Since this recurrence relation is not of the form $T(n) = aT(n/b) + n^d$ and therefore the master theorem does not apply, I'm confused about where to begin in solving it.

Could anyone give me an idea of what method(s) might be useful in solving this? I'm flipping through our textbook and don't see anything on solving general recurrence relations where master's theorem is unusable.

asked Feb 03 '15 at 21:40

Ryan

7

You could definitely apply the Master Theorem here: $n^2 + \log n$ is just $O(n^2)$. – Tom van der Zanden Feb 03 '15 at 21:57
The version of the Master Theorem you have seen may be simplified, the wikipedia version is sufficient for this recurrence. – Luke Mathieson Feb 03 '15 at 23:45
@TomvanderZanden Make an answer? – Yuval Filmus Feb 04 '15 at 05:00
1

You can apply the more general Master Theorem or stick to your version in class and notice that $7T(n/2) + n^2 < T(n) < 7T(n/2) + n^{2+\epsilon}$. – hengxin Feb 04 '15 at 05:11

Solving $T(n) = 7T(n/2) + n^2 + \log(n)$

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