Is Master's theorem applicable on $T(n) = 2 T(\frac{n}{2})+n\log n$ ?

I got this doubt from here:


  • 2
    $\begingroup$ Wikipedia has a good description of the master theorem: en.wikipedia.org/wiki/Master_theorem_(analysis_of_algorithms). I suggest ignoring all other online sources. $\endgroup$ Sep 17, 2018 at 7:01
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    $\begingroup$ Unregistered users cannot see the contents of the website. Could you please quote relevant part here? $\endgroup$
    – xskxzr
    Sep 17, 2018 at 8:57

1 Answer 1


Yes, Master's theorem is applicable to equations of type:

$$T(n) = aT(\frac{n}{b}) + \Theta(n^k log^pn)$$

where $a \geq 1$, $b \gt 1$, $k \geq 0$ and for some real number p.

This is slightly modified and we can apply it more easily. The results are as follows:

  1. if $a \gt b^k$, then $$T(n) = \Theta(n^{log_ba})$$

  2. if $a = b^k$, then

    a) if $p > -1$, $$T(n) = \Theta(n^{log_ba} log^{p+1}n)$$

    b) if $p = -1$, $$T(n) = \Theta(n^{log_ba} loglogn)$$

    c) if $p < -1$, $$T(n) = \Theta(n^{log_ba})$$

  3. if $a < b^k$, then

    a) if $p \geq 0$, $$T(n) = \Theta(n^klog^pn)$$

    b) if $p < 0$, $$T(n) = O(n^k)$$


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