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Given $T(n) = 9T(n/3) + n^3$,

I know that $a =9$, $b=3$, and $f(n) = n^3$

and $n^{\log_{3}9} = n^2$

thus Case 3 applies: $n^{\log_{b}a} < f(n)$, $n^2 < n^3$.

Can someone explain how to apply the regularity condition and how to check the regularity condition?

$af(n/b) \le cf(n)$ where $c < 1$

I appreciate your help!

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  • $\begingroup$ Apply (equivalent) regularity condition as $a\cdot f(n/b) \leq f(n)$ and retry. $\endgroup$ – Shreesh Sep 9 '20 at 3:40
  • $\begingroup$ Welcome to COMPUTERSCIENCE @SE. I thing you did a good job pinpointing just exactly one position on the way to the solution. $\endgroup$ – greybeard Sep 10 '20 at 7:32
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Your statement of the regularity condition has a typo: it suffices to show that there exists a constant $c < 1$ such that for large enough $n$, $$ af(n/b) \leq cf(n). $$

Substituting $a,b,f$, we have to show that for some constant $c < 1$, $$ 9(n/3)^3 \le c n^3. $$ You take it from here.

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  • $\begingroup$ How did you transform $af(n/b) \le cf(n)$ to $9(n/3)^3 \le cfn^3$> I see the $cf(n)$ but not $9(n/3)^3$ $\endgroup$ – dairyknight86 Sep 9 '20 at 13:28
  • $\begingroup$ Since $a = 9$ and $b = 3$, $af(n/b) = 9f(n/3) = 9(n/3)^3$. $\endgroup$ – Yuval Filmus Sep 9 '20 at 14:32

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