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I am revising for my algorithms exam and I have come across one topic in particular that I do not quite understand; What I would like to ask, if there is a certain way to find out O-Notation? Actually I know that the purpose of O-Notation, it helps us to estimate the effciency of algorithm. But in a specific questions like O(2^n) * O(n^10) I assumed that O(2^n * n^10) as in the worst case but learned that I was false, it should be 2^n and I don't know why? Shouldn't we think that which term would go faster to eternity, and in this case I thought that when I multiply both then it can go faster than 2^n, can't it? Could anyone explain please ? Just short and with beginner sight:)

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  • $\begingroup$ Yes, your are right that $O(2^n) * O(n^{10}) = O(2^n * n^{10})$. $\endgroup$
    – John L.
    Dec 16 '20 at 21:18
  • $\begingroup$ Our Professor said that O(2^n) must be. Because O(n^10) can be ignored due to very slow growth rate compare to 2^n. $\endgroup$ Dec 16 '20 at 21:32
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    $\begingroup$ This is correct when talking about addition. When talking about multiplication, this kind of thing cannot be ignored $\endgroup$
    – nir shahar
    Dec 16 '20 at 21:42
  • $\begingroup$ I am really confused right now:) I knew that I was right but then checked my classmate and he said also this is 2^n.But anyway, thanks for you all for helping $\endgroup$ Dec 16 '20 at 23:03
  • $\begingroup$ If I have to guess, the question was about some algorithm, which consists of two parts. Since the parts are not nested, their running times should be added, not multiplied. $\endgroup$
    – user114966
    Dec 17 '20 at 12:33
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We can show that $O(2^n\cdot n^{10}) \not\subset O(2^n)$:

As $2^n\cdot n^{10} \in O(2^n\cdot n^{10})$, then, if also holds $2^n\cdot n^{10} \in O(2^n)$ then we will have $2^n\cdot n^{10} \leqslant C 2^n$, which gives false $n^{10} \leqslant C$.

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