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Suppose $f_1(n)=n\log^*n$,$f_2(n)=n\log h$ that $h$ is number of vertices of convex hull.

Can we conclude that $$f_1+f_2=O(f_2)?$$ Edited:

Note that, $h$ is a function accroding to $n$ that $h\leq n$, and according to the input $n$ it can be less than $n$ such as constant.

$n\log h$ is running time of an ouput sensetive algorithm for finding convex hull.

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  • $\begingroup$ You need info about $h=h(n)$. Different cases implies different answers. $\endgroup$
    – zkutch
    Oct 30, 2021 at 9:35

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$h$ is the number of vertices of what convex hull?

If $h$ is a constant, then $f_2 = \mathcal{O}(f_1)$ because $\log^*n\underset{n\rightarrow +\infty}{\longrightarrow}+\infty$.

If I try to interpret more than you wrote, and consider $h$ to be the number of vertices of the convex hull of a set of $n$ points, then you could still have $f_2 = \mathcal{O}(f_1)$, because $h$ could be equal to $3$, even with $n$ arbitrary big.

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  • $\begingroup$ Thank you, please review my question, I edit my question. $\endgroup$
    – Ahmad
    Oct 30, 2021 at 9:54

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