We define theta notation as follows: $\Theta(g(n))$ = {f(n): there are exist $c_1, c_2$ > 0 and $n_0$ such that 0 $\leq$ $c_1$g(n) $\leq$ f(n) $\leq$ $c_2$g(n) for all n > $n_0$}.

I found an illustration of this statement on Wikipedia, but I cannot figure out what exactly point $n_0$ means. How should we choose $n_0$ in practice? If I found, how should I use it?


It just means that the condition $0 \le c_1 g(n) \le f(n) \le c_2 g(n)$ only needs to hold for large enough integers.

This allows you to ignore all integers that are smaller than $n_0$ so you can just pick $n_0$ as large as needed for the inequalities to hold.

Notice that you do not need to pick $n_0$ as the smallest integer with that property, so you might as well pick one that makes proving the inequalities as easy as possible.

  • $\begingroup$ That makes sense, since not all values of n hold the statement above due to constants. Thank you! $\endgroup$ Jul 8 '21 at 10:37
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    $\begingroup$ For strictly positive functions on the integers domain (such as functions representing running time), the smallest $n_0$ is actually 0: just choose a constant $c$ big enough to overcome the first $n_0$ values. So I think its important to also note that $n_0$ isn't always needed, and when it isn't it can just be ommited altogether :) (to save your precious time from writing that $n_0$) $\endgroup$
    – nir shahar
    Jul 8 '21 at 11:07

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