I designed an algorithm, and the Time Analysis provided next time complexity:

$\mathcal{O}(2^p \cdot (\frac{t}{2}/2^p) \cdot \frac{n}{4})$

what must be simplified to:

$\mathcal{O}(2^p \cdot \frac{n^2}{16})$ = $\mathcal{O}(2^p \cdot n^2)$

That constant factor is quite relevant in this case, and the only reason I'm considering keeping it, is because represents an important fact of the algorithm behavior.

  • $\begingroup$ If that constant is important, it means it's not a constant, it's a variable. Be careful while picking basic operations. $\endgroup$ – ferit Sep 23 '18 at 23:18

In asymptotic notation constant factors are "modded out". For example, an algorithm runs in time $O(n)$ iff it runs in time $O(2n)$ iff it runs in time $O(n/2)$.

If you want to capture such a difference meaningfully, you have (at least) two options:

  1. Add another parameter. There could be a parameter $t$ such that your algorithm runs in time $O(tn)$. One way to think about it is that the algorithm runs in time $O_t(n)$, i.e., with a big O constant that depends on $t$, and the dependence on the big O constant on $t$ is $O(t)$.

  2. Count a specific operation. In some cases you might be able to count a specific type of operation more accurately. For example, perhaps one sorting algorithm performs $(2 \pm o(1)) n\log n$ comparisons on average, and another performs $(3 \pm o(1)) n\log n$ comparisons on average.

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