# How to properly describe this behavior (in terms of time complexity)

After completing the time analysis for an algorithm I have been working on, I need some help to write its time complexity properly, let me describe how it works...

The algorithm has two inputs $x$ and $y$.

When $x < 15$, steps to solve are $2^y$, so I know this is $O(2^y)$.

However when $x >= 15$, the steps to solve begin ramping down (from the $2^y$) at a rate of $-1\%$ as $x$ increases, up to $x < 50$ (don't know how to express this part)

And then, when $x>=50$ it flats out, and keep the same cost, that is $O(2^y*35\%)$, and doesn't matter what is the value of $x$ (as long as it is greater than $50$)

How to write a time complexity that has 3 delimited phases depending on the inputs?

You're interested in asymptotic behavior as $x$ increases without bound.

$O(.35 \times 2^y)$ is still $O(2^y)$.

When $x<15$, steps to solve are $2^y$, so I know this is $O(2^y)$.

It's also O(1).

Go back to the definition of $O$: small $x$ don't matter. In your case, pick $x_0 \geq 50$ and ignore the other cases.

That said, your question reads as if you looked at plots, not done a proper analysis. Be aware that plots (or other means of looking at finite samples) never prove any asymptotic property. You may want to peruse our reference questions.

• described behavior is consistent with the algorithm implementation, there is an exact point that a optimization engages (depends on $x$), as $x$ grows, the optimization reduce the steps required by the algorithm, however, there is a balance point where the optimization reaches a maximum, and from there the step count stay flat for any $y$. (Not using plotting / guessing). I simplified the info to raise the questions, might be possible I simplified too much. – Jesus Salas Oct 9 '17 at 17:57
• @JesusSalas Never mind my last paragraph, then. (Leaving it here for Googlers.) – Raphael Oct 10 '17 at 7:28