Understanding multi-variable big O (time complexity)

Question

I have something of this kind:

$$(n-1) d m + m + 2m*v+2v^2 + v$$

Where all n,d,m,v are variables.

My little knowledge of computational complexity leads to do this kind of approximation:

$$ O( (n-1) d m + mv + v^2 )$$

However I feel I could further simplify into

$$ O( n d m + mv + v^2 )$$

and

$$ O( n d m + v^2 )$$

hence

$$ O(v^2)$$

Are my assumptions correct?

• can I remove the $-1$ from $(n-1)$?
• can I remove $mv$ since $v$ will grow $v^2$ any way?

Answer 1

You can remove the $-1$ from $(n-1)$, since $(n-1)dm=ndm-dm\leq ndm$. On the other hand, you can not simplify beyond $O(ndm+mv+v^2)$ unless you know more about the relations between the variables. $mv$ and $v^2$ are incomparable, depending on the ratio between $m$ and $v$ one may grow faster than the other. Similarly, your last step (removing $ndm$ in favor of $v^2$) is also incorrect.