I have an algorithm with three variables affecting the time complexity: $k$, $L$, and $n$. I have come up with the following that expresses the complexity:
$O(kn + k^2L + k^2nL + knL)$
I think I should be able to simplify this to:
$O(k^2nL)$
Am I correct? I'm a little fuzzy on how to simplify things when working with multiple variables, but it seems right that since every other term is a factor of $k^2nL$, it should dominate the other terms?
Suppose that there exists a strictly positive constant $c > 0$ such that $k,n,L \geq c$; this happens for example if $k,n,L$ are all positive integers. In this case, a function is in $O(kn + k^2L + k^2nL + knL)$ iff it is in $O(k^2nL)$, where for our purposes $f(k,n,L) = O(g(k,n,L))$ (read: $f(k,n,L)$ is in $O(g(k,n,L))$) if there exists a constant $C>0$ such that for all $k,n,L$ "in range" we have $f(k,n,L) \leq C g(k,n,L)$. Here "in range" is the common domain of the functions $f,g$, such as all positive integers or all positive reals which are at least $c$, for some positive $c>0$.
