I had an algorithm with time-complexity of $O(h\times w)$, knowing $h$ is the height and $w$ is the width of an image being processed (or a simple matrix of size $h\times w$).

I managed to reduce the range that the algorithm process. So rather than dealing with $h\times w$ elements, it is dealing with $n\times m$ elements, where $n<h$ and $m<w$.

To recapitulate the optimization :

• Time-complexity of old algorithm is $O(h\times w)$
• Time-complexity of new algorithm is $O(h) + O(n\times m)$

Now my question is : how to express this time complexity optimization in terms of $h$ and $w$ ? is it a real optimization ?