my new sorting algorithm, what should my upper bound be?
It looks like the algorithm you are describing is called counting sort.
This algorithm works in $O(n+k)$ where $n$ is the number of items in the input, and $k$ would be their range: $k=max - min$.
This algorithm is indeed better in computational time complexity than mergesort when you restrict $k$ to be small enough. For your examples, you &quot;fixed&quot; a value for $n$ and $k$, which makes no sense to compute the big-O notation on. The big-O describes the algorithm in its full behaviour, and is an upper bound for any &quot;fixed&quot; value you would give.
As a side note, this algorithm is not a comparison-based sorting (sorting only by comparing numbers), and can only sort natural numbers. Actual comparison-based sorting algorithms (like mergesort) must take at least $\Omega(n\log(n))$, but are capable of sorting any list, and not just natural numbers.