Question

for the above 2 big O’s, what happens if n>>m . How does the big O change? Does it become O(n) in the first case. If yes ,why?

Answer 1

It depends on what you know about what the maximum value of m could be (depending on n).

If both m and n are independent variables O(mn) is O(mn) and cannot be further simplified. If you know that m will never be greater than n, but nothing else, you can also write it as O(n^2). If you know for example that m will never be greater than log n (which would satisfy n >> m), O(mn) can be written as O(n log n).