Best case Big O complexity

Question

The question:
how can you limit the input data to achieve a better Big O complexity? Describe an algorithm for handling this limited data to find if there are an

Answer 1

Assume that sorting is our problem.

We know that sorting with only comparisons requires Ω(n*log(n)) time and we can do it in O(n*log(n)) by for example a merge sort.

However, if we limit n to some constant, for example let n < 10^6, then we can do it for any input in O(10^6 * log(10^6)) which is O(1) in terms of Big-O.

The bottom line is, if you want to measure a performance in terms of the Big-O notation, you can not assume any size limitation on the input.