Validation of Minimal Worst-Case Time Complexity by Stirling’s, Ramanujan’s, and Mortici’s Approximation

We have a lot of sorting algorithms in practice in this modern informatica world. Most of them are operating in quadratic time, some have managed to get them down to logarithmic time combined with linear, but there is no sorting algorithm that is efficient enough to sort data in linear time alone. In this paper, we will try to devise some mathematical support to the failure of minimalizing the sorting algorithms in linear time by comparing three well-known approximations of ! with detailed mathematical proofs associated with it. Further, this paper deals with detailed mathematical analysis to support the fact the minimum time complexity that can be attained by a sorting algorithm is of order O( ) without taking into account any extrapolated system advancement or computer architecture modification. There are various sorting algorithms like Merge sort, Heap sort, Interpolation sort, etc. which attains this minimum possible time complexity (worst case). But if we consider some biased cases then better time complexity can be obtained. If the elements of the array are in arithmetic progression, then interpolation search takes lesser time and there are many such biased cases where a particular algorithm gives better output but this paper deals with the general case i.e., no conditions are imposed on the elements of the array.