Merge sort

In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the relative order of equal elements is the same in the input and output. Merge sort is a divide-and-conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up merge sort appeared in a report by Goldstine and von Neumann as early as 1948.

Merge sort

An example of merge sort. First, divide the list into the smallest unit (1 element), then compare each element with the adjacent list to sort and merge the two adjacent lists. Finally, all the elements are sorted and merged.
Class	Sorting algorithm
Data structure	Array
Worst-case performance	${\displaystyle O(n\log n)}$
Best-case performance	${\displaystyle \Omega (n\log n)}$ typical, ${\displaystyle \Omega (n)}$ natural variant
Average performance	${\displaystyle \Theta (n\log n)}$
Worst-case space complexity	${\displaystyle O(n)}$ total with ${\displaystyle O(n)}$ auxiliary, ${\displaystyle O(1)}$ auxiliary with linked lists