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In computer science, a **merge sort** (also commonly spelled **mergesort**) is an *O*(*n* log *n*) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Merge sort is a divide and conquer algorithm that was invented by John von Neumann in 1945.^{[1]} A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948

## Algorithm

Each recursive call has O(n) runtime, and a total of O(log n) recursions are required, thus the runtime of this algorithm is O(n * log n). A merge sort can also be modified for performance on lists that are nearly sorted to begin with. After sorting each half of the data, if the highest element in one list is less than the lowest element in the other half, then the merge step is unnecessary. (The Java API implements this particular optimization, for instance.)

Conceptually, a merge sort works as follows

- Divide the unsorted list into
*n*sublists, each containing 1 element (a list of 1 element is considered sorted). - Repeatedly merge sublists to produce new sublists until there is only 1 sublist remaining. This will be the sorted list.

Each recursive call has O(n) runtime, and a total of O(log n) recursions are required, thus the runtime of this algorithm is O(n * log n). A merge sort can also be modified for performance on lists that are nearly sorted to begin with. After sorting each half of the data, if the highest element in one list is less than the lowest element in the other half, then the merge step is unnecessary. (The Java API implements this particular optimization, for instance.)