Merge-Insertion Sort a.k.a. Ford-Johnson Algorithm

The Ford-Johnson algorithm[1], also known as the merge-insertion sort[2,3] uses the minimum number of possible comparisons for lists of 22 items or less, and at the time of writing has the fewest comparisons known for lists of 46 items or less. It is therefore very well suited for cases where comparisons are expensive, such as user input, and the API is implemented to take an async comparator function for this reason.

>>> from merge_insertion import merge_insertion_sort
>>> # A Comparator must return 0 if the first item is larger, or 1 if the second item is larger.
>>> # It can use any criteria for comparison, in this example we'll use user input:
>>> async def comparator(ab :tuple[str,str]):
...     choice = None
...     while choice not in ab:
...         choice = input(f"Please choose {ab[0]!r} or {ab[1]!r}: ")
...     return 0 if choice == ab[0] else 1
...
>>> # Sort five items in ascending order with a maximum of only seven comparisons:
>>> sorted = merge_insertion_sort('DABEC', comparator)
>>> # Since we can't `await` in the REPL, use asyncio to run the coroutine here:
>>> import asyncio
>>> asyncio.run(sorted)  
Please choose 'D' or 'A': D
...
Please choose 'B' or 'A': B
['A', 'B', 'C', 'D', 'E']

References

1. Ford, L. R., & Johnson, S. M. (1959). A Tournament Problem. The American Mathematical Monthly, 66(5), 387-389. https://doi.org/10.1080/00029890.1959.11989306
2. Knuth, D. E. (1998). The Art of Computer Programming: Volume 3: Sorting and Searching (2nd ed.). Addison-Wesley. https://cs.stanford.edu/~knuth/taocp.html#vol3
3. https://en.wikipedia.org/wiki/Merge-insertion_sort

See Also

• JavaScript / TypeScript version: https://www.npmjs.com/package/merge-insertion
• This algorithm in action: https://haukex.github.io/pairrank/ (select “Efficient”)

API

class merge_insertion.T

A type of object that can be compared by a Comparator and therefore sorted by merge_insertion_sort(). Must have sensible support for the equality operators.

alias of TypeVar(‘T’)

merge_insertion.Comparator

A user-supplied async function to compare two items. The single argument is a tuple of the two items to be compared; they must not be equal. Must return 0 if the first item is ranked higher, or 1 if the second item is ranked higher.

async merge_insertion.merge_insertion_sort(array: Sequence[T], comparator: Callable[[tuple[T, T]], Awaitable[Literal[0, 1]]]) → Sequence[T]

Merge-Insertion Sort (Ford-Johnson algorithm) with async comparison.

• Parameters:
  • array – Array to sort. Duplicate items are not allowed.
  • comparator – Async comparison function as described in Comparator.
• Returns: A shallow copy of the array sorted in ascending order.

merge_insertion.merge_insertion_max_comparisons(n: int) → int

Returns the maximum number of comparisons that merge_insertion_sort() will perform depending on the input length.

• Parameters: n – The number of items in the list to be sorted.
• Returns: The expected maximum number of comparisons.

Author, Copyright and License

Copyright © 2025 Hauke Dämpfling (haukex@zero-g.net)

Permission to use, copy, modify, and/or distribute this software for any purpose with or without fee is hereby granted, provided that the above copyright notice and this permission notice appear in all copies.

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