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.

Example

import { mergeInsertionSort, Comparator } from 'merge-insertion'

// 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, including user input, this is just a simple example:
const comp :Comparator<string> = async ([a, b]) => a > b ? 0 : 1

// Sort five items in ascending order with a maximum of only seven comparisons:
const sorted = await mergeInsertionSort(['D', 'A', 'B', 'E', 'C'], comp)

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

• Python version: https://pypi.org/project/merge-insertion/

• This algorithm in action: https://haukex.github.io/pairrank/ (select "Efficient")

Type Aliases

Comparable

Comparable = NonNullable<unknown>

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

Comparator()

Comparator<T> = (ab) => Promise<0 | 1>

A user-supplied function to compare two items.

Type Parameters

T

T extends Comparable

Parameters

ab

Readonly<[T, T]>

A tuple of the two items to be compared; they must not be equal.

Returns

Promise<0 | 1>

Must return a Promise resolving to 0 if the first item is ranked higher, or 1 if the second item is ranked higher.

Functions

mergeInsertionMaxComparisons()

mergeInsertionMaxComparisons(n): number

Returns the maximum number of comparisons that mergeInsertionSort will perform depending on the input length.

Parameters

n

number

The number of items in the list to be sorted.

Returns

number

The expected maximum number of comparisons.

mergeInsertionSort()

mergeInsertionSort<T>(array, comparator): Promise<T[]>

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

Type Parameters

T

T extends object

The type of the items to sort.

Parameters

array

readonly T[]

Array to sort. Duplicate items are not allowed.

comparator

Comparator<T>

Async comparison function.

Returns

Promise<T[]>

A Promise resolving to a shallow copy of the array sorted in ascending order.

Author, Copyright and License

Copyright © 2025 Hauke Dämpfling ([email protected])

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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