Incomputer science,merge-insertion sort or theFord–Johnson algorithm is acomparison sorting algorithm published in 1959 byL. R. Ford Jr. andSelmer M. Johnson.^[1][2][3][4] It uses fewer comparisons in theworst case than the best previously known algorithms,binary insertion sort andmerge sort,^[1] and for 20 years it was the sorting algorithm with the fewest known comparisons.^[5] Although not of practical significance, it remains of theoretical interest in connection with the problem of sorting with a minimum number of comparisons.^[3] The same algorithm may have also been independently discovered byStanisław Trybuła and Czen Ping.^[4]

Merge-insertion sort performs the following steps, on an input${\displaystyle X}$ of${\displaystyle n}$ elements:^[6]

1. Group the elements of${\displaystyle X}$ into${\displaystyle \lfloor n/2\rfloor }$ pairs of elements, arbitrarily, leaving one element unpaired if there is an odd number of elements.
2. Perform${\displaystyle \lfloor n/2\rfloor }$ comparisons, one per pair, to determine the larger of the two elements in each pair.
3. Recursively sort the${\displaystyle \lfloor n/2\rfloor }$ larger elements from each pair, creating a sorted sequence${\displaystyle S}$ of${\displaystyle \lfloor n/2\rfloor }$ of the input elements, in ascending order, using the merge-insertion sort.
4. Insert at the start of${\displaystyle S}$ the element that was paired with the first and smallest element of${\displaystyle S}$.
5. Insert the remaining${\displaystyle \lceil n/2\rceil -1}$ elements of${\displaystyle X\setminus S}$ into${\displaystyle S}$, one at a time, with a specially chosen insertion ordering described below. Usebinary search in subsequences of${\displaystyle S}$ (as described below) to determine the position at which each element should be inserted.

The algorithm is designed to take advantage of the fact that the binary searches used to insert elements into${\displaystyle S}$ are most efficient (from the point of view of worst case analysis) when the length of the subsequence that is searched is one less than apower of two. This is because, for those lengths, all outcomes of the search use the same number of comparisons as each other.^[1] To choose an insertion ordering that produces these lengths, consider the sorted sequence${\displaystyle S}$ after step 4 of the outline above (before inserting the remaining elements),and let${\displaystyle x_i}$ denote the${\displaystyle i}$th element of this sorted sequence. Thus,

${\displaystyle S=(x_1,x_2,x_3,\dots ),}$

where each element${\displaystyle x_i}$ with${\displaystyle i\ge 3}$ is paired with an element that has not yet been inserted. (There are no elements${\displaystyle y_1}$ or${\displaystyle y_2}$ because${\displaystyle x_1}$ and${\displaystyle x_2}$ were paired with each other.) If${\displaystyle n}$ is odd, the remaining unpaired element should also be numbered as${\displaystyle y_i}$ with${\displaystyle i}$ larger than the indexes of the paired elements.Then, the final step of the outline above can be expanded into the following steps:^[1][2][3][4]

• Partition the uninserted elements${\displaystyle y_i}$ into groups with contiguous indexes. There are two elements${\displaystyle y_3}$ and${\displaystyle y_4}$ in the first group, and the sums of sizes of every two adjacent groups form a sequence of powers of two. Thus, the sizes of groups are: 2, 2, 6, 10, 22, 42, ...
• Order the uninserted elements by their groups (smaller indexes to larger indexes), but within each group order them from larger indexes to smaller indexes. Thus, the ordering becomes

${\displaystyle y_4,y_3,y_6,y_5,y_{12},y_{11},y_{10},y_9,y_8,y_7,y_{22},y_{21}\dots }$

• Use this ordering to insert the elements${\displaystyle y_i}$ into${\displaystyle S}$. For each element${\displaystyle y_i}$, use a binary search from the start of${\displaystyle S}$ up to but not including${\displaystyle x_i}$ to determine where to insert${\displaystyle y_i}$.

Let${\displaystyle C(n)}$ denote the number of comparisons that merge-insertion sort makes, in the worst case, when sorting${\displaystyle n}$ elements.This number of comparisons can be broken down as the sum of three terms:

• ${\displaystyle \lfloor n/2\rfloor }$ comparisons among the pairs of items,
• ${\displaystyle C(\lfloor n/2\rfloor )}$ comparisons for the recursive call, and
• some number of comparisons for the binary insertions used to insert the remaining elements.

In the third term, the worst-case number of comparisons for the elements in the first group is two, because each is inserted into a subsequence of${\displaystyle S}$ of length at most three. First,${\displaystyle y_4}$ is inserted into the three-element subsequence${\displaystyle (x_1,x_2,x_3)}$. Then,${\displaystyle y_3}$ is inserted into some permutation of the three-element subsequence${\displaystyle (x_1,x_2,y_4)}$, or in some cases into the two-element subsequence${\displaystyle (x_1,x_2)}$. Similarly, the elements${\displaystyle y_6}$ and${\displaystyle y_5}$ of the second group are each inserted into a subsequence of length at most seven, using three comparisons. More generally, the worst-case number of comparisons for the elements in the${\displaystyle i}$th group is${\displaystyle i+1}$, because each is inserted into a subsequence of length at most${\displaystyle 2^{i+1}-1}$.^[1][2][3][4] By summing the number of comparisons used for all the elements and solving the resultingrecurrence relation,this analysis can be used to compute the values of${\displaystyle C(n)}$, giving the formula^[7]

${\displaystyle C(n)=\sum _{i=1}^n\lceil {\log }_2\frac{3i}{4}\rceil \approx n{\log }_{2}n-1.415n}$

or, inclosed form,^[8]

${\displaystyle C(n)=n\lceil {\log }_2\frac{3n}{4}\rceil -\lfloor \frac{2^{\lfloor {\log }_{2}6n\rfloor }}{3}\rfloor +\lfloor \frac{{\log }_{2}6n}{2}\rfloor .}$

For${\displaystyle n=1,2,\dots }$ the numbers of comparisons are^[1]

0, 1, 3, 5, 7, 10, 13, 16, 19, 22, 26, 30, 34, ... (sequenceA001768 in theOEIS)

The algorithm is called merge-insertion sort because the initial comparisons that it performs before its recursive call (pairing up arbitrary items and comparing each pair) are the same as the initial comparisons ofmerge sort,while the comparisons that it performs after the recursive call (using binary search to insert elements one by one into a sorted list) follow the same principle asinsertion sort. In this sense, it is ahybrid algorithm that combines both merge sort and insertion sort.^[9]

For small inputs (up to${\displaystyle n=11}$) its numbers of comparisons equal thelower bound on comparison sorting of${\displaystyle \lceil {\log }_{2}n!\rceil \approx n{\log }_{2}n-1.443n}$. However, for larger inputs the number of comparisons made by the merge-insertion algorithm is bigger than this lower bound.Merge-insertion sort also performs fewer comparisons than thesorting numbers, which count the comparisons made by binary insertion sort or merge sort in the worst case. The sorting numbers fluctuate between${\displaystyle n{\log }_{2}n-0.915n}$ and${\displaystyle n{\log }_{2}n-n}$, with the same leading term but a worse constant factor in the lower-order linear term.^[1]

Merge-insertion sort is the sorting algorithm with the minimum possible comparisons for${\displaystyle n}$ items whenever${\displaystyle n\le 22}$, and it has the fewest comparisons known for${\displaystyle n\le 46}$.^[10][11]For 20 years, merge-insertion sort was the sorting algorithm with the fewest comparisons known for all input lengths.However, in 1979 Glenn Manacher published another sorting algorithm that used even fewer comparisons, for large enough inputs.^[3][5]It remains unknown exactly how many comparisons are needed for sorting, for all${\displaystyle n}$, but Manacher's algorithmand later record-breaking sorting algorithms have all used modifications of the merge-insertion sort ideas.^[3]

• Comparison sorts
• 1959 in computing

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