Merge-insertion sort: Difference between revisions

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{{Short description|Type of comparison sorting algorithm}}

In [[computer science]], '''merge-insertion sort''' or the '''Ford–Johnson algorithm''' is a [[Comparisoncomparison sort|comparison sorting]]ing algorithm published in 1959 by [[L. R. Ford Jr.]] and [[Selmer M. Johnson]].{{r|fj|c4cs|distrib|knuth}} It uses fewer comparisons in the [[Worstworst case analysis|worst case]] than the best previously known algorithms, [[Insertioninsertion sort|binary insertion sort]] and [[merge sort]],.{{r|fj}} Forand for 20 years, it was the sorting algorithm with the fewest known comparisons.{{r|nonopt}} Although not of practical significance, it remains of theoretical interest in connection with the problem of sorting with a minimum number of comparisons.{{r|distrib}} StanisławThe Trybułasame and Czen Pingalgorithm may have also been independently discovered theby same[[Stanisław algorithmTrybuła]] and Czen Ping.{{r|knuth}}

[[File:Ford-janson.gif|thumb|An animation of the [[Merge algorithm|merge-algorithm]] sorting an array of randomized values.]]

==Algorithm==

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#Group the elements of <math>X</math> into <math>\lfloor n/2\rfloor</math> pairs of elements, arbitrarily, leaving one element unpaired if there is an odd number of elements.

#Perform <math>\lfloor n/2\rfloor</math> comparisons, one per pair, to determine the larger of the two elements in each pair.

#Recursively sort the <math>\lfloor n/2\rfloor</math> larger elements from each pair, creating a sorted sequence <math>S</math> of <math>\lfloor n/2\rfloor</math> of the input elements, in ascending order, using the merge-insertion sort.

#Insert at the start of <math>S</math> the element that was paired with the first and smallest element of <math>S</math>.

#Insert the remaining <math>\lceil n/2\rceil-1</math> elements of <math>X\setminus S</math> into <math>S</math>, one at a time, with a specially chosen insertion ordering described below. Use [[binary search]] in subsequences of <math>S</math> (as described below) to determine the position at which each element should be inserted.

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Merge-insertion sort also performs fewer comparisons than the [[sorting number]]s, which count the comparisons made by binary insertion sort or merge sort in the worst case. The sorting numbers fluctuate between <math>n\log_2 n - 0.915n</math> and <math>n\log_2 n - n</math>, with the same leading term but a worse constant factor in the lower-order linear term.{{r|fj}}

Merge-insertion sort is the sorting algorithm with the minimum possible comparisons for <math>n</math> items whenever <math>n\le 15</math> or <math>20\le n\le 22</math>, and it has the fewest comparisons known for <math>n\le 46</math>.{{r|pec|pec2}}

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.{{r|distrib|nonopt}}

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| pages = 441–456

| title = The Ford-Johnson Sorting Algorithm Is Not Optimal

| volume = 26}}</ref>| doi-access = free

}}</ref>

<ref name=pec>{{citation

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[[Category:Comparison sorts]]

[[Category:1959 in computing]]

[[Category:Sorting algorithms]]