Merge-insertion sort: Difference between revisions

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

In [[computer science]], '''merge-insertinsertion sort''' or the '''Ford–Johnson algorithm''' is a [[comparison sort]]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 [[worst case analysis|worst case]] than the best previously known algorithms, [[insertion sort|binary insertion sort]] and [[merge sort]],{{r|fj}} and 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}} The same algorithm may have also been independently discovered by S.[[Stanisław TrybulaTrybuła]] and P. Czen Ping.{{r|knuth}}

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

==Algorithm==

Merge-insertinsertion sort performs the following steps, on an input <math>X</math> of <math>n</math> elements:<ref>The original description by {{harvtxt|Ford|Johnson|1959}} sorted the elements in descending order. The steps listed here reverse the output, following the description in {{harvtxt|Knuth|1998}}. The resulting algorithm makes the same comparisons but produces ascending order instead.</ref>

#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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:<math>S=(x_1,x_2,x_3,\dots),</math>

where each element <math>x_i</math> with <math>i\ge 3</math> is paired with an element <math>y_i < x_i</math> that has not yet been inserted. (There are no elements <math>y_1</math> or <math>y_2</math> because <math>x_1</math> and <math>x_2</math> were paired with each other.) If <math>n</math> is odd, the remaining unpaired element should also be numbered as <math>y_i</math> with <math>i</math> larger than the indexes of the paired elements.

Then, the final step of the outline above can be expanded into the following steps:{{r|fj|c4cs|distrib|knuth}}

*Partition the uninserted elements <math>y_i</math> into groups with contiguous indexes. There are two elements <math>y_3</math> and <math>y_4</math> in the first group, and the sizesums of eachsizes subsequentof groupevery equalstwo theadjacent numbergroups ofform elementsa insequence allof previouspowers of groupstwo. Thus, the sizes of the groups form a sequence of powers of twoare: 2, 2, 46, 10, 822, 1642, ...

*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

::<math>y_4,y_3,y_6,y_5,y_{12},y_{11},y_{10},y_9,y_8,y_7,y_{1822},y_{21}\dots</math>

*Use this ordering to insert the elements <math>y_i</math> into <math>S</math>. For each element <math>y_i</math>, use a binary search from the start of <math>S</math> up to but not including <math>x_i</math> to determine where to insert <math>y_i</math>.

==Analysis==

Let <math>C(n)</math> denote the number of comparisons that merge-insertinsertion sort makes, in the worst case, when sorting <math>n</math> elements.

This number of comparisons can be broken down as the sum of three terms:

*<math>\lfloor n/2\rfloor</math> comparisons among the pairs of items,

*<math>C(\lfloor n/2\rfloor)</math> comparisons for the recursive call, and

*Somesome 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 <math>S</math> of length at most three. First, <math>y_4</math> is inserted into the three-element subsequence <math>(x_1,x_2,x_3)</math>. Then, <math>y_3</math> is inserted into some permutation of the three-element subsequence <math>(x_1,x_2,y_4)</math>, or in some cases into the two-element subsequence <math>(x_1,x_2)</math>. Similarly, the elements <math>y_6</math> and <math>y_5</math> 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 <math>i</math>th group is <math>i+1</math>, because each is inserted into a subsequence of length at most <math>2^{i+1}-1</math>.{{r|fj|c4cs|distrib|knuth}} By summing the number of comparisons used for all the elements and solving the resulting [[recurrence relation]],

this analysis can be used to compute the values of <math>C(n)</math>, giving the formula<ref>{{harvtxt|Knuth|1998}} credits thisthe summation formula to the 1960 Ph.D. thesis of A. Hadian. The approximation formula was already given by {{harvtxt|Ford|Johnson|1959}}.</ref>

:<math>C(n)=\sum_{i=1}^n \left\lceil \log_2 \frac{3i}{4} \right\rceil \approx n\log_2 n - 1.415n</math>

or, in [[closed-form expression|closed form]],{{r|gn}}

:<math>C(n)=n\biggl\lceil\log_2\frac{3n}{4}\biggr\rceil-\biggl\lfloor\frac{2^{\lfloor \log_2 6n\rfloor}}{3}\biggr\rfloor+\biggl\lfloor\frac{\log_2 6n}{2}\biggr\rfloor.</math>

For <math>n=1,2,\dots</math> the numbers of comparisons are{{r|fj}}

:0, 1, 3, 5, 7, 10, 13, 16, 19, 22, 26, 30, 34, ... {{OEIS|A001768}}

[[Asymptotic analysis|Asymptotically]], these numbers are approximately{{r|fj}}

:<math>n\log_2 n - 1.415n</math>

For small inputs (up to <math>n=11</math>) these numbers of comparisons equal the [[lower bound]] on comparison sorting of <math>\lceil\log_2 n!\rceil\approx n\log_2 n - 1.443n</math>. However, for larger inputs the number of comparisons made by the merge-insert algorithm is bigger than this lower bound.

Merge-insert 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}}

==Relation to other comparison sorts==

The algorithm is called merge-insertinsertion 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 of [[merge 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 principalprinciple as [[insertion sort]]. In this sense, it is a [[hybrid algorithm]] that combines both merge sort and insertion sort.<ref>{{harvtxt|Knuth|1998}}, p. 184: "Since it involves some aspects of merging and some aspects of insertion, we call it ''merge insertion''."</ref>

For small inputs (up to <math>n=11</math>) theseits numbers of comparisons equal the [[lower bound]] on comparison sorting of <math>\lceil\log_2 n!\rceil\approx n\log_2 n - 1.443n</math>. However, for larger inputs the number of comparisons made by the merge-insertinsertion algorithm is bigger than this lower bound.

Merge insert sort is the sorting algorithm with the fewest possible comparisons for <math>n</math> items whenever <math>n\le 14</math>, and for <math>20\le n\le 22</math>.{{r|pec}}

Merge-insertinsertion 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}}

For 20 years, merge-insert sort was the sorting algorithm with the fewest comparisons known for all input lengths.

Merge insert-insertion sort is the sorting algorithm with the fewestminimum possible comparisons for <math>n</math> items whenever <math>n\le 1422</math>, and it has the fewest comparisons known for <math>20\le n\le 2246</math>.{{r|pec|pec2}}

For 20 years, merge-insertinsertion 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}}

It remains unknown exactly how many comparisons are needed for sorting, for all <math>n</math>, but Manacher's algorithm

and later record-breaking sorting algorithms have all used modifications of the merge-insertinsertion sort ideas.{{r|distrib}}

==References==

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| volume = 66

| year = 1959}}</ref>

<ref name=gn>{{citation

| last1 = Guy | first1 = Richard K. | author1-link = Richard K. Guy

| last2 = Nowakowski | first2 = Richard J.

| date = December 1995

| doi = 10.2307/2975272

| issue = 10

| journal = [[American Mathematical Monthly]]

| pages = 921–926

| title = ''Monthly'' Unsolved Problems, 1969-1995

| volume = 102}}</ref>

<ref name=knuth>{{citation

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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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| volume = 40

| year = 2004}}</ref>

<ref name=pec2>{{citation

| last = Peczarski | first = Marcin

| doi = 10.1016/j.ipl.2006.09.001

| issue = 3

| journal = Information Processing Letters

| mr = 2287331

| pages = 126–128

| title = The Ford-Johnson algorithm still unbeaten for less than 47 elements

| volume = 101

| year = 2007}}</ref>

}}

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{{Sorting}}

[[Category:Comparison sorts]]

[[Category:1959 in computing]]