Merge sort

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O(n log n) typical,

Merge sort is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, meaning that the implementation preserves the input order of equal elements in the sorted output. It is a divide and conquer algorithm. Merge sort was invented by John von Neumann in 1945.[1]

Contents

Algorithm

Conceptually, a merge sort works as follows

Merge sort incorporates two main ideas to improve its runtime:

Example: Using merge sort to sort a list of integers contained in an array:

Suppose we have an array A with n indices ranging from A0 to An − 1. We apply merge sort to A(A0..Ac − 1) and A(Ac..An − 1) where c is the integer part of n / 2. When the two halves are returned they will have been sorted. They can now be merged together to form a sorted array.

In a simple pseudocode form, the algorithm could look something like this:

function merge_sort(m)
    if length(m) ≤ 1
        return m
    var list left, right, result

var integer middle = length(m) / 2 for each x in m up to middle add x to left for each x in m after middle add x to right left = merge_sort(left) right = merge_sort(right) result = merge(left, right) return result

Following writing merge_sort function, then it is required to merge both the left and right lists created above. There are several variants for the merge() function; one possibility is this:

function merge(left,right)
    var list result
    while length(left) > 0 or length(right) > 0
        if length(left) > 0 and length(right) > 0
            if first(left) ≤ first(right)
                append first(left) to result
                left = rest(left)
            else
                append first(right) to result
                right = rest(right)
        else if length(left) > 0
            append first(left) to result
            left = rest(left)             
        else if length(right) > 0
            append first(right) to result
            right = rest(right)
    end while
    return result

[edit] Analysis

A recursive merge sort algorithm used to sort an array of 7 integer values. These are the steps a human would take to emulate merge sort (top-down).

In sorting n objects, merge sort has an average and worst-case performance of O(n log n). If the running time of merge sort for a list of length n is T(n), then the recurrence T(n) = 2T(n/2) + n follows from the definition of the algorithm (apply the algorithm to two lists of half the size of the original list, and add the n steps taken to merge the resulting two lists). The closed form follows from the master theorem.

In the worst case, merge sort does an amount of comparisons equal to or slightly smaller than (n ⌈lg n⌉ - 2⌈lg n + 1), which is between (n lg n - n + 1) and (n lg n + n + O(lg n)).[2]

For large n and a randomly ordered input list, merge sort's expected (average) number of comparisons approaches α·n fewer than the worst case where \alpha = -1 + \sum_{k=0}^\infty \frac1{2^k+1} \approx 0.2645.

In the worst case, merge sort does about 39% fewer comparisons than quicksort does in the average case; merge sort always makes fewer comparisons than quicksort, except in extremely rare cases, when they tie, where merge sort's worst case is found simultaneously with quicksort's best case. In terms of moves, merge sort's worst case complexity is O(n log n)—the same complexity as quicksort's best case, and merge sort's best case takes about half as many iterations as the worst case.[citation needed]

Recursive implementations of merge sort make 2n − 1 method calls in the worst case, compared to quicksort's n, thus merge sort has roughly twice as much recursive overhead as quicksort. However, iterative, non-recursive implementations of merge sort, avoiding method call overhead, are not difficult to code. Merge sort's most common implementation does not sort in place; therefore, the memory size of the input must be allocated for the sorted output to be stored in (see below for versions that need only n/2 extra spaces).

Merge sort as described here also has an often overlooked, but practically important, best-case property. If the input is already sorted, its complexity falls to O(n). Specifically, n-1 comparisons and zero moves are performed, which is the same as for simply running through the input, checking if it is pre-sorted.

Sorting in-place is possible (e.g., using lists rather than arrays) but is very complicated, and will offer little performance gains in practice, even if the algorithm runs in O(n log n) time. (Katajainen, Pasanen & Teuhola 1996) In these cases, algorithms like heapsort usually offer comparable speed, and are far less complex. Additionally, unlike the standard merge sort, in-place merge sort is not a stable sort. In the case of linked lists the algorithm does not use more space than that the already used by the list representation, but the O(log(k)) used for the recursion trace.

Merge sort is more efficient than quick sort for some types of lists if the data to be sorted can only be efficiently accessed sequentially, and is thus popular in languages such as Lisp, where sequentially accessed data structures are very common. Unlike some (efficient) implementations of quicksort, merge sort is a stable sort as long as the merge operation is implemented properly.

As can be seen from the procedure merge sort, there are some demerits. One complaint we might raise is its use of 2n locations; the additional n locations were needed because one couldn't reasonably merge two sorted sets in place. But despite the use of this space the algorithm must still work hard: The contents of m are first copied into left and right and later into the list result on each invocation of merge_sort (variable names according to the pseudocode above). An alternative to this copying is to associate a new field of information with each key (the elements in m are called keys). This field will be used to link the keys and any associated information together in a sorted list (a key and its related information is called a record). Then the merging of the sorted lists proceeds by changing the link values; no records need to be moved at all. A field which contains only a link will generally be smaller than an entire record so less space will also be used.

Another alternative for reducing the space overhead to n/2 is to maintain left and right as a combined structure, copy only the left part of m into temporary space, and to direct the merge routine to place the merged output into m. With this version it is better to allocate the temporary space outside the merge routine, so that only one allocation is needed. The excessive copying mentioned in the previous paragraph is also mitigated, since the last pair of lines before the return result statement (function merge in the pseudo code above) become superfluous.

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