 An example of merge sort. First, divide the list into the smallest unit (1 element), then compare each element with the adjacent list to sort and merge the two adjacent lists. Finally, all the elements are sorted and merged. | |
| Class | Sorting algorithm |
|---|---|
| Data structure | Array |
| Worst-caseperformance | |
| Best-caseperformance | typical, natural variant |
| Averageperformance | |
| Worst-casespace complexity | total with auxiliary, auxiliary with linked lists[1] |
Incomputer science,merge sort (also commonly spelled asmergesort and asmerge-sort[2]) is an efficient, general-purpose, andcomparison-basedsorting algorithm. Most implementations of merge sort arestable, which means that the relative order of equal elements is the same between the input and output. Merge sort is adivide-and-conquer algorithm that was invented byJohn von Neumann in 1945.[3] A detailed description and analysis of bottom-up merge sort appeared in a report byGoldstine and von Neumann as early as 1948.[4]
Conceptually, a merge sort works as follows:
ExampleC-like code using indices for top-down merge sort algorithm that recursively splits the list (calledruns in this example) into sublists until sublist size is 1, then merges those sublists to produce a sorted list. The copy back step is avoided with alternating the direction of the merge with each level of recursion (except for an initial one-time copy, that can be avoided too).
As a simple example, consider an array with two elements. The elements are copied toB[], then merged back toA[]. If there are four elements, when the bottom of the recursion level is reached, single element runs fromA[] are merged toB[], and then at the next higher level of recursion, those two-element runs are merged toA[]. This pattern continues with each level of recursion.
// Array A[] has the items to sort; array B[] is a work array.voidTopDownMergeSort(A[],B[],n){CopyArray(A,0,n,B);// one time copy of A[] to B[]TopDownSplitMerge(A,0,n,B);// sort data from B[] into A[]}// Split A[] into 2 runs, sort both runs into B[], merge both runs from B[] to A[]// iBegin is inclusive; iEnd is exclusive (A[iEnd] is not in the set).voidTopDownSplitMerge(B[],iBegin,iEnd,A[]){if(iEnd-iBegin<=1)// if run size == 1return;// consider it sorted// split the run longer than 1 item into halvesiMiddle=(iEnd+iBegin)/2;// iMiddle = mid point// recursively sort both runs from array A[] into B[]TopDownSplitMerge(A,iBegin,iMiddle,B);// sort the left runTopDownSplitMerge(A,iMiddle,iEnd,B);// sort the right run// merge the resulting runs from array B[] into A[]TopDownMerge(B,iBegin,iMiddle,iEnd,A);}// Left source half is A[ iBegin:iMiddle-1].// Right source half is A[iMiddle:iEnd-1 ].// Result is B[ iBegin:iEnd-1 ].voidTopDownMerge(B[],iBegin,iMiddle,iEnd,A[]){i=iBegin,j=iMiddle;// While there are elements in the left or right runs...for(k=iBegin;k<iEnd;k++){// If left run head exists and is <= existing right run head.if(i<iMiddle&&(j>=iEnd||A[i]<=A[j])){B[k]=A[i];i=i+1;}else{B[k]=A[j];j=j+1;}}}voidCopyArray(A[],iBegin,iEnd,B[]){for(k=iBegin;k<iEnd;k++)B[k]=A[k];}
Sorting the entire array is accomplished byTopDownMergeSort(A, B, length(A)).
Example C-like code using indices for bottom-up merge sort algorithm which treats the list as an array ofn sublists (calledruns in this example) of size 1, and iteratively merges sub-lists back and forth between two buffers:
// array A[] has the items to sort; array B[] is a work arrayvoidBottomUpMergeSort(A[],B[],n){// Each 1-element run in A is already "sorted".// Make successively longer sorted runs of length 2, 4, 8, 16... until the whole array is sorted.for(width=1;width<n;width=2*width){// Array A is full of runs of length width.for(i=0;i<n;i=i+2*width){// Merge two runs: A[i:i+width-1] and A[i+width:i+2*width-1] to B[]// or copy A[i:n-1] to B[] ( if (i+width >= n) )BottomUpMerge(A,i,min(i+width,n),min(i+2*width,n),B);}// Now work array B is full of runs of length 2*width.// Copy array B to array A for the next iteration.// A more efficient implementation would swap the roles of A and B.CopyArray(B,A,n);// Now array A is full of runs of length 2*width.}}// Left run is A[iLeft :iRight-1].// Right run is A[iRight:iEnd-1 ].voidBottomUpMerge(A[],iLeft,iRight,iEnd,B[]){i=iLeft,j=iRight;// While there are elements in the left or right runs...for(k=iLeft;k<iEnd;k++){// If left run head exists and is <= existing right run head.if(i<iRight&&(j>=iEnd||A[i]<=A[j])){B[k]=A[i];i=i+1;}else{B[k]=A[j];j=j+1;}}}voidCopyArray(B[],A[],n){for(i=0;i<n;i++)A[i]=B[i];}
Pseudocode for top-down merge sort algorithm which recursively divides the input list into smaller sublists until the sublists are trivially sorted, and then merges the sublists while returning up the call chain.
function merge_sort(list m)is //Base case. A list of zero or one elements is sorted, by definition.if length of m ≤ 1thenreturn m //Recursive case. First, divide the list into equal-sized sublists //consisting of the first half and second half of the list. //This assumes lists start at index 0.var left := empty listvar right := empty listfor each xwith index iin mdoif i < (length of m)/2then add x to leftelse add x to right //Recursively sort both sublists. left := merge_sort(left) right := merge_sort(right) // Then merge the now-sorted sublists.return merge(left, right)
In this example, themerge function merges the left and right sublists.
function merge(left, right)isvar result := empty listwhile left is not emptyand right is not emptydoif first(left) ≤ first(right)then append first(left) to result left := rest(left)else append first(right) to result right := rest(right) //Either left or right may have elements left; consume them. //(Only one of the following loops will actually be entered.)while left is not emptydo append first(left) to result left := rest(left)while right is not emptydo append first(right) to result right := rest(right)return result
Pseudocode for bottom-up merge sort algorithm which uses a small fixed size array of references to nodes, wherearray[i] is either a reference to a list of size 2i ornil.node is a reference or pointer to a node. Themerge() function would be similar to the one shown in the top-down merge lists example, it merges two already sorted lists, and handles empty lists. In this case,merge() would usenode for its input parameters and return value.
function merge_sort(node head)is // return if empty listif head = nilthenreturn nilvarnode array[32]; initially all nilvarnode resultvarnode nextvarint i result := head // merge nodes into arraywhile result ≠ nildo next := result.next; result.next := nilfor (i = 0; (i < 32) && (array[i] ≠ nil); i += 1)do result := merge(array[i], result) array[i] := nil // do not go past end of arrayif i = 32then i -= 1 array[i] := result result := next // merge array into single list result := nilfor (i = 0; i < 32; i += 1)do result := merge(array[i], result)return result
Haskell-like pseudocode, showing how merge sort can be implemented in such a language using constructs and ideas fromfunctional programming.
mergeSort::Orda=>[a]->[a]mergeSort[]=[]mergeSort[x]=[x]mergeSortxs=merge(mergeSortl,mergeSortr)where(l,r)=splitAt(lengthxs`div`2)xsmerge::Orda=>([a],[a])->[a]merge([],xs)=xsmerge(xs,[])=xsmerge(x:xs,y:ys)|x<=y=x:merge(xs,y:ys)|otherwise=y:merge(x:xs,ys)
In sortingn objects, merge sort has anaverage andworst-case performance ofO(n log n) comparisons. If the running time (number of comparisons) of merge sort for a list of lengthn isT(n), then therecurrence relationT(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 then steps taken to merge the resulting two lists).[5] The closed form follows from themaster theorem for divide-and-conquer recurrences.
The number of comparisons made by merge sort in the worst case is given by thesorting numbers. These numbers are 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)).[6] Merge sort's best case takes about half as many iterations as its worst case.[7]
For largen and a randomly ordered input list, merge sort's expected (average) number of comparisons approachesα·n fewer than the worst case, where
In theworst case, merge sort uses approximately 39% fewer comparisons thanquicksort does in itsaverage case, and in terms of moves, merge sort's worst case complexity isO(n log n) - the same complexity as quicksort's best case.[7]
Merge sort is more efficient than quicksort for some types of lists if the data to be sorted can only be efficiently accessed sequentially, and is thus popular in languages such asLisp, where sequentially accessed data structures are very common. Unlike some (efficient) implementations of quicksort, merge sort is a stable sort.
Merge sort's most common implementation does not sort in place;[8] therefore, the memory size of the input must be allocated for the sorted output to be stored in (see below for variations that need onlyn/2 extra spaces).
A natural merge sort is similar to a bottom-up merge sort except that any naturally occurringruns (sorted sequences) in the input are exploited. Both monotonic and bitonic (alternating up/down) runs may be exploited, with lists (or equivalently tapes or files) being convenient data structures (used asFIFO queues orLIFO stacks).[9] In the bottom-up merge sort, the starting point assumes each run is one item long. In practice, random input data will have many short runs that just happen to be sorted. In the typical case, the natural merge sort may not need as many passes because there are fewer runs to merge. In the best case, the input is already sorted (i.e., is one run), so the natural merge sort need only make one pass through the data. In many practical cases, long natural runs are present, and for that reason natural merge sort is exploited as the key component ofTimsort. Example:
Start : 3 4 2 1 7 5 8 9 0 6Select runs : (3 4)(2)(1 7)(5 8 9)(0 6)Merge : (2 3 4)(1 5 7 8 9)(0 6)Merge : (1 2 3 4 5 7 8 9)(0 6)Merge : (0 1 2 3 4 5 6 7 8 9)
Formally, the natural merge sort is said to beRuns-optimal, where is the number of runs in, minus one.
Tournament replacement selection sorts are used to gather the initial runs for external sorting algorithms.
Instead of merging two blocks at a time, a ping-pong merge merges four blocks at a time. The four sorted blocks are merged simultaneously to auxiliary space into two sorted blocks, then the two sorted blocks are merged back to main memory. Doing so omits the copy operation and reduces the total number of moves by half. An early public domain implementation of a four-at-once merge was by WikiSort in 2014, the method was later that year described as an optimization forpatience sorting and named a ping-pong merge.[10][11] Quadsort implemented the method in 2020 and named it a quad merge.[12]
One drawback of merge sort, when implemented on arrays, is itsO(n) working memory requirement. Several methods to reduce memory or make merge sort fullyin-place have been suggested:
Anexternal merge sort is practical to run usingdisk ortape drives when the data to be sorted is too large to fit intomemory.External sorting explains how merge sort is implemented with disk drives. A typical tape drive sort uses four tape drives. All I/O is sequential (except for rewinds at the end of each pass). A minimal implementation can get by with just two record buffers and a few program variables.
Naming the four tape drives as A, B, C, D, with the original data on A, and using only two record buffers, the algorithm is similar tothe bottom-up implementation, using pairs of tape drives instead of arrays in memory. The basic algorithm can be described as follows:
Instead of starting with very short runs, usually ahybrid algorithm is used, where the initial pass will read many records into memory, do an internal sort to create a long run, and then distribute those long runs onto the output set. The step avoids many early passes. For example, an internal sort of 1024 records will save nine passes. The internal sort is often large because it has such a benefit. In fact, there are techniques that can make the initial runs longer than the available internal memory. One of them, the Knuth's 'snowplow' (based on abinary min-heap), generates runs twice as long (on average) as a size of memory used.[18]
With some overhead, the above algorithm can be modified to use three tapes.O(n logn) running time can also be achieved using twoqueues, or astack and a queue, or three stacks. In the other direction, usingk > two tapes (andO(k) items in memory), we can reduce the number of tape operations inO(logk) times by using ak/2-way merge.
A more sophisticated merge sort that optimizes tape (and disk) drive usage is thepolyphase merge sort.
On modern computers,locality of reference can be of paramount importance insoftware optimization, because multilevelmemory hierarchies are used.Cache-aware versions of the merge sort algorithm, whose operations have been specifically chosen to minimize the movement of pages in and out of a machine's memory cache, have been proposed. For example, thetiled merge sort algorithm stops partitioning subarrays when subarrays of size S are reached, where S is the number of data items fitting into a CPU's cache. Each of these subarrays is sorted with an in-place sorting algorithm such asinsertion sort, to discourage memory swaps, and normal merge sort is then completed in the standard recursive fashion. This algorithm has demonstrated better performance[example needed] on machines that benefit from cache optimization. (LaMarca & Ladner 1997)
Merge sort parallelizes well due to the use of thedivide-and-conquer method. Several different parallel variants of the algorithm have been developed over the years. Some parallel merge sort algorithms are strongly related to the sequential top-down merge algorithm while others have a different general structure and use theK-way merge method.
The sequential merge sort procedure can be described in two phases, the divide phase and the merge phase. The first consists of many recursive calls that repeatedly perform the same division process until the subsequences are trivially sorted (containing one or no element). An intuitive approach is the parallelization of those recursive calls.[19] Following pseudocode describes the merge sort with parallel recursion using thefork and join keywords:
//Sort elements lo through hi (exclusive) of array A.algorithm mergesort(A, lo, hi)isif lo+1 < hithen //Two or more elements. mid := ⌊(lo + hi) / 2⌋fork mergesort(A, lo, mid) mergesort(A, mid, hi)join merge(A, lo, mid, hi)
This algorithm is the trivial modification of the sequential version and does not parallelize well. Therefore, its speedup is not very impressive. It has aspan of, which is only an improvement of compared to the sequential version (seeIntroduction to Algorithms). This is mainly due to the sequential merge method, as it is the bottleneck of the parallel executions.
Better parallelism can be achieved by using a parallelmerge algorithm.Cormen et al. present a binary variant that merges two sorted sub-sequences into one sorted output sequence.[19]
In one of the sequences (the longer one if unequal length), the element of the middle index is selected. Its position in the other sequence is determined in such a way that this sequence would remain sorted if this element were inserted at this position. Thus, one knows how many other elements from both sequences are smaller and the position of the selected element in the output sequence can be calculated. For the partial sequences of the smaller and larger elements created in this way, the merge algorithm is again executed in parallel until the base case of the recursion is reached.
The following pseudocode shows the modified parallel merge sort method using the parallel merge algorithm (adopted from Cormen et al.).
/** * A: Input array * B: Output array * lo: lower bound * hi: upper bound * off: offset */algorithm parallelMergesort(A, lo, hi, B, off)is len := hi - lo + 1if len == 1then B[off] := A[lo]else let T[1..len] be a new array mid := ⌊(lo + hi) / 2⌋ mid' := mid - lo + 1fork parallelMergesort(A, lo, mid, T, 1) parallelMergesort(A, mid + 1, hi, T, mid' + 1)join parallelMerge(T, 1, mid', mid' + 1, len, B, off)
In order to analyze arecurrence relation for the worst case span, the recursive calls of parallelMergesort have to be incorporated only once due to their parallel execution, obtaining
For detailed information about the complexity of the parallel merge procedure, seeMerge algorithm.
The solution of this recurrence is given by
This parallel merge algorithm reaches a parallelism of, which is much higher than the parallelism of the previous algorithm. Such a sort can perform well in practice when combined with a fast stable sequential sort, such asinsertion sort, and a fast sequential merge as a base case for merging small arrays.[20]
It seems arbitrary to restrict the merge sort algorithms to a binary merge method, since there are usually p > 2 processors available. A better approach may be to use aK-way merge method, a generalization of binary merge, in which sorted sequences are merged. This merge variant is well suited to describe a sorting algorithm on aPRAM.[21][22]
Given an unsorted sequence of elements, the goal is to sort the sequence with availableprocessors. These elements are distributed equally among all processors and sorted locally using a sequentialSorting algorithm. Hence, the sequence consists of sorted sequences of length. For simplification let be a multiple of, so that for.
These sequences will be used to perform a multisequence selection/splitter selection. For, the algorithm determines splitter elements with global rank. Then the corresponding positions of in each sequence are determined withbinary search and thus the are further partitioned into subsequences with.
Furthermore, the elements of are assigned to processor, means all elements between rank and rank, which are distributed over all. Thus, each processor receives a sequence of sorted sequences. The fact that the rank of the splitter elements was chosen globally, provides two important properties: On the one hand, was chosen so that each processor can still operate on elements after assignment. The algorithm is perfectlyload-balanced. On the other hand, all elements on processor are less than or equal to all elements on processor. Hence, each processor performs thep-way merge locally and thus obtains a sorted sequence from its sub-sequences. Because of the second property, no furtherp-way-merge has to be performed, the results only have to be put together in the order of the processor number.
In its simplest form, given sorted sequences distributed evenly on processors and a rank, the task is to find an element with a global rank in the union of the sequences. Hence, this can be used to divide each in two parts at a splitter index, where the lower part contains only elements which are smaller than, while the elements bigger than are located in the upper part.
The presentedsequential algorithm returns the indices of the splits in each sequence, e.g. the indices in sequences such that has a global rank less than and.[23]
algorithm msSelect(S : Array of sorted Sequences [S_1,..,S_p], k : int)isfor i = 1to pdo (l_i, r_i) = (0, |S_i|-1)while there exists i: l_i < r_ido// pickPivot Element in S_j[l_j], .., S_j[r_j], chose random j uniformlyv := pickPivot(S, l, r)for i = 1to pdo m_i = binarySearch(v, S_i[l_i, r_i]) // sequentiallyif m_1 + ... + m_p >= kthen // m_1+ ... + m_p is the global rank of v r := m // vector assignmentelse l := mreturn l
For the complexity analysis thePRAM model is chosen. If the data is evenly distributed over all, the p-fold execution of thebinarySearch method has a running time of. The expected recursion depth is as in the ordinaryQuickselect. Thus the overall expected running time is.
Applied on the parallel multiway merge sort, this algorithm has to be invoked in parallel such that all splitter elements of rank for are found simultaneously. These splitter elements can then be used to partition each sequence in parts, with the same total running time of.
Below, the complete pseudocode of the parallel multiway merge sort algorithm is given. We assume that there is a barrier synchronization before and after the multisequence selection such that every processor can determine the splitting elements and the sequence partition properly.
/** * d: Unsorted Array of Elements * n: Number of Elements * p: Number of Processors * return Sorted Array */algorithm parallelMultiwayMergesort(d : Array, n : int, p : int)is o :=new Array[0, n] // the output arrayfor i = 1to pdo in parallel // each processor in parallel S_i := d[(i-1) * n/p, i * n/p] // Sequence of length n/psort(S_i) // sort locallysynchv_i := msSelect([S_1,...,S_p], i * n/p) // element with global rank i * n/psynch(S_i,1, ..., S_i,p) := sequence_partitioning(si, v_1, ..., v_p) // split s_i into subsequences o[(i-1) * n/p, i * n/p] := kWayMerge(s_1,i, ..., s_p,i) // merge and assign to output arrayreturn o
Firstly, each processor sorts the assigned elements locally using a sorting algorithm with complexity. After that, the splitter elements have to be calculated in time. Finally, each group of splits have to be merged in parallel by each processor with a running time of using a sequentialp-way merge algorithm. Thus, the overall running time is given by
.
The multiway merge sort algorithm is very scalable through its high parallelization capability, which allows the use of many processors. This makes the algorithm a viable candidate for sorting large amounts of data, such as those processed incomputer clusters. Also, since in such systems memory is usually not a limiting resource, the disadvantage ofspace complexity of merge sort is negligible. However, other factors become important in such systems, which are not taken into account when modelling on aPRAM. Here, the following aspects need to be considered:Memory hierarchy, when the data does not fit into the processors cache, or the communication overhead of exchanging data between processors, which could become a bottleneck when the data can no longer be accessed via the shared memory.
Sanders et al. have presented in their paper abulk synchronous parallel algorithm for multilevel multiway mergesort, which divides processors into groups of size. All processors sort locally first. Unlike single level multiway mergesort, these sequences are then partitioned into parts and assigned to the appropriate processor groups. These steps are repeated recursively in those groups. This reduces communication and especially avoids problems with many small messages. The hierarchical structure of the underlying real network can be used to define the processor groups (e.g.racks,clusters,...).[22]
Merge sort was one of the first sorting algorithms where optimal speed up was achieved, with Richard Cole using a clever subsampling algorithm to ensureO(1) merge.[24] Other sophisticated parallel sorting algorithms can achieve the same or better time bounds with a lower constant. For example, in 1991 David Powers described a parallelizedquicksort (and a relatedradix sort) that can operate inO(logn) time on aCRCWparallel random-access machine (PRAM) withn processors by performing partitioning implicitly.[25] Powers further shows that a pipelined version of Batcher'sBitonic Mergesort atO((logn)2) time on a butterflysorting network is in practice actually faster than hisO(logn) sorts on a PRAM, and he provides detailed discussion of the hidden overheads in comparison, radix and parallel sorting.[26]
Althoughheapsort has the same time bounds as merge sort, it requires only Θ(1) auxiliary space instead of merge sort's Θ(n). On typical modern architectures, efficientquicksort implementations generally outperform merge sort for sorting RAM-based arrays.[27] Quicksorts are preferred when the data size to be sorted is lesser, since the space complexity for quicksort is O(logn), it helps in utilizing cache locality better than merge sort (with space complexity O(n)).[27] On the other hand, merge sort is a stable sort and is more efficient at handling slow-to-access sequential media. Merge sort is often the best choice for sorting alinked list: in this situation it is relatively easy to implement a merge sort in such a way that it requires only Θ(1) extra space, and the slow random-access performance of a linked list makes some other algorithms (such as quicksort) perform poorly, and others (such as heapsort) completely impossible.
As ofPerl 5.8, merge sort is its default sorting algorithm (it was quicksort in previous versions of Perl).[28] InJava, theArrays.sort() methods use merge sort or a tuned quicksort depending on the datatypes and for implementation efficiency switch toinsertion sort when fewer than seven array elements are being sorted.[29] TheLinux kernel uses merge sort for its linked lists.[30]
Timsort, a tuned hybrid of merge sort and insertion sort is used in variety of software platforms and languages including the Java and Android platforms[31] and is used byPython since version 2.3; since version 3.11, Timsort's merge policy was updated toPowersort.[32]
