Balanced K-way Merge Sort

Merge algorithms are a family of algorithms that take multiple sorted lists as input and produce a single list as output, containing all the elements of the inputs lists in sorted order. These algorithms are used as subroutines in various sorting algorithms, most famously merge sort.

Application

The merge algorithm plays a critical role in the merge sort algorithm, a comparison-based sorting algorithm. Conceptually, the merge sort algorithm consists of two steps:

# Recursively divide the list into sublists of (roughly) equal length, until each sublist contains only one element, or in the case of iterative (bottom up) merge sort, consider a list of ''n'' elements as ''n'' sub-lists of size 1. A list containing a single element is, by definition, sorted.
# Repeatedly merge sublists to create a new sorted sublist until the single list contains all elements. The single list is the sorted list.

The merge algorithm is used repeatedly in the merge sort algorithm.

An example merge sort is given in the illustration. It starts with an unsorted array of 7 integers. The array is divided into 7 partitions; each partition contains 1 element and is sorted. The sorted partitions are then merged to produce larger, sorted, partitions, until 1 partition, the sorted array, is left.

Merging two lists

Merging two sorted lists into one can be done in linear time and linear or constant space (depending on the data access model). The following pseudocode demonstrates an algorithm that merges input lists (either linked lists or Array data structure, arrays) and into a new list . The function yields the first element of a list; "dropping" an element means removing it from its list, typically by incrementing a pointer or index.

algorithm merge(A, B) is
inputs A, B : list
returns list

C := new empty list
while A is not empty and B is not empty do
if head(A) ≤ head(B) then
append head(A) to C
drop the head of A
else
append head(B) to C
drop the head of B

''// By now, either A or B is empty. It remains to empty the other input list.''
while A is not empty do
append head(A) to C
drop the head of A
while B is not empty do
append head(B) to C
drop the head of B

return C

When the inputs are linked lists, this algorithm can be implemented to use only a constant amount of working space; the pointers in the lists' nodes can be reused for bookkeeping and for constructing the final merged list.

In the merge sort algorithm, this subroutine is typically used to merge two sub-arrays , of a single array . This can be done by copying the sub-arrays into a temporary array, then applying the merge algorithm above. The allocation of a temporary array can be avoided, but at the expense of speed and programming ease. Various in-place merge algorithms have been devised, sometimes sacrificing the linear-time bound to produce an algorithm; see for discussion.

K-way merging

-way merging generalizes binary merging to an arbitrary number of sorted input lists. Applications of -way merging arise in various sorting algorithms, including patience sorting and an external sorting algorithm that divides its input into blocks that fit in memory, sorts these one by one, then merges these blocks.

Several solutions to this problem exist. A naive solution is to do a loop over the lists to pick off the minimum element each time, and repeat this loop until all lists are empty:

* Input: a list of lists.
* While any of the lists is non-empty:
** Loop over the lists to find the one with the minimum first element.
** Output the minimum element and remove it from its list.

Best, worst and average case, In the worst case, this algorithm performs element comparisons to perform its work if there are a total of elements in the lists.
It can be improved by storing the lists in a priority queue (heap (data structure), min-heap) keyed by their first element:

* Build a min-heap of the lists, using the first element as the key.
* While any of the lists is non-empty:
** Let .
** Output the first element of list and remove it from its list.
** Re-heapify .

Searching for the next smallest element to be output (find-min) and restoring heap order can now be done in time (more specifically, comparisons), and the full problem can be solved in time (approximately comparisons).

A third algorithm for the problem is a divide and conquer algorithm, divide and conquer solution that builds on the binary merge algorithm:

* If , output the single input list.
* If , perform a binary merge.
* Else, recursively merge the first lists and the final lists, then binary merge these.

When the input lists to this algorithm are ordered by length, shortest first, it requires fewer than comparisons, i.e., less than half the number used by the heap-based algorithm; in practice, it may be about as fast or slow as the heap-based algorithm.

Parallel merge

A task parallelism, parallel version of the binary merge algorithm can serve as a building block of a Merge sort#Parallel merge sort, parallel merge sort. The following pseudocode demonstrates this algorithm in a fork–join model, parallel divide-and-conquer style (adapted from Cormen ''et al.''). It operates on two sorted arrays and and writes the sorted output to array . The notation denotes the part of from index through , exclusive.

algorithm merge(A[i...j], B[k...ℓ], C[p...q]) is
inputs A, B, C : array
i, j, k, ℓ, p, q : indices

let m = j - i,
n = ℓ - k

if m < n then
swap A and B ''// ensure that A is the larger array: i, j still belong to A; k, ℓ to B''
swap m and n

if m ≤ 0 then
return ''// base case, nothing to merge''

let r = ⌊(i + j)/2⌋
let s = binary-search(A[r], B[k...ℓ])
let t = p + (r - i) + (s - k)
C[t] = A[r]

in parallel do
merge(A[i...r], B[k...s], C[p...t])
merge(A[r+1...j], B[s...ℓ], C[t+1...q])

The algorithm operates by splitting either or , whichever is larger, into (nearly) equal halves. It then splits the other array into a part with values smaller than the midpoint of the first, and a part with larger or equal values. (The binary search subroutine returns the index in where would be, if it were in ; that this always a number between and .) Finally, each pair of halves is merged Divide and conquer algorithm, recursively, and since the recursive calls are independent of each other, they can be done in parallel. Hybrid approach, where serial algorithm is used for recursion base case has been shown to perform well in practice

The Analysis of parallel algorithms#Overview, work performed by the algorithm for two arrays holding a total of elements, i.e., the running time of a serial version of it, is . This is optimal since elements need to be copied into . To calculate the Analysis of parallel algorithms#Overview, span of the algorithm, it is necessary to derive a Recurrence relation. Since the two recursive calls of ''merge'' are in parallel, only the costlier of the two calls needs to be considered. In the worst case, the maximum number of elements in one of the recursive calls is at most $\frac 3 4 n$ since the array with more elements is perfectly split in half. Adding the $\Theta\left( \log(n)\right)$ cost of the Binary Search, we obtain this recurrence as an upper bound:

$T_^\text(n) = T_^\text\left(\frac n\right) + \Theta\left( \log(n)\right)$

The solution is $T_^\text(n) = \Theta\left(\log(n)^2\right)$, meaning that it takes that much time on an ideal machine with an unbounded number of processors.

Note: The routine is not Sorting algorithm#Stability, stable: if equal items are separated by splitting and , they will become interleaved in ; also swapping and will destroy the order, if equal items are spread among both input arrays. As a result, when used for sorting, this algorithm produces a sort that is not stable.

Parallel merge of two lists

There are also algorithms that introduce parallelism within a single instance of merging of two sorted lists. These can be used in field-programmable gate arrays (FPGAs), specialized sorting circuits, as well as in modern processors with single-instruction multiple-data (SIMD) instructions.

Existing parallel algorithms are based on modifications of the merge part of either the bitonic sorter or odd-even mergesort. In 2018, Saitoh M. et al. introduced MMS for FPGAs, which focused on removing a multi-cycle feedback datapath that prevented efficient pipelining in hardware. Also in 2018, Papaphilippou P. et al. introduced FLiMS that improved the hardware utilization and performance by only requiring $\log_2(P)+1$ pipeline stages of compare-and-swap units to merge with a parallelism of elements per FPGA cycle.

Language support

Some computer languages provide built-in or library support for merging sorted Collection (abstract data type), collections.

C++

The C++'s Standard Template Library has the function , which merges two sorted ranges of iterators, and , which merges two consecutive sorted ranges ''in-place''. In addition, the (linked list) class has its own method which merges another list into itself. The type of the elements merged must support the less-than () operator, or it must be provided with a custom comparator.

C++17 allows for differing execution policies, namely sequential, parallel, and parallel-unsequenced.

Python

Python (programming language), Python's standard library (since 2.6) also has a function in the module, that takes multiple sorted iterables, and merges them into a single iterator.

See also

* Merge (revision control)
* Join (relational algebra)
* Join (SQL)
* Join (Unix)

References

Further reading

* Donald Knuth. ''The Art of Computer Programming'', Volume 3: ''Sorting and Searching'', Third Edition. Addison-Wesley, 1997. . Pages 158–160 of section 5.2.4: Sorting by Merging. Section 5.3.2: Minimum-Comparison Merging, pp. 197–207.

External links

High Performance Implementation
of Parallel and Serial Merge in C Sharp (programming language), C# with source i
GitHub
and in C++]
GitHub

{{DEFAULTSORT:Merge Algorithm
Articles with example pseudocode
Sorting algorithms