Batcher odd–even mergesort

Batcher odd–even mergesort
Visualization of the odd–even mergesort network with eight inputs
Class	Sorting algorithm
Data structure	Array
Worst-case performance	$O(\log ^{2}(n))$ parallel time
Best-case performance	$O(\log ^{2}(n))$ parallel time
Average performance	$O(\log ^{2}(n))$ parallel time
Worst-case space complexity	$O(n\log ^{2}(n))$ non-parallel time
Optimal	No

Batcher's odd–even mergesort is a generic construction devised by Ken Batcher for sorting networks of size O(n (log n)) and depth O((log n)), where n is the number of items to be sorted. Although it is not asymptotically optimal, Knuth concluded in 1998, with respect to the AKS network that "Batcher's method is much better, unless n exceeds the total memory capacity of all computers on earth!"

It is popularized by the second GPU Gems book, as an easy way of doing reasonably efficient sorts on graphics-processing hardware.

Pseudocode

Various recursive and iterative schemes are possible to calculate the indices of the elements to be compared and sorted. This is one iterative technique to generate the indices for sorting n elements:

# note: the input sequence is indexed from 0 to (n-1)
for p = 1, 2, 4, 8, ... # as long as p < n
  for k = p, p/2, p/4, p/8, ... # as long as k >= 1
    for j = mod(k,p) to (n-1-k) with a step size of 2k
      for i = 0 to min(k-1, n-j-k-1) with a step size of 1
        if floor((i+j) / (p*2)) == floor((i+j+k) / (p*2))
          compare and sort elements (i+j) and (i+j+k)

Non-recursive calculation of the partner node index is also possible.

References

Batcher, Ken (1968), "Sorting Networks and their Applications", Proceedings of the April 30--May 2, 1968, spring joint computer conference on - AFIPS '68 (Spring), Atlantic City, New Jersey: Association for Computing Machinery, pp. 307–314, doi:10.1145/1468075.1468121, S2CID 207171031, archived from the original on 2020-10-24
D.E. Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Second Edition. Addison-Wesley, 1998. ISBN 0-201-89685-0. Section 5.3.4: Networks for Sorting, pp. 219–247.
"Chapter 46. Improved GPU Sorting".
"Sorting network from Batcher's Odd-Even merge: partner calculation". Renat Bekbolatov. Retrieved 7 May 2015.

External links

Odd–even mergesort at hs-flensburg.de
Odd-even mergesort network generator Interactive Batcher's Odd-Even merge-based sorting network generator.

v t e Sorting algorithms
Theory	Computational complexity theory Big O notation Total order Lists Inplacement Stability Comparison sort Adaptive sort Sorting network Integer sorting X + Y sorting Transdichotomous model Quantum sort
Exchange sorts	Bubble sort Cocktail shaker sort Odd–even sort Comb sort Gnome sort Proportion extend sort Quicksort
Selection sorts	Selection sort Heapsort Smoothsort Cartesian tree sort Tournament sort Cycle sort Weak-heap sort
Insertion sorts	Insertion sort Shellsort Splaysort Tree sort Library sort Patience sorting
Merge sorts	Merge sort Cascade merge sort Oscillating merge sort Polyphase merge sort
Distribution sorts	American flag sort Bead sort Bucket sort Burstsort Counting sort Interpolation sort Pigeonhole sort Proxmap sort Radix sort Flashsort
Concurrent sorts	Bitonic sorter Batcher odd–even mergesort Pairwise sorting network Samplesort
Hybrid sorts	Block merge sort Kirkpatrick–Reisch sort Timsort Introsort Spreadsort Merge-insertion sort
Other	Topological sorting Pre-topological order Pancake sorting Spaghetti sort
Impractical sorts	Stooge sort Slowsort Bogosort

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Sorting algorithms

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