Batcher odd–even mergesort

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Batcher odd–even mergesort
Visualization of the odd–even mergesort network with eight inputs
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 case performance O(\log^2(n)) parallel time
Worst case space complexity O(n\log^2(n)) comparators

Batcher's odd–even mergesort is a generic construction devised by Ken Batcher for sorting networks of size O(n (log n)2) and depth O((log n)2), 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!"[1]

It is popularized by the second GPU Gems book,[2] as an easy way of doing reasonably efficient sorts on graphics-processing hardware.

Example code[edit]

The following is an implementation of odd–even mergesort algorithm in Python. The input is a list x of length a power of 2. The output is a list sorted in ascending order.

def oddeven_merge(lo, hi, r):
    step = r * 2
    if step < hi - lo:
        yield from oddeven_merge(lo, hi, step)
        yield from oddeven_merge(lo + r, hi, step)
        yield from [(i, i + r) for i in range(lo + r, hi - r, step)]
    else:
        yield (lo, lo + r)
 
def oddeven_merge_sort_range(lo, hi):
    """ sort the part of x with indices between lo and hi.
 
    Note: endpoints (lo and hi) are included.
    """
    if (hi - lo) >= 1:
        # if there is more than one element, split the input
        # down the middle and first sort the first and second
        # half, followed by merging them.
        mid = lo + ((hi - lo) // 2)
        yield from oddeven_merge_sort_range(lo, mid)
        yield from oddeven_merge_sort_range(mid + 1, hi)
        yield from oddeven_merge(lo, hi, 1)
 
def oddeven_merge_sort(length):
    """ "length" is the length of the list to be sorted.
    Returns a list of pairs of indices starting with 0 """
    yield from oddeven_merge_sort_range(0, length - 1)
 
def compare_and_swap(x, a, b):
    if x[a] > x[b]:
        x[a], x[b] = x[b], x[a]
>>> data = [2, 4, 3, 5, 6, 1, 7, 8]
>>> pairs_to_compare = list(oddeven_merge_sort(len(data)))
>>> pairs_to_compare
[(0, 1), (2, 3), (0, 2), (1, 3), (1, 2), (4, 5), (6, 7), (4, 6), (5, 7), (5, 6), (0, 4), (2, 6), (2, 4), (1, 5), (3, 7), (3, 5), (1, 2), (3, 4), (5, 6)]
>>> for i in pairs_to_compare: compare_and_swap(data, *i)
>>> data
[1, 2, 3, 4, 5, 6, 7, 8]

See also[edit]

References[edit]

  1. ^ D.E. Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1998. ISBN 0-201-89685-0. Section 5.3.4: Networks for Sorting, pp. 219–247.
  2. ^ http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter46.html

External links[edit]