Headline

The Merge sort sorting algorithm

Citation

(http://en.wikipedia.org/wiki/Merge_sort, 21 April 2013)

Conceptually, a merge sort works as follows

  1. Divide the unsorted list into n sublists, each containing 1 element (a list of 1 element is considered sorted).
  2. Repeatedly merge sublists to produce new sublists until there is only 1 sublist remaining. This will be the sorted list.

Illustration

See the visualization of Merge sort on Wikipedia:

http://en.wikipedia.org/wiki/Merge_sort

See various illustrations of Merge sort as available on YouTube, e.g.:

Merge-sort with Transylvanian-saxon (German) folk dance

Recursive merge sort in Java

	public static void mergeSort(int[] a) {
		int[] temp = new int[a.length];
		mergeSort(a, temp, 0, a.length - 1);
	}

	public static void mergeSort(int[] a, int[] temp, int min, int max) {
		// Base case
		if (!(min < max)) return;

		// Split array and solve sub-problems recursively
		int middle = (min + max) / 2;
		mergeSort(a, temp, min, middle);
		mergeSort(a, temp, middle + 1, max);

		// Merge via temporary array
		merge(a, temp, min, middle, max);
	}

	public static void merge(int[] a, int[] temp, int min, int middle, int max) {
		int i = min; // loop over left half
		int j = middle + 1; // loop over right half
		int k = min; // loop over merged result
		while (k <= max)
			temp[k++] = (i <= middle && (j > max || a[i] < a[j])) ?
					  a[i++] // copy from left half
					: a[j++]; // copy from right half
		// Override array by merged tempory result
		for (k = min; k <= max; k++)
			a[k] = temp[k];
	}

Recursive merge sort in Haskell

-- Polymorphic sorting
sort :: Ord a => [a] -> [a]
sort [] = []
sort [x] = [x]
sort xs = merge (sort ys) (sort zs)
  where
    (ys,zs) = split xs

-- Split a list into halves
split :: [a] -> ([a],[a])
split xs = (take len xs, drop len xs)
  where
    len = length xs `div` 2

-- Merge sorted sublists
merge :: Ord a => [a] -> [a] -> [a]
merge [] ys = ys
merge xs [] = xs
merge (x:xs) (y:ys) =
  if x<=y 
    then x : merge xs (y:ys)
    else y : merge (x:xs) ys

The main sorting function relies on helpers for splitting the input lists into halves and for merging sorted sublists. An empty list as much as a singleton lists are immediately sorted, as modeled by the first two equations of sort. The split helper uses list-processing goodies take and drop to extract two halves (+/- one element for a list of odd length). The merge help offers two base cases for a merge to trivially complete, if either of the two operands is an empty list; the recursive case compares the heads of both operands to decide on which of them goes first into the merged result.

The implementation is exercised as follows:

main = do
  let input = [2,4,3,1,4]
  print $ sort input -- [1,2,3,4,4]

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