## Concept:

# Merge sort

## Headline

The Merge sort sorting algorithm

## Citation

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

Conceptually, a merge sort works as follows

- Divide the unsorted list into n sublists, each containing 1 element (a list of 1 element is considered sorted).
- 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.:

### 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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