Note: This guide is not intended as a fully comprehensive guide
to sorting, only a glimpse at how recursion can be used to
effectively sort. For more information on the sorting
algorithms described within (as well as other algorithms not
mentioned), please see the SparkNote guide to sorting
algorithms.

Recursive techniques can be utilized in sorting algorithms,
allowing for the sorting of n elements in O(nlogn) time
(compared with the O(n^{2}) efficiency of bubble sort. Two
such algorithms which will be examined here are Mergesort
and Quicksort.

Mergesort

To discuss mergesort, we must first discuss the merge operation, the
process of combining to sorted data sets into one sorted data set.
The merge operation can be accomplished in O(n) time.

Given two sorted data sets to merge, we start at the beginning
of each:

We take the smallest element from the two we're comparing
(these being the two elements at the front of the sets), and we
move it into the new data set. This repetition is done until
all of the elements have been moved, or until one of the lists
is empty, at which point all of the elements in the non-empty
list are moved to the new list, leaving them in the same order.

The following code implements the merge operation. It merges
a1[] and a2[], and stores the merged list back into
a1[] (therefore a1[] must be large enough to hold both
lists):

void merge(int a1[], int a1_len, int a2[], int a2_len)
{
int joint_size;
int a1_index, a2_index, joint_index;
int *tempp;
/* Create a temporary array */
joint_size = (a1_len + a2_len) * sizeof(int);
tempp = (int *) malloc(joint_size);
if (tempp == NULL) {
printf("Unable to malloc space.\n");
return;
}
/* Do the merge pass */
joint_index = 0;
a1_index = 0;
a2_index = 0;
while (a1_index < a1_len || a2_index < a2_len) {
if (a1_index < a1_len && a2_index < a2_len) {
if (a1[a1_index] < a2[a2_index]) {
tempp[joint_index] = a1[a1_index];
} else {
tempp[joint_index] = a2[a2_index];
}
} else if (a1_index < a1_len) {
tempp[joint_index] = a1[a1_index];
} else {
tempp[joint_index] = a2[a2_index];
}
}
/* Copy the temporary back into the argument array */
for (joint_index = 0; joint_index < a1_len + a2_len; joint_index++) {
a1[joint_index] = tempp[joint_index];
}
/* Free the temporary array */
free(tempp);
}

This merge operation is key to the mergesort algorithm.

Mergesort is a divide-and-conquer algorithm, meaning that it
accomplishes its task by dividing up the data in order to
better handle it. Mergesort has the following algorithm: split
the list in half, sort each side, then merge the two sides
together. See the recursive aspect? The second step of the
mergesort algorithm is to sort each half. What algorithm might
we use to sort each half of the set? In the spirit of
recursion, we'll use mergesort.

void mergesort(int arr[], int n)
{
int a1_len;
int a2_len;
if (n <= 1) {
return;
} else {
a1_len = n / 2;
a2_len = n - a1_len;
mergesort(arr, a1_len);
mergesort(&arr[a1_len], a2_len);
merge(arr, a1_len, &arr[a1_len], a2_len);
}
}

Just as with binary search, mergesort continually splits the
data set in half, doing O(n) operations at each level of
recursion. There are O(logn) splits of the data set.
Therefore, mergesort() runs in O(nlogn) time, the provably
best efficiency for a comparison-based sort.

Quicksort

Quicksort, an algorithm developed by C.A.R. Hoare in the 1960s,
is one of the most efficient sorting algorithms; for a large,
random data set, it is often considered to be the fastest sort.
Like mergesort(), it is also a divide-and-conquer algorithm
resulting in an average case running time of O(nlogn).

Like mergesort, quicksort partitions the data into two sets.
The algorithm for quicksort is as follows: choose a pivot value
(a value to which we'll compare the rest of the data in the
set), put all values smaller than that pivot on one side of the
set and all values greater than that pivot on the other side of
the set, then sort each half. Again, we'll recursively sort
each half of the data set using the same algorithm, quicksort.

void swap_elements_ptr(int *a, int *b)
{
int temp = *a; *a = *b; *b = temp;
}
void quick_sort(int arr[], int n)
{
int num_equal, num_on_left, num_on_right;
int val, *ip, *equalp, *less_thanp, *greater_thanp;
if (n <= 1) return;
val = arr[0];
equalp = arr;
less_thanp = &arr[1];
greater_thanp = &arr[n - 1];
while (less_thanp <= greater_thanp) {
if (*less_thanp == val) {
equalp;
swap_elements_ptr(less_thanp, equalp);
less_thanp;
} else if (*less_thanp > val) {
swap_elements_ptr(less_thanp, greater_thanp);
greater_thanp--;
} else less_thanp;
}
less_thanp--;
greater_thanp;
for (ip = arr; ip <= equalp; ip++) {
swap_elements_ptr(ip, less_thanp);
less_thanp--;
}
num_equal = equalp - arr + 1;
num_on_left = less_thanp - arr + 1;
num_on_right = n - num_equal - num_on_left;
quick_sort(arr, num_on_left);
quick_sort(greater_thanp, num_on_right);
}

Sometimes when the size of the partition gets small enough, a
programmer will use another non-recursive sorting algorithm,
like selection sort or bubble sort (see the SparkNote guide
on sorting if you are unfamiliar with this sort),
to sort the small sets; this often counters the inefficiency of
many recursive calls.

void swap_elements_ptr(int *a, int *b)
{
int temp = *a; *a = *b; *b = temp;
}
void quick_sort(int arr[], int n)
{
int num_equal, num_on_left, num_on_right;
int val, *ip, *equalp, *less_thanp, *greater_thanp;
int i, j;
/* Change base case to do bubblesort after threshold has been reached */
if (n <= 6) {
for (i = 0; i < n; i) {
for(j = 0; j < n-1; j) {
if (arr[j] > arr[j+1]) {
swap_elements_ptr(arr+j, arr+j+1);
}
}
}
return;
}
val = arr[0];
equalp = arr;
less_thanp = &arr[1];
greater_thanp = &arr[n - 1];
while (less_thanp <= greater_thanp) {
if (*less_thanp == val) {
equalp;
swap_elements_ptr(less_thanp, equalp);
less_thanp;
} else if (*less_thanp > val) {
swap_elements_ptr(less_thanp, greater_thanp);
greater_thanp--;
} else less_thanp;
}
less_thanp--;
greater_thanp;
for (ip = arr; ip <= equalp; ip++) {
swap_elements_ptr(ip, less_thanp);
less_thanp--;
}
num_equal = equalp - arr + 1;
num_on_left = less_thanp - arr + 1;
num_on_right = n - num_equal - num_on_left;
quick_sort(arr, num_on_left);
quick_sort(greater_thanp, num_on_right);
}

There are many variants of the basic quicksort algorithm, such
as different methods for picking a pivot value (most of which
are better than the one used above), methods for partitioning
the data, different thresholds for stopping the recursion, etc.
For more information, please refer to the SparkNote guide to
sorting.