Counting sort

In the most general case, the input to counting sort consists of a collection of n items, each of which has a non-negative integer key whose maximum value is at most k.[3] In some descriptions of counting sort, the input to be sorted is assumed to be more simply a sequence of integers itself,[1] but this simplification does not accommodate many applications of counting sort. For instance, when used as a subroutine in radix sort, the keys for each call to counting sort are individual digits of larger item keys; it would not suffice to return only a sorted list of the key digits, separated from the items.

In applications such as in radix sort, a bound on the maximum key value k will be known in advance, and can be assumed to be part of the input to the algorithm. However, if the value of k is not already known then it may be computed, as a first step, by an additional loop over the data to determine the maximum key value that actually occurs within the data.

The output is an array of the items, in order by their keys. Because of the application to radix sorting, it is important for counting sort to be a stable sort: if two items have the same key as each other, they should have the same relative position in the output as they did in the input.[1][2]

In pseudocode, the algorithm may be expressed as:

count = array of k+1 zeros
for x in input do
    count[key(x)] += 1

total = 0
for i in 0, 1, ... k do
    count[i], total = total, count[i] + total

output = array of the same length as input
for x in input do
    output[count[key(x)]] = x
    count[key(x)] += 1 

return output

Here input is the input array to be sorted, key returns the numeric key of each item in the input array, count is an auxiliary array used first to store the numbers of items with each key, and then (after the second loop) to store the positions where items with each key should be placed, k is the maximum value of the non-negative key values and output is the sorted output array.

In summary, the algorithm loops over the items in the first loop, computing a histogram of the number of times each key occurs within the input collection. After that, it then performs a prefix sum computation on count to determine, for each key, the position range where the items having that key should be placed in; i.e. items of key ${\displaystyle i}$ should be placed starting in position count[${\displaystyle i}$]. This is done through the second loop. Finally, it loops over the items again in the third loop, moving each item into its sorted position in the output array.[1][2][3] The relative order of items with equal keys is preserved here; i.e., this is a stable sort.

Because the algorithm uses only simple for loops, without recursion or subroutine calls, it is straightforward to analyze. The initialization of the count array, and the second for loop which performs a prefix sum on the count array, each iterate at most k + 1 times and therefore take O(k) time. The other two for loops, and the initialization of the output array, each take O(n) time. Therefore, the time for the whole algorithm is the sum of the times for these steps, O(n + k).[1][2]

Because it uses arrays of length k + 1 and n, the total space usage of the algorithm is also O(n + k).[1] For problem instances in which the maximum key value is significantly smaller than the number of items, counting sort can be highly space-efficient, as the only storage it uses other than its input and output arrays is the Count array which uses space O(k).[5]

If each item to be sorted is itself an integer, and used as key as well, then the second and third loops of counting sort can be combined; in the second loop, instead of computing the position where items with key i should be placed in the output, simply append Count[i] copies of the number i to the output.

This algorithm may also be used to eliminate duplicate keys, by replacing the Count array with a bit vector that stores a one for a key that is present in the input and a zero for a key that is not present. If additionally the items are the integer keys themselves, both second and third loops can be omitted entirely and the bit vector will itself serve as output, representing the values as offsets of the non-zero entries, added to the range's lowest value. Thus the keys are sorted and the duplicates are eliminated in this variant just by being placed into the bit array.

For data in which the maximum key size is significantly smaller than the number of data items, counting sort may be parallelized by splitting the input into subarrays of approximately equal size, processing each subarray in parallel to generate a separate count array for each subarray, and then merging the count arrays. When used as part of a parallel radix sort algorithm, the key size (base of the radix representation) should be chosen to match the size of the split subarrays.[6] The simplicity of the counting sort algorithm and its use of the easily parallelizable prefix sum primitive also make it usable in more fine-grained parallel algorithms.[7]

As described, counting sort is not an in-place algorithm; even disregarding the count array, it needs separate input and output arrays. It is possible to modify the algorithm so that it places the items into sorted order within the same array that was given to it as the input, using only the count array as auxiliary storage; however, the modified in-place version of counting sort is not stable.[3]

Although radix sorting itself dates back far longer, counting sort, and its application to radix sorting, were both invented by Harold H. Seward in 1954.[1][4][8]

The Wikibook Algorithm implementation has a page on the topic of: Counting sort

• Counting Sort html5 visualization
• Demonstration applet from Cardiff University
• Kagel, Art S. (2 June 2006), "counting sort", in Black, Paul E. (ed.), Dictionary of Algorithms and Data Structures, U.S. National Institute of Standards and Technology, retrieved 2011-04-21.
• A simple Counting Sort implementation.