Counting sort

Counting sort
Class	Sorting Algorithm
Data structure	Array
Worst-case performance	, where k is the range of the non-negative key values.
Worst-case space complexity

In computer science, counting sort is an algorithm for sorting a collection of objects according to keys that are small positive integers; that is, it is an integer sorting algorithm. It operates by counting the number of objects that possess distinct key values, and applying prefix sum on those counts to determine the positions of each key value in the output sequence. Its running time is linear in the number of items and the difference between the maximum key value and the minimum key value, so it is only suitable for direct use in situations where the variation in keys is not significantly greater than the number of items. It is often used as a subroutine in radix sort, another sorting algorithm, which can handle larger keys more efficiently.^[1]^[2]^[3]

Counting sort is not a comparison sort; it uses key values as indexes into an array and the $Ω (n log n)$ lower bound for comparison sorting will not apply.^[1] Bucket sort may be used in lieu of counting sort, and entails a similar time analysis. However, compared to counting sort, bucket sort requires linked lists, dynamic arrays, or a large amount of pre-allocated memory to hold the sets of items within each bucket, whereas counting sort stores a single number (the count of items) per bucket.^[4]

^ ^a ^b Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), "8.2 Counting Sort", Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, pp. 168–170, ISBN 0-262-03293-7. See also the historical notes on page 181.
^ Edmonds, Jeff (2008), "5.2 Counting Sort (a Stable Sort)", How to Think about Algorithms, Cambridge University Press, pp. 72–75, ISBN 978-0-521-84931-9.
^ Sedgewick, Robert (2003), "6.10 Key-Indexed Counting", Algorithms in Java, Parts 1-4: Fundamentals, Data Structures, Sorting, and Searching (3rd ed.), Addison-Wesley, pp. 312–314.
^ Cite error: The named reference knuth was invoked but never defined (see the help page).

[clrs-1] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), "8.2 Counting Sort", Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, pp. 168–170, ISBN 0-262-03293-7. See also the historical notes on page 181.

[edmonds-2] Edmonds, Jeff (2008), "5.2 Counting Sort (a Stable Sort)", How to Think about Algorithms, Cambridge University Press, pp. 72–75, ISBN 978-0-521-84931-9.

[sedgewick-3] Sedgewick, Robert (2003), "6.10 Key-Indexed Counting", Algorithms in Java, Parts 1-4: Fundamentals, Data Structures, Sorting, and Searching (3rd ed.), Addison-Wesley, pp. 312–314.

[knuth-4] Cite error: The named reference knuth was invoked but never defined (see the help page).

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