- Counting sort
In computer science, counting sort is an algorithm for sorting a collection of objects according to keys that are small integers; that is, it is an integer sorting algorithm. It operates by counting the number of objects that have each distinct key value, and using arithmetic 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 maximum key value, so it is only suitable for use directly in situations where the keys are not significantly larger than the number of items. However, it is often used as a subroutine in another sorting algorithm, radix sort, that can handle larger keys more efficiently.
Because counting sort uses key values as indexes into an array, it is not a comparison sort, and the Ω(n log n) lower bound for comparison sorting is inapplicable to it. Bucket sort may be used for many of the same tasks as counting sort, with a similar time analysis; however, compared to counting sort, bucket sort requires linked lists, dynamic arrays or a large amount of preallocated memory to hold the sets of items within each bucket, whereas counting sort instead stores a single number (the count of items) per bucket.
Input and output assumptions
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. In some descriptions of counting sort, the input to be sorted is assumed to be more simply a sequence of integers itself, 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 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.
In summary, the algorithm loops over the items, computing a histogram of the number of times each key occurs within the input collection. It then performs a prefix sum computation (a second loop, over the range of possible keys) to determine, for each key, the starting position in the output array of the items having that key. Finally, it loops over the items again, moving each item into its sorted position.
In pseudocode, this may be expressed as follows:
allocate an array Count[0..k]; initialize each array cell to zero for each input item x: Count[key(x)] = Count[key(x)] + 1 total = 0 for i = 0, 1, ... k: c = Count[i] Count[i] = total total = total + c allocate an output array Output[0..n-1] for each input item x: store x in Output[Count[key(x)]] Count[key(x)] = Count[key(x)] + 1 return Output
After the first for loop, Count[i] stores the number of items with key equal to i. After the second for loop, it instead stores the number of items with key less than i, which is the same as the first index at which an item with key i should be stored. Throughout the third loop, Count[i] always stores the next position into which an item with key i should be stored, so each item is moved into its correct position in the output array.
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).
Because it uses arrays of length k + 1 and n, the total space usage of the algorithm is also O(n + k). 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).
If each item to be sorted is itself an integer, and the keys are the same as the items, 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.
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. 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.
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.
- ^ a b c d e f g h i 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.
- ^ a b c d e Edmonds, Jeff (2008), "5.2 Counting Sort (a Stable Sort)", How to Think about Algorithms, Cambridge University Press, pp. 72–75, ISBN 9780521849319 .
- ^ a b c d e 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 .
- ^ a b Knuth, D. E. (1998), The Art of Computer Programming, Volume 3: Sorting and Searching (2nd ed.), Addison-Wesley, ISBN 0-201-89685-0 . Section 5.2, Sorting by counting, pp. 75–80, and historical notes, p. 170.
- ^ Burris, David S.; Schember, Kurt (1980), "Sorting sequential files with limited auxiliary storage", Proceedings of the 18th annual Southeast Regional Conference, New York, NY, USA: ACM, pp. 23–31, doi:10.1145/503838.503855 .
- ^ Zagha, Marco; Blelloch, Guy E. (1991), "Radix sort for vector multiprocessors", Proceedings of Supercomputing '91, November 18-22, 1991, Albuquerque, NM, USA, IEEE Computer Society / ACM, pp. 712–721, doi:10.1145/125826.126164 .
- ^ Reif, John H. (1985), "An optimal parallel algorithm for integer sorting", Proc. 26th Annual Symposium on Foundations of Computer Science (FOCS 1985), pp. 496–504, doi:10.1109/SFCS.1985.9 .
- ^ Seward, H. H. (1954), "2.4.6 Internal Sorting by Floating Digital Sort", Information sorting in the application of electronic digital computers to business operations, Master's thesis, Report R-232, Massachusetts Institute of Technology, Digital Computer Laboratory, pp. 25–28, http://bitsavers.org/pdf/mit/whirlwind/R-series/R-232_Information_Sorting_in_the_Application_of_Electronic_Digital_Computers_to_Business_Operations_May54.pdf .
- Demonstration applet from Cardiff University
- Efficient Counting Sort in Haskell
- Kagel, Art S. (2 June 2006), "counting sort", in Black, Paul E., Dictionary of Algorithms and Data Structures, U.S. National Institute of Standards and Technology, http://www.nist.gov/dads/HTML/countingsort.html, retrieved 2011-04-21 .
- A simple Counting Sort implementation.
Sorting algorithms Theory Exchange sorts Selection sorts Insertion sorts Merge sorts Distribution sorts Concurrent sorts Hybrid sorts Quantum sorts Other
Wikimedia Foundation. 2010.
См. также в других словарях:
Counting Crows discography — Counting Crows discography Adam Duritz, lead singer of Counting Crows Releases ↙Studio albums 5 … Wikipedia
Counting Crows — Adam Duritz, chanteur de Counting Crows Pays d’origine San Francisco … Wikipédia en Français
Counting Bodies Like Sheep to the Rhythm of the War Drums — Front Cover of the single Single by A Perfect Circle from the album eMOTIVe … Wikipedia
Counting Single Transferable Votes — The single transferable vote (STV) is a voting system based on proportional representation and preferential voting. Under STV, an elector s vote is initially allocated to his or her most preferred candidate. After candidates have been either… … Wikipedia
Radix sort — In computer science, radix sort is a sorting algorithm that sorts integers by processing individual digits. Because integers can represent strings of characters (e.g., names or dates) and specially formatted floating point numbers, radix sort is… … Wikipedia
Pigeonhole sort — Class Sorting algorithm Data structure Array Worst case performance O(N + n), where N is the range of key values and n is the input size Worst case space complexity O(N * n) … Wikipedia
Bucket sort — Bucket sort, or bin sort, is a sorting algorithm that works by partitioning an array into a number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm, or by recursively applying the bucket sorting… … Wikipedia
Cycle sort — Example of cycle sort sorting a list of random numbers. Class Sorting algorithm Data structure Array Worst case performance Θ(n2) … Wikipedia
Merge sort — Example of merge sort sorting a list of random dots. Class Sorting algorithm Data structure Array Worst case performance O(n log n) … Wikipedia
Selection sort — Infobox Algorithm class=Sorting algorithm data=Array time= О(n²) space= О(n) total, O(1) auxiliary optimal=Not usuallySelection sort is a sorting algorithm, specifically an in place comparison sort. It has O( n 2) complexity, making it… … Wikipedia