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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 difference between the maximum and minimum key values, so it is only suitable for direct use in situations where the variation in keys is not significantly greater 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.〔. See also the historical notes on page 181.〕〔.〕〔.〕 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 does not apply 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 items, each of which has a non-negative integer key whose maximum value is at most .〔 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 will be known in advance, and can be assumed to be part of the input to the algorithm. However, if the value of 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.〔〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Counting sort」の詳細全文を読む スポンサード リンク
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