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selection algorithm
In computer science, a selection algorithm is an algorithm for finding the ''k''th smallest number in a list or array; such a number is called the ''k''th ''order statistic''. This includes the cases of finding the minimum, maximum, and median elements. There are O(''n'') (worst-case linear time) selection algorithms, and sublinear performance is possible for structured data; in the extreme, O(1) for an array of sorted data. Selection is a subproblem of more complex problems like the nearest neighbor and shortest path problems. Many selection algorithms are derived by generalizing a sorting algorithm, and conversely some sorting algorithms can be derived as repeated application of selection.
The simplest case of a selection algorithm is finding the minimum (or maximum) element by iterating through the list, keeping track of the running minimum – the minimum so far – (or maximum) and can be seen as related to the selection sort. Conversely, the hardest case of a selection algorithm is finding the median, and this necessarily takes ''n''/2 storage. In fact, a specialized median-selection algorithm can be used to build a general selection algorithm, as in median of medians. The best-known selection algorithm is quickselect, which is related to quicksort; like quicksort, it has (asymptotically) optimal average performance, but poor worst-case performance, though it can be modified to give optimal worst-case performance as well.
== Selection by sorting ==
By sorting the list or array then selecting the desired element, selection can be reduced to sorting. This method is inefficient for selecting a single element, but is efficient when many selections need to be made from an array, in which case only one initial, expensive sort is needed, followed by many cheap selection operations – O(1) for an array, though selection is O(''n'') in a list, even if sorted, due to lack of random access. In general, sorting requires O(''n'' log ''n'') time, where ''n'' is the length of the list, although a lower bound is possible with non-comparative sorting algorithms like radix sort and counting sort.
Rather than sorting the whole list or array, one can instead use partial sorting to select the ''k'' smallest or ''k'' largest elements. The ''k''th smallest (resp., ''k''th largest element) is then the largest (resp., smallest element) of the partially sorted list – this then takes O(1) to access in an array and O(''k'') to access in a list. This is more efficient than full sorting, but less efficient than simply selecting, and takes O(''n'' + ''k'' log ''k'') time, due to the sorting of the ''k'' elements. Partial sorting algorithms can often be derived from (total) sorting algorithms. As with total sorting, partial sorting means that further selections (below the ''k''th element) can be done in O(1) time for an array and O(''k'') time for a list. Further, if the partial sorting also partitions the original data into "sorted" and "unsorted", as with an in-place sort, the partial sort can be extended to a larger partial sort by only sorting the incremental portion, and if this is done, further selections above the ''k''th element can also be done relatively cheaply.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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