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→Sorting and heapselect: sorted order may be useful later |
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* [[Sorting|Sort]] the collection
* If the output of the sorting algorithm is an array, jump to its <math>k</math>th element; otherwise, scan the sorted sequence to find the <math>k</math>th element.
The time for this method is dominated by the sorting step, which requires <math>\Theta(n\log n)</math> time using a [[comparison sort]].{{r|clrs|skiena}} Even when [[integer sorting]] algorithms may be used, these are generally slower than the linear time that may be achieved using specialized selection algorithms. Nevertheless, the simplicity of this approach makes it attractive, especially when a highly-optimized sorting routine is provided as part of a runtime library, but a selection algorithm is not. This method also produces a sorted version of the collection, which may be useful for other later computations, and in particular for selection with other choices of <math>k</math>.{{r|skiena}}
For a sorting algorithm that generates one item at a time, such as [[selection sort]], the scan can be done in tandem with the sort, and the sort can be terminated once the <math>k</math> element has been found. One possible design of a consolation bracket in a [[single-elimination tournament]], in which the teams who lost to the eventual winner play another mini-tournament to determine second place, can be seen as an instance of this method.{{r|bfprt}} Applying this optimization to [[heapsort]] produces the [[heapselect]] algorithm, which can select the <math>k</math>th smallest value in time <math>O(n+k\log n)</math>. This is fast when <math>k</math> is small relative to <math>n</math>, but degenerates to <math>O(n\log n)</math> for larger values of <math>k</math>, such as the choice <math>k=n/2</math> used for median finding.
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