Selection algorithm: Difference between revisions

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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. For inputs of moderate size, sorting can be faster than non-random selection algorithms, because of the smaller constant factors in its running time.{{r|erickson}} 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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Many methods for selection are based on choosing a special "pivot" element from the input, and using comparisons with this element to divide the remaining <math>n-1</math> input values into two subsets: the set <math>L</math> of elements less than the pivot, and the set <math>R</math> of elements greater than the pivot. The algorithm can then determine where the <math>k</math>th smallest value is to be found, based on a comparison of <math>k</math> with the sizes of these sets. In particular, if <math>k\le|L|</math>, the <math>k</math>th smallest value is in <math>L</math>, and can be found recursively by applying the same selection algorithm to <math>L</math>. If <math>k=|L|+1</math>, then the <math>k</math>th smallest value is the pivot, and it can be returned immediately. In the remaining case, the <math>k</math>th smallest value is in <math>R</math>, and more specifically it is the element in position <math>k-|L|-1</math> of <math>R</math>. It can be found by applying a selection algorithm recursively, seeking the value in this position in <math>R</math>.{{r|kletar}}

As with the related pivoting-based [[quicksort]] algorithm, the partition of the input into <math>L</math> and <math>R</math> may be done by making new collections for these sets, or by a method that partitions a given list or array data type in-place. Details vary depending on how the input collection is represented.<ref>For instance, Cormen et al. use an in-place array partition, while Kleinberg and Tardos describe the input as a set and use a method that partitions it into two new sets.</ref> The time to compare the pivot against all the other values is <math>O(n)</math>.{{r|kletar}} However, pivoting methods differ in how they choose the pivot, which affects how big the subproblems in each recursive call will be. The efficiency of these methods depends greatly on the choice of the pivot. If the pivot is chosen badly, the running time of this method can be as slow as <math>O(n^2)</math>.{{r|erickson}}

*If the pivot were exactly at the median of the input, then each recursive call would have at most half as many values as the previous call, and the total times would add in a [[geometric series]] to <math>O(n)</math>. However, finding the median is itself a selection problem, on the entire original input. Trying to find it by a recursive call to a selection algorithm would lead to an infinite recursion, because the problem size would not decrease in each call.{{r|kletar}}

*[[Quickselect]] chooses the pivot uniformly at random from the input values. It can be described as a variant of [[quicksort]], with the same pivoting strategy, but where quicksort makes two recursive calls to sort the two subcollections <math>L</math> and <math>R</math>, quickselect only makes one of these two calls. Its [[expected time]] is <math>O(n)</math>.{{r|clrs|kletar}}

*The [[Floyd–Rivest algorithm]], a variation of quickselect, chooses a pivot by randomly sampling a subset of <math>r</math> data values, for some sample size <math>r</math>, and then recursively selecting two elements somewhat above and below position <math>rk/n</math> of the sample to use as pivots. With this choice, it is likely that <math>k</math> is sandwiched between the two pivots, so that after pivoting only a small number of data values between the pivots are left for a recursive call. This method can achieve an expected number of comparisons that is <math>n+\min(k,n-k)+o(n)</math>.{{r|floriv}} In their original work, Floyd and Rivest claimed that the <math>o(n)</math> term could be made as small as <math>O(\sqrt n)</math> by a recursive sampling scheme, but the correctness of their analysis has been questioned.{{r|brown|prt}} Instead, more rigorous analysis has shown that a version of their algorithm achieves <math>O(\sqrt{n\log n})</math> for this term.{{r|knuth}}

*The [[median of medians]] method partitions the input into sets of five elements, and then uses some other method (rather than a recursive call) to find the median of each of these sets in constant time per set. It then recursively calls the same selection algorithm to find the median of these <math>n/5</math> medians, using the result as its pivot. It can be shown that, for this choice of pivot, <math>\max(|L|,|R|)\le 7n/10</math>. Thus, a problem on <math>n</math> elements is reduced to two recursive problems on <math>n/5</math> and at most <math>7n/10</math> elements. The total size of these two recursive subproblems is at most <math>9n/10</math>, allowing the total time to be analyzed as a geometric series adding to <math>O(n)</math>. Unlike quickselect, this algorithm is deterministic, not randomized.{{r|clrs|erickson|bfprt}} It was the first linear-time deterministic selection algorithm known,{{r|bfprt}} and is commonly taught in undergraduate algorithms classes as an example of a [[divide and conquer]] algorithm that does not divide into two equal subproblems. However, the high constant factors in its <math>O(n)</math> time bound make it slower than quickselect in practice.,{{r|skiena}} and slower even than sorting for inputs of moderate size.{{r|erickson}}

*Hybrid algorithms such as [[introselect]] can be used to achieve the practical performance of quickselect with a fallback to medians of medians guaranteeing worst-case <math>O(n)</math> time.{{r|musser}}

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| volume = 14

| year = 2001}}</ref>

<ref name=erickson>{{cite book|title=Algorithms|url=https://jeffe.cs.illinois.edu/teaching/algorithms/|first=Jeff|last=Erickson|date=June 2019|chapter=1.8: Linear-time selection|pages=35–39}}</ref>

<ref name=floriv>{{cite journal