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*[[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> {{nowrap|and <math>R</math>,}} quickselect only makes one of these two calls. Its [[expected time]] {{nowrap|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 {{nowrap|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 {{nowrap|<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 {{nowrap|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 {{nowrap|term.{{r|knuth}}}}
[[File:Mid-of-mid.png|thumb|upright=1.35|Visualization of pivot selection for the [[median of medians]] method. Each set of five elements is shown as a column of dots in the figure, sorted in increasing order from top to bottom. If their medians (the green and purple dots in the middle row) are sorted in increasing order from left to right, and the median of medians is chosen as the pivot, then the <math>3n/10</math>
*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, {{nowrap|<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 {{nowrap|most <math>9n/10</math>,}} allowing the total time to be analyzed as a geometric series adding {{nowrap|to <math>O(n)</math>.}} Unlike quickselect, this algorithm is deterministic, not {{nowrap|randomized.{{r|clrs|erickson|bfprt}}}} It was the first linear-time deterministic selection algorithm {{nowrap|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 {{nowrap|practice,{{r|skiena}}}} and slower even than sorting for inputs of moderate {{nowrap|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> {{nowrap|time.{{r|musser}}}}
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