Editing Selection algorithm (section)

==Problem statement==
An algorithm for the selection problem takes as input a collection of values, and a {{nowrap|number <math>k</math>.}} It outputs the {{nowrap|<math>k</math>th}} smallest of these values, or, in some versions of the problem, a collection of the <math>k</math> smallest values. For this to be well-defined, it should be possible to [[Sorting|sort]] the values into an order from smallest to largest; for instance, they may be [[Integer (computer science)|integers]], [[floating-point number]]s, or some other kind of [[Object (computer science)|object]] with a numeric key. However, they are not assumed to have been already sorted. Often, selection algorithms are restricted to a comparison-based [[model of computation]], as in [[comparison sort]] algorithms, where the algorithm has access to a [[Relational operator|comparison operation]] that can determine the relative ordering of any two values, but may not perform any other kind of [[Arithmetic|arithmetic operations]] on these values.{{r|cunmun}}

To simplify the problem, some works on this problem assume that the values are all distinct from each {{nowrap|other,{{r|clrs}}}} or that some consistent tie-breaking method has been used to assign an ordering to pairs of items with the same value as each other. Another variation in the problem definition concerns the numbering of the ordered values: is the smallest value obtained by {{nowrap|setting <math>k=0</math>,}} as in [[zero-based numbering]] of arrays, or is it obtained by {{nowrap|setting <math>k=1</math>,}} following the usual English-language conventions for the smallest, second-smallest, etc.? This article follows the conventions used by Cormen et al., according to which all values are distinct and the minimum value is obtained from {{nowrap|<math>k=1</math>.{{r|clrs}}}}

With these conventions, the maximum value, among a collection of <math>n</math> values, is obtained by {{nowrap|setting <math>k=n</math>.}} When <math>n</math> is an [[odd number]], the [[median]] of the collection is obtained by {{nowrap|setting <math>k=(n+1)/2</math>.}} When <math>n</math> is even, there are two choices for the median, obtained by rounding this choice of <math>k</math> down or up, respectively: the ''lower median'' with <math>k=n/2</math> and the ''upper median'' with {{nowrap|<math>k=n/2+1</math>.{{r|clrs}}}}