Selection algorithm


In computer science, a selection algorithm is an algorithm for finding the th smallest value in a collection of orderable values, such as numbers. The value that it finds is called the order statistic. Selection includes as special cases the problems of finding the minimum, median, and maximum element in the collection. Selection algorithms include quickselect, and the median of medians algorithm. When applied to a collection of values, these algorithms take linear time, as expressed using big O notation. For data that is already structured, faster algorithms may be possible; as an extreme case, selection in an already-sorted array takes

Problem statement

An algorithm for the selection problem takes as input a collection of values, and a It outputs the smallest of these values, or, in some versions of the problem, a collection of the smallest values. For this to be well-defined, it should be possible to sort the values into an order from smallest to largest; for instance, they may be integers, floating-point numbers, or some other kind of 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 comparison operation that can determine the relative ordering of any two values, but may not perform any other kind of arithmetic operations on these values.
To simplify the problem, some works on this problem assume that the values are all distinct from each 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 as in zero-based numbering of arrays, or is it obtained by 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
With these conventions, the maximum value, among a collection of values, is obtained by When is an odd number, the median of the collection is obtained by When is even, there are two choices for the median, obtained by rounding this choice of down or up, respectively: the lower median with and the upper median with

Algorithms

Sorting and heapselect

As a baseline algorithm, selection of the smallest value in a collection of values can be performed by the following two steps:
  • Sort the collection
  • If the output of the sorting algorithm is an array, retrieve its element; otherwise, scan the sorted sequence to find the element.
The time for this method is dominated by the sorting step, which requires time using a 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 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
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 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 Applying this optimization to heapsort produces the heapselect algorithm, which can select the smallest value in This is fast when is small relative but degenerates to for larger values such as the choice used for median finding.

Pivoting

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 input values into two subsets: the set of elements less than the pivot, and the set of elements greater than the pivot. The algorithm can then determine where the smallest value is to be found, based on a comparison of with the sizes of these sets. In particular, the smallest value is and can be found recursively by applying the same selection algorithm then the smallest value is the pivot, and it can be returned immediately. In the remaining case, the smallest value is and more specifically it is the element in position It can be found by applying a selection algorithm recursively, seeking the value in this position
As with the related pivoting-based quicksort algorithm, the partition of the input into and 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 The time to compare the pivot against all the other values 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
  • 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 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
  • Quickselect chooses the pivot uniformly at random from the input values. It can be described as a prune and search algorithm, a variant of quicksort, with the same pivoting strategy, but where quicksort makes two recursive calls to sort the two subcollections quickselect only makes one of these two calls. Its expected time For any constant, the probability that its number of comparisons exceeds is superexponentially small
  • The Floyd–Rivest algorithm, a variation of quickselect, chooses a pivot by randomly sampling a subset of data values, for some sample and then recursively selecting two elements somewhat above and below position of the sample to use as pivots. With this choice, it is likely that 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 In their original work, Floyd and Rivest claimed that the term could be made as small as by a recursive sampling scheme, but the correctness of their analysis has been Instead, more rigorous analysis has shown that a version of their algorithm achieves for this Although the usual analysis of both quickselect and the Floyd–Rivest algorithm assumes the use of a true random number generator, a version of the Floyd–Rivest algorithm using a pseudorandom number generator seeded with only logarithmically many true random bits has been proven to run in linear time with high probability.
  • The median of medians method partitions the input into sets of five elements, and uses some other non-recursive method to find the median of each of these sets in constant time per set. It then recursively calls itself to find the median of these medians. Using the resulting median of medians as the pivot produces a partition with Thus, a problem on elements is reduced to two recursive problems on elements and at most elements. The total size of these two recursive subproblems is at allowing the total time to be analyzed as a geometric series adding Unlike quickselect, this algorithm is deterministic, not It was the first linear-time deterministic selection algorithm and is commonly taught in undergraduate algorithms classes as an example of a divide and conquer that does not divide into two equal However, the high constant factors in its time bound make it slower than quickselect in and slower even than sorting for inputs of moderate
  • 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

    Factories

The deterministic selection algorithms with the smallest known numbers of comparisons, for values of that are far from are based on the concept of factories, introduced in 1976 by Arnold Schönhage, Mike Paterson, and These are methods that build partial orders of certain specified types, on small subsets of input values, by using comparisons to combine smaller partial orders. As a very simple example, one type of factory can take as input a sequence of single-element partial orders, compare pairs of elements from these orders, and produce as output a sequence of two-element totally ordered sets. The elements used as the inputs to this factory could either be input values that have not been compared with anything yet, or "waste" values produced by other factories. The goal of a factory-based algorithm is to combine together different factories, with the outputs of some factories going to the inputs of others, in order to eventually obtain a partial order in which one element is larger than some other elements and smaller than another others. A careful design of these factories leads to an algorithm that, when applied to median-finding, uses at most comparisons. For other values the number of comparisons is

Parallel algorithms

s for selection have been studied since 1975, when Leslie Valiant introduced the parallel comparison tree model for analyzing these algorithms, and proved that in this model selection using a linear number of comparisons requires parallel steps, even for selecting the minimum or Researchers later found parallel algorithms for selection in steps, matching this In a randomized parallel comparison tree model it is possible to perform selection in a bounded number of steps and a linear number of On the more realistic parallel RAM model of computing, with exclusive read exclusive write memory access, selection can be performed in time with processors, which is optimal both in time and in the number of With concurrent memory access, slightly faster parallel time is possible in and the term in the time bound can be replaced