Selection algorithm

In computer science, a selection algorithm is an algorithm for finding the kth smallest number in a list (such a number is called the kth order statistic). This includes the cases of finding the minimum, maximum, and median elements. There are O(n), worst-case linear time, selection algorithms. Selection is a subproblem of more complex problems like the nearest neighbor problem and shortest path problems.

The term "selection" is used in other contexts in computer science, including the stage of a genetic algorithm in which genomes are chosen from a population for later breeding; see Selection (genetic algorithm). This article addresses only the problem of determining order statistics.

Selection by sorting

Selection can be reduced to sorting by sorting the list and then extracting the desired element. This method is efficient when many selections need to be made from a list, in which case only one initial, expensive sort is needed, followed by many cheap extraction operations. In general, this method requires O(n log n) time, where n is the length of the list.

Linear minimum/maximum algorithms

Linear time algorithms to find minimums or maximums work by iterating over the list and keeping track of the minimum or maximum element so far.

Nonlinear general selection algorithm

Using the same ideas used in minimum/maximum algorithms, we can construct a simple, but inefficient general algorithm for finding the kth smallest or kth largest item in a list, requiring O(kn) time, which is effective when k is small. To accomplish this, we simply find the most extreme value and move it to the beginning until we reach our desired index. This can be seen as an incomplete selection sort. Here is the minimum-based algorithm:

 function select(list[1..n], k)
     for i from 1 to k
         minIndex = i
         minValue = list[i]
         for j from i+1 to n
             if list[j] < minValue
                 minIndex = j
                 minValue = list[j]
         swap list[i] and list[minIndex]
     return list[k]

Other advantages of this method are:

• After locating the jth smallest element, it requires only O(j + (k-j)²) time to find the kth smallest element, or only O(k) for k ≤ j.
• It can be done with linked list data structures, whereas the one based on partition requires random access.

Partition-based general selection algorithm

A general selection algorithm that is efficient in practice, but has poor worst-case performance, was conceived by the inventor of quicksort, C.A.R. Hoare, and is known as Hoare's selection algorithm or quickselect.

In quicksort, there is a subprocedure called partition that can, in linear time, group a list (ranging from indices left to right) into two parts, those less than a certain element, and those greater than or equal to the element. Here is pseudocode that performs a partition about the element list[pivotIndex]:

 function partition(list, left, right, pivotIndex)
     pivotValue := list[pivotIndex]
     swap list[pivotIndex] and list[right]  // Move pivot to end
     storeIndex := left
     for i from left to right
         if list[i] < pivotValue
             swap list[storeIndex] and list[i]
             increment storeIndex
     swap list[right] and list[storeIndex]  // Move pivot to its final place
     return storeIndex

In quicksort, we recursively sort both branches, leading to best-case Ω(n log n) time. However, when doing selection, we already know which partition our desired element lies in, since the pivot is in its final sorted position, with all those preceding it in sorted order and all those following it in sorted order. Thus a single recursive call locates the desired element in the correct partition:

 function select(list, left, right, k)
     if left = right // If the list contains only one element
         return list[left]  // Return that element
     select pivotIndex between left and right
     pivotNewIndex := partition(list, left, right, pivotIndex)
     pivotDist := pivotNewIndex - left + 1 
     // The pivot is in its final sorted position, 
     // so pivotDist reflects its 1-based position if list were sorted
     if pivotDist = k 
         return list[pivotNewIndex]
     else if k < pivotDist 
         return select(list, left, pivotNewIndex - 1, k)
     else
         return select(list, pivotNewIndex + 1, right, k - pivotDist)

Note the resemblance to quicksort: just as the minimum-based selection algorithm is a partial selection sort, this is a partial quicksort, generating and partitioning only O(log n) of its O(n) partitions. This simple procedure has expected linear performance, and, like quicksort, has quite good performance in practice. It is also an in-place algorithm, requiring only constant memory overhead, since the tail recursion can be eliminated with a loop like this:

 function select(list, left, right, k)
     loop
         select pivotIndex between left and right
         pivotNewIndex := partition(list, left, right, pivotIndex)
         pivotDist := pivotNewIndex - left + 1
         if pivotDist = k
             return list[pivotNewIndex]
         else if k < pivotDist
             right := pivotNewIndex - 1
         else
             k := k - pivotDist
             left := pivotNewIndex + 1

Like quicksort, the performance of the algorithm is sensitive to the pivot that is chosen. If bad pivots are consistently chosen, this degrades to the minimum-based selection described previously, and so can require as much as O(n²) time. David Musser describes a "median-of-3 killer" sequence that can force the well-known median-of-three pivot selection algorithm to fail with worst-case behavior (see Introselect section below).

Linear general selection algorithm - Median of Medians algorithm

Median of Medians

Class	Selection algorithm
Data structure	Array
Worst case performance	O(n)
Best case performance	O(n)
Worst case space complexity	O(1) auxiliary

A worst-case linear algorithm for the general case of selecting the kth largest element was published by Blum, Floyd, Pratt, Rivest and Tarjan in their 1973 paper "Time bounds for selection", sometimes called BFPRT after the last names of the authors. It is based on the quickselect algorithm and is also known as the median-of-medians algorithm.

Although quickselect is linear-time on average, it can require quadratic time with poor pivot choices (consider the case of pivoting around the smallest element at each step). The solution to make it O(n) in the worst case is to consistently find "good" pivots. A good pivot is one for which we can establish that a constant proportion of elements fall both below and above it.

The Select algorithm divides the list into groups of five elements. (Left over elements are ignored for now.) Then, for each group of five, the median is calculated (an operation that can potentially be made very fast if the five values can be loaded into registers and compared). (If sorting in-place, then these medians are moved into one contiguous block in the list.) Select is then called recursively on this sublist of n/5 elements to find their true median. Finally, the "median of medians" is chosen to be the pivot.

Properties of pivot

The chosen pivot is both less than and greater than half of the elements in the list of medians, which is around n / 10 elements (1 / 2 * (n / 5)) for each half. Each of these elements is a median of 5, making it less than 2 other elements and greater than 2 other elements outside the block. Hence, the pivot is less than 3(n / 10) elements outside the block, and greater than another 3(n / 10) elements outside the block. Thus the chosen median splits the elements somewhere between 30%/70% and 70%/30%, which assures worst-case linear behavior of the algorithm. To visualize:

One iteration on the list {0,1,2,3,...99}

	12		15		11		2		9		5		0		7		3		21		44		40		1		18		20		32		19		35		37		39
	13		16		14		8		10		26		6		33		4		27		49		46		52		25		51		34		43		56		72		79
Medians	17		23		24		28		29		30		31		36		42		47		50		55		58		60		63		65		66		67		81		83
	22		45		38		53		61		41		62		82		54		48		59		57		71		78		64		80		70		76		85		87
	96		95		94		86		89		69		68		97		73		92		74		88		99		84		75		90		77		93		98		91

(red = "(one of the two possible) median of medians", gray = "number < red", white = "number > red")

5-tuples are shown here sorted by median, for clarity. Sorting the tuples is not necessary because we only need the median for use as pivot element.

Note that all elements above/left of the red (30% of the 100 elements) are less, and all elements below/right of the red (another 30% of the 100 elements) are greater.

Proof of O(n) running time

The median-calculating recursive call does not exceed worst-case linear behavior because the list of medians is 20% of the size of the list, while the other recursive call recurse on at most 70% of the list, making the running time

 T(n) ≤ T(n/5) + T(7n/10) + O(n)

The O(n) is for the partitioning work (we visited each element a constant number of times, in order to form them into O(n) groups and take each median in O(1) time). From this, one can then show that T(n) ≤ c*n*(1 + (9/10) + (9/10)² + ...) = O(n).

Important notes

Although this approach optimizes quite well, it is typically outperformed in practice by the expected linear algorithm with random pivot choices^{[citation needed]}.

The worst-case algorithm can construct a worst-case O(n log n) quicksort algorithm, by using it to find the median at every step.

Introselect

David Musser's well-known introsort achieves practical performance comparable to quicksort while preserving O(n log n) worst-case behavior by creating a hybrid of quicksort and heapsort. In the same paper, Musser introduced an "introspective selection" algorithm, popularly called introselect, which combines Hoare's algorithm with the worst-case linear algorithm described above to achieve worst-case linear selection with performance similar to Hoare's algorithm.^[1] It works by optimistically starting out with Hoare's algorithm and only switching to the worst-time linear algorithm if it recurses too many times without making sufficient progress. Simply limiting the recursion to constant depth is not good enough, since this would make the algorithm switch on all sufficiently large lists. Musser discusses a couple of simple approaches:

• Keep track of the list of sizes of the subpartitions processed so far. If at any point k recursive calls have been made without halving the list size, for some small positive k, switch to the worst-case linear algorithm.
• Sum the size of all partitions generated so far. If this exceeds the list size times some small positive constant k, switch to the worst-case linear algorithm. This sum is easy to track in a single scalar variable.

Both approaches limit the recursion depth to O(klog n), which is O(log n) since k is a predetermined constant. The paper suggested that more research on introselect was forthcoming, but as of 2007 it has not appeared.

Selection as incremental sorting

One of the advantages of the sort-and-index approach, as mentioned, is its ability to amortize the sorting cost over many subsequent selections. However, sometimes the number of selections that will be done is not known in advance, and may be either small or large. In these cases, we can adapt the algorithms given above to simultaneously select an element while partially sorting the list, thus accelerating future selections.

Both the selection procedure based on minimum-finding and the one based on partitioning can be seen as a form of partial sort. The minimum-based algorithm sorts the list up to the given index, and so clearly speeds up future selections, especially of smaller indexes. The partition-based algorithm does not achieve the same behaviour automatically, but can be adapted to remember its previous pivot choices and reuse them wherever possible, avoiding costly partition operations, particularly the top-level one. The list becomes gradually more sorted as more partition operations are done incrementally; no pivots are ever "lost." If desired, this same pivot list could be passed on to quicksort to reuse, again avoiding many costly partition operations.

Using data structures to select in sublinear time

Given an unorganized list of data, linear time (Ω(n)) is required to find the minimum element, because we have to examine every element (otherwise, we might miss it). If we organize the list, for example by keeping it sorted at all times, then selecting the kth largest element is trivial, but then insertion requires linear time, as do other operations such as combining two lists.

The strategy to find an order statistic in sublinear time is to store the data in an organized fashion using suitable data structures that facilitate the selection. Two such data structures are tree-based structures and frequency tables.

When only the minimum (or maximum) is needed, a good approach is to use a heap, which is able to find the minimum (or maximum) element in constant time, while all other operations, including insertion, are O(log n) or better. More generally, a self-balancing binary search tree can easily be augmented to make it possible to both insert an element and find the kth largest element in O(log n) time. We simply store in each node a count of how many descendants it has, and use this to determine which path to follow. The information can be updated efficiently since adding a node only affects the counts of its O(log n) ancestors, and tree rotations only affect the counts of the nodes involved in the rotation.

Another simple strategy is based on some of the same concepts as the hash table. When we know the range of values beforehand, we can divide that range into h subintervals and assign these to h buckets. When we insert an element, we add it to the bucket corresponding to the interval it falls in. To find the minimum or maximum element, we scan from the beginning or end for the first nonempty bucket and find the minimum or maximum element in that bucket. In general, to find the kth element, we maintain a count of the number of elements in each bucket, then scan the buckets from left to right adding up counts until we find the bucket containing the desired element, then use the expected linear-time algorithm to find the correct element in that bucket.

If we choose h of size roughly sqrt(n), and the input is close to uniformly distributed, this scheme can perform selections in expected O(sqrt(n)) time. Unfortunately, this strategy is also sensitive to clustering of elements in a narrow interval, which may result in buckets with large numbers of elements (clustering can be eliminated through a good hash function, but finding the element with the kth largest hash value isn't very useful). Additionally, like hash tables this structure requires table resizings to maintain efficiency as elements are added and n becomes much larger than h². A useful case of this is finding an order statistic or extremum in a finite range of data. Using above table with bucket interval 1 and maintaining counts in each bucket is much superior to other methods. Such hash tables are like frequency tables used to classify the data in descriptive statistics.

Selecting k smallest or largest elements

Another fundamental selection problem is that of selecting the k smallest or k largest elements, which is particularly useful where we want to present just the "top k" of an unsorted list, such as the top 100 corporations by gross sales.

Application of simple selection algorithms

We can use the linear-time solution discussed above to select the "k"-th largest element, then run through the list in linear time and choose all elements less-than-or-equal-to "k". If the list needs to be sorted, then this can be done in O(k log k) after the fact.

Direct application of the quicksort-based selection algorithm

The quicksort-based selection algorithm can be used to find the k smallest or the k largest elements. To find the k smallest elements, find the kth smallest element using the median of medians quicksort-based algorithm. After the partition that finds the kth smallest element, all elements smaller than the kth smallest element will be to the left of the kth element and all elements larger will be to the right. Thus all elements from the 1st to the kth element inclusive constitute the k smallest elements. The time complexity is linear in n, the total number of elements.

Data structure-based solutions

Another simple method is to add each element of the list into an ordered set data structure, such as a heap or self-balancing binary search tree, with at most k elements. Whenever the data structure has more than k elements, we remove the largest element, which can be done in O(log k) time. Each insertion operation also takes O(log k) time, resulting in O(nlog k) time overall.

It is possible to transform the list into a heap in Θ(n) time, and then traverse the heap using a modified Breadth-first search algorithm that places the elements in a Priority Queue (instead of the ordinary queue that is normally used in a BFS), and terminate the scan after traversing exactly k elements. As the queue size remains O(k) throughout the traversal, it would require O(k log k) time to complete, leading to a time bound of O(n + k log k) on this algorithm.

We can achieve an O(log n) time solution using skip lists. Skip lists are sorted data structures that allow insertion, deletion and indexed retrieval in O(log n) time. Thus, for any given percentile, we can insert a new element into (and possibly delete an old element from) the list in O(log n), calculate the corresponding index(es) and finally access the percentile value in O(log n) time. See, for example, this Python-based implementation for calculating running median.

Optimised sorting algorithms

More efficient than any of these are specialized partial sorting algorithms based on mergesort and quicksort. The simplest is the quicksort variation: there is no need to recursively sort partitions which only contain elements that would fall after the kth place in the end (starting from the "left" boundary). Thus, if the pivot falls in position k or later, we recurse only on the left partition:

 function quicksortFirstK(list, left, right, k)
     if right > left
         select pivotIndex between left and right
         pivotNewIndex := partition(list, left, right, pivotIndex)
         quicksortFirstK(list, left, pivotNewIndex-1, k)
         if pivotNewIndex < left + k
             quicksortFirstK(list, pivotNewIndex+1, right, k)

The resulting algorithm requires an expected time of only O(n + k log k), and is quite efficient in practice, especially if we substitute selection sort when k becomes small relative to n. However, the worst-case time complexity is still very bad, in the case of a bad pivot selection. Pivot selection along the lines of the worst-case linear time selection algorithm could be used to get better worst-case performance.

Even better is if we don't require those k items to be themselves sorted. Losing that requirement means we can ignore all partitions that fall entirely before or after the kth place. We recurse only into the partition that actually contains the kth element itself.

 function quickfindFirstK(list, left, right, k)
     if right > left
         select pivotIndex between left and right
         pivotNewIndex := partition(list, left, right, pivotIndex)
         if pivotNewIndex > left + k  // new condition
             quickfindFirstK(list, left, pivotNewIndex-1, k)
         if pivotNewIndex < left + k
             quickfindFirstK(list, pivotNewIndex+1, right, k+left-pivotNewIndex-1)

The resulting algorithm requires an expected time of only O(n), which is the best such an algorithm can hope for.

A simpler formulation of a worst-case O(n) algorithm, is as follows:

• simply use the #Linear general selection algorithm - Median of Medians algorithm described in above sections to find the kth element in O(n) time worst-case
• use the Quicksort#Algorithm partition operation (which is O(n)) from Quicksort to partition into the elements less than and greater than the kth element

Tournament Algorithm

Another method is tournament algorithm. The idea is to conduct a knockout minimal round tournament to decide the ranks. It first organises the games (comparisons) between adjacent pairs and moves the winners to next round until championship (the first best) is decided. It also constructs the tournament tree along the way. Now the second best element must be among the direct losers to winner and these losers can be found out by walking in the binary tree in O(log n) time. It organises another tournament to decide the second best among these potential elements. The third best must be one among the losers of the second best in either of the two tournament trees. The approach continues until we find k elements. This algorithm takes O(n + k log n) complexity, which for any fixed k independent of n is O(n).

Lower bounds

In his seminal The Art of Computer Programming, Donald E. Knuth discussed a number of lower bounds for the number of comparisons required to locate the k smallest entries of an unorganized list of n items (using only comparisons). There's a trivial lower bound of n − 1 for the minimum or maximum entry. To see this, consider a tournament where each game represents one comparison. Since every player except the winner of the tournament must lose a game before we know the winner, we have a lower bound of n − 1 comparisons.

The story becomes more complex for other indexes. To find the k smallest values requires at least this many comparisons:

$n - k + \sum_{n+1-k < j \leq n} \lceil{\operatorname{lg}\, j}\rceil.$

This bound is achievable for k=2 but better, more complex bounds exist for larger k.

Language support

Very few languages have built-in support for general selection, although many provide facilities for finding the smallest or largest element of a list. A notable exception is C++, which provides a templated nth_element method with a guarantee of expected linear time. It is implied but not required that it is based on Hoare's algorithm by its requirement of expected linear time. (Ref section 25.3.2 of ISO/IEC 14882:2003(E) and 14882:1998(E), see also SGI STL description of nth_element)

C++ also provides the partial_sort algorithm, which solves the problem of selecting the smallest k elements (sorted), with a time complexity of O(n log k). No algorithm is provided for selecting the greatest k elements since this should be done by inverting the ordering predicate.

For Perl, the module Sort::Key::Top, available from CPAN, provides a set of functions to select the top n elements from a list using several orderings and custom key extraction procedures.

Python's standard library (since 2.4) includes heapq.nsmallest() and nlargest(), returning sorted lists in O(n + k log n) time.

Because language support for sorting is more ubiquitous, the simplistic approach of sorting followed by indexing is preferred in many environments despite its disadvantage in speed. Indeed for lazy languages, this simplistic approach can even get you the best complexity possible for the k smallest/greatest sorted (with maximum/minimum as a special case) if your sort is lazy enough.

Online selection algorithm

In certain selection problems, selection must be online, that is, an element can only be selected from a sequential input at the instance of observation and each selection, respectively refusal, is irrevocable. The problem is to select, under these constraints, a specific element of the input sequence (as for example the largest or the smallest value) with largest probability. This problem can be tackled by the Odds algorithm which is known to be optimal under an independence condition. The algorithm is also optimal itself with the number of operations being linear in the length of input.

Notes

1. ^ David R. Musser. Introspective Sorting and Selection Algorithms. Software: Practice and Experience, vol. 27, no. 8, pp.983–993. 1997. Section: Introspective Selection Algorithms.

References

• M. Blum, R.W. Floyd, V. Pratt, R. Rivest and R. Tarjan, "Time bounds for selection," J. Comput. System Sci. 7 (1973) 448-461.
• K. C. Kiwiel. On Floyd and Rivest’s SELECT Algorithm, Theoretical Computer Sci. 347 (2005) 214-238.
• Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Section 5.3.3: Minimum-Comparison Selection, pp.207–219.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 9: Medians and Order Statistics, pp.183–196. Section 14.1: Dynamic order statistics, pp.302–308.
• Paul E. Black, Select at the NIST Dictionary of Algorithms and Data Structures.

External links

• Design and Analysis of Algorithms, for a detailed explanation of the recurrence relation for the median-of-medians