Algorithm
In quicksort, there is a subprocedure calledpartition
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 ivotIndex/code>:
function partition(list, left, right, pivotIndex) is
pivotValue := list ivotIndex swap list ivotIndexand list ight ''// Move pivot to end''
storeIndex := left
for i from left to right − 1 do
if list < pivotValue then
swap list toreIndexand list increment storeIndex
swap list ightand list toreIndex ''// Move pivot to its final place''
return storeIndex
This is known as the Lomuto partition scheme, which is simpler but less efficient than Hoare's original partition scheme.
In quicksort, we recursively sort both branches, leading to best-case 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 an unsorted order and all those following it in an unsorted order. Therefore, a single recursive call locates the desired element in the correct partition, and we build upon this for quickselect:
''// Returns the k-th smallest element of list within left..right inclusive''
''// (i.e. left <= k <= right).''
function select(list, left, right, k) is
if left = right then ''// If the list contains only one element,''
return listeft
A newt is a salamander in the subfamily Pleurodelinae. The terrestrial juvenile phase is called an eft. Unlike other members of the family Salamandridae, newts are semiaquatic, alternating between aquatic and terrestrial habitats. Not all aqu ...
''// return that element''
pivotIndex := ... ''// select a pivotIndex between left and right,''
''// e.g.,'' left + floor(rand() % (right − left + 1))
pivotIndex := partition(list, left, right, pivotIndex)
''// The pivot is in its final sorted position''
if k = pivotIndex then
return list else if k < pivotIndex then
return select(list, left, pivotIndex − 1, k)
else
return select(list, pivotIndex + 1, right, k)
----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 of its partitions. This simple procedure has expected linear performance, and, like quicksort, has quite good performance in practice. It is also an in-place algorithm
In computer science, an in-place algorithm is an algorithm which transforms input using no auxiliary data structure. However, a small amount of extra storage space is allowed for auxiliary variables. The input is usually overwritten by the output ...
, requiring only constant memory overhead if tail call
In computer science, a tail call is a subroutine call performed as the final action of a procedure. If the target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recur ...
optimization is available, or if eliminating the tail recursion
In computer science, a tail call is a subroutine call performed as the final action of a procedure. If the target of a tail is the same subroutine, the subroutine is said to be tail recursive, which is a special case of direct recursion. Tail recur ...
with a loop:
function select(list, left, right, k) is
loop
if left = right then
return listeft
A newt is a salamander in the subfamily Pleurodelinae. The terrestrial juvenile phase is called an eft. Unlike other members of the family Salamandridae, newts are semiaquatic, alternating between aquatic and terrestrial habitats. Not all aqu ...
pivotIndex := ... ''// select pivotIndex between left and right''
pivotIndex := partition(list, left, right, pivotIndex)
if k = pivotIndex then
return list else if k < pivotIndex then
right := pivotIndex − 1
else
left := pivotIndex + 1
Time complexity
Like quicksort, quickselect has good average performance, but is sensitive to the pivot that is chosen. If good pivots are chosen, meaning ones that consistently decrease the search set by a given fraction, then the search set decreases in size exponentially and by induction (or summing the geometric series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
:\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots
is geometric, because each succ ...
) one sees that performance is linear, as each step is linear and the overall time is a constant times this (depending on how quickly the search set reduces). However, if bad pivots are consistently chosen, such as decreasing by only a single element each time, then worst-case performance is quadratic: This occurs for example in searching for the maximum element of a set, using the first element as the pivot, and having sorted data. The probability of the worst-case occurring decreases exponentially with so quickselect has almost certain
In probability theory, an event (probability theory), event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty ...
time complexity.
Variants
The easiest solution is to choose a random pivot, which yields almost certain
In probability theory, an event (probability theory), event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty ...
linear time. Deterministically, one can use median-of-3 pivot strategy (as in the quicksort), which yields linear performance on partially sorted data, as is common in the real world. However, contrived sequences can still cause worst-case complexity; David Musser David "Dave" Musser is a professor emeritus of computer science at the Rensselaer Polytechnic Institute in Troy, New York, United States.
He is known for his work in generic programming, particularly as applied to C++, and his collaboration with Al ...
describes a "median-of-3 killer" sequence that allows an attack against that strategy, which was one motivation for his introselect
In computer science, introselect (short for "introspective selection") is a selection algorithm that is a hybrid of quickselect and median of medians which has fast average performance and optimal worst-case performance. Introselect is related ...
algorithm.
One can assure linear performance even in the worst case by using a more sophisticated pivot strategy; this is done in the median of medians
In computer science, the median of medians is an approximate (median) selection algorithm, frequently used to supply a good pivot for an exact selection algorithm, mainly the quickselect, that selects the ''k''th smallest element of an initially u ...
algorithm. However, the overhead of computing the pivot is high, and thus this is generally not used in practice. One can combine basic quickselect with median of medians as fallback to get both fast average case performance and linear worst-case performance; this is done in introselect
In computer science, introselect (short for "introspective selection") is a selection algorithm that is a hybrid of quickselect and median of medians which has fast average performance and optimal worst-case performance. Introselect is related ...
.
Finer computations of the average time complexity yield a worst case of for random pivots (in the case of the median; other ''k'' are faster).Blum-style analysis of Quickselect
David Eppstein
David Arthur Eppstein (born 1963) is an American computer scientist and mathematician. He is a Distinguished Professor of computer science at the University of California, Irvine. He is known for his work in computational geometry, graph algorit ...
, October 9, 2007. The constant can be improved to 3/2 by a more complicated pivot strategy, yielding the Floyd–Rivest algorithm
In computer science, the Floyd-Rivest algorithm is a selection algorithm developed by Robert W. Floyd and Ronald L. Rivest that has an optimal expected number of comparisons within lower-order terms. It is functionally equivalent to quickselect, ...
, which has average complexity of for median, with other ''k'' being faster.
See also
* Floyd–Rivest algorithm
In computer science, the Floyd-Rivest algorithm is a selection algorithm developed by Robert W. Floyd and Ronald L. Rivest that has an optimal expected number of comparisons within lower-order terms. It is functionally equivalent to quickselect, ...
* Introselect
In computer science, introselect (short for "introspective selection") is a selection algorithm that is a hybrid of quickselect and median of medians which has fast average performance and optimal worst-case performance. Introselect is related ...
* Median of medians
In computer science, the median of medians is an approximate (median) selection algorithm, frequently used to supply a good pivot for an exact selection algorithm, mainly the quickselect, that selects the ''k''th smallest element of an initially u ...
References
{{reflist
External links
*
qselect
, ''Quickselect algorithm in Matlab,'' Manolis Lourakis
Selection algorithms