Sorting algorithm

Sorting algorithm 1
Sorting algorithm
A sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are
numerical order and lexicographical order. Efficient sorting is important for optimizing the use of other algorithms
(such as search and merge algorithms) which require input data to be in sorted lists; it is also often useful for
canonicalizing data and for producing human-readable output. More formally, the output must satisfy two
conditions:
1. The output is in nondecreasing order (each element is no smaller than the previous element according to the
desired total order);
2. The output is a permutation (reordering) of the input.
Further, the data is often taken to be in an array, which allows random access, rather than a list, which only allows
sequential access, though often algorithms can be applied with suitable modification to either type of data.
Since the dawn of computing, the sorting problem has attracted a great deal of research, perhaps due to the
complexity of solving it efficiently despite its simple, familiar statement. For example, bubble sort was analyzed as
early as 1956.
[1]
A fundamental limit of comparison sorting algorithms is that they require linearithmic time – O(n
log n) – in the worst case, though better performance is possible on real-world data (such as almost-sorted data), and
algorithms not based on comparison, such as counting sort, can have better performance. Although many consider
sorting a solved problem – asymptotically optimal algorithms have been known since the mid-20th century – useful
new algorithms are still being invented, with the now widely used Timsort dating to 2002, and the library sort being
first published in 2006.
Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the
problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide and
conquer algorithms, data structures such as heaps and binary trees, randomized algorithms, best, worst and average
case analysis, time-space tradeoffs, and upper and lower bounds.
Classification
Sorting algorithms are often classified by:
• Computational complexity (worst, average and best behavior) of element comparisons in terms of the size of the
list (n). For typical serial sorting algorithms good behavior is O(n log n), with parallel sort in O(log
2
n), and bad
behavior is O(n
2
). (See Big O notation.) Ideal behavior for a serial sort is O(n), but this is not possible in the
average case, optimal parallel sorting is O(log n). Comparison-based sorting algorithms, which evaluate the
elements of the list via an abstract key comparison operation, need at least O(n log n) comparisons for most
inputs.
• Computational complexity of swaps (for "in place" algorithms).
• Memory usage (and use of other computer resources). In particular, some sorting algorithms are "in place".
Strictly, an in place sort needs only O(1) memory beyond the items being sorted; sometimes O(log(n)) additional
memory is considered "in place".
•• Recursion. Some algorithms are either recursive or non-recursive, while others may be both (e.g., merge sort).
•• Stability: stable sorting algorithms maintain the relative order of records with equal keys (i.e., values).
• Whether or not they are a comparison sort. A comparison sort examines the data only by comparing two elements
with a comparison operator.
• General method: insertion, exchange, selection, merging, etc. Exchange sorts include bubble sort and quicksort.
Selection sorts include shaker sort and heapsort. Also whether the algorithm is serial or parallel. The remainder of
this discussion almost exclusively concentrates upon serial algorithms and assumes serial operation.

Sorting algorithm 2
• Adaptability: Whether or not the presortedness of the input affects the running time. Algorithms that take this into
account are known to be adaptive.
Stability
An example of stable sorting on playing cards.
When the cards are sorted by rank with a stable
sort, the two 5s must remain in the same order in
the sorted output that they were originally in.
When they are sorted with a non-stable sort, the
5s may end up in the opposite order in the sorted
output.
When sorting some kinds of data, only part of the data is examined
when determining the sort order. For example, in the card sorting
example to the right, the cards are being sorted by their rank, and their
suit is being ignored. This allows the possibility of multiple different
correctly sorted versions of the original list. Stable sorting algorithms
choose one of these, according to the following rule: if two items
compare as equal, like the two 5 cards, then their relative order will be
preserved, so that if one came before the other in the input, it will also
come before the other in the output.
More formally, the data being sorted can be represented as a record or
tuple of values, and the part of the data that is used for sorting is called
the key. In the card example, cards are represented as a record (rank,
suit), and the key is the rank. A sorting algorithm is stable if whenever
there are two records R and S with the same key, and R appears before
S in the original list, then R will always appear before S in the sorted
list.
When equal elements are indistinguishable, such as with integers, or
more generally, any data where the entire element is the key, stability
is not an issue. Stability is also not an issue if all keys are different.
Unstable sorting algorithms can be specially implemented to be stable.
One way of doing this is to artificially extend the key comparison, so
that comparisons between two objects with otherwise equal keys are
decided using the order of the entries in the original input list as a
tie-breaker. Remembering this order, however, may require additional
time and space.
One application for stable sorting algorithms is sorting a list using a primary and secondary key. For example,
suppose we wish to sort a hand of cards such that the suits are in the order clubs (♣), diamonds (♦), hearts (♥),
spades (♠), and within each suit, the cards are sorted by rank. This can be done by first sorting the cards by rank
(using any sort), and then doing a stable sort by suit:

Sorting algorithm 3
Within each suit, the stable sort preserves the ordering by rank that was already done. This idea can be extended to
any number of keys, and is leveraged by radix sort. The same effect can be achieved with an unstable sort by using a
lexicographic key comparison, which e.g. compares first by suits, and then compares by rank if the suits are the
same.
Comparison of algorithms
In this table, n is the number of records to be sorted. The columns "Average" and "Worst" give the time complexity
in each case, under the assumption that the length of each key is constant, and that therefore all comparisons, swaps,
and other needed operations can proceed in constant time. "Memory" denotes the amount of auxiliary storage needed
beyond that used by the list itself, under the same assumption. The run times and the memory requirements listed
below should be understood to be inside big O notation. Logarithms are of any base; the notation means
.
These are all comparison sorts, and so cannot perform better than O(n log n) in the average or worst case.
Comparison sorts
Name Best Average Worst Memory Stable Method Other notes
Quicksort on
average, worst
case is ;
Sedgewick
variation is
worst
case
typical
in-place
sort is not
stable;
stable
versions
exist
Partitioning Quicksort is usually done in place
with O(log n) stack
space.Wikipedia:Citation needed
Most implementations are unstable,
as stable in-place partitioning is more
complex. Naïve variants use an O(n)
space array to store the
partition.Wikipedia:Citation needed
Quicksort variant using three-way
(fat) partitioning takes O(n)
comparisons when sorting an array of
equal keys.
Merge sort worst case Yes Merging Highly parallelizable (up to O(log n)
using the Three Hungarian's
AlgorithmWikipedia:Please clarify
or, more practically, Cole's parallel
merge sort) for processing large
amounts of data.

Sorting algorithm 4
In-place
merge sort
— — Yes Merging
Can be implemented as a stable sort
based on stable in-place merging.
[2]
Heapsort No Selection
Insertion sort Yes Insertion O(n + d),Wikipedia:Please clarify
where d is the number of inversions.
Introsort No Partitioning
& Selection
Used in several STL
implementations.
Selection sort No Selection
Stable with O(n) extra space, for
example using lists.
[3]
Timsort Yes Insertion &
Merging
Makes n comparisons when the data
is already sorted or reverse sorted.
Shell sort
or
Depends on gap
sequence;
best known is
No Insertion Small code size, no use of call stack,
reasonably fast, useful where
memory is at a premium such as
embedded and older mainframe
applications.
Bubble sort Yes Exchanging Tiny code size.
Binary tree
sort
Yes Insertion When using a self-balancing binary
search tree.
Cycle sort — No Insertion In-place with theoretically optimal
number of writes.
Library sort — Yes Insertion
Patience
sorting
— — No Insertion &
Selection
Finds all the longest increasing
subsequences in O(n log n).
Smoothsort No Selection An adaptive sort: comparisons
when the data is already sorted, and 0
swaps.
Strand sort Yes Selection
Tournament
sort
— [4] ? Selection
Cocktail sort Yes Exchanging
Comb sort No Exchanging Small code size.
Gnome sort Yes Exchanging Tiny code size.
UnShuffle
Sort
In place for
linked lists.
N*sizeof(link)
for array.
Can be
made
stable by
appending
the input
order to the
key.
Distribution
and Merge
No exchanges are performed.
Performance is independent of data
size. The constant 'k' is proportional
to the entropy in the input. K = 1 for
ordered or ordered by reversed input
so runtime is equivalent to checking
the order O(N).
Franceschini's
method
— Yes ?
Block sort Yes Insertion &
Merging
Combine a block-based O(n) in-place
merge algorithm with a bottom-up
merge sort.
Bogosort No Random
shuffling

Sorting algorithm 5
The following table describes integer sorting algorithms and other sorting algorithms that are not comparison sorts.
As such, they are not limited by a lower bound. Complexities below assume n items to be sorted, with
keys of size k, digit size d, and r the range of numbers to be sorted. Many of them are based on the assumption that
the key size is large enough that all entries have unique key values, and hence that n << 2
k
, where << means "much
less than."
Non-comparison sorts
Name Best Average Worst Memory Stable
n <<
2
k
Notes
Pigeonhole sort — Yes Yes
Bucket sort
(uniform keys)
— Yes No Assumes uniform distribution of elements from the domain
in the array.
Bucket sort (integer
keys)
— Yes Yes If r is O(n), then Average is O(n).
Counting sort — Yes Yes If r is O(n), then Average is O(n).
LSD Radix Sort — Yes No
MSD Radix Sort — Yes No Stable version uses an external array of size n to hold all of
the bins.
MSD Radix Sort
(in-place)
— No No
recursion levels, 2
d
for count array.
Spreadsort — No No
Asymptotics are based on the assumption that n << 2
k
, but
the algorithm does not require this.
The following table describes some sorting algorithms that are impractical for real-life use due to extremely poor
performance or specialized hardware requirements.
Name Best Average Worst Memory Stable Comparison Other notes
Bead sort — N/A N/A — N/A No Requires specialized hardware.
Simple
pancake
sort
— No Yes Count is number of flips.
Spaghetti
(Poll) sort
Yes Polling This a linear-time, analog algorithm for sorting a
sequence of items, requiring O(n) stack space, and the
sort is stable. This requires n parallel processors. See
Spaghetti sort#Analysis.
Sorting
networks
Varies (stable sorting
networks require more
comparisons)
Yes Order of comparisons are set in advance based on a
fixed network size. Impractical for more than 32
items.
Theoretical computer scientists have detailed other sorting algorithms that provide better than O(n log n) time
complexity assuming additional constraints, including:
• Han's algorithm, a deterministic algorithm for sorting keys from a domain of finite size, taking O(n log log n)
time and O(n) space.
[5]
• Thorup's algorithm, a randomized algorithm for sorting keys from a domain of finite size, taking O(n log log n)
time and O(n) space.
[6]
• A randomized integer sorting algorithm taking expected time and O(n) space.
[7]

Sorting algorithm 6
Algorithms not yet compared above include:
•• Odd-even sort
•• Flashsort
•• Burstsort
•• Postman sort
•• Stooge sort
•• Samplesort
•• Bitonic sorter
Popular sorting algorithms
While there are a large number of sorting algorithms, in practical implementations a few algorithms predominate.
Insertion sort is widely used for small data sets, while for large data sets an asymptotically efficient sort is used,
primarily heap sort, merge sort, or quicksort. Efficient implementations generally use a hybrid algorithm, combining
an asymptotically efficient algorithm for the overall sort with insertion sort for small lists at the bottom of a
recursion. Highly tuned implementations use more sophisticated variants, such as Timsort (merge sort, insertion sort,
and additional logic), used in Android, Java, and Python, and introsort (quicksort and heap sort), used (in variant
forms) in some C++ sort implementations and in .NET.
For more restricted data, such as numbers in a fixed interval, distribution sorts such as counting sort or radix sort are
widely used. Bubble sort and variants are rarely used in practice, but are commonly found in teaching and theoretical
discussions.
When physically sorting objects, such as alphabetizing papers (such as tests or books), people intuitively generally
use insertion sorts for small sets. For larger sets, people often first bucket, such as by initial letter, and multiple
bucketing allows practical sorting of very large sets. Often space is relatively cheap, such as by spreading objects out
on the floor or over a large area, but operations are expensive, particularly moving an object a large distance –
locality of reference is important. Merge sorts are also practical for physical objects, particularly as two hands can be
used, one for each list to merge, while other algorithms, such as heap sort or quick sort, are poorly suited for human
use. Other algorithms, such as library sort, a variant of insertion sort that leaves spaces, are also practical for physical
use.
Simple sorts
Two of the simplest sorts are insertion sort and selection sort, both of which are efficient on small data, due to low
overhead, but not efficient on large data. Insertion sort is generally faster than selection sort in practice, due to fewer
comparisons and good performance on almost-sorted data, and thus is preferred in practice, but selection sort uses
fewer writes, and thus is used when write performance is a limiting factor.
Insertion sort
Main article: Insertion sort
Insertion sort is a simple sorting algorithm that is relatively efficient for small lists and mostly sorted lists, and often
is used as part of more sophisticated algorithms. It works by taking elements from the list one by one and inserting
them in their correct position into a new sorted list. In arrays, the new list and the remaining elements can share the
array's space, but insertion is expensive, requiring shifting all following elements over by one. Shell sort (see below)
is a variant of insertion sort that is more efficient for larger lists.

Sorting algorithm 7
Selection sort
Main article: Selection sort
Selection sort is an in-place comparison sort. It has O(n
2
) complexity, making it inefficient on large lists, and
generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity, and also has
performance advantages over more complicated algorithms in certain situations.
The algorithm finds the minimum value, swaps it with the value in the first position, and repeats these steps for the
remainder of the list. It does no more than n swaps, and thus is useful where swapping is very expensive.
Efficient sorts
Practical general sorting algorithms are almost always based on an algorithm with average complexity (and generally
worst-case complexity) O(n log n), of which the most common are heap sort, merge sort, and quicksort. Each has
advantages and drawbacks, with the most significant being that simple implementation of merge sort uses O(n)
additional space, and simple implementation of quicksort has O(n
2
) worst-case complexity. These problems can be
solved or ameliorated at the cost of a more complex algorithm.
While these algorithms are asymptotically efficient on random data, for practical efficiency on real-world data
various modifications are used. First, the overhead of these algorithms becomes significant on smaller data, so often
a hybrid algorithm is used, commonly switching to insertion sort once the data is small enough. Second, the
algorithms often perform poorly on already sorted data or almost sorted data – these are common in real-world data,
and can be sorted in O(n) time by appropriate algorithms. Finally, they may also be unstable, and stability is often a
desirable property in a sort. Thus more sophisticated algorithms are often employed, such as Timsort (based on
merge sort) or introsort (based on quicksort, falling back to heap sort).
Merge sort
Main article: Merge sort
Merge sort takes advantage of the ease of merging already sorted lists into a new sorted list. It starts by comparing
every two elements (i.e., 1 with 2, then 3 with 4...) and swapping them if the first should come after the second. It
then merges each of the resulting lists of two into lists of four, then merges those lists of four, and so on; until at last
two lists are merged into the final sorted list. Of the algorithms described here, this is the first that scales well to very
large lists, because its worst-case running time is O(n log n). It is also easily applied to lists, not only arrays, as it
only requires sequential access, not random access. However, it has additional O(n) space complexity, and involves a
large number of copies in simple implementations.
Merge sort has seen a relatively recent surge in popularity for practical implementations, due to its use in the
sophisticated algorithm Timsort, which is used for the standard sort routine in the programming languages Python
[8]
and Java (as of JDK7
[9]
). Merge sort itself is the standard routine in Perl,
[10]
among others, and has been used in Java
at least since 2000 in JDK1.3.
[11][12]
Heapsort
Main article: Heapsort
Heapsort is a much more efficient version of selection sort. It also works by determining the largest (or smallest)
element of the list, placing that at the end (or beginning) of the list, then continuing with the rest of the list, but
accomplishes this task efficiently by using a data structure called a heap, a special type of binary tree. Once the data
list has been made into a heap, the root node is guaranteed to be the largest (or smallest) element. When it is
removed and placed at the end of the list, the heap is rearranged so the largest element remaining moves to the root.
Using the heap, finding the next largest element takes O(log n) time, instead of O(n) for a linear scan as in simple
selection sort. This allows Heapsort to run in O(n log n) time, and this is also the worst case complexity.

Sorting algorithm 8
Quicksort
Main article: Quicksort
Quicksort is a divide and conquer algorithm which relies on a partition operation: to partition an array an element
called a pivot is selected. All elements smaller than the pivot are moved before it and all greater elements are moved
after it. This can be done efficiently in linear time and in-place. The lesser and greater sublists are then recursively
sorted. This yields average time complexity of O(n log n), with low overhead, and thus this is a popular algorithm.
Efficient implementations of quicksort (with in-place partitioning) are typically unstable sorts and somewhat
complex, but are among the fastest sorting algorithms in practice. Together with its modest O(log n) space usage,
quicksort is one of the most popular sorting algorithms and is available in many standard programming libraries.
The important caveat about quicksort is that its worst-case performance is O(n
2
); while this is rare, in naive
implementations (choosing the first or last element as pivot) this occurs for sorted data, which is a common case. The
most complex issue in quicksort is thus choosing a good pivot element, as consistently poor choices of pivots can
result in drastically slower O(n
2
) performance, but good choice of pivots yields O(n log n) performance, which is
asymptotically optimal. For example, if at each step the median is chosen as the pivot then the algorithm works in
O(n log n). Finding the median, such as by the median of medians selection algorithm is however an O(n) operation
on unsorted lists and therefore exacts significant overhead with sorting. In practice choosing a random pivot almost
certainly yields O(n log n) performance.
Bubble sort and variants
Bubble sort, and variants such as the cocktail sort, are simple but highly inefficient sorts. They are thus frequently
seen in introductory texts, and are of some theoretical interest due to ease of analysis, but they are rarely used in
practice, and primarily of recreational interest. Some variants, such as the Shell sort, have open questions about their
behavior.
Bubble sort
A bubble sort, a sorting algorithm that
continuously steps through a list, swapping items
until they appear in the correct order.
Main article: Bubble sort
Bubble sort is a simple sorting algorithm. The algorithm starts at the
beginning of the data set. It compares the first two elements, and if the
first is greater than the second, it swaps them. It continues doing this
for each pair of adjacent elements to the end of the data set. It then
starts again with the first two elements, repeating until no swaps have
occurred on the last pass. This algorithm's average and worst-case
performance is O(n
2
), so it is rarely used to sort large, unordered data
sets. Bubble sort can be used to sort a small number of items (where its
asymptotic inefficiency is not a high penalty). Bubble sort can also be
used efficiently on a list of any length that is nearly sorted (that is, the
elements are not significantly out of place). For example, if any
number of elements are out of place by only one position (e.g.
0123546789 and 1032547698), bubble sort's exchange will get them in
order on the first pass, the second pass will find all elements in order, so the sort will take only 2n time.

Sorting algorithm 9
Shell sort
A Shell sort, different from bubble sort in that it
moves elements to numerous swapping positions
Main article: Shell sort
Shell sort was invented by Donald Shell in 1959. It improves upon
bubble sort and insertion sort by moving out of order elements more
than one position at a time. One implementation can be described as
arranging the data sequence in a two-dimensional array and then
sorting the columns of the array using insertion sort.
Comb sort
Main article: Comb sort
Comb sort is a relatively simple sorting algorithm originally designed
by Wlodzimierz Dobosiewicz in 1980.
[13]
Later it was rediscovered
and popularized by Stephen Lacey and Richard Box with a Byte
Magazine article published in April 1991. Comb sort improves on bubble sort. The basic idea is to eliminate turtles,
or small values near the end of the list, since in a bubble sort these slow the sorting down tremendously. (Rabbits,
large values around the beginning of the list, do not pose a problem in bubble sort)
Distribution sort
See also: External sorting
Distribution sort refers to any sorting algorithm where data are distributed from their input to multiple intermediate
structures which are then gathered and placed on the output. For example, both bucket sort and flashsort are
distribution based sorting algorithms. Distribution sorting algorithms can be used on a single processor, or they can
be a distributed algorithm, where individual subsets are separately sorted on different processors, then combined.
This allows external sorting of data too large to fit into a single computer's memory.
Counting sort
Main article: Counting sort
Counting sort is applicable when each input is known to belong to a particular set, S, of possibilities. The algorithm
runs in O(|S| + n) time and O(|S|) memory where n is the length of the input. It works by creating an integer array of
size |S| and using the ith bin to count the occurrences of the ith member of S in the input. Each input is then counted
by incrementing the value of its corresponding bin. Afterward, the counting array is looped through to arrange all of
the inputs in order. This sorting algorithm cannot often be used because S needs to be reasonably small for it to be
efficient, but the algorithm is extremely fast and demonstrates great asymptotic behavior as n increases. It also can
be modified to provide stable behavior.
Bucket sort
Main article: Bucket sort
Bucket sort is a divide and conquer sorting algorithm that generalizes Counting sort by partitioning an array into a
finite number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm, or by
recursively applying the bucket sorting algorithm.
Due to the fact that bucket sort must use a limited number of buckets it is best suited to be used on data sets of a
limited scope. Bucket sort would be unsuitable for data that have a lot of variation, such as social security numbers.

Sorting algorithm 10
Radix sort
Main article: Radix sort
Radix sort is an algorithm that sorts numbers by processing individual digits. n numbers consisting of k digits each
are sorted in O(n · k) time. Radix sort can process digits of each number either starting from the least significant digit
(LSD) or starting from the most significant digit (MSD). The LSD algorithm first sorts the list by the least significant
digit while preserving their relative order using a stable sort. Then it sorts them by the next digit, and so on from the
least significant to the most significant, ending up with a sorted list. While the LSD radix sort requires the use of a
stable sort, the MSD radix sort algorithm does not (unless stable sorting is desired). In-place MSD radix sort is not
stable. It is common for the counting sort algorithm to be used internally by the radix sort. A hybrid sorting
approach, such as using insertion sort for small bins improves performance of radix sort significantly.
Memory usage patterns and index sorting
When the size of the array to be sorted approaches or exceeds the available primary memory, so that (much slower)
disk or swap space must be employed, the memory usage pattern of a sorting algorithm becomes important, and an
algorithm that might have been fairly efficient when the array fit easily in RAM may become impractical. In this
scenario, the total number of comparisons becomes (relatively) less important, and the number of times sections of
memory must be copied or swapped to and from the disk can dominate the performance characteristics of an
algorithm. Thus, the number of passes and the localization of comparisons can be more important than the raw
number of comparisons, since comparisons of nearby elements to one another happen at system bus speed (or, with
caching, even at CPU speed), which, compared to disk speed, is virtually instantaneous.
For example, the popular recursive quicksort algorithm provides quite reasonable performance with adequate RAM,
but due to the recursive way that it copies portions of the array it becomes much less practical when the array does
not fit in RAM, because it may cause a number of slow copy or move operations to and from disk. In that scenario,
another algorithm may be preferable even if it requires more total comparisons.
One way to work around this problem, which works well when complex records (such as in a relational database) are
being sorted by a relatively small key field, is to create an index into the array and then sort the index, rather than the
entire array. (A sorted version of the entire array can then be produced with one pass, reading from the index, but
often even that is unnecessary, as having the sorted index is adequate.) Because the index is much smaller than the
entire array, it may fit easily in memory where the entire array would not, effectively eliminating the disk-swapping
problem. This procedure is sometimes called "tag sort".
[14]
Another technique for overcoming the memory-size problem is to combine two algorithms in a way that takes
advantages of the strength of each to improve overall performance. For instance, the array might be subdivided into
chunks of a size that will fit in RAM, the contents of each chunk sorted using an efficient algorithm (such as
quicksort), and the results merged using a k-way merge similar to that used in mergesort. This is faster than
performing either mergesort or quicksort over the entire list.
Techniques can also be combined. For sorting very large sets of data that vastly exceed system memory, even the
index may need to be sorted using an algorithm or combination of algorithms designed to perform reasonably with
virtual memory, i.e., to reduce the amount of swapping required.

Inefficient/humorous sorts
Some algorithms are slow compared to those discussed above, such as the bogosort O(n⋅n!) and the stooge sort
O(n
2.7
).
Related algorithms
Related problems include partial sorting (sorting only the k smallest elements of a list, or alternatively computing the
k smallest elements, but unordered) and selection (computing the kth smallest element). These can be solved
inefficiently by a total sort, but more efficient algorithms exist, often derived by generalizing a sorting algorithm.
The most notable example is quickselect, which is related to quicksort. Conversely, some sorting algorithms can be
derived by repeated application of a selection algorithm; quicksort and quickselect can be seen as the same pivoting
move, differing only in whether one recurses on both sides (quicksort, divide and conquer) or one side (quickselect,
decrease and conquer).
A kind of opposite of a sorting algorithm is a shuffling algorithm. These are fundamentally different because they
require a source of random numbers. Interestingly, shuffling can also be implemented by a sorting algorithm, namely
by a random sort: assigning a random number to each element of the list and then sorting based on the random
numbers. This is generally not done in practice, however, and there is a well-known simple and efficient algorithm
for shuffling: the Fisher–Yates shuffle.
References
[1][1] Demuth, H. Electronic Data Sorting. PhD thesis, Stanford University, 1956.
[2] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.8381
[3] http://www.algolist.net/Algorithms/Sorting/Selection_sort
[4] http://dbs.uni-leipzig.de/skripte/ADS1/PDF4/kap4.pdf
[5] Y. Han. Deterministic sorting in time and linear space. Proceedings of the thirty-fourth annual ACM symposium on theory of computing,
Montreal, Quebec, Canada, 2002, pp. 602-608.
[6] M. Thorup. Randomized Sorting in Time and Linear Space Using Addition, Shift, and Bit-wise Boolean Operations. Journal of Algorithms,
Volume 42, Number 2, February 2002, pp. 205-230(26).
[7] Han, Y. and Thorup, M. 2002. Integer Sorting in UNIQ-math-0-a83a145a02b7d084-QINU Expected Time and Linear Space. In Proceedings
of the 43rd Symposium on Foundations of Computer Science (November 16–19, 2002). FOCS. IEEE Computer Society, Washington, DC, pp.
135-144.
[8] Tim Peters's original description of timsort (http://svn.python.org/projects/python/trunk/Objects/listsort.txt)
[9] http://hg.openjdk.java.net/jdk7/tl/jdk/rev/bfd7abda8f79
[10] Perl sort documentation (http://perldoc.perl.org/functions/sort.html)
[11] Merge sort in Java 1.3 (http://java.sun.com/j2se/1.3/docs/api/java/util/Arrays.html#sort(java.lang.Object[])), Sun.
[12][12] Java 1.3 live since 2000
[13] Brejová, Bronislava. "Analyzing variants of Shellsort" (http://www.sciencedirect.com/science/article/pii/S0020019000002234)
[14] Definition of "tag sort" according to PC Magazine (http://www.pcmag.com/encyclopedia_term/0,2542,t=tag+sort&i=52532,00.asp)
• D. E. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching.
External links
The Wikibook Algorithm implementation has a page on the topic of: Sorting algorithms
The Wikibook A-level Mathematics has a page on the topic of: Sorting algorithms
Wikimedia Commons has media related to Sorting algorithms.

• Sorting Algorithm Animations (http://www.sorting-algorithms.com/) - Graphical illustration of how different
algorithms handle different kinds of data sets.
• Sequential and parallel sorting algorithms (http://www.iti.fh-flensburg.de/lang/algorithmen/sortieren/algoen.
htm) - Explanations and analyses of many sorting algorithms.
• Dictionary of Algorithms, Data Structures, and Problems (http://www.nist.gov/dads/) - Dictionary of
algorithms, techniques, common functions, and problems.
• Slightly Skeptical View on Sorting Algorithms (http://www.softpanorama.org/Algorithms/sorting.shtml)
Discusses several classic algorithms and promotes alternatives to the quicksort algorithm.
• 15 Sorting Algorithms in 6 Minutes (Youtube) (http://www.youtube.com/watch?v=kPRA0W1kECg)
Visualization and "audibilization" of 15 Sorting Algorithms in 6 Minutes.
• A036604 sequence in OEIS database titled "Sorting numbers: minimal number of comparisons needed to sort n
elements" (https://oeis.org/A036604) which performed by Ford-Johnson algorithm.

Article Sources and Contributors 13
Article Sources and Contributors
Sorting algorithm Source: http://en.wikipedia.org/w/index.php?oldid=616190619 Contributors: -OOPSIE-, 124Nick, 132.204.27.xxx, A3 nm, A5b, A930913, AManWithNoPlan, Aaron
Rotenberg, Abce2, Abhishekupadhya, Accelerometer, Adair2324, AdamProcter, Advance512, Aeons, Aeonx, Aeriform, Agateller, Agorf, Aguydude, Ahoerstemeier, Ahshabazz, Ahy1, Alain
Amiouni, Alansohn, AlexPlank, Alksub, AllyUnion, Altenmann, Alvestrand, Amirmalekzadeh, Anadverb, Andre Engels, Andy M. Wang, Ang3lboy2001, Angela, Arpi0292, Artoonie, Arvindn,
Astronouth7303, AxelBoldt, Ayushyogi, BACbKA, Bachrach44, Balabiot, Baltar, Gaius, Barometz, Bartoron2, Bbi5291, Beland, Ben Standeven, BenFrantzDale, BenKovitz, Bender2k14,
Bento00, Bidabadi, Bkell, Bobo192, Boleyn, BonzaiThePenguin, Booyabazooka, Bradyoung01, Brendanl79, Bryan Derksen, BryghtShadow, Bubba73, BurtAlert, C. A. Russell, C7protal, CJLL
Wright, Caesura, Calculuslover, Calixte, CambridgeBayWeather, Canthusus, Carey Evans, Ccn, Charles Matthews, Chenopodiaceous, Chinju, Chris the speller, Ciaccona, Circular17,
ClockworkSoul, Codeman38, Cole Kitchen, Compfreak7, Conversion script, Cpl Syx, Crashmatrix, Crumpuppet, Cuberoot31, Cwolfsheep, Cyan, Cybercobra, Cymbalta, Cyrius, DHN, DIY,
DaVinci, Daiyuda, Damian Yerrick, Danakil, Daniel Quinlan, DarkFalls, Darkwind, DarrylNester, Darth Panda, David Eppstein, Dcirovic, Dcljr, Dcoetzee, Deanonwiki, Debackerl, Decrypt3,
Deepakjoy, Deskana, DevastatorIIC, Dgse87, Diannaa, Dihard, Domingos, Doradus, Duck1123, Duvavic1, Dybdahl, Dysprosia, EdC, Eddideigel, Efansoftware, Egomes jason, Elassint, Eliz81,
Energy Dome, Erebus Morgaine, Etopocketo, Excirial, Fagstein, Falcon8765, Fastily, Fawcett5, FenixFeather, Firsfron, Foobarnix, Foot, Fragglet, Frankenviking, Fred Bauder, Fredrik,
Frencheigh, Fresheneesz, Fuzzy, GDallimore, GanKeyu, GateKeeper, Gavenkoa, Gavia immer, Gdr, Giftlite, Glrx, Grafen, Graham87, Graue, GregorB, H3nry, HJ Mitchell, Hadal, Hagerman,
Hairhorn, Hamaad.s, Hannes Hirzel, Hashar, Hede2000, Hgranqvist, Hirzel, Hobart, HolyCookie, Hpa, I am One of Many, I dream of horses, IMalc, InverseHypercube, Iridescent,
Itsameen-bc103112, J.delanoy, JBakaka, JLaTondre, JRSpriggs, JTN, Jacektomas, Jachto, Jaguaraci, Jamesday, Japo, Jay Litman, Jbonneau, Jeffq, Jeffrey Mall, Jeronimo, Jesin, Jirka6, Jj137, Jll,
Jmw02824, John of Reading, Jokes Free4Me, JonGinny, Jonadab, Jonas Kölker, Josh Kehn, Joshk, Jprg1966, Jthemphill, Justin W Smith, Jwoodger, Kalraritz, Kevinsystrom, Khazar2, Kievite,
KiloByte, Kingjames iv, KlappCK, Knivd, Knutux, Kragen, KyuubiSeal, LC, Ldoron, Lee J Haywood, LilHelpa, Lowercase Sigma, Luna Santin, Lzap, Magioladitis, Makeemlighter, Malcolm
Farmer, Mandarax, Mark Renier, MarkisLandis, MartinHarper, Marvon7Newby, MarvonNewby, Mas.morozov, Materialscientist, Mathmensch, MattGiuca, Matthew0028, Mav, Maximus Rex,
Mbernard707, Mdd4696, Mdtr, Medich1985, Michael Greiner, Michael Hardy, Michaelbluejay, Mike Rosoft, Mike00500, Mina86, Mindmatrix, Mjs1991, Mountain, Mr Elmo,
Mrck@charter.net, Mrjeff, Mudd1, Musiphil, Myanw, NTF, Nanshu, Navstar55, Nayuki, Nbarth, Nevakee11, NewEnglandYankee, NickGarvey, NickT988, Nicolaum, Nikai, Nish0009, Nixdorf,
Nknight, Nomen4Omen, NostinAdrek, OfekRon, Olathe, Olivier, Omegatron, Ondra.pelech, OoS, Oskar Sigvardsson, Oğuz Ergin, Pablo.cl, Pajz, Pakaraki, Pamulapati, Panarchy, Panu-Kristian
Poiksalo, PatPeter, Patrick, Paul Murray, Pbassan, Pcap, Pce3@ij.net, Pelister, Perstar, Pete142, Peter Flass, Petri Krohn, Pfalstad, Philomathoholic, PierreBoudes, Piet Delport, Populus, Pparent,
PsyberS, Pyfan, Quaeler, Qwertyus, RHaworth, RJFJR, RapidR, Rasinj, Raul654, RaulMetumtam, RazorICE, Rcbarnes, Rcgldr, Reyk, Riana, Rich Smith, Roadrunner, Robert L, Robin S,
RobinK, Rodspade, Roman V. Odaisky, Rsathish, Rursus, Ruud Koot, Ryguasu, SVoid, Scalene, Schnozzinkobenstein, Shadowjams, Shanes, Shredwheat, SimonP, SiobhanHansa, Sir Nicholas
de Mimsy-Porpington, Slashme, Sligocki, Smartech, Smjg, Snickel11, Sophus Bie, Soultaco, SouthernNights, Spoon!, Ssd, StanfordProgrammer, Staplesauce, Starwiz, Staszek Lem, Stephen
Howe, Stephenb, StewieK, Suanshsinghal, Summentier, Sven nestle2, Swamp Ig, Swift, T4bits, TBingmann, TakuyaMurata, Tamfang, Taw, Tawker, Techie007, Teles, Templatetypedef, The
Anome, The Thing That Should Not Be, TheKMan, TheRingess, Thefourlinestar, Thinking of England, This lousy T-shirt, ThomasTomMueller, Thumperward, Timwi, Titodutta, Tobias
Bergemann, Tortoise 74, TowerDragon, Travuun, Trixter, Twikir, Tyler McHenry, UTSRelativity, Udirock, Ulfben, UpstateNYer, User A1, VTBassMatt, Vacation9, Valenciano, Veganfanatic,
VeryVerily, Veryangrypenguin, Verycuriousboy, Vrenator, Wantnot, Wazimuko, Wchargin, Wei.cs, Wfunction, Widr, Wiki.ryansmith, Wikipelli, Wikiwonky, WillNess, Wimt, Worch,
Writtenonsand, Ww, Yansa, Yintan, Yuval madar, Zawersh, Zipcodeman, Ztothefifth, Zundark, 971 anonymous edits
Image Sources, Licenses and Contributors
File:Sorting stability playing cards.svg Source: http://en.wikipedia.org/w/index.php?title=File:Sorting_stability_playing_cards.svg License: Creative Commons Zero Contributors:
User:Dcoetzee, User:WDGraham
File:Sorting playing cards using stable sort.svg Source: http://en.wikipedia.org/w/index.php?title=File:Sorting_playing_cards_using_stable_sort.svg License: Creative Commons Zero
Contributors: User:Dcoetzee, User:WDGraham
File:Bubblesort-edited-color.svg Source: http://en.wikipedia.org/w/index.php?title=File:Bubblesort-edited-color.svg License: Creative Commons Zero Contributors: User:Pmdumuid
File:Shellsort-edited.png Source: http://en.wikipedia.org/w/index.php?title=File:Shellsort-edited.png License: Public domain Contributors: by crashmatrix (talk)
Image:Wikibooks-logo-en-noslogan.svg Source: http://en.wikipedia.org/w/index.php?title=File:Wikibooks-logo-en-noslogan.svg License: Creative Commons Attribution-Sharealike 3.0
Contributors: User:Bastique, User:Ramac et al.
Image:Commons-logo.svg Source: http://en.wikipedia.org/w/index.php?title=File:Commons-logo.svg License: logo Contributors: Anomie
License
Creative Commons Attribution-Share Alike 3.0
//creativecommons.org/licenses/by-sa/3.0/

Sorting algorithm

Recommended

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to Sorting algorithm (20)

Recently uploaded (20)

Sorting algorithm