In computer science and mathematics, 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 to optimizing the use of other algorithms (such as search and merge algorithms) that require sorted lists to work correctly; it is also often useful for canonicalizing data and for producing human-readable output. More formally, the output must satisfy two conditions: Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... Look up list in Wiktionary, the free dictionary. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ... Sorting refers to a process of arranging items in some sequence and/or in different sets, and accordingly, it has two common, yet distinct meanings: ordering: aranging items of the same kind, class, nature, etc. ... In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions. ... Merge algorithms are a family of algorithms that run sequentially over multiple sorted lists, typically producing more sorted lists as output. ... Canonicalization is the process of converting data that has more than one possible representation into a standard canonical representation. ...

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, or reordering, of the input.

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] Although many consider it a solved problem, useful new sorting algorithms are still being invented to this day (for example, library sort was first published in 2004). 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, randomized algorithms, best, worst and average case analysis, time-space tradeoffs, and lower bounds. In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X. The relation is transitive, antisymmetric, and total. ... Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... Bubble sort is a simple sorting algorithm. ... Bubble sort, or gapped insertion sort is a sorting algorithm much like insertion sort but with gaps left in the sorted array to accelerate subsequent insertions. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... In computer science, divide and conquer (D&C) is an important algorithm design paradigm. ... A binary tree, a simple type of branching linked data structure. ... A randomized algorithm or probabilistic algorithm is an algorithm which employs a degree of randomness as part of its logic. ... In computer science, best, worst and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. ... In computer science, a space-time or time-memory trade-off is a situation where the programmer can reduce memory use at the cost of slower program execution, or can reduce computation time at the cost of increased memory use. ...

Classification

Sorting algorithms used in computer science are often classified by: Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

• Computational complexity (worst, average and best behaviour) of element comparisons in terms of the size of the list (n). For typical sorting algorithms good behavior is O(n log n) and bad behavior is Ω(n²). (See Big O notation) Ideal behavior for a sort is O(n). Sort algorithms which only use an abstract key comparison operation always need Ω(n log n) comparisons in the worst case.
• Computational complexity of swaps (for "in place" algorithms).
• Memory usage (and use of other computer resources). In particular, some sorting algorithms are "in place", such that only O(1) or O(log n) memory is needed beyond the items being sorted, while others need to create auxiliary locations for data to be temporarily stored.
• 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). See below for more information.
• 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.

As a branch of the theory of computation in computer science, computational complexity theory investigates the problems related to the amounts of resources required for the execution of algorithms (e. ... In computer science, best and worst cases in a given algorithm express what the resource usage is at least and at most, respectively. ... In computer science, best and worst cases in a given algorithm express what the resource usage is at least and at most, respectively. ... The term best-case performance is used in computer science to describe the way an algorithm behaves under optimal conditions. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... As a branch of the theory of computation in computer science, computational complexity theory investigates the problems related to the amounts of resources required for the execution of algorithms (e. ... In computer science, an in-place algorithm is an algorithm which transforms a data structure using a small, constant amount of extra storage space. ... A comparison sort is a particular type of sorting algorithm; a number of well-known algorithms are comparison sorts. ...

Stability

Stable sorting algorithms maintain the relative order of records with equal keyshttp://en.wikipedia.org/wiki/Strict_weak_ordering (i.e., sort key values). That is, a sorting algorithm is stable if whenever there are two records R and S with the same key and with R appearing before S in the original list, R will appear before S in the sorted list. A strict weak ordering is a binary relation that defines an equivalence relation and has the properties stated below. ...

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. However, assume that the following pairs of numbers are to be sorted by their first component:

 (4, 1) (3, 7) (3, 1) (5, 6) 

In this case, two different results are possible, one which maintains the relative order of records with equal keys, and one which does not:

 (3, 7) (3, 1) (4, 1) (5, 6) (order maintained) (3, 1) (3, 7) (4, 1) (5, 6) (order changed) 

Unstable sorting algorithms may change the relative order of records with equal keys, but stable sorting algorithms never do so. 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 data order as a tie-breaker. Remembering this order, however, often involves an additional space cost.

Sorting based on a primary, secondary, tertiary, etc. sort key can be done by any sorting method, taking all sort keys into account in comparisons (in other words, using a single composite sort key). If a sorting method is stable, it is also possible to sort multiple times, each time with one sort key. In that case the keys need to be applied in order of increasing priority.

Example: sorting pairs of numbers as above by first, then second component:

 (4, 1) (3, 7) (3, 1) (4, 6) (original) 

 (4, 1) (3, 1) (4, 6) (3, 7) (after sorting by second component) (3, 1) (3, 7) (4, 1) (4, 6) (after sorting by first component) 

On the other hand:

 (3, 7) (3, 1) (4, 1) (4, 6) (after sorting by first component) (3, 1) (4, 1) (4, 6) (3, 7) (after sorting by second component, order by first component is disrupted) 

List of sorting 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. These are all comparison sorts. A comparison sort is a particular type of sorting algorithm; a number of well-known algorithms are comparison sorts. ...

This table describes sorting algorithms that are not comparison sorts. As such, they are not limited by a O(nlog n) lower bound. Complexities below are in terms of n, the number of items to be sorted, k, the size of each key, and s, the chunk size used by the implementation. 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. Bubble sort is a simple sorting algorithm. ... Cocktail sort, also known as bidirectional bubble sort, cocktail shaker sort, shaker sort, ripple sort, shuttle sort or happy hour sort, is a stable sorting algorithm that varies from bubble sort in that instead of repeatedly passing through the list from top to bottom, it passes alternately from top to... In computer science, comb sort is a relatively simplistic sort algorithm designed by Stephen Lacey and Richard Box, who described it in Byte Magazine in April 1991. ... Gnome sort is a sort algorithm which is similar to insertion sort except that moving an element to its proper place is accomplished by a series of swaps, as in bubble sort. ... Selection sort is a sorting algorithm, specifically an in-place comparison sort. ... Example of insertion sort sorting a list of random numbers. ... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... Shell sort is a sorting algorithm that, with its original implementation, requires O(n2) comparisons and exchanges in the worst case. ... In computer science, a binary search tree (BST) is a binary tree where every node has a value, every nodes left subtree has values less than or equal to the nodes value, and every right subtree has values greater. ... In computing, a self-balancing binary search tree or height-balanced binary search tree is a binary search tree that attempts to keep its height, or the number of levels of nodes beneath the root, as small as possible at all times, automatically. ... Bubble sort, or gapped insertion sort is a sorting algorithm much like insertion sort but with gaps left in the sorted array to accelerate subsequent insertions. ... A merge sort algorithm used to sort an array of 7 integer values. ... In computer science, algorithms work-in-place if they transform a data structure using a minimal, constant amount of extra memory (or disk) space. ... A merge sort algorithm used to sort an array of 7 integer values. ... A run of the heapsort algorithm sorting an array of randomly permuted values. ... The smoothsort sorting algorithm is a variation of heapsort developed by Edsger Dijkstra in 1981. ... Q sort redirects here. ... A naïve algorithm is a very simple solution to a problem that has a very high time- or memory-complexity. ... Introsort or introspective sort is a sorting algorithm designed by David Musser in 1997. ... [[Im[[Image:Example. ... Patience sorting is a sorting algorithm, based on a solitaire card game, that has the property of being able to efficiently compute the length of the longest increasing subsequence in a given array. ... The longest increasing subsequence problem is to find the longest increasing subsequence of a given sequence. ... A comparison sort is a particular type of sorting algorithm; a number of well-known algorithms are comparison sorts. ...

This table describes some sorting algorithms that are impractical for real-life use due to extremely poor performance or a requirement for specialized hardware. Pigeonhole sorting, also known as count sort, is a sorting algorithm that takes linear time (Θ(n)), which is the best possible performance for a sorting algorithm since one must inevitably look at each of the elements being sorted at least once, regardless of the sorting algorithm. ... Bucket sort is a sort algorithm that works by partitioning an array into a finite number of buckets and then sorting each bucket. ... Counting sort is a sorting algorithm which (like bucket sort) takes advantage of knowing the range of the numbers in the array to be sorted (array A). ... A radix sort is a sorting algorithm that can rearrange integer representations based on the processing of individual digits in such a way that the integer representations are eventually in either ascending or descending order. ... A radix sort is a sorting algorithm that can rearrange integer representations based on the processing of individual digits in such a way that the integer representations are eventually in either ascending or descending order. ... Spreadsort is a relatively new sorting algorithm invented by Steven J. Ross in 2002. ...

Bogosort is a particularly ineffective sorting algorithm. ... Bogosort is a particularly ineffective sorting algorithm. ... Stooge sort is a recursive sorting algorithm with a time complexity of about O(n2. ... The Bead sort is a natural sorting algorithm, developed by Joshua J. Arulanandham, Cristian S. Calude and Michael J. Dinneen in 2002, and published in The Bulletin of the European Association for Theoretical Computer Science. ... Pancake sorting is a variation of the sorting problem in which the only allowed operation is to reverse the elements of some prefix of the sequence. ... A sorting network is a circuit with n input connections for data items to be presented in an arbitrary order, and n output connections on which the items are to be output in sorted order. ...

Summaries of popular sorting algorithms

Bubble sort

Main article: Bubble sort

Bubble sort is a straightforward and simplistic method of sorting data that is used in computer science education. 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. While simple, this algorithm is highly inefficient and is rarely used except in education. A slightly better variant, cocktail sort, works by inverting the ordering criteria and the pass direction on alternating passes. Its average case and worst case are both O(n²). Bubble sort is a simple sorting algorithm. ... Cocktail sort, also known as bidirectional bubble sort, cocktail shaker sort, shaker sort, ripple sort, shuttle sort or happy hour sort, is a stable sorting algorithm that varies from bubble sort in that instead of repeatedly passing through the list from top to bottom, it passes alternately from top to...

Selection sort

Main article: Selection sort

Selection sort is a simple sorting algorithm that improves on the performance of bubble sort. It works by first finding the smallest element using a linear scan and swapping it into the first position in the list, then finding the second smallest element by scanning the remaining elements, and so on. Selection sort is unique compared to almost any other algorithm in that its running time is not affected by the prior ordering of the list: it performs the same number of operations because of its simple structure. Selection sort also requires only n swaps, and hence just Θ(n) memory writes, which is optimal for any sorting algorithm. Thus it can be very attractive if writes are the most expensive operation, but otherwise selection sort will usually be outperformed by insertion sort or the more complicated algorithms. Selection sort is a sorting algorithm, specifically an in-place comparison sort. ...

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. The insertion sort works just like its name suggests - it inserts each item into its proper place in the final list. The simplest implementation of this requires two list structures - the source list and the list into which sorted items are inserted. To save memory, most implementations use an in-place sort that works by moving the current item past the already sorted items and repeatedly swapping it with the preceding item until it is in place. Shell sort (see below) is a variant of insertion sort that is more efficient for larger lists. This method is much more efficient than the bubble sort, though it has more constraints. Example of insertion sort sorting a list of random numbers. ... Shell sort is a sorting algorithm that, with its original implementation, requires O(n2) comparisons and exchanges in the worst case. ...

Shell sort

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. Although this method is inefficient for large data sets, it is one of the fastest algorithms for sorting small numbers of elements (sets with less than 1000 or so elements). Another advantage of this algorithm is that it requires relatively small amounts of memory. Shell sort is a sorting algorithm that, with its original implementation, requires O(n2) comparisons and exchanges in the worst case. ... Donald L. Shell (1924-) was an American computer scientist best known as the author of the Shell sort, a sorting algorithm. ... Year 1959 (MCMLIX) was a common year starting on Thursday (link will display full calendar) of the Gregorian calendar. ...

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. A merge sort algorithm used to sort an array of 7 integer values. ...

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 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. A run of the heapsort algorithm sorting an array of randomly permuted values. ... Selection sort is a sorting algorithm, specifically an in-place comparison sort. ... Example of a complete binary max heap In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: if B is a child node of A, then key(A) ≥ key(B). ... In computer science, a binary tree is a tree data structure in which each node has at most two children. ...

Quicksort

Main article: Quicksort

Quicksort is a divide and conquer algorithm which relies on a partition operation: to partition an array, we choose an element, called a pivot, move all smaller elements before the pivot, and move all greater elements after it. This can be done efficiently in linear time and in-place. We then recursively sort the lesser and greater sublists. 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, this makes quicksort one of the most popular sorting algorithms, available in many standard libraries. The most complex issue in quicksort is choosing a good pivot element; consistently poor choices of pivots can result in drastically slower (O(n²)) performance, but if at each step we choose the median as the pivot then it works in O(n log n). Q sort redirects here. ... In computer science, divide and conquer (D&C) is an important algorithm design paradigm. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In computer science, an in-place algorithm is an algorithm which transforms a data structure using a small, constant amount of extra storage space. ...

Bucket sort

Main article: Bucket sort

Bucket sort is a sorting algorithm that works 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. A variation of this method called the single buffered count sort is faster than the quick sort and takes about the same time to run on any set of data. More information is available at http://www.mcky.net/hsrto.htm Bucket sort is a sort algorithm that works by partitioning an array into a finite number of buckets and then sorting each bucket. ...

Radix sort

Main article: Radix sort

Radix sort is an algorithm that sorts a list of fixed-size numbers of length k in O(n · k) time by treating them as bit strings. We first sort the list by the least significant bit while preserving their relative order using a stable sort. Then we sort them by the next bit, and so on from right to left, and the list will end up sorted. Most often, the counting sort algorithm is used to accomplish the bitwise sorting, since the number of values a bit can have is small. A radix sort is a sorting algorithm that can rearrange integer representations based on the processing of individual digits in such a way that the integer representations are eventually in either ascending or descending order. ... Counting sort is a sorting algorithm which (like bucket sort) takes advantage of knowing the range of the numbers in the array to be sorted (array A). ...

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. In computer architecture, a bus is a subsystem that transfers data or power between computer components inside a computer or between computers and typically is controlled by device driver software. ... CPU redirects here. ...

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. Q sort redirects here. ...

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".^[2] A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how that data is arranged). ...

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 easily in RAM (say, a few thousand elements), the chunks sorted using an efficient algorithm (such as quicksort or heapsort), and the results merged as per mergesort. This is more efficient than just doing mergesort in the first place, but it requires less physical RAM (to be practical) than a full quicksort on the whole array. Q sort redirects here. ... A run of the heapsort algorithm sorting an array of randomly permuted values. ... In computer science, merge sort or mergesort is a sort algorithm for rearranging lists (or any other data structure that can only be accessed sequentially, e. ...

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. How virtual memory maps to physical memory Virtual memory is an addressing scheme implemented in hardware and software that allows non-contiguous memory to be addressed as if it were contiguous. ...

Graphical representations

• Hobart and William Smith Colleges has a graphical demonstration of various sorting algorithms. It includes explanations of each step in the sort, plus a timing test. Thus is a good tool for the novice computer science student.
• University of British Columbia has a graphical demonstration of various in-place sorts.
• University of Linköping has a similar demonstration in their course material.
• The sort algorithm visualizer provides graphical animation of 11 comparison-based sorts in a variety of initial configurations.
• A Java applet visually demonstrates several sorting algorithms.
• Boston College has a graphical demonstration of various sorting algorithms with various initial conditions.

Microsoft's "Quick" programming languages (such as QuickBASIC and QuickPascal) have a file named "sortdemo" (with extension BAS and PAS for QB and QP, respectively) in the examples folder that provides a graphical representation of several of the various sort procedures described here, as well as performance ratings of each. Hobart and William Smith Colleges, located in Geneva, New York, are together a liberal arts college. ... The University of British Columbia (UBC) is a Canadian public research university with campuses in Vancouver and Kelowna. ... Linköping University Linköping University (LiU) or Linköpings universitet is a state university in Linköping, Sweden. ... For similarly-named academic institutions, see Boston (disambiguation)#Education. ... Microsoft QuickBASIC (also QB or sometimes, QBasic, which is also a different system) is an Integrated Development Environment (or IDE) and Compiler for the BASIC programming language that was developed by Microsoft. ... Microsoft Pascal was an implementation of the Pascal programming language that was developed by the Microsoft Corporation for compiling programs for running on its MS-DOS operating system and, in later versions, on OS/2 (albeit that it was only capable of generating 16-bit programs for the latter). ...

See also

• Big O notation
• External sorting
• Sorting networks (compare)
• Collation
• Schwartzian transform
• Shuffling algorithms
• Search algorithms
• Wikibooks: Algorithms: Uses sorting a deck of cards with many sorting algorithms as an example

For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... External sorting is a generic term for a class of sorting algorithms that can handle massive amounts of data. ... A sorting network is a circuit with n input connections for data items to be presented in an arbitrary order, and n output connections on which the items are to be output in sorted order. ... Alphabetical redirects here. ... The Schwartzian transform is a technique in computer programming used to efficiently sort an array by using the result of a computation-heavy operation without re-computing that result for every comparison performed by the sort subroutine. ... The riffle Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. ... In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions. ...

Notes and references

• D. E. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching.

Donald Ervin Knuth ( or Ka-NOOTH[1]) (b. ... Cover of books The Art of Computer Programming[1] is a comprehensive monograph written by Donald Knuth which covers many kinds of programming algorithms and their analysis. ...

External links

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Wikibooks Algorithm implementation has a page on the topic of

Sorting algorithms

• Sequential and parallel sorting algorithms - has explanations and analyses of many of these algorithms.
• Ricardo Baeza-Yates' sorting algorithms on the Web
• 'Dictionary of Algorithms, Data Structures, and Problems'
• Slightly Skeptical View on Sorting Algorithms Softpanorama page that discusses several classic algorithms and promotes alternatives to quicksort.
• Sorting Revisited
• Sorting small arrays C++ code and analysis looking at the best (fastest) way to sort small (ie < 64) items, including special attention on sorting networks.
• For a repository of algorithms with source code and lectures, see The Stony Brook Algorithm Repository
• Graphical Java illustrations of the Bubble sort, Insertion sort, Quicksort, and Selection sort
• xSortLab - An interactive Java demonstration of Bubble, Insertion, Quick, Select and Merge sorts, which displays the data as a bar graph with commentary on the workings of the algorithm printed below the graph.
• Sorting Algorithms Demo - Java applets that chart the progress of several common sorting algorithms while sorting an array of data using in-place algorithms.
• Sorting contest - An applet visually demonstrating a contest between a number of different sorting algorithms
• (Italian) Java Applet that show some sorting algorithms
• The Three Dimensional Bubble Sort- A method of sorting in three or more dimensions (of questionable merit)
• Sorting Algorithms Visualized - Java applet visualizing the different approaches of 22 sorting algorithms (configurable input set size and value distribution)
• Sort huge amounts of data by doing a multi-phase sorting on temporary file
• AniSort - Java applet visualizing 6 different sorting algorithms
• Naturalorder - Natural Order Numerical Sorting
• Pointers to sorting visualizations
• Monkey Sort discussion (another name for Bozo-sort)
• Extensive collection of animated sorting algorithms with source code.
• Several sorting algorithms demonstrated in Java
• OPL booklet of the main sorting algorithms by Michael Lamont
• QiSort - A new O(n log n) algorithm for 2007