Sorting algorithm

Merge sort

In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending order or descending order. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and merge algorithms) that require input data to be in sorted lists. Sorting is also often useful for canonicalizing data and for producing human-readable output.

Formally, the output of any sorting algorithm must satisfy two conditions:

1. The output is in monotonic order (each element is no smaller/larger than the previous element, according to the required order).
2. The output is a permutation (a reordering, yet retaining all of the original elements) of the input.

Although some algorithms are designed for sequential access, the highest-performing algorithms assume data is stored in a data structure which allows random access.

nlog⁡n−1.4427n+O(log⁡n) comparisons. Algorithms not based on comparisons, such as counting sort, can have better performance.

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.

Sorting small arrays optimally (in the fewest comparisons and swaps) or fast (i.e. taking into account machine-specific details) is still an open research problem, with solutions only known for very small arrays (fewer than 20 elements). Similarly optimal (by various definitions) sorting on a parallel machine is an open research topic.

Sorting algorithms can be classified by:

• Computational complexity
  • Best, worst and average case behavior in terms of the size of the list. For typical serial sorting algorithms, good behavior is O(n log n), with parallel sort in O(log n), and bad behavior is O(n). 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).
  • 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 typically recursive or typically non-recursive, while others may typically 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 cycle sort and heapsort.
• Whether the algorithm is serial or parallel. The remainder of this discussion almost exclusively concentrates on serial algorithms and assumes serial operation.
• Adaptability: Whether or not the presortedness of the input affects the running time. Algorithms that take this into account are known to be adaptive.
• Online: An algorithm such as Insertion Sort that is online can sort a constant stream of input.

An example of stable sort 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.

Stable sorting algorithms sort equal elements in the same order that they appear in the input. 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, i.e. if one comes before the other in the input, it will come before the other in the output.

Stability is important to preserve order over multiple sorts on the same data set. For example, say that student records consisting of name and class section are sorted dynamically, first by name, then by class section. If a stable sorting algorithm is used in both cases, the sort-by-class-section operation will not change the name order; with an unstable sort, it could be that sorting by section shuffles the name order, resulting in a nonalphabetical list of students.

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:

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 utilised 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 suit, and then compares by rank if the suits are the same.

This analysis assumes that the length of each key is constant and that all comparisons, swaps and other operations can proceed in constant time.

Legend:

• (log n).

Below is a table of comparison sorts. Mathematical analysis demonstrates a comparison sort cannot perform better than