Quicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961. It is still a commonly used algorithm for sorting. Overall, it is slightly faster than merge sort and heapsort for randomized data, particularly on larger distributions. Quicksort is a divide-and-conquer algorithm. It works by selecting a "pivot" element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. For this reason, it is sometimes called partition-exchange sort. The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to perform the sorting. Quicksort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total order) is defined. It is a comparison-based sort since elements a and b are only swapped in case their relative order has been obtained in the transitive closure of prior comparison-outcomes. Most implementations of quicksort are not stable, meaning that the relative order of equal sort items is not preserved. Mathematical analysis of quicksort shows that, on average, the algorithm takes O ( n log n ) {\displaystyle O(n\log {n})} comparisons to sort n items. In the worst case, it makes O ( n 2 ) {\displaystyle O(n^{2})} comparisons.
History The quicksort algorithm was developed in 1959 by Tony Hoare while he was a visiting student at Moscow State University. At that time, Hoare was working on a machine translation project for the National Physical Laboratory. As a part of the translation process, he needed to sort the words in Russian sentences before looking them up in a Russian-English dictionary, which was in alphabetical order on magnetic tape. After recognizing that his first idea, insertion sort, would be slow, he came up with a new idea. He wrote the partition part in Mercury Autocode but had trouble dealing with the list of unsorted segments. On return to England, he was asked to write code for Shellsort. Hoare mentioned to his boss that he knew of a faster algorithm, and his boss bet a sixpence that he did not. His boss ultimately accepted that he had lost the bet. Hoare published a paper about his algorithm, including a theoretical analysis, in The Computer Journal Volume 5, Issue 1, 1962, Pages 10–16. Later, Hoare learned about ALGOL and its ability to do recursion, which enabled him to publish an improved version of the algorithm in ALGOL in Communications of the Association for Computing Machinery, the premier computer science journal of the time. The ALGOL code is published in Communications of the ACM (CACM), Volume 4, Issue 7 July 1961, pp 321 Algorithm 63: partition and Algorithm 64: Quicksort. Quicksort gained widespread adoption, appearing, for example, in Unix as the default library sort subroutine. Hence, it lent its name to the C standard library subroutine qsort and in the reference implementation of Java. Robert Sedgewick's PhD thesis in 1975 is considered a milestone in the study of Quicksort where he resolved many open problems related to the analysis of various pivot selection schemes including Samplesort, adaptive partitioning by Van Emden as well as derivation of expected number of comparisons and swaps. Jon Bentley and Doug McIlroy in 1993 incorporated various improvements for use in programming libraries, including a technique to deal with equal elements and a pivot scheme known as pseudomedian of nine, where a sample of nine elements is divided into groups of three and then the median of the three medians from three groups is chosen. Bentley described another simpler and compact partitioning scheme in his book Programming Pearls that he attributed to Nico Lomuto. Later Bentley wrote that he used Hoare's version for years but never really understood it but Lomuto's version was simple enough to prove correct. Bentley described Quicksort as the "most beautiful code I had ever written" in the same essay. Lomuto's partition scheme was also popularized by the textbook Introduction to Algorithms although it is inferior to Hoare's scheme because it does three times more swaps on average and degrades to O(n2) runtime when all elements are equal. McIlroy would further produce an AntiQuicksort (aqsort) function in 1998, which consistently drives even his 1993 variant of Quicksort into quadratic behavior by producing adversarial data on-the-fly.
Algorithm
Quicksort is a type of divide-and-conquer algorithm for sorting an array, based on a partitioning routine; the details of this partitioning can vary somewhat, so that quicksort is really a family of closely related algorithms. Applied to a range of at least two elements, partitioning produces a division into two consecutive non-empty sub-ranges, in such a way that no element of the first sub-range is greater than any element of the second sub-range. After applying this partition, quicksort then recursively sorts the sub-ranges, possibly after excluding from them an element at the point of division that is at this point known to be already in its final location. Due to its recursive nature, quicksort (like the partition routine) has to be formulated so as to be callable for a range within a larger array, even if the ultimate goal is to sort a complete array. The steps for in-place quicksort are:
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