Sorting algorithm

In computer science, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most frequently used orders are numerical order and lexicographical order. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and merge algorithms) which require input data to be in sorted lists. Sorting is also often useful for canonicalizing data and for producing human-readable output. More formally, the output of any sorting algorithm must satisfy two conditions: In computer science, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most frequently used orders are numerical order and lexicographical order. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and merge algorithms) which require input data to be in sorted lists. Sorting is also often useful for canonicalizing data and for producing human-readable output. More formally, the output of any sorting algorithm must satisfy two conditions: Further, the input data is often stored in an array, which allows random access, rather than a list, which only allows sequential access; though many algorithms can be applied to either type of data after suitable modification. Sorting algorithms are often referred to as a word followed by the word 'sort,' and grammatically are used in English as noun phrases, for example in the sentence, 'it is inefficient to use insertion sort on large lists,' the phrase insertion sort refers to the insertion sort sorting algorithm. From the beginning 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. Among the authors of early sorting algorithms around 1951 was Betty Holberton (née Snyder), who worked on ENIAC and UNIVAC. Bubble sort was analyzed as early as 1956. Comparison sorting algorithms have a fundamental requirement of Ω(n log n) comparisons (some input sequences will require a multiple of n log n comparisons); algorithms not based on comparisons, such as counting sort, can have better performance. 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.