Sorting  (Page 21/20)

6. sorting

6.1. basic sort algorithms

(From Wikipedia, the free encyclopedia)

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:

1. The output is in non-decreasing 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. 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.

Classification

Sorting algorithms used in computer science are often classified by:

• 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 at least Ω(n log n) comparisons on average.
• 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.

Stability

Stable sorting algorithms maintain the relative order of records with equal keys (External Link) (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.