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Concept# Comparison sort

Summary

A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator or a three-way comparison) that determines which of two elements should occur first in the final sorted list. The only requirement is that the operator forms a total preorder over the data, with:
if a ≤ b and b ≤ c then a ≤ c (transitivity)
for all a and b, a ≤ b or b ≤ a (connexity).
It is possible that both a ≤ b and b ≤ a; in this case either may come first in the sorted list. In a stable sort, the input order determines the sorted order in this case.
A metaphor for thinking about comparison sorts is that someone has a set of unlabelled weights and a balance scale. Their goal is to line up the weights in order by their weight without any information except that obtained by placing two weights on the scale and seeing which one is heavier (or if they weigh the same).
Some of the most well-known comparison sorts include:
Quicksort
Heapsort
Shellsort
Merge sort
Introsort
Insertion sort
Selection sort
Bubble sort
Odd–even sort
Cocktail shaker sort
Cycle sort
Merge-insertion sort
Smoothsort
Timsort
Block sort
There are fundamental limits on the performance of comparison sorts. A comparison sort must have an average-case lower bound of Ω(n log n) comparison operations, which is known as linearithmic time. This is a consequence of the limited information available through comparisons alone — or, to put it differently, of the vague algebraic structure of totally ordered sets. In this sense, mergesort, heapsort, and introsort are asymptotically optimal in terms of the number of comparisons they must perform, although this metric neglects other operations. Non-comparison sorts (such as the examples discussed below) can achieve O(n) performance by using operations other than comparisons, allowing them to sidestep this lower bound (assuming elements are constant-sized).
Comparison sorts may run faster on some lists; many adaptive sorts such as insertion sort run in O(n) time on an already-sorted or nearly-sorted list.

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Selection sort

In computer science, selection sort is an in-place comparison sorting algorithm. It has an O(n2) time complexity, which makes it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity and has performance advantages over more complicated algorithms in certain situations, particularly where auxiliary memory is limited. The algorithm divides the input list into two parts: a sorted sublist of items which is built up from left to right at the front (left) of the list and a sublist of the remaining unsorted items that occupy the rest of the list.

Quicksort

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.

Sorting network

In computer science, comparator networks are abstract devices built up of a fixed number of "wires", carrying values, and comparator modules that connect pairs of wires, swapping the values on the wires if they are not in a desired order. Such networks are typically designed to perform sorting on fixed numbers of values, in which case they are called sorting networks. Sorting networks differ from general comparison sorts in that they are not capable of handling arbitrarily large inputs, and in that their sequence of comparisons is set in advance, regardless of the outcome of previous comparisons.

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