Insertion sort is a simple sorting algorithm, a comparison sort in which the sorted array (or list) is built one entry at a time. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort, but it has various advantages: Image File history File links No higher resolution available. ... In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a list in a certain order. ... A comparison sort is a particular type of sorting algorithm; a number of well-known algorithms are comparison sorts. ... Quicksort in action on a list of random numbers. ... A run of the heapsort algorithm sorting an array of randomly permuted values. ... A merge sort algorithm used to sort an array of 7 integer values. ...

• Simple to implement
• Efficient on (quite) small data sets
• Efficient on data sets which are already substantially sorted: it runs in O(n + d) time, where d is the number of inversions
• More efficient in practice than most other simple O(n²) algorithms such as selection sort or bubble sort: the average time is n²/4 and it is linear in the best case
• Stable (does not change the relative order of elements with equal keys)
• In-place (only requires a constant amount O(1) of extra memory space)
• It is an online algorithm, in that it can sort a list as it receives it.

In abstract terms, every iteration of an insertion sort removes an element from the input data, inserting it at the correct position in the already sorted list, until no elements are left in the input. The choice of which element to remove from the input is arbitrary and can be made using almost any choice algorithm. In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. ... Selection sort is a sorting algorithm, specifically an in-place comparison sort. ... Bubble sort, sometimes shortened to bubblesort, is a simple sorting algorithm. ... In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a list in a certain order. ... In computer science, an in-place algorithm is an algorithm which transforms a data structure using a small, constant amount of extra storage space. ... In computer science, an online algorithm is one that can process its input piece-by-piece, without having the entire input available from the start. ...

Sorting is typically done in-place. The resulting array after k iterations contains the first k entries of the input array and is sorted. In each step, the first remaining entry of the input is removed, inserted into the result at the right position, thus extending the result:

Insertion sort before image, for insertion sort page, made by Derrick Coetzee in Illustrator, see Image:insertionsort-after. ...

becomes:

Insertion sort after image, for insertion sort page, made by Derrick Coetzee in Illustrator, see Image:insertionsort-before. ...

with each element > x copied to the right as it is compared against x.

The most common variant, which operates on arrays, can be described as:

1. Suppose we have a method called insert designed to insert a value into a sorted sequence at the beginning of an array. It operates by starting at the end of the sequence and shifting each element one place to the right until a suitable position is found for the new element. It has the side effect of overwriting the value stored immediately after the sorted sequence in the array.
2. To perform insertion sort, start at the left end of the array and invoke insert to insert each element encountered into its correct position. The ordered sequence into which we insert it is stored at the beginning of the array in the set of indexes already examined. Each insertion overwrites a single value, but this is okay because it's the value we're inserting.

A simple pseudocode version of the complete algorithm follows, where the arrays are zero-based:

 insert(array a, int length, value) { int i = length - 1; while (i >= 0 && a[i] > value) { a[i + 1] = a[i]; i--; } a[i + 1] := value; } insertionSort(array a, int length) { for (int i = 1; i < length; i++) insert(a, i, a[i]); } 

Good and bad input cases

In the best case of an already sorted array, this implementation of insertion sort takes O(n) time: in each iteration, the first remaining element of the input is only compared with the last element of the sorted subsection of the array. This same case provides worst-case behavior for non-randomized quicksort, which will take O(n²) time to sort an already-sorted list. Thus, if an array is sorted or nearly sorted, insertion sort will significantly outperform quicksort. Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. ... Quicksort in action on a list of random numbers. ... Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. ...

The worst case is an array sorted in reverse order, as every execution of the inner loop will have to scan and shift the entire sorted section of the array before inserting the next element. Insertion sort takes O(n²) time in this worst case as well as in the average case, which makes it impractical for sorting large numbers of elements. However, insertion sort's inner loop is very fast, which often makes it one of the fastest algorithms for sorting small numbers of elements, typically less than 10 or so.

Comparisons to other sorts

Insertion sort is very similar to selection sort. Just like in selection sort, after k passes through the array, the first k elements are in sorted order. For selection sort, these are the k smallest elements, while in insertion sort they are whatever the first k elements were in the unsorted array. Insertion sort's advantage is that it only scans as many elements as it needs to in order to place the k + 1st element, while selection sort must scan all remaining elements to find the absolute smallest element. Selection sort is a sorting algorithm, specifically an in-place comparison sort. ...

Simple calculation shows that insertion sort will therefore usually perform about half as many comparisons as selection sort. Assuming the k + 1st element's rank is random, it will on the average require shifting half of the previous k elements over, while selection sort always requires scanning all unplaced elements. If the array is not in a random order, however, insertion sort can perform just as many comparisons as selection sort (for a reverse-sorted list). It will also perform far fewer comparisons, as few as n - 1, if the data is pre-sorted, thus insertion sort is much more efficient if the array is already sorted or "close to sorted." It can be seen as an advantage for some real-time applications that selection sort will perform identically regardless of the order of the array, while insertion sort's running time can vary considerably. Wikipedia does not have an article with this exact name. ...

While insertion sort typically makes fewer comparisons than selection sort, it requires more writes because the inner loop can require shifting large sections of the sorted portion of the array. In general, insertion sort will write to the array O(n²) times while selection sort will write only O(n) times. For this reason, selection sort may be better in cases where writes to memory are significantly more expensive than reads, such as EEPROM or Flash memory. Selection sort is a sorting algorithm, specifically an in-place comparison sort. ... An EEPROM (also called an E2PROM)[] or Electronically Erasable Programmable Read-Only Memory, is a non-volatile storage chip used in computers and other devices to store small amounts of volatile (configuration) data. ... A USB flash drive. ...

Some divide-and-conquer algorithms such as quicksort and mergesort sort by recursively dividing the list into smaller sublists which are then sorted. A useful optimization in practice for these algorithms is to switch to insertion sort for "sorted enough" sublists on which insertion sort outperforms the more complex algorithms. The size of list for which insertion sort has the advantage varies by environment and implementation, but is typically around 8 to 20 elements. In computer science, divide and conquer (D&C) is an important algorithm design paradigm. ... Quicksort in action on a list of random numbers. ... In computer science, merge sort or mergesort is a sort algorithm for rearranging lists (or any other data structure that can only be accessed sequentially, e. ...

Variants

D.L. Shell made substantial improvements to the algorithm, and the modified version is called Shell sort. It compares elements separated by a distance that decreases on each pass. Shell sort has distinctly improved running times in practical work, with two simple variants requiring O(n^3/2) and O(n^4/3) time. Donald L. Shell (1924-) was an American computer scientist best known as the author of the Shell sort, a sorting algorithm. ... Shell sort is a sorting algorithm that, with its original implementation, requires O(n2) comparisons and exchanges in the worst case. ...

If comparisons are very costly compared to swaps, as is the case for example with string keys stored by reference or with human interaction (such as choosing one of a pair displayed side-by-side), then using binary insertion sort can be a good strategy. Binary insertion sort employs binary search to find the right place to insert new elements, and therefore performs comparisons in the worst case, which is Θ(n log n). The algorithm as a whole still takes Θ(n²) time on average due to the series of swaps required for each insertion, and since it always uses binary search, the best case is no longer Ω(n) but Ω(n log n). In computer science, binary search or binary chop is a search algorithm for finding a particular value in a linear array, by ruling out half of the data at each step. ...

To avoid having to make a series of swaps for each insertion, we could instead store the input in a linked list, which allows us to insert and delete elements in constant time. Unfortunately, binary search on a linked list is impossible, so we still spend O(n²) time searching. If we instead replace it by a more sophisticated data structure such as a heap or binary tree, we can significantly decrease both search and insert time. This is the essence of heap sort and binary tree sort. In computer science, a linked list is one of the fundamental data structures used in computer programming. ... A binary tree, a simple type of branching linked data structure. ... Example of a complete binary max heap In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: if B is a child node of A, then key(A) ≥ key(B). ... In computer science, a binary tree is a tree data structure in which each node has at most two children. ... Heapsort is one of the best general-purpose sort algorithms, and is part of the selection sort family. ... In computer science, a binary search tree (BST) is a binary tree where every node has a value, every nodes left subtree has values less than or equal to the nodes value, and every right subtree has values greater. ...

In 2004, Bender, Farach-Colton, and Mosteiro published a new variant of insertion sort called library sort or gapped insertion sort that leaves a small number of unused spaces ("gaps") spread throughout the array. The benefit is that insertions need only shift elements over until a gap is reached. Surprising in its simplicity, they show that this sorting algorithm runs with high probability in O(n log n) time. [1] Bubble sort, or gapped insertion sort is a sorting algorithm much like insertion sort but with gaps left in the sorted array to accelerate subsequent insertions. ...

External links

Wikibooks Algorithm implementation has a page on the topic of

Insertion sort

• Analyze Insertion Sort in an online Javascript IDE
• Insertion Sort Java Applet
• NGenerics - implementation in C#
• An animated Java applet showing a step-by-step insertion sort.
• Literate implementations of insertion sort in various languages on LiteratePrograms
• Pointers to insertion sort visualizations
• Insertion sort explained and C++ code

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References

• Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Second Edition. Addison-Wesley, 1998. ISBN 0-201-89685-0. Section 5.2.1: Sorting by Insertion, pp.80–105.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 2.1: Insertion sort, pp.15–21.