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Insertion sort
Example of insertion sort sorting a list of random numbers.
General Data
Class: Sorting algorithm
Data Structure: Array
Time Complexity: О(n²)
Space Complexity: О(n) total, O(1) auxiliary
Optimal: Not usually


Insertion sort is a simple sorting algorithm, a comparison sort in which the sorted array (or list) is built one entry at a time. 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 array is a Data structure consisting of a group of elements that are accessed by indexing. 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 type of Sorting algorithm that only reads the list elements through a single abstract comparison operation (often a "less than or equal to" operator It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort, but it has various advantages:

Contents

Algorithm

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.

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:

The array right before insertion of x

becomes:

The array right after insertion of x

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 and the for loop includes both the top and bottom limits (like in Pascal):

insertionSort(array A)
    for i = 1 to length[A]-1 do
    begin
        value = A[i]
        j = i-1
        while j >= 0 and A[j] > value do
        begin
            A[j + 1] = A[j]
            j = j-1
        end
        A[j+1] = value
    end

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. Pseudocode is a compact and informal high-level description of a Computer programming Algorithm that uses the structural conventions of some Programming language Pascal is an influential imperative and procedural Programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments This same case provides worst-case behavior for non-randomized and poorly implemented quicksort, which will take O(n2) time to sort an already-sorted list. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments

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. Selection sort is a Sorting algorithm, specifically an in-place Comparison 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.

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. "

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. Selection sort is a Sorting algorithm, specifically an in-place Comparison sort. 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. EEPROM (also written E2PROM and pronounced e-e-prom or simply e-squared which stands for E lectrically E rasable P rogrammable Flash memory is non-volatile computer memory that can be electrically erased and reprogrammed

Some divide-and-conquer algorithms such as quicksort and mergesort sort by recursively dividing the list into smaller sublists which are then sorted. In Computer science, divide and conquer ( D&C) is an important Algorithm design Paradigm. In Computer science, merge sort or mergesort is an O ( n log n) comparison-based Sorting algorithm. A useful optimization in practice for these algorithms is to switch to insertion sort for small 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.

Variants

D.L. Shell made substantial improvements to the algorithm, and the modified version is called Shell sort. Donald L Shell (born March 1, 1924) is a retired American computer scientist who designed the Shell sort Sorting algorithm Shell sort is a Sorting algorithm that is a generalization of Insertion sort, with two observations insertion sort is efficient if the input is "almost 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(n3/2) and O(n4/3) time.

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 \lceil log_2(n!) \rceil comparisons in the worst case, which is Θ(n log n). A binary search algorithm (or binary chop) is a technique for locating a particular value in a sorted list of values 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).

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. In Computer science, a linked list is one of the fundamental Data structures and can be used to implement other data structures 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. A data structure in Computer science is a way of storing Data in a computer so that it can be used efficiently In Computer science, a heap is a specialized tree -based Data structure that satisfies the heap property if B is a In Computer science, a binary tree is a tree data structure in which each node has at most two children. This is the essence of heap sort and binary tree sort. Heapsort (method is a comparison-based Sorting algorithm, and is part of the Selection sort family A tree sort is a Sort algorithm that builds a Binary search tree from the keys to be sorted and then traverses the tree so that the keys come out in sorted order

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. Library sort, or gapped insertion sort is a Sorting algorithm that uses an Insertion sort, but with gaps in the array to accelerate subsequent insertions 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]


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References

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