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:
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:
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 and poorly implemented quicksort, which will take O(n2) time to sort an already-sorted list.
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.
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.
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. 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.
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 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.
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(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 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).
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 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.