Insertion sort

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. However, insertion sort provides several advantages:

Most humans when sorting—ordering a deck of cards, for example—use a method that is similar to insertion sort.

Algorithm

Sorting is typically done in-place. The resulting array after k iterations has the property where the first k + 1 entries are sorted. In each iteration the first remaining entry of the input is removed, inserted into the result at the correct position, thus extending the result:

becomes

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

The most common variant of insertion sort, which operates on arrays, can be described as follows:

# Suppose there exists a function called Insert designed to insert a value into a sorted sequence at the beginning of an array. It operates by beginning 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. The function has the side effect of overwriting the value stored immediately after the sorted sequence in the array. # To perform an insertion sort, begin at the left-most element of the array and invoke Insert to insert each element encountered into its correct position. The ordered sequence into which the element is inserted is stored at the beginning of the array in the set of indices already examined. Each insertion overwrites a single value: the value being inserted.

Pseudocode of the complete algorithm follows, where the arrays are zero-based and the for-loop includes both the top and bottom limits (as in Pascal): insertionSort(array A)

{ This procedure sorts in ascending order. } begin for i := 1 to length[A]-1 do begin value := A[i]; j := i - 1; done := false; repeat { To sort in descending order simply reverse the operator i.e. A[j] < value } if A[j] > value then begin A[j + 1] := A[j]; j := j - 1; if j < 0 then done := true; end else done := true; until done; A[j + 1] := value; end; end;

Below is the pseudocode for insertion sort for a zero-based array (as in C): 1. for j ←1 to length(A)-1 2. key ← A[ j ] 3. > A[ j ] is added in the sorted sequence A[1, .. j-1] 4. i ← j - 1 5. while i >= 0 and A [ i ] > key 6. A[ i +1 ] ← A[ i ] 7. i ← i -1 8. A [i +1] ← key

Best, worst, and average cases

The worst case input is an array sorted in reverse order. In this case every iteration of the inner loop will scan and shift the entire sorted subsection of the array before inserting the next element. For this case insertion sort has a quadratic running time (i.e., O(n²)).

The average case is also quadratic, which makes insertion sort impractical for sorting large arrays. However, insertion sort is one of the fastest algorithms for sorting very small arrays, even faster than quick sort; indeed, good quick sort implementations use insertion sort for arrays smaller than a certain threshold, also when arising as subproblems; the exact threshold must be determined experimentally and depends on the machine, but is commonly around ten.

Example: The following table shows the steps for sorting the sequence {5, 7, 0, 3, 4, 2, 6, 1}. For each iteration, the number of positions the inserted element has moved is shown in parentheses. Altogether this amounts to 17 steps.

5 7 0 3 4 2 6 1 (0)

5 7 0 3 4 2 6 1 (0)

0 5 7 3 4 2 6 1 (2)

0 3 5 7 4 2 6 1 (2)

0 3 4 5 7 2 6 1 (2)

0 2 3 4 5 7 6 1 (4)

0 2 3 4 5 6 7 1 (1)

0 1 2 3 4 5 6 7 (6)

Comparisons to other sorting algorithms

Calculations show that insertion sort will usually perform about half as many comparisons as selection sort. Assuming the k+1^th element's rank is random, insertion sort will on average require shifting half of the previous k elements, while selection sort always requires scanning all unplaced elements. If the input array is reverse-sorted, insertion sort performs as many comparisons as selection sort. If the input array is already sorted, insertion sort performs as few as n-1 comparisons, thus making insertion sort more efficient when given sorted or "nearly-sorted" arrays.

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, whereas selection sort will write only O(n) times. For this reason selection sort may be preferable in cases where writing to memory is significantly more expensive than reading, such as with 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 use insertion sort for sorting small sublists, where insertion sort outperforms these more complex algorithms. The size of list for which insertion sort has the advantage varies by environment and implementation, but is typically between eight and twenty elements.

Variants

If the cost of comparisons exceeds the cost of 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 may yield better performance. Binary insertion sort employs a binary search to determine the correct location to insert new elements, and therefore performs $\backslash lceil\; \backslash log\_2(n)\; \backslash rceil$ comparisons in the worst case, which is Θ(n log n). The algorithm as a whole still has a running time of Θ(n²) on average because of the series of swaps required for each insertion.

The number of swaps can be reduced by calculating the position of multiple elements before moving them. For example, if the target position of two elements is calculated before they are moved into the right position, the number of swaps can be reduced by about 25% for random data. In the extreme case, this variant works similar to merge sort.

To avoid having to make a series of swaps for each insertion, the input could be stored in a linked list, which allows elements to be inserted and deleted in constant-time. However, performing a binary search on a linked list is impractical because random access is not directly supported by a linked list and can only be effectively achieved by sequentially following the link chain to each desired element, which is relatively very slow; therefore, the running time required for searching is O(n²). If a more sophisticated data structure (e.g., heap or binary tree) is used, the time required for searching and insertion can be reduced significantly; 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 (i.e., "gaps") spread throughout the array. The benefit is that insertions need only shift elements over until a gap is reached. The authors show that this sorting algorithm runs with high probability in O(n log n) time.

References

External links

Category:Sorting algorithms Category:Comparison sorts Category:Stable sorts Category:Articles with example pseudocode Category:Online sorts