Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time by comparisons. 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:
Simple implementation: Jon Bentley shows a version that is three lines in C-like pseudo-code, and five lines when optimized. Efficient for (quite) small data sets, much like other quadratic (i.e., O(n2)) sorting algorithms May be more efficient in practice than most other simple quadratic algorithms such as selection sort or bubble sort – but relative incoming data order and read/write costs matter – with high exchange costs and randomly ordered data, selection sort is faster Adaptive, i.e., efficient for data sets that are already substantially sorted: the time complexity is O(kn) when each element in the input is no more than k places away from its sorted position Stable; i.e., does not change the relative order of elements with equal keys In-place; i.e., only requires a constant amount O(1) of additional memory space Online; i.e., can sort a list as it receives it When people manually sort cards in a bridge hand, most use a method that is similar to insertion sort.
Algorithm
Insertion sort iterates, consuming one input element each repetition, and grows a sorted output list. At each iteration, insertion sort removes one element from the input data, finds the correct location within the sorted list, and inserts it there. It repeats until no input elements remain. Sorting is typically done in-place, by iterating up the array, growing the sorted list behind it. At each array-position, it checks the value there against the largest value in the sorted list (which happens to be next to it, in the previous array-position checked). If larger, it leaves the element in place and moves to the next. If smaller, it finds the correct position within the sorted list, shifts all the larger values up to make a space, and inserts into that correct position. The resulting array after k iterations has the property where the first k + 1 entries are sorted ("+1" because the first entry is skipped). In each iteration the first remaining entry of the input is removed, and 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:
i ← 1 while i < length(A) j ← i while j > 0 and A[j-1] > A[j] swap A[j] and A[j-1] j ← j - 1 end while i ← i + 1 end while
The outer loop runs over all the elements except the first one, because the single-element prefix A[0:1] is trivially sorted, so the invariant that the first i entries are sorted is true from the start. The inner loop moves element A[i] to its correct place so that after the loop, the first i+1 elements are sorted. Note that the and-operator in the test must use short-circuit evaluation, otherwise the test might result in an array bounds error, when j=0 and it tries to evaluate A[j-1] > A[j] (i.e. accessing A[-1] fails). After expanding the swap operation in-place as x ← A[j]; A[j] ← A[j-1]; A[j-1] ← x (where x is a temporary variable), a slightly faster version can be produced that moves A[i] to its position in one go and only performs one assignment in the inner loop body:
i ← 1 while i < length(A) x ← A[i] j ← i while j > 0 and A[j-1] > x A[j] ← A[j-1] j ← j - 1 end while A[j] ← x i ← i + 1 end while
The new inner loop shifts elements to the right to clear a spot for x = A[i]. The algorithm can also be implemented in a recursive way. The recursion just replaces the outer loop, calling itself and storing successively smaller values of n on the stack until n equals 0, where the function then returns up the call chain to execute the code after each recursive call starting with n equal to 1, with n increasing by 1 as each instance of the function returns to the prior instance. The initial call would be insertionSortR(A, length(A)-1).
function insertionSortR(array A, int n) if n > 0 insertionSortR(A, n-1) x ← A[n] j ← n-1 while j >= 0 and A[j] > x A[j+1] ← A[j] j ← j-1 end while A[j+1] ← x end if end function
It does not make the code any shorter, it also does not reduce the execution time, but it increases the additional memory consumption from O(1) to O(N) (at the deepest level of recursion the stack contains N references to the A array, each with accompanying value of variable n from N down to 1).
Implementation Below is an implementation in C.
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