0.6 Sorting  (Page 3/20)

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:

insertionSort(array A)

for i<- 1 to length[A]-1 do

value<- A[i]

j<- i-1

while j>= 0 and A[j]>value do

A[j + 1] = A[j];

j<- j-1

A[j+1]<- value

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 and poorly implemented 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.

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.

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.