Adaptive

An Adaptive Sorting algorithm is a sorting algorithm that decreases in comparisons as the number of inversions in the original array decreases:

An example is Insertion sort, a prototypical example of a adaptive sorting algorithm: In fact, the number of "inversions" in an array is usually defined as the number of swaps that it takes for insertion sort to sort the array, being approximately $$n^2/4 $$ for random data.

= Significance = Adaptive sorting algorithms are significant because of the large amount of data in nature that is say: already sorted, reverse sorted, partially sorted, etc.

With an adaptive algorithm, you can expect to sort already sorted and nearly sorted elements in ~$$O(n) $$ (for most cases), and in a few cases, reverse sorted lists to be reversed in ~$$O(n) $$ as well.

Some examples include: Insertion sort, Smoothsort, and Optimized Bubble Sort.