Comparison-based sorting algorithms have traditionally dealt with achieving an optimal bound of O(n log n) when dealing with time complexity.
Adaptive sort takes advantage of the existing order of the input to try to achieve better times, so that the time taken by the algorithm to sort is a smoothly growing function of the size of the sequence and the disorder in the sequence.
In other words, the more presorted the input is, the faster it should be sorted.
Most worst-case sorting algorithms that do optimally well in the worst-case, notably heap sort and merge sort, do not take existing order within their input into account, although this deficiency is easily rectified in the case of merge sort by checking if the last element of the left-hand group is less than (or equal) to the first element of the righthand group, in which case a merge operation may be replaced by simple concatenation – a modification that is well within the scope of making an algorithm adaptive.
[1] In this sorting algorithm, the input is scanned from left to right, repeatedly finding the position of the current item, and inserting it into an array of previously sorted items.