How does the intro sort algorithm function?

IntroSort is a hybrid sorting algorithm that provides both fast average performance and optimal worst-case performance.

IntroSort, short for Introspective Sort, is a sorting algorithm that was first implemented in the C++ Standard Library. It was designed to overcome the limitations of QuickSort and HeapSort. QuickSort is fast and efficient on average but can degrade to O(n^2) performance in the worst-case scenario. HeapSort, on the other hand, always guarantees O(n log n) performance but is typically slower on average than QuickSort.

IntroSort begins with QuickSort and switches to HeapSort when the recursion depth exceeds a level based on (the logarithm of) the number of elements being sorted. This switching behaviour is where the 'introspective' part of the name comes from. The algorithm 'introspects' or 'looks into' the amount of work done and switches strategy if QuickSort is doing poorly.

The QuickSort phase uses a variation of QuickSort called 'median of three' QuickSort, which chooses the pivot as the median of the first, middle, and last element of the partition. This pivot selection method helps to avoid the worst-case performance of QuickSort.

When the recursion depth exceeds a certain limit (usually 2 * log2(n)), the algorithm switches to HeapSort. HeapSort is not a recursive algorithm, so it doesn't suffer from the potential stack overflow issues that can plague QuickSort. It also guarantees O(n log n) performance, which is better than the worst-case performance of QuickSort.

In summary, IntroSort combines the average-case efficiency of QuickSort with the worst-case performance guarantee of HeapSort. It introspects or monitors its own performance and dynamically switches between two different sorting strategies to ensure both fast average performance and optimal worst-case performance. This makes it a reliable and efficient sorting algorithm for many practical sorting tasks.