Introsort

If a heapsort implementation and partitioning functions of the type discussed in the quicksort article are available, the introsort can be described succinctly as

procedure sort(A : array):
    let maxdepth = ⌊log(length(A))⌋ × 2
    introsort(A, maxdepth)

procedure introsort(A, maxdepth):
    n ← length(A)
    if n ≤ 1:
        return  // base case
    else if maxdepth = 0:
        heapsort(A)
    else:
        p ← partition(A)  // assume this function does pivot selection, p is the final position of the pivot
        introsort(A[0:p-1], maxdepth - 1)
        introsort(A[p+1:n], maxdepth - 1)

The factor 2 in the maximum depth is arbitrary; it can be tuned for practical performance. A[i:j] denotes the array slice of items i to j.

In quicksort, one of the critical operations is choosing the pivot: the element around which the list is partitioned. The simplest pivot selection algorithm is to take the first or the last element of the list as the pivot, causing poor behavior for the case of sorted or nearly sorted input. Niklaus Wirth's variant uses the middle element to prevent these occurrences, degenerating to O(n²) for contrived sequences. The median-of-3 pivot selection algorithm takes the median of the first, middle, and last elements of the list; however, even though this performs well on many real-world inputs, it is still possible to contrive a median-of-3 killer list that will cause dramatic slowdown of a quicksort based on this pivot selection technique.

Musser reported that on a median-of-3 killer sequence of 100,000 elements, introsort's running time was 1/200 that of median-of-3 quicksort. Musser also considered the effect on caches of Sedgewick's delayed small sorting, where small ranges are sorted at the end in a single pass of insertion sort. He reported that it could double the number of cache misses, but that its performance with double-ended queues was significantly better and should be retained for template libraries, in part because the gain in other cases from doing the sorts immediately was not great.

Introsort or some variant is used in a number of standard library sort functions, including some C++ sort implementations.

The June 2000 SGI C++ Standard Template Library stl_algo.h implementation of unstable sort uses the Musser introsort approach with the recursion depth to switch to heapsort passed as a parameter, median-of-3 pivot selection and the Knuth final insertion sort pass for partitions smaller than 16.

The GNU Standard C++ library is similar: uses introsort with a maximum depth of 2×log₂ n, followed by an insertion sort on partitions smaller than 16.[3]

The Microsoft .NET Framework Class Library, starting from version 4.5 (2012), uses introsort instead of simple quicksort.[4]

Go uses introsort with small modification: for slices of 12 or less elements it uses Shellsort instead of insertion sort, and more advanced median of three medians of three pivot selection for quicksort.