Faster to the quick.

Even though it’s fast, Quicksort can be made faster. We’ll consider two simple ways to increase it’s speed in the next entry. In this entry, we’ll discuss how when Quicksort is faced with entries containing equal keys, the speed falls to quadratic time or O(n2).

That’s a problem.

Edgster Dijkstra created a variation of Quicksort that deals with this. It’s called the Dijkstra 3-way partitioning. And with this discussion we are finally getting close to a sort algorithm that is similar to the system Sort function for Swift (Timsort).

The method creates a new stretch of equal array values during partitioning. Because this stretch of equal values is created during partitioning (and not during sorting), it is a stretch of sorted elements! And therefore it actually increases the efficiency of sorting tremendously. So much so that for arrays containing large numbers of equal keys, we approach linear speeds during sorting.

A nicely illustrated explanation of this is available in Algorithms by Sedgewick and Wayne — using Java.

The Swift code is below. You can see that it is marginally simpler than the Quicksort code.

func DQuickSort(array: [Double]) -> [Double] {

    var a = knuthShuffle(a: array)

    DijkstraQuickSort(a: &a, lo: 0, hi: a.count – 1)

    return a

}

func DijkstraQuickSort(a: inout [Double], lo: Int, hi: Int) {

    if hi <= lo {

        return

    }

    var lt = lo

    var gt = hi

    let v = a[lo]

    var i = lo

    while i <= gt {

        if a[i] < v {

            a = exchange(a: a, i: lt, j: i)

            lt += 1

            i += 1

        }

        else if a[i] > v {

            a = exchange(a: a, i: i, j: gt)

            gt -= 1

        }

        else {

            i += 1

        }

    }

    DijkstraQuickSort(a: &a, lo: lo, hi: lt – 1)

    DijkstraQuickSort(a: &a, lo: gt + 1, hi: hi)

}

And as an example of how Quicksort and the Dijkstra sort compare in a real world array of doubles with many duplicated keys, I’ve included an array with many duplicate keys. On my M1 the sort takes 729 ms with the normal Quicksort and 396 ms with the Dijkstra sort. The array count is 13970.

var a1: [Double] = [23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,12,54,3,7,4,4,2,9,8,16,10,32,23,350,54,44,7,4,2,99,6.0,23,1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And in case you were wondering how many duplicates there are, here is that same array with all the keys in sorted order:

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